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1, then the DFE of (3.10) is unstable.
3.6
Global stability of the disease free equilibrium
In the case of proportional incidence, a comparison theorem argument can be used to show that if no < 1, then the DFE is globally asymptotically stable.
3.6.1
Simple SEIRS
In the case of system (3.2), the local asymptotic stability result for 1 is readily strengthened to a global result.
no
00. Substituting this large time limit value N;p for Nsp in (3.30) gives d Esp dt
* =~ ~ f3sjp (Nsp
I jp - Esp - Isp - Rsp ) N*
j=1
lP
n
- (dsp
+ €sp)Esp + L
n
m!,qESq -
q=1
(3.32)
L m~pEsp. q=1
Therefore, system (3.31) is asymptotically autonomous, with limit equation (3.33) x' = g(x). To show that 0 is a globally asymptotically stable equilibrium for the limit system (3.33), remark that the linear system
x'
=
Lx,
(3.34)
where x is the 3sp dimensional vector consisting ofthe Esp, Isp and R sp ,
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Julien Arino
but with the equation for Esp taking the form
p
+L
p
m1fr,q E Sq -
q=l
(3.35)
L m~pEsp q=l
is such that g(x) ::; Lx for all x E lR.~n. In system (3.34), the equations for Esp and Isp do not involve R sp , and can thus be considered independently from the latter. Let x be the part of the vector x corresponding to the variables Esp and I sp , and L be the corresponding submatrix of L. The term (3sjp N;p/ Njp corresponds to the (s, j) entry of the matrix Gp used in Theorem 3.9, since Ssp ---t N;p under the current assumptions. (See the remark following Theorem 3.9.) Therefore, the method used in Section 3.5 to prove local stability can also be applied to study the stability of the x = 0 equilibrium of the subsystem x' = Lx, with L = F - V. Therefore, if no < 1, then the equilibrium x = 0 of the subsystem x' = Lx is stable. When x = 0, the conclusion of Theorem 3.7 holds, and limt--+ooRs(t) = 0, with Rs = (Rsl, ... ,Rsp)T. Thus the equilibrium Rs = 0 of this linear system in Rs is stable. As a consequence, the equilibrium x = 0 of (3.34) is stable when no < 1. Using a standard comparison theorem (see, e.g., [38, Theorem 1.5.4]), it follows that 0 is a globally asymptotically stable equilibrium of (3.33). For no < 1, the linear system (3.35) and (3.5c) has a unique equilibrium (the DFE) since its coefficient matrix F - V is nonsingular. The proof of global stability is completed using results on asymptotically autonomous equations; see, e.g., [58, Theorem 4.1] and [19]. D As in the simple SEIRS case (Theorem 3.11), any incidence function
1 for all p = 1, ... ,p, then the DFE is unstable. If, additionally, 1 would be interesting steps in that direction.
3.9.2
Understand the effect of movement
Theorem 3.17 establishes that in the case of a relatively homogeneous system, the movement matrix plays a role only insofar as it determines the value at the DFE. If disease transmission is also homogeneous within each patch, then Corollary 3.18 proves that the situation is even more constrained. In particular, in that case, it is impossible, for example, for movement to stabilize an unstable situation, or to destabilize a stable situation. Indeed, consider a system consisting of two connected patches,
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and suppose that both are such that their ng < 1 when taken in isolation. If the conditions of Corollary 3.18 hold, then movement cannot change this situation. The same holds true if both patches are such that ng > 1: movement cannot stabilize such a situation. In a less restrictive setting, movement can either stabilize an unstable situation, or destabilize a stable one. This has been investigated, for example, in [1, 49, 50].
4 4.1
Diseases in metapopulations -
A review
Focus of the review
Our definition of metapopulations de facto excludes pure group models, that consider strict interactions between groups. Such models have been considered for about the same amount of time as metapopulation models. The first such models are due to Rushton and Mautner [47] and Haskey [29]. Other well known examples are due to Lajmanovich and Yorke [37], Hethcote [33], Hethcote and Thieme [31]. While these models are conceptually quite similar to metapopulation models, they make the assumption that there is no exchange of individuals between the subpopulations. Their analysis can be quite similar to the analysis of metapopulation models. Also, the focus is on deterministic models that have been mathematically analyzed. Simulation work as well as stochastic models will be evoked when they give insight into the mechanisms. Finally, we focus on time continuous models. There are some very interesting works that are formulated in discrete time (see, e.g., [2, 16]), but the theory is quite different.
4.2
Early works
Bartlett, 1956 The first work that we are aware of that uses a patch approach is due to Bartlett [11]. He considers the following model on two patches, 8~ = -«(31h
I~ 8~
I~
+ (32h)8 1 + b + ms(82 - 8 1 ),
= Uhh + (32h)81 - (d + p)J.lh + mI(I2 - II), = -«(31h + (32h)82 + b + ms(81 - 82 ), = «(31h + (3212)82 - (d + p)I2 + mI(h - h).
(4.1a) (4.1b) (4.1c) (4.1d)
The rate d + p incorporates the natural death rate d as well as the rate p of occurence of any other event leading to an individual leaving the infected class (disease specific death, recovery, etc.). b is the birth rate.
Diseases in Metapopulations
109
Note that this model is a hybrid of metapopulation and group models. Indeed, there is an exchange of individuals between patches through migration, but there is also cross patch infection. Baroyan and R vachev, late 60s, and directly related articles Following the work of Bartlett comes works by Baroyan, Rvachev and collaborators [9, 10]. They consider the spatial spread of influenza between cities in the Soviet Union. In their approach, a large geographic region (country) is partitioned into smaller sub-regions (cities). Migration and transportation between these cities are explicitly incorporated, and within a given city, transmission is modeled by a discrete deterministic compartmental SIR model. In [48], the parameters of the model are estimated using Hong Kong as a reference; the model is then used to simulate the spread of the Hong Kong influenza pandemic between 52 world cities. In [40], an epidemic threshold theorem is obtained. An SEIR version of this model was used recently [26]. Using the framework of Rvachev and Longini, Hyman and LaForce [34] formulate a multy-city transmission model for the spread of influenza between cities (patches) with the assumption that people continue to travel when they are infectious and there is no death due to influenza. Because influenza is more likely to spread in the winter than in the summer, they assume that the infection rate has a periodic component. In addition, they introduce a new disease state P in which people have partial immunity to the current strain of influenza. Thus they have an SIRPS model in which both susceptible and partially immune individuals can be infected, but this is more likely for susceptibles. A symmetric travel matrix M = [mij] with mij = mji is assumed, thus the population of each city remains constant. Their model for p cities is formulated as a 4p system of non autonomous ODEs. Epidemic parameters appropriate for influenza virus are used, in particular for strains of H3N2 in the 1996-2001 influenza seasons with an infectious period of l/ex = 4.1 days in all cities. Parameters modeling the number of adequate contacts per person per day and the seasonal change of infectivity are estimated by a least squares fit to data. The populations of the largest 33 cities in the US are taken from 2000 census data, and migration between cities is approximated by airline flight data. A sensitivity analysis reveals that 1/ ex, the average duration of infection, is the most important parameter.
4.3
Kermack-McKendrick-type models
The model known as the Kermack-McKendrick (KMK) model takes the form [36]
d
-8 = -f38I,
dt
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110
d -I dt d dtR
=
f3S1 - "II,
= "II,
that is, an SIR model without demography. The parameter "I represents here the rate of removal from the I class, it aggregating disease induced death and recovery from the disease. This system has the advantage that an explicit solution can be found (see, e.g., [13]). Several authors have used KMK-type models in a metapopulation context. Faddy, 1986 model
In a short note, Faddy [22] introduces a KMK-type SI n
S:
=
-Si ~f3jilj,
(4.2a)
j=l
n
I:
= Si L j=l
f3ji l j - 'Yili
+ ~ mij I j Ji-i
L mjih
(4.2b)
Ji-i
where "Ii represents the sum of all removals from the I class. As in the case of system (4.1), this system mixes group models with migration. An interesting remark made by Faddy is that there exists a sort of conservation law, since the quantity
does not change over time. The interest here is on the final size of the epidemic, for which an expression is obtained. In the spatially homogeneous case where Si(O) = S(O), a given value in all patches, mi = 2:Ji-i mij = m, f3ij = f3 and "Ii = "I, he obtains that
where Si( 00) is the final number of susceptibles remaining uninfected in patch i. Clancy, 1996 Clancy [20] introduces a Kermack-McKendrick type model on patches, but that describes the dynamics of a very simple epidemic with carriers, whose numbers is denoted C. The carriers are
Diseases in Metapopulations
111
subject to specific removal (either through treatment of death), at a rate "(. The system includes a removed class that we do not show here as it bears no influence on the dynamics of the system. The latter takes the form (4.3a) n
C~ = -,,(Ci +
L m~Cj.
(4.3b)
j=l
He also introduces a corresponding stochastic version. The focus here is on the ultimate size of the epidemic; more precisely, estimates of Si as t --+ 00 are sought. Rodriguez and Torres-Sorando, 2001, consider in [44] a direct transmission model and a model of malaria. The malaria model tracks the evolution of the numbers Ii and Yi of infectious humans and infectious mosquitoes, respectively, on patch i. The total population of both species is assumed constant on each patch, and denoted Ni and M i , respectively, for humans and mosquitoes. Thus, the numbers of susceptibles are obtained by Si = Ni - Ii and Zi = Mi - Yi. The system takes the form II
= /3SiYi
- "(Ii
+ /3Si L
mij Yj,
(4.4a)
jf-i
~'
=
/3ZJi - d M Yi
+ /3Zi L
mjiIj.
(4.4b)
#i
/3 is the rate of transmission of the disease when a contact occurs. They consider the effect of different migration patterns and of the environment heterogeneity on the dynamics of the system, and in particular on the possibility of the disease becoming established. To study this, they consider the jacobian matrix at the DFE, and study the sign of the dominant eigenvalue. 4.4
Migration models
Wang and Mulone, 2003 Wang and Mulone [63] consider the following model in the case p = 2 patches,
S~ = di(Ni -
Si) - /3i S i ~. ,
j5
+ "(iIi + L m~Sj, j=l
(4.5a)
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(4.5b)
They establish a series of interesting results concerning the conditions under which the disease is persistent in the system. One particular conclusion that they draw is that, provided they are positive, the migration rates of susceptibles mr2, m~1 do not playa role in the permanence conditions. Wang and Zhao, 2004 form
Wang and Zhao [64] consider a model of the n
S~ = Bi(Ni)Ni - /liSi - (3i S Ji
+ fili + L m~Sj,
(4.6a)
j=1
n
I~
=
(3iSi1i - (/li
+ fi)Ii + L
mIjlj .
(4.6b)
j=1
With this more general birth function B i , even finding a disease free equilibrium is a difficult task. It is shown that, in this case, population movement can either intensify or reduce the spread of disease. Salmani and van den Driessche, 2006 Salmani and van den Driessche introduce in [50] a single species SEIRS model with status dependent movement and disease induced death, from which (3.2) is derived. In a first part, following the approach of [6, 7], they establish a basic reproduction number for the system, and as in [4], the global stability of the DFE when Ro < 1. They then proceed to a more detailed study of an SIS particular case in two patches. They establish that with different movement rates, different situations can prevail, with for example a global Ro < 1 and individual Rg in the patches less than 1 (it will be established in the next chapter that this cannot be the case when the movement rates are identical for all epidemiological states). Fulford, Roberts and Heesterbeek, 2002 The spread of bovine tuberculosis amongst the common brushtail possum in New Zealand, is modeled by Fulford et al [25]. Since only maturing possums (1 to 2 year old males) travel large distances, a two-age class metapopulation model is formulated, with juvenile and adult possums. As this disease is fatal, an SEI model is appropriate. In addition to horizontal transmission between both age-classes, pseudo-vertical transmission is included since juveniles may become infected by their mothers. Susceptible and exposed juveniles (but not infective juveniles) travel between patches as they
Diseases in Metapopulations
113
mature. For p patches, the authors formulate a system of 6p ODEs to describe the disease dynamics. Using the next generation matrix method [21], the authors explicitly calculate Ro for p = 1 and for p = 2, and give the structures of the next generation matrices for p = 4 and three spatial topologies, namely a spider, chain and loop. The design of control strategies (culling) based on these three spatial topologies is considered. The critical culling rates (where Ro = 1) are calculated and the spatial aspects are shown to be important.
4.5
Model including residency patch
Sattenspiel and coauthors In [55], Sattenspiel and Simon introduced a model for the interaction between individuals in p neighborhoods, taking into account that some individuals only have contacts in their neighborhoods. Although this is a strict group model, since individuals do not move explicitly between neighborhoods, it is mentioned here because it is an obvious prequel to the models with residency patch. Also, it contains some interesting matrix-based analysis. Sattenspiel and Dietz [51] introduced a single species, multi-patch model that describes the travel of individuals, and keeps track of the patch where an individual is born and usually resides as well as the patch where an individual is at a given time. Hence this type of model describes human travel rather than migration. This framework was subsequently used numerically by Sattenspiel and others to describe various situations linked to the spread of influenza in the Canadian subartic [52], the effect of quarantining [53] and the influence of the mobility patterns [54]. Arino and van den Driessche We studied the model of [51] in [6, 7], giving some analytical results and calculating the basic reproduction number in the SIS [7] and SEIRS [6] cases, giving the first example of application of the method of [60] to such high dimensional models. These models have a unique DFE. Numerical simulations show that a change in travel rates can lead to a bifurcation at Ro = 1; thus travel can stabilize or destabilize the disease free equilibrium. This model is the basis for [8], which extends the model of [51] by allowing individuals to travel between two patches that are not their residency patch. The resulting model is system (3.10), analyzed in detail in Section 3. Ruan, Wang and Levin, 2006 Using the framework of [51]' Ruan, Wang and Levin [45] study the global spread of SARS. The system takes the form of an SEIRS model with an additional class for quarantined individuals, denoted Q, that do not travel. They study the existence of
Julien Arino
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a DFE, and establish the basic reproduction number R o, deducing the local asymptotic stability of the DFE when Ro < 1 using the result of [60], as detailed in the SEIRS case in Section 3.5. The basic reproduction number depends explicitly on quarantining parameters. A particular case for two cities (Honk Kong and Toronto) is then considered.
4.6
New directions
To conclude this brief review of diseases in metapopulations, a few directions that appear promising to the author are now listed. It is hoped that, although necessarily biased by the author's opinions, this will encourage readers to study more in detail some of the aspects.
4.6.1
True patch heterogeneity
Metapopulations have been introduced in the context of epidemic diseases to take into account spatial heterogeneities. However, in all the models discussed so far, the spatial heterogeneity is not 'true'. Indeed, it is assumed that in the different patches, parameters are different, but that the incidence function is similar. The effect of the contact structure (the nature of the incidence function) on the dynamics is determinant. A model of Fromont, Pontier and Langlais [24] is the first we know that breaks this homogeneity. They consider a model appropriate for Feline Leukemia Virus among a population of domestic cats. There are p patches called farms or villages depending on the magnitude of the patch carrying capacity. Dispersal (which depends on disease state) can take place between any pair of patches or int%ut of non-specified populations surrounding the patches (representing transient feral males). Infected cats become either infectious or immune and remain so for life, thus the model is of SIR type, but a proportion of cats go directly from the susceptible to the immune state. A density dependent mortality function is assumed, as well as different incidence functions depending on the population density (mass action for cats on farms, standard incidence for cats in villages). The model consists of 3p ODEs and is analyzed for the case p = 2, taking data appropriate for the virus with one patch being a village and one patch being a farm, or both patches being farms. For a set of parameters such that in isolation the virus develops in the village but goes extinct on the farm, travel between the patches of either susceptible and immune cats or of infective cats can result in the virus persisting in both patches. Thus results show that, in general, spatial heterogeneity promotes disease persistence.
Diseases in Metapopulations 4.6.2
115
Models with infinite dimensional aspects
Wang, Fergola and Tenneriello, 2003 Wang, Fergola and Tennierello [62] study a model for the diffusion of innovation in a two patch environment. Although not strictly an epidemic model, the spread of innovation can easily be reread in terms of disease propagation. They first formulate the model in ODE and show that it has a globally asymptotically stable equilibrium (note that this is different from classical epidemic models, in the sense that there is no bifurcation from a disease free equilibrium to an endemic equilibrium). They then incorporate delay, in the form of product duration. Written in epidemiological terms, the model then takes the form d
dt 51 = III - (a1 + f31h)5 1 - d 15 1 + m2(52 + (1 - k 2)h) - m 15 1 + e- dIT1 (11 51(t - Td + f3 15 1(t - TdJ(t - Td), (4.7a) d
dt h
=
(a1
+ f31h)51 -
d1h - m1h
- e- d1T1 ('"Yl51 (t - Td d
dt 52
=
II2 - (a2
+ f32h)5 2 -
+ e-d2T2(1252(t d
dth
=
+ k2m2h
+ f3 15(t -
T1)J(t - Td),
d25 2 + m1 (51
+ (1 -
(4. 7b)
k 1)h) - m252
T2) + f32S2(t - T2)J(t - T2)),
(4.7c)
(a2 + f32lz)52 - d2lz - m2J2 + k1m1h - e-d2T2(1252(t - T2)
+ f3 25(t -
T2)J(t - T2)).
(4.7d)
In each patch i = 1,2, besides the usual parameters, IIi is the birth rate (a constant), ai is the intensity of advertisement for the products (this additional recruitment term is the main difference from classical epidemic models) and Ti is the duration of the product (that is, of infection) in each patch. The migration is slightly different from the other models seen so far, in the sense that individuals can change status when they move from one patch to the other: m1 is the rate of movement from 1 to 2, m2 is the rate of movement from 2 to 1, k1 is the fraction of infected from patch 1 that remain infected when moving to 2 and k2 is the fraction of infected from patch 2 that remain infected when moving to patch 1. Here again, there is a unique positive equilibrium, which is shown to be globally asymptotically stable under some conditions on parameter values. The paper concludes with a study of periodic solutions in the case where advertisement, i.e., ai, is periodic (in the delayed case). It is shown that there exists parameter values for which a periodic solution exists and is globally stable.
Julien Arino
116
Wang and Zhao, 2005 Wang and Zhao [65J formulated a model for an SIS model on patches, with a class of juveniles (denoted by J). On each patch p = 1, ... ,p,
(4.8a)
(4.8b)
(4.8c) where Ap = Sp + Ip (the population of adults), dt is the death rate of juveniles and d p is the death rate of adults, and Rp(t) is rate of recruitment of juveniles into the susceptible adult class. It is assumed that B(Ap) > 0 for Ap > 0, Bp continuously differentiable for Ai > 0 and B~(Ap) < 0 for all Ap > O. Recruitment into the adult class has to take into account that juveniles can be born in a given patch, and become adults in another patch. Let r be the age of recruitment into the adult class (assumed the same in each patch), and J(t, a) := (h(t, a), ... , Jp(t, a))T, with Jp(t, a) the number of juveniles in patch p at time t that are of age a. The recruitment R(t) then satisfies R(t) := (Rl (t), ... ,Rp(t) f = J(t, r). The age-space dynamics is described by
(at + oa)Jp(t, a) =
t,
mtkJk(t, a) -
(~m"£p + d~) Jp(t, a)
p
=
L m~kJk(t, a) -
d~ Jp(t, a),
k=l with J(t,O) = B(A(t)) := (Bl(Al(t))Al(t), ... ,Bp(Ap(t))Ap(t))T. Then, after some computations,
R(t) where
=
J(t,r)
-d{ + mil
~~l
CJ =
(
J m pl
=
exp(CJr)B(A(t - r)), C12
-d£ +m{2··· J m p2
Using R(t), the equations for S and I decouple from the equations for J, giving a system of 2p delay differential equations.
Diseases in Metapopulations
117
The authors then establish the existence of a unique disease free equilibrium under a certain number of assumptions. They then derive a basic reproduction number for the system, and consider the global stability of the disease free equilibrium, as well as the persistence of the system when this equilibrium is unstable, and the existence of an endemic equilibrium. The paper concludes with a study of a two patch particular case.
5
Cond usion
My aim here was to show that metapopulation models are usable in the context of epidemiology, to provide an extensive overview of the mathematical problems that arise when studying such models, and to illustrate some of the solutions that can be given to these problems. This was done through two classes of models that van den Driessche and I have considered, with a simple single population SEIRS also used to illustrate the most simple properties. I hope to have convinced the reader, at the cost of maybe a little too much detail, that the mathematical complications arising in these models can be dealt with, and that there is a pattern to these solutions that allows to envision a general theory of metapopulation models in epidemiology. This theory is barely sketched here.
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[6] J. Arino and P. van den Driessche. The basic reproduction number in a multi-city compartmental epidemic model. Lecture Notes in Control and Information Science, 294: 135-142, 2003. [7] J. Arino and P. van den Driessche. A multi-city epidemic model. Mathematical Population Studies, 10(3): 175-193,2003. [8] J. Arino and P. van den Driessche. Metapopulation epidemic models. A survey. Fields Institute Communications, 48: 1-13, 2006. [9] V. 0 Baroyan and L.A Rvachev. Deterministic epidemic models for a territory with a transport network. Kibernetica, 3: 67-73,1967. [10] V. O. Baroyan, L.A. Rvachev, U.V. Basilevsky, V.V. Ezmakov, K.D. Frank, M.A. Rvachev, and V.A. Shaskov. Computer modeling of influenza epidemics for the whole country (USSR). Adv. App. Prob., 3: 224-226, 1971. [11] M.S. Bartlett. Deterministic and stochastic models for recurrent epidemics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume IV, pages 81-109. University of California Press, 1956. [12] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, volume 9 of Classics in Applied Mathematics. SIAM, 1994. [13] F. Brauer. The Kermack-McKendrick epidemic model revisited. Math. Biosci., 198: 119-131, 2005. [14] F. Brauer and C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology. Springer, 200l. [15] S. Busenberg and K.L. Cooke. Springer-Verlag, 1993.
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[16] C. Castillo-Chavez and Yakubu A.-A. Intraspecific competition, dispersal and disease dynamics in discrete-time patchy invironments. In C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and Yakubu A.-A., editors, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, volume 125 of IMA Series on Mathematics and its Applications. Springer, 2002. [17] C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu, editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, volume 125 of IMA Series on Mathematics and its Applications. Springer, 200l. [18] C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu, editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory,
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volume 126 of IMA Series on Mathematics and its Applications. Springer, 200l. [19] C. Castillo-Chavez and H. Thieme. Asymptotically autonomous epidemic models. In O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, editors. Mathematical Population Dynamics: Analysis of Heterogeneity, 33-49, Wuerz, Winnipeg, 1995. [20] D. Clancy. Carrier-borne epidemic models incorporating population mobility. Math. Biosci., 132: 185-204, 1996. [21] O. Diekmann and J.A.P. Heesterbeek. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, 2000. [22] M.J. Faddy. A note on the behavior of deterministic spatial epidemics. Math. Biosci., 80: 19-22, 1986. [23] M. Fiedler. Special Matrices and their Applications in Numerical Mathematics. Martinus Nijhoff Publishers, 1986. [24] E. Fromont, D. Pontier, and M. Langlais. Disease propagation in connected host populations with density-dependent dynamics: the case of the Feline Leukemia Virus. J. Theor. Biol., 223: 465-475, 2003. [25] G.R. Fulford, M.G. Roberts, and J.A.P. Heesterbeek. The metapopulation dynamics of an infectious disease: tuberculosis in possums. Theoretical Population Biology, 61: 15-29, 2002. [26] R.F. Grais, J.H. Ellis, and G.E. Glass. Assessing the impact of airline travel on the geographic spread of pandemic influenza. European Journal of Epidemiology, 18: 1065-1072, 2003. [27] B.T. Grenfell and A.P. Dobson, editors. Ecology of Infectious Diseases in Natural Populations. Cambridge University Press, Cambridge, 1995. [28] 1. A. Hanski and M.E. Gilpin. Metapopulation Biology: Ecology, Genetics, and Evolution. Academic Press, 1997. [29] H.W. Haskey. Stochastic cross-infection between two otherwise isolated groups. Biometrika, 44(1/2): 193-204, 1957. [30] H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4): 599-653, 2000. [31] H. W. Hethcote and H. Thieme. Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci., 75: 205227,1985. [32] H. W. Hethcote and J.A. Yorke. Gonorrhea Transmission Dynamics and Control, volume 56 of Lecture Notes in Biomathematics. Springer-Verlag, 1984.
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Modeling the Start of a Disease Out break* Fred Brauer Department of Mathematics, University of British Columbia Vancouver, BC V6T lZ2, Canada E-mail: [email protected]
1
Introduction
The Kermack-McKendrick compartmental epidemic model assumes that the sizes of the compartments are large enough that the mixing of members is homogeneous, or at least that there is homogeneous mixing in each subgroup if the population is stratified by activity levels. However, at the beginning of a disease outbreak, there are a very small number of infective individuals and the transmission of infection is a stochastic event depending on the pattern of contacts between members of the population; a description should take this pattern into account. It has often been observed in epidemics that there is a small number of "superspreaders" who transmit infection to many other members of the population, while most infectives do not transmit infections at all or transmit infections to very few others [17]. This suggests that homogeneous mixing at the beginning of an epidemic may not be a good approximation. The SARS epidemic of 2002-2003 spread much more slowly than would have been expected on the basis of the data on disease spread at the start of the epidemic. Early in the SARS epidemic of 2002-2003 it was estimated that Ro had a value between 2.2 and 3.6. At the beginning of an epidemic, the exponential rate of growth of the number of infectives is approximately (Ro - l)/a, where l/a is the generation time of the epidemic, estimated to be approximately 10 days for SARS. This would have predicted at least 30,000 cases of SARS in China during the first four months of the epidemic. In fact, there were fewer than 800 cases reported in this time. An explanation for this discrepancy is that the estimates were based on transmission data in hospitals and crowded apartment complexes. It was observed that there was intense activity in some locations and very little in others. This suggests that the actual reproduction number (averaged over the whole population) was much -This work has been supported by NSERC and MITACS.
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Fred Brauer
lower, perhaps in the range 1.2-1.6, and that heterogeneous mixing was a very important aspect of the epidemic.
2
The network corresponding to a disease outbreak
Our approach will be to give a stochastic branching process description of the beginning of a disease outbreak to be applied so long as the number of infectives remains small, distinguishing a (minor) disease outbreak confined to this stage from a (major) epidemic which occurs if the number of infectives begins to grow at an exponential rate. Once an epidemic has started we may switch to a deterministic compartmental model, arguing that in a major epidemic contacts would tend to be more homogeneously distributed. However, if we continue to follow the network model we would obtain a somewhat different estimate of the final size of the epidemic. Simulations indicate that the assumption of homogeneous mixing in a compartmental model would lead to a higher estimate of the final size of the epidemic than the prediction of the network model. We describe the network of contacts between individuals by a graph with members of the population represented by vertices and with contacts between individuals represented by edges. The study of graphs originated with the abstract theory of Erdos and Renyi of the 1950's and 1960's [3, 4, 5], and has become important more recently in many areas, including social contacts, computer networks, as well as the spread of communicable diseases. We will think of networks as bi-directional, with disease transmission possible in either direction along an edge. An edge is a contact between vertices that can transmit infection. The number of edges of a graph at a vertex is called the degree of the vertex. The degree distribution of a graph is {pd, where Ph is the fraction of vertices having degree k. The degree distribution is fundamental in the description of the spread of disease. Initially, we proceed as if all contacts between an infective and a susceptible transmit infection, but we will not make this assumption when we study the course of a disease outbreak in Section 3. We think of a small number of infectives in a population of susceptibles large enough that in the initial stage we may neglect the decrease in the susceptible population. Our development begins along the lines of that of [7] and then develops along the lines of [6, 14, 16]. We assume that the infectives make contacts independently of one another and let Ph denote the probability that the number of contacts by a randomly chosen individual is exactly k, with L~o Ph = 1. In other words, {Ph}
Modeling the Start of a Disease Outbreak
125
is the degree distribution of the vertices of the graph corresponding to the population network. For convenience, we define the generating function 00
Go(z) =
2::>k Zk . k=O
Since L~o Pk = 1, this power series converges for 0 ~ be differentiated term by term. Thus
_ G~k)(O) k! '
Pk -
k
=
z ~ 1, and may
0,1,2, ....
It is easy to verify that the generating function has the properties
Go(O) = Po,
Go(1) = 1,
G~(z)
> 0,
G~(z)
> O.
< k >, is
The mean degree, which we denote by 00
< k >= :L kPk = G~(1). k=l
More generally, we define the moments 00
< k j >= :L kjPk,
j
= 1,2,···
,00.
k=l
When a disease is introduced into a network, we think of it as starting at a vertex (patient zero) who may transmit infection to every individual to whom this individual is connected, that is, along every edge of the graph from the vertex corresponding to this individual. For transmission of disease after the first generation we need to use the excess degree of a vertex. If we follow an edge to a vertex, the excess degree of this vertex is one less than the degree. We use the excess degree because infection can not be transmitted back along the edge whence it came. The probability of reaching a vertex of degree k, or excess degree (k - 1), by following a random edge is proportional to k, and thus the probability that a vertex at the end of a random edge has excess degree (k - 1) is a constant multiple of kPk with the constant chosen to make the sum over k of the probabilities equal to 1. Then the probability that a vertex has excess degree (k - 1) is kPk qk-l = < k >. This leads to a generating function G1(z) for the excess degree
Fred Brauer
126
and the mean excess degree, which we denote by 1
< ke >, is
00
2:
k (k -l)Pk < ke > = - k < > k=l 1
=
1
00
k=l
00
- - L k 2 Pk- - - L k p k
k=l
2
< k > _ 1 = G' (1).
3
1
Transmissibility
Contacts do not necessarily transmit infection. For each contact between individuals of whom one has been infected and the other is susceptible there is a probability that infection will actually be transmitted. This probability depends on such factors as the closeness of the contact, the infectivity of the member who has been infected, and the susceptibility of the susceptible member. We assume that there is a mean probability T, called the transmissibility, of transmission of infection. The transmissibility depends on the rate of contacts, the probability that a contact will transmit infection, the duration time of the infection, and the susceptibility. In this section, we will continue to assume that there is a network describing the contacts between members of the population whose degree distribution is given by the generating function Go (z), but we will assume in addition that there is a mean transmissibility T. If all contacts transmit infection, then T = 1. When disease begins in a network, it spreads to some of the vertices of the network. Edges that are infected during a disease outbreak are called occupied, and the size of the disease outbreak is the cluster of vertices connected to the initial vertex by a continuous chain of occupied edges. The probability that exactly m infections are transmitted by an infective vertex of degree k is
We define r 0 (z, T) be the generating function for the distribution of the number of occupied edges attached to a randomly chosen vertex, which is the same as the distribution of the infections transmitted by a randomly
Modeling the Start of a Disease Outbreak
127
chosen individual for any (fixed) transmissibility T. Then
r" (z, T)
~ %;, [~ p, (,~,) 1~'(1 - T),'-m)1zm
~
t, [t (!) p,
(zT)m(l - T),'-m)
1
(3, 1)
00
= LPk[zT + (1 -
T)]k
=
G o(l
+ (z -
l)T).
k=O
In this calculation we have used the binomial theorem to see that
Note that
ro(o, T) = G o(1- T),
ro(l, T) = G o(l) = 1,
r~(z, T) = TG~(l
+ (z -
l)T).
For secondary infections beyond the first generation we need the generating function rl(z,T) for the distribution of occupied edges leaving a vertex reached by following a randomly chosen edge. This is obtained from the excess degree distribution in the same way,
and rdO,T) = G I (l- T), r~ (z, T)
r l (l,T) =
= TG~ (1 + (z -
G I (l)
=
1,
l)T).
We let RI = r~ (1, T) = TG~ (1), the mean number of occupied edges. Although Ro = r~(l, T), the mean number of secondary cases infected by patient zero is the basic reproduction number as usually defined, the mean excess number of occupied edges is a more accurate description for the spread of disease and the threshold for an epidemic is determined by R I . If the incidence is mass action, the degree distribution is a Poisson distribution and Go(z) = GI(z), so that RI = Ro. For other degree distributions the values of RI and Ro could be quite different. Our next goal is to calculate the probability that the infection will die out and will not develop into a major epidemic. We begin by assuming that patient zero is a vertex of degree k. Suppose patient zero transmits infection to a vertex of degree j. We let Zn (T) denote the probability that
Fred Brauer
128
this infection dies out within the next n generations. The probability that there are m infections caused by a secondary vertex of degree j is
If there are m secondary infections coming from this vertex, for the infec-
tion to die out in n generations each of these secondary infections must die out within (n - 1) generations. The probability of this is Zn-l (T) for each secondary infection, and the probability that all secondary infections will die out in (n - 1) generations is [zn_l(T)]m. Thus the probability that all secondary infections from this vertex die out within (n - 1) generations is
Now zn(T) is the sum over j of these probabilities, weighted by the probability qj of j secondary infections. Thus
Since f1(z, T) is an increasing function of z, the sequence zn(T) is an increasing sequence and has a limit zoo(T), which is the probability that this infection will die out eventually. Then ZOO (T) is the limit as n ~ 00 of the solution of the difference equation
Thus Zoo (T) must be an equilibrium of this difference equation, that is, a solution of Z = f1(z, T). Let w be the smallest positive solution of Z = f1(z,T). Then, because f1(z,T) is an increasing function of z, k Z ~ f1(z,T) ~ f1(w,T) = w for 0 ~ Z ~ w. Since za ) = 0 < wand Z~~-;.l) (T) ~ w implies
it follows by induction that
z~k) (T) ~ w,
n
= 0,1, ...
,00.
k
= 1,2""
,00.
From this we deduce that
z~)(T) = w,
Modeling the Start of a Disease Outbreak
129
The equation r 1 (Z, T) = Z has a root Z = 1 since r 1 (1, T) = 1. Because the function r 1 (Z, T) - Z has a positive second derivative, its derivative r~ (z, T) - 1 is increasing and can have at most one zero. This implies that the equation r 1 (z, T) = z has at most two roots in 0 ~ z ~ 1. If Ro < 1 the function r1(z, T) - z has a negative first derivative r~(z,T) -1 ~ r~(l,T) -1
= TG~(l) -1 = Rl
-1 < 0
and the equation r1(z,T) = z has only one root, namely z = 1. On the other hand, if Rl > 1 the function r 1 (z, T) - z is positive for z = 0 and negative near z = 1 since it is zero at z = 1 and its derivative is positive for z < 1 and z near 1. Thus in this case the equation r 1 (z, T) = z has a second root ZOO (T) < 1. In either case, the limit z~:,l (T) is independent of k, and we denote it by zoo(T). The probability that the disease outbreak will die out eventually is the sum over k of the probabilities that the initial infection in a vertex of degree k will die out, weighted by the degree distribution {pd of the original infection, and this is 00
LPkZ~(T) = Go(zoo(T)). k=O
To summarize this analysis, we see that if Rl = TG~ (1) < 1 the probability that the infection will die out is 1. If Rl > 1 there is a solution ZOO (T) < 1 of
r1(z, T)
=
z,
and a probability 1 - ro(zoo(T), T) > 0 that the infection will persist, and will lead to an epidemic. However, there is a positive probability r 1 (zoo (T), T) that the infection will increase initially but will produce only a minor outbreak and will die out before triggering a major epidemic. Another interpretation of the basic reproduction number is that there is a critical transmissibility Tc defined by
In other words, the critical transmissibility is the transmissibility that makes the basic reproduction number equal to 1. If the mean transmissibility can be decreased below the critical transmissibility, then an epidemic can be prevented. The measures used to try to control an epidemic may include contact interventions, that is, measures affecting the network such as avoidance of public gatherings and rearrangement of the patterns of interaction between caregivers and patients in a hospital, and transmission interventions such as careful hand washing or face masks to decrease the probability that a contact will lead to disease transmission.
130
Fred Brauer
More sophisticated network analysis makes it possible to predict such quantities as the size of an epidemic, the probability that an individual will set off an epidemic, the risk for an individual of becoming infected, the probability that a cluster of infections will set off an epidemic small disease outbreak when the transmissibility is less than the critical transmissibility and how the probability of an epidemic depends on the degree of patient zero, the initial disease case [12, 14].
4
Some examples of contact networks
The above analysis assumes that there is a known generating function Go(z) or, equivalently, a degree distribution {Pk}. In studying a disease outbreak, we need to know the degree distribution of the network. If we know the degree distribution we can calculate the basic reproduction number and also the probability of an epidemic. What kinds of networks are observed in practice in social interactions? There are some standard examples. If contacts between members of the population are random, corresponding to the assumption of mass action in the transmission of disease, then the probabilities Pk are given by the Poisson distribution
e-cc k Pk =
-;;;!
for some constant c. To show this, we think of a probability of contact cD.t in a time interval D.t, and we let n
=
1 D.t.
Then the probability of k contacts in a time interval ,6.t is
where
(nk)=~n! k!(n - k)!
is the binomial coefficient. We rewrite this probability as n(n - l)(n - 2)··· (n - k nk
We let D.t
----+
0, or n
----+ 00.
+ 1) c k (1 -
Since
n(n - l)(n - 2)··· (n - k
+ 1)
~----'--.::.---'-:---'------'- ----+ nk
-;;,-)n
k!(l--;;'-)k·
1,
Modeling the Start of a Disease Outbreak and (1-
~r
->
e-
c
131
,
the limiting probability that there are k contacts is
Then the generating function is
L 00
Go(z)
= e- c
k
~! zk = e-ce cz =
ec(z-l) ,
k=O
and G~(z) = cec(z-l),
G~(l)
= c.
The generating function for the Poisson distribution is ec(z-l) and no TG~(l) = cT. We note also that for the Poisson distribution G1(z)
Go(z), n 1
=
= =
n ,.
The commonly observed situation that most infectives do not pass on infection but there are a few "superspreading events" [17] corresponds to a probability distribution that is quite different from a Poisson distribution, and could give a quite different probability that an epidemic will occur. For example, taking T = 1 for simplicity, if no = 2.5 the assumption of a Poisson distribution gives Zoo = 0.107 and Go(zoo) = 0.107, so that the probability of an epidemic is 0.893. The assumption that nine out of ten infectives do not transmit infection while the tenth transmits 25 infections gives
from which we see that the probability of an epidemic is 0.1. Another example, possibly more realistic, is to assume that a fraction (1 - p) of the population follows a Poisson distribution with constant r while the remaining fraction p consists of superspreaders each of whom makes L contacts. This would give a generating function
Go(z) = (1 - p)er(Z-l) so that no
+ pzL
= r(l - p) + pL, G1(z)
=
and
n1 =
r(l - p)er(z-l) + pLzL-l r(l-p)+pL ' r2(1 - p) + pL(L - 1) . r(l- p) + pL
Fred Brauer
132 For example, if r simulation gives
= 2.2, L = 10, P = 0.01, so that Ro = 2.278 numerical Rl
= 2.5,
Zoo
= 0.146,
so that the probability of an epidemic is 0.849. For network models, Rl is a better description of the spread of a disease out break than Ro. These examples demonstrate that the probability of a major epidemic depends strongly on the nature of the contact network. Simulations suggest that for a given value of the basic reproduction number the Poisson distribution is the one with the maximum probability of a major epidemic. It has been observed in many situations that there are a small number of long range conections in the graph, allowing rapid spread of infection. There is a high degree of clustering (some vertices with many edges) and there are short path lengths. Such a situation may arise if a disease is spread to a distant location by an air traveller. This type of network is called a small world network. Long range connections in a network can increase the likelihood of an epidemic dramatically. A third kind of network frequently observed is a scale free network. In a random network, the quantity Pk approaches zero very rapidly (exponentially) as k -+ 00. A scale free network has a "fatter tail", with Pk approaching zero as k -+ 00 but more slowly than in a random network. In an epidemic setting it corresponds to a situation in which there is an active core group but there are also "superspreaders" making many contacts. In a scale free network, Pk is proportional to k- a with a a constant. In practice, a is usually between 2 and 3. Often an exponential cutoff is introduced in applications of scale free networks in order to make G~(l) < 00 for every choice of a, so that Pk
= Ck-ae- k / e.
The constant C, chosen so that '2:.':=0 Pk = 1, can be expressed in terms of logarithmic integrals. These examples indicate that the probability of an epidemic depends strongly on the contact network at the beginning of a disease outbreak. The study of complex networks is a field which is developing very rapidly. Some basic references are [15, 18], and other references to particular kinds of networks include [1, 2, 13, 19]. Examination of the contact network in a disease outbreak situation may lead to an estimate of the probability distribution for the number of contacts [11, 12], and thus to a prediction of the course of the disease outbreak. A recent development in the study of networks in epidemic modeling is the construction of very detailed networks by observation of particular locations. The data that goes into such a network includes household sizes, age distributions, travel to schools, workplaces, and other public
Modeling the Start of a Disease Outbreak
133
locations. The networks constructed are very complex but may offer a great deal of realism. However, it is very difficult to estimate how sensitive the predictions obtained from a model using such a complex network will be to small changes in the network. Nevertheless, simulations based on complicated networks are the primary models currently being used for developing strategies to cope with a potential influenza pandemic. This approach has been followed in [8,9, 10]. An alternative to simulations based on a very detailed network would be to analyze the behaviour of a model based on a simpler network, such as a random network or a scale-free network with parameters chosen to match the reproduction number corresponding to the detailed network. A truncated scale free network would have superspreaders and thus may be closer than a random network to what is often observed in actual epidemics.
5
Conclusions
We have described the beginning of a disease outbreak in terms of the degree distribution of a branching process, and have related this to a contact network. There is a developing theory of network epidemic models which is not confined to the early stages [12, 14]. This involves more complicated considerations, such as the way in which a contact network may change over the course of an epidemic. We have restricted our attention to the beginning of an epidemic in order not to have to examine these complications. Another extension would be to semi-directional networks with disease transmission in only one direction for some edges. For example, person A may go to a hospital only if infected, and may transmit infection to a health care worker B in hospital, but if A is not infected and never goes to the hospital to meet B, then A can not infect B. There are many aspects of network models for epidemics that have not yet been studied. While we have suggested using a deterministic compartmental model once an epidemic is underway, it may be reasonable to go beyond the simplest Kermack-McKendrick epidemic model. Heterogeneity of contact rates, age structure, and other aspects of an actual epidemic can be modeled. Ideally, for the initial stages of an epidemic we would like to use a network somewhere between the over-simplification of a random network and the extreme complication of an individual-based model.
References [lJ R. Albert, A.-L. Barabasi: Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47-97 (2002).
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[2] A.-L. Barabasi, R Albert: Emergence of scaling in random networks, Science 286, 509-512 (1999). [3] P. Erdos, A. Renyi: On random graphs, Publicationes Mathematicae 6, 290-297 (1959). [4] P. Erdos, A. Renyi: On the evolution of random graphs, Pub. Math. Inst. Hung. Acad. Science 5, 17-61 (1960). [5] P. Erdos, A. Renyi: On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12,261-267 (1961). [6] D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Network robustness and fragility: Percolation on random graphs, Phys. Rev. Letters 85,5468-5471 (2000). [7] O. Diekmann, J.A.P. Heesterbeek: Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester (2000). [8] N.M. Ferguson, D.A.T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, D.S. Burke: Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature 437, 209-214 (2005). [9] LM. Longini, M.E. Halloran, A. Nizam, Y. Yang: Containing pandemic influenza with antiviral agents, Am. J. Epidem. 159, 623633 (2004). [10] LM. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D.A.T. Cummings, M.E. Halloran: Containing pandemic influenza at the source, Science 309, 1083-1087 (2005). [11] M.J. Keeling, K.T.D. Eames: Networks and epidemic models J. Roy. Soc. Interface 2, 295-307 (2006). [12] L.A. Meyers, B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, RC. Brunham: Network theory and SARS: predicting outbreak diversity. J. Theor. BioI. 232, 71-81 (2005). [13] A.L. Lloyd, RM. May: Epedemiology: How viruses spread among computers and people, Science 292,1316-1317 (2001). [14] M.E.J. Newman: The spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002). [15] M.E.J. Newman: The structure and function of complex networks. SIAM Review 45, 167-256 (2003). [16] M.E.J. Newman, S.H. Strogatz, D.J. Watts: Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001). [17] S. Riley, C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Raddad, A.J. Hedley, G.M. Leung, L-M Ho, T-H Lam, T.Q. Thach, P.
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Chau, K-P Chan, S-V Lo, P-Y Leung, T. Tsang, W. Ho, K-H Lee, E.M.C. Lau, N.M. Ferguson, RM. Anderson: Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science 300, 1961-1966 (2003). [18] S.H. Strogatz: Exploring complex networks. Nature 410, 268-276 (2001). [19] D.J. Watts, S.H. Strogatz: Collective dynamics of 'small world' networks, Nature 393, 440-442 (1998).
136
Mathematical Techniques in the Evolutionary Epidemiology of Infectious Diseases Troy Day Departments of Mathematics/Statistics fj Biology, Queen's University Kingston, Ontario, K7L 3N6 Canada E-mail: [email protected]
Abstract I provide a brief introduction to two complimentary mathematical approaches for incorporating evolution into epidemiological models. These are referred to as the invasion-analysis technique and the Price-equation technique.
1
Introduction
The epidemiology of infectious diseases is a vibrant and growing area of research. Mathematics has come to play a central role in this field because it allows one to better understand disease dynamics and to assess the utility of different potential control measures. There have been many new developments and extensions of epidemiological models in recent years, including the development of models that account for pathogen evolution. In this chapter I provide a brief overview of two different ways in which the dynamics of pathogen evolution can be incorporated into epidemiological models. The need for incorporating evolution into epidemiological models of infectious disease stems, in part, from the high levels of genetic variation that are often generated through mutation and recombination during individual infections. These different genetic strains can have very different epidemiological characteristics, and therefore an accurate prediction of the epidemiological dynamics often cannot be made without accounting for this variation. Most epidemiological models follow the seminal work of Kermack and McKendrick [12] and are referred to as compartment models [l1J. In such models, the host population is divided into mutually exclusive classes (e.g., susceptible, infected, recovered, etc), and dynamical equa-
Mathematical Techniques in the Evolutionary Epidemiology· . .
137
tions are developed to model the flow of individuals among the different classes. In this way, the dynamics of the pathogen itself are not explicitly modeled, but rather the epidemiological consequences at the level of the host population are tracked, and related to various aspects of infection at the level of the individual host (e.g., transmission rate between hosts, parasite induced mortality rate, recovery rate, etc). Most epidemiological models of pathogen evolution have been developed from compartment models, because the ultimate interest again is typically the epidemiological dynamics at the level of the host population. The challenge, therefore, has been to bridge the scale of pathogen replication and evolution within hosts (which is where all genetic variation is generated), to the scale of pathogen evolution and replication between hosts. Some strains of pathogens might be very good competitors within hosts but very poor at transmitting to new hosts and vice versa. In evolutionary terms, selection on the pathogen population acts at different levels of biological organization (i.e., within-host level and the between-host level). One of the key obstacles to developing a complete theory for the evolution of infectious diseases is this complexity of "multi-level selection" . To make any progress, some simplifications are necessary. Because this chapter is meant to be an introductory overview, I will make an extreme simplification and assume that selection acts only at the between-host level. In other words, I will suppose that any given infected host harbours only a single strain of pathogen at any given time. This strain type might change due to mutation, but I assume that if a mutant strain ever does arise within a host, it either dies out, or it displaces the original strain instantaneously [2, 18, 4]. Consequently, the evolution of the pathogen population occurs solely as a result of differences among strains in their ability to transmit from host-to-host, as well as differences in the mortality they induce and their susceptibility to clearance by host immunological responses. More realistic extensions of the techniques to be presented here have been developed [19, 18, 4, 16, 5], but this simple case is sufficient for introductory purposes.
2
Mathematical models of pathogen evolution
The two approaches for modeling pathogen evolution to be presented will be referred to as the 'Invasion-analysis' technique and the 'Priceequation' technique. The invasion-analysis approach is based on an assumption that evolutionary change is very slow relative to the timescale of the epidemiological dynamics. More specifically, it assumes that the epidemiological dynamics always reach their limiting behavior (typically
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c
assumed to be a point equilibrium) in between the appearance of successive mutations. As a result, there is only ever at most two-strains circulating in the population at any given time. The other key feature of this ~pproach is that it typically seeks to identify the endpoints of evolution only (as will be described below) but not to provide any information about the evolutionary dynamics that occur along the way. These simplifications allow for a relatively complete mathematical analysis of evolution in many contexts. The Price-equation technique is more complex but allows for any time scale of evolutionary change. It supposes that there are multiple strains present at any given time, and that mutations continually occur among these. It then tracks the simultaneous epidemiological and evolutionary dynamics. This is clearly a more realistic approach, but it comes with the drawback that rarely can a complete mathematical analysis be conducted. Rather, this approach is best suited towards providing qualitative insights into the epidemiological and evolutionary dynamics, although numerical analysis can also sometimes be used to gain quantitative insights as well. The above distinction is a bit artificial, and one can readily imagine constructing models with elements of both approaches. Nevertheless this distinction is useful because, in practice, most published models tend to use one or the other approach. In what follows I will use a very simple toy epidemiological model to highlight the main ingredients of each of these two approaches. To give the examples a concrete grounding in some epidemiological question of interest, I will focus on the issue of virulence evolution. This has been a question of considerable interest in evolutionary epidemiology, and is one for which both of the approaches have been used.
2.1
The underlying epidemiological model
Many pathogens infect their hosts without causing substantial damage (e.g., many rhinoviruses that cause the common cold) while others induce higher levels of mortality (e.g., flavivurses that cause dengue fever). One explanation for this variation in pathogen virulence is that the costs and benefits of pathogen-induced mortality vary among pathogens, and this has resulted in the evolution of different levels of virulence. I will work with a simple epidemiological model that is meant to explore this possibility, and to allow predictions of the level of virulence that we expect to evolve. Consider a simple 81 model, where S and I are the numbers of sus-
Mathematical Techniques in the Evolutionary Epidemiology. . .
139
ceptible and infected hosts respectively: dS -=edt
dI
II
B-S(31 ,
~
dt = S(31 -
(2.1)
(J.L + lI)I.
The parameter e is the immigration rate of susceptible hosts, J.L is the per capita background mortality rate of hosts, 1I is the increased mortality rate of hosts due to infection (the virulence), and (3 is the transmission rate (assuming mass-action transmission). This model has two equilibria, one in which the pathogen is absentE1 := (e / J.L, 0) and the other in which the pathogen is endemic E2 := ((J.L+1I)/(3, ~). The diseasefree equilibrium, E 1 , is always biologically feasibl:' whereas the endemic equilibrium, E 2 , is feasible if and only if Ro > 1 where Ro = !l.L. FurJ.LJ.L+v thermore, it can be shown (e.g., by the method of Lyapunov functions; [13]; Appendix) that, when E2 is feasible, it is globally asymptotically stable. Otherwise, El is globally asymptotically stable. It is this model that will be used in presenting the two techniques below.
!v -
2.2
Invasion analysis technique
The technique of invasion analysis a quite general approach for modeling evolution (see [20]) but here I will discuss it within the context of the above epidemiological model. Suppose that all pathogen strains can be characterized by their transmission rate, (3, as well as the level of virulence, 1I, that they induce. Thus, there is also a value of Ro specific to each strain. I will restrict attention to those strains having a value of Ro larger than one. The underlying logic of invasion analysis is as follows. Suppose that a single strain is currently present in the population, and that it has reached its endemic steady state of (2.1). Now imagine that a small number of individuals carrying a second strain are introduced into the population. An invasion analysis seeks to determine whether this new strain invades or dies out. More specifically, it seeks to determine if there is a strain that, once present at endemic levels, can resist invasion by all possible mutants that might arise. If so, it is reasoned that this is a plausible endpoint of evolution because, once here, no further evolutionary change can occur. Such strains are said to be evolutionarily stable (ES). To look for an ES strain for model (2.1) we need to augment this model to allow for a second strain. Using a subscript 'm' to denote the
140
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mutant parameter values, we have
~~ = () dI dt
=
dIm dt
p,8 - 8(31 - 8(3mIm,
8(31 - (11 + v)I,
(2.2)
= 8(3m Im - (11 + vm)Im.
System (2.2) implicitly assumes that the only way in which the two strains interact is through competition for the infection of a common pool of susceptible hosts. As expected, one equilibrium of the augmented system (2.2) is S = fl+ V , i = JL!v -~, and im = 0, and it is the stability of this equilibrium tgat determines whether or not the mutant strain can invade. If and only if this equilibrium is asymptotically stable for all possible mutants is the resident strain ES. Model (2.2) is simple enough that a complete, global analysis is possible (Appendix). For most models, however, only a local analysis can be done and therefore I will focus on such local results here to illustrate the general approach. In this case we are looking for locally evolutionarily stable (LES) strains. A linear stability analysis of system (2.2) at the equilibrium S = flt, i = JL!v -~, im = 0 yields the following Jacobian matrix:
(
-I1,-i(3
-8(3
1(3
-11-v+8(3
o
0
(2.3a)
To appreciate its structure, rewrite matrix (2.3a) in a way that emphasizes its blocktriangular form: (2.3b)
where -0 := (0 0),
u :=
(-8(3m) 0 ' J mut := -11 - Vm
~ + 8(3m,
and J res
is given by J res := (-P,I'-(3i(3
- 8(3 ) -11- v +8(3 .
(2.4)
Thus, the eigenvalues of (2.3a) are simply the eigenvalues of the diagonal blocks, J res and J mut . The notation J res emphasizes the fact that, a linear stability analysis of system (2.1) at the endemic equilibrium E2 yields a Jacobian matrix
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that is exactly (2.4). Because we have assumed that the epidemiological dynamics reach a stable equilibrium when only the resident strain is present, we know that the eigenvalues of (2.4) have negative real parts (throughout I will focus only on hyperbolic equilibria). Therefore, the stability of the equilibrium when we introduce the mutant type is completely determined by the leading eigenvalue of the submatrix J mut . In this case, this submatrix is simply a single element and thus the eigenvalue is trivially equal to r := -p, - Vm + S(3m. Now, as mentioned, the resident strain is LES if and only if this equilibrium is locally asymptotically stable. Thus, we have
+ S(3m < 0, >~,
LES {::} -p, - Vm LES {::}; S
p,+vm
(2.5)
LES {::} R > R m , where LES is shorthand for the statement "The resident strain is locally evolutionarily stable", and where I have defined R = (3/(p, + v) and Rm = (3m/(P, + vm ). Thus, the resident strain is locally evolutionarily stable if and only if it has the largest value of (3d(p, + Vi) of all possible strains, i. These results can be shown to hold globally for this particular model as well (Appendix). This general approach has been used in a very wide variety of epidemiological models to characterize evolutionarily stable strains (see references in [5]). In the present case, we have seen that evolution maximizes a quantity (R in this case) and we can use this fact to elucidate important properties of pathogen evolution. In other more complex models, however, there need not be a simple maximization principle such as this. Rather, the inequality in (2.5) for more complex models often cannot be separated into terms solely involving mutant parameters on one side and resident parameters on the other. In this case, slightly more sophisticated analyses are required that are beyond the scope of this overview (interested readers should consult Otto and Day 2007). Let us now see how the above maximization principle can be used to understand pathogen evolution. To begin, we can immediately see that strains with very high transmission and very low virulence will be best (i.e., they have the largest value of R). For some pathogens, however, it is not possible for a strain to have a high transmission rate without also inducing a high mortality rate [6, 1, 15, 17, 14, 7]. The simplest way to account for this constraint is to suppose that transmission rate is an increasing function of virulence. In this case we can then seek the level of virulence that maximizes R = (3(v)/(p, + v). So long as the function (3(v) increases at a diminishing rate (i.e., d 2 (3/dv 2 < 0), Rwill be maximized at an intermediate level of virulence, v*. Furthermore,
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under these conditions it is easy to see that the ES level of virulence is an increasing function of host mortality rate, JL. In particular, v* must satisfy the following, first order necessary condition: d,6 dv
,6(v) JL
(2.6)
+v 2
Since d,6/dv decreases with increasing v (Le., d ,6/dv2 < 0), a simple implicit differentiation argument using (2.6) shows that that value of v satisfying (2.6) is larger, for larger values of JL. Thus, host species with high levels of natural mortality are predicted to harbour highly virulence pathogens.
2.3
Price equation technique
Price's equation has been widely used in evolutionary biology to model the dynamics of allele frequencies [22, 3, 8, 21]. Recently, however, it has been adapted to model the dynamics of the frequency of different pathogen strains in epidemiological models as well [4, 5]. To describe this approach let us return to the toy model (2.1) and first extend it to allow for n pathogen strains. We have dS
cit
= () -
dI· dt'
= S,6Ji - (JL + Vi) h
JLS - S L-i ,6ih (2.7) Vi E {I, 2, ... , n}.
System (2.7) consists of n+l equations; one for each type of infected host, and one for the dynamics of susceptible hosts. To maintain genetic variation in strains within the population, we now further extend system (2.7) to allow for the occurrence of mutation. Biologically, when mutations arise, they do so in a host that is already infected with some other genotype of pathogen. This then creates a host harbouring more than one type of pathogen. In order to maintain a simplified model in which hosts only ever contain a single pathogen, we therefore assume that such mutations either die out or supplant the original strain instantaneously. 'Mutation' in the model therefore really represents a change from one genotype of infection to another. Thus, as is common in many models of evolutionary epidemiology, we assume that a polymorphism is never maintained within a host [2, 19, 9, 10, 18]. Extending system (2.7) to incorporate this type of mutation, we have dS
cit = () - JLS - S L-i ,6i Ii,
~i = S,6i Ii -
(JL + Vi) Ii - rtIi
Vi,j E {1,2, ... ,n}.
+ rt L-j mjiIj,
(2.8)
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Here ry is the rate at which infections change genotype through mutation, and mji is the probability that, given such a change, an infection of genotype j changes to one of genotype i. System (2.8) completely specifies the dynamics of the different strain types. From an evolutionary standpoint, however, it is often more useful to change variables and to track these dynamics in terms of the frequencies of the different strain types. Defining qi := Id Ir as the frequency of strain type i, where Ir = L::i Ii, we have
dqi dt
-
dIddt dIT/dt Ir Ir = qi (Ti - r) - ryqi + ry L:: j mjiqj,
= ---qi---
(2.9)
where Ti = S(3i - f.l - Vi is referred to as the 'fitness' of strain i, and r := L::i qiTi is the mean fitness of all strains. Let's now step back for a moment and see what we have done. Given n strains of pathogen, we now have n - 1 equations for the dynamics of their frequencies (since L::i qi = 1). To completely specify system (2.8) in these new variables, however, we must also track the dynamics of the total number of infected individuals, as well as the number of susceptible individuals. Using (2.8) this gives
dS
cit = dIT
cit =
B - f.lS - S(3IT , S(3Ir - (f.l + v) Ir,
(2.10)
where x := L::i qiXi, and where I have made use of the fact that L::i mji = 1. System (2.10) provides the final two equations required to completely specify the dynamics (bringing the total number of equations back to n
+1). At this stage it might seem strange to employ this change of variables because it has resulted in a system of equations that is more complex than the original system (2.7). In fact, if we were primarily interested in the dynamics of the number of each strain type this would not be a useful change of variables. If, however, we are primarily interested in the evolution of some characteristic of the pathogen (e.g., its transmission rate or its virulence) then this change of variables does prove to be useful because we can now readily derive equations for the dynamics of the average value of these characteristics across all pathogens. Furthermore, this change of variables has also separated the epidemiological dynamics of the system (given by equations (2.10)) from the evolutionary dynamics of the system (given by equations (2.9)). As we will see, this is also a useful thing to do.
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To derive equations for the evolutionary dynamics of the mean level of virulence and transmission, we simply need to differentiate D = I:i qiVi and 13 = I:i qi(3i with respect to time. Doing so, and using (2.9) to simplify the result, we get [4]
(2.11a)
or (2.11b)
Here O'xy is the covariance between x and y across the pathogen strains that are circulating in the population, and xm := I:i,j Ximjiqj is the average value of trait x among all new mutations. Equations (2.11) are versions of Price's equation [22, 3, 8, 21] and each has a useful interpretation. The average trait value in the population changes as a result of two processes. First, the average trait value changes in a direction given by the sign of the covariance between the trait and fitness. For example, the average value of transmission is driven upward by the fact that strains with large values of transmission, (3, tend to have higher fitness (the term SO'{3{3), but it is also affected by the fact that strains with high virulence have lower fitness, and virulence might be genetically correlated with transmission across parasite strains (the term -a(3v)' Second, the average trait value changes in a direction governed by any mutational bias that might occur (e.g., the term -"1(13 -13m) for the dynamics of
13).
The variances and covariances in system (2.11), O'xy, will also change through time, and equations for these dynamics will typically depend on higher moments of the strain distribution. So in this sense, equations (2.11) cannot be immediately solved to obtain the evolutionary dynamics of the average values of v and (3. Nevertheless, system (2.11) can be used to gain some important insights into pathogen evolution without requiring a full solution. To see how, it is useful to write equations (2.11) in matrix notation:
(2.12)
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145
where G is the genetic (co )variance matrix and ( -1 S) T is termed the selection gradient. The product of G with the selection gradient in equation (2.12) describes the way in which natural selection changes the average level of virulence and transmission in the pathogen population. Natural selection always favours reduced virulence with a strength of -1. On the other hand, natural selection always favours an increased transmission rate with a strength that is proportional to the density of susceptible hosts, S. At equilibrium the force of mutation must balance the force of natural selection, as mediated through the genetic covariance structure of the pathogen population [4]. Interestingly, this formulation also separates the effects of the epidemiological dynamics on evolution (represented here by the selection gradient vector) and the genetic structure of the pathogen population (represented here by the genetic covariance matrix). In the invasion analysis approach, we were only able to infer properties of the endpoint of evolution, and therefore we needed to make an assumption of a tradeoff between virulence and transmission for such an endpoint to exist. Here, however, we can make predictions about the evolutionary dynamics regardless of whether such tradeoffs exist. Of course, we still do not expect there to be an intermediate equilibrium level of virulence unless, ultimately, some sort of constraint between ever increasing transmission and ever decreasing virulence occurs. In the Price-equation approach, this would be manifest as a positive covariance between the two traits. Let us now suppose such a constraint occurs, and return to the question of how natural host mortality rate affects virulence evolution using the Price equation approach. We saw previously (using an invasion analysis) that high mortality selects for the evolution of high virulence. An examination of equations (2.11) however, reveals that host mortality /1 does not enter directly into the evolutionary dynamics. Therefore, such mortality affects virulence evolution only if it indirectly affects either the genetics of the pathogen population, or the selection gradient (-1 S)T. We typically do not expect host mortality to significantly alter the genetics of the pathogen population, and therefore the only way in which host mortality affects virulence evolution is indirectly through the epidemiological dynamics. For example, if we assume that the epidemiological dynamics are always approximately at equilibrium (as in the invasion analysis), then from (2.10) we have S ~ (/1 + D) / jJ. Thus, higher host mortality rates lead, indirectly, to a higher number of susceptible hosts. This, in turn, increases the advantage of strains with higher transmission rate, and to the extent that transmission and virulence are positively correlated with one another, this leads to the evolution of higher virulence. Thus, the Price equation approach provides a more mechanistic picture
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of the factors that govern the evolution of pathogen populations. For example, if we were to test these predictions by experimentally elevating host mortality rate and measuring evolutionary changes in pathogen virulence, the invasion analysis would lead us to believe that higher virulence is always expected. The Price equation approach, however, reveals that this will only be true if our experimental manipulation allows this change in host mortality to indirectly feed back, through the epidemiological dynamics, to elevate the number of susceptible hosts (something that might well not occur in all experimental systems).
3
Discussion
This chapter has presented a brief overview of two different approaches to modeling evolutionary epidemiology. I have concentrated on a very simple toy epidemiological model that is not meant to represent the dynamics of any particular disease. Rather, it is simply meant as a tool to elucidate the similarities and differences of the two techniques. The interested reader should consult [4, 5] for more complex and realistic examples. The invasion analysis technique presented is somewhat restricted in its scope and does not provide complete information about the mechanistic details driving evolution in pathogen populations. It main advantage, however, is that it provides an analytically tractable approach for modeling pathogen evolution. The Price equation approach is more general in the sense that it makes fewer restrictive assumptions, and it also provides a more complete description of the mechanistic details of evolution. Its primary drawback, however, is that it rarely allows for a complete, analytical treatment of pathogen evolution. Rather, it is best at providing qualitative insights. As such these two approaches are best viewed as complimentary techniques.
Appendix Consider the following function
V(8, 1)
=8
-
8 In (818) + 1 -
j In (111),
(A1)
where Sand j are the equilibrium values of 8 and 1 at the endemic equilibrium, E 2 · Expression (A1) is a Lyapunov function for system (2.1) [13]. To see this, note that this function has a unique minimum in 8 and 1 at the equilibrium values Sand 1. Therefore, all we need to show is that dV/dt :::;; 0 along all trajectories of system (2.1), with
Mathematical Techniques in the Evolutionary Epidemiology··· equality holding only when S using equations (2.1) gives
d V dt
=
I
= 1.
dS + (1-!) ( S8) - ~
(1-~) S
=e
= 8,
dt
I
147
Differentiating (A1), and
dI dt (A2)
J-l S + J-l S - j3 S 1+ (J.l + 1/)1.
1-
A
A
Now, using the fact that j3Si = j38i~ = (e-J-l8)~ and e = J-l8+(J-l+I/)i, (A2) can be re-written as
dV dt
= e (2 -
§.) :;(
~S _ 8 '" 0,
(A3)
s.
with equality holding only when S = We can also use an extension of the Lyapunov function provided by [13] to demonstrate that the local results given in (2.5) of the text, hold globally as well. Consider the following function
V(S, I, 1m) = S -
8 In (SI8) + I
- i In (I I i)
+ 1m ,
(A4)
where Sand i are the equilibrium values of S and I at the endemic equilibrium, E 2 • Expression (A4) has a unique minimum in S, I, and 1m at (8, i, 0). The time derivative of (A4) along trajectories of system (2.2) is
(A5)
Now suppose that R > Rm. Expression (A5) is then clearly negative everywhere except when S = 8, 1m = 0, in which case it is zero. Thus, (A4) is a Lyapunov function for equilibrium (S, i, 0) of system (~.~) if R> Rm. If R < R m , then we already know that the equilibrium (S, I, 0) is unstable. Thus, result (2.5) of the text holds globally.
References [1] R.M. Anderson, R.M. May, Coevolution Of Hosts and Parasites, Parasitology, 85 (1982), 411-426.
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[2] S. Bonhoeffer, M.A. Nowak, Mutation and the Evolution of Virulence, Proceedings of the Royal Society of London Series BBiological Sciences, 258 (1994), 133-140. [3] M. Bulmer, Theoretical evolutionary ecology, Sinauer Associates, Sunderland Massachusetts, 1994. [4] T. Day, S. Gandon, Insights from Price's equation into evolutionary epidemiology, in Feng, Dieckmann and Levin, eds., Disease Evolution: Models, Concepts, and Data Analysis, AMS, 2006. [5] T. Day, S.R Proulx, A general theory for the evolutionary dynamics of virulence, American Naturalist, 163 (2004), E40-E63. [6J D. Ebert, Virulence and Local Adaptation of a Horizontally Transmitted Parasite, Science, 265 (1994), 1084-1086. [7] D. Ebert, K.L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite, Evolution, 51 (1997),1828-1837. [8] S.A. Frank, George Price's Contributions to Evolutionary Genetics, Journal of Theoretical Biology, 175 (1995), 373-388. [9] S. Gandon, V.A.A. Jansen, M. van Baalen, Host life history and the evolution of parasite virulence, Evolution, 55 (2001), 1056-1062. [10] S. Gandon, M. van Baalen, V.A.A. Jansen, The evolution of parasite virulence, superinfection, and host resistance, American Naturalist, 159 (2002), 658-669.
[11] H.W. Hethcote, The mathematics of infectious diseases, Siam Review, 42 (2000), 599-653. [12] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics, part 1, Proceedings of the Royal Society of London, Series A (1927), 700-72l. [13J A. Korobeinikov, G.C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Applied Mathematics Letters (2002), 955-960. [14] M. Lipsitch, E.R Moxon, Virulence and transmissibility of pathogens: What is the relationship?, Trends in Microbiology, 5 (1997), 31-37. [15] M.J. Mackinnon, A.F. Read, Selection for high and low virulence in the malaria parasite Plasmodium chabaudi, Proceedings of the Royal Society of London Series B-Biological Sciences, 266 (1999), 741-748. [16] RM. May, M.A. Nowak, Coinfection and the Evolution of Parasite Virulence, Proceedings of the Royal Society of London Series BBiological Sciences, 261 (1995), 209-215.
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[17] S.L. Messenger, 1.J. Molineux, J.J. Bull, Virulence evolution in a virus obeys a trade-off, Proceedings Of the Royal Society Of London Series B-Biological Sciences, 266 (1999), 397-404. [18] J. Mosquera, F.R. Adler, Evolution of virulence: a unified framework for coinfection and superinfection, Journal of Theoretical Biology, 195 (1998), 293-313. [19] M.A. Nowak, R.M. May, Superinfection and the Evolution of Parasite Virulence, Proceedings of the Royal Society of London Series B-Biological Sciences, 255 (1994), 81-89. [20] S.P. Otto, T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press, Princeton, N.J., U.S.A., 2007. [21] G. Price, Selection and covariance, Nature, 227 (1970), 520-52l. [22] G. Price, Extension of covariance selection mathematics, Annals of Human Genetics, 35 (1972), 485-490.
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The Uses of Epidemiological Models in the Study of Disease Control Zhilan Feng* Department of Mathematics, Purdue University West Lafayette, IN 47907, USA E-mail: [email protected]
Dashun Xu Department of Mathematics, Southern Illinois University Carbondale, IL 62901, USA E-mail: [email protected]
Haiyun Zhao Department of Mathematics, Purdue University West Lafayette, IN 47907, USA E-mail: [email protected]
Abstract Recently many mathematical models have been used to study the effectiveness of quarantine and isolation as control measures for the spread of infectious diseases. Most of the deterministic models have focused on the use of ordinary differential equation (ODE) models with the assumption of exponentially distributed disease stages. In this paper we demonstrate that some of these models may generate misleading predictions. We formulate a general integral equation model which assumes an arbitrarily distributed disease stage for both the latent and the infectious stages, and show that it reduces to the commonly used SEIR model (ODE) with quarantine and isolation when the stage distributions are exponential. The general model reduces to another ODE model when the disease stages are assumed to have a gamma distribution. The control reproductive number Rc for each model is calculated. The effect of control strategies of these models is compared in terms of both Rc and the final epidemic size. We demonstrate that the two ODE models can produce conflict pre-This author is partly supported by NSF DMS-0314575.
The Uses of Epidemiological Models in the Study of . . .
151
dictions regarding the effectiveness of disease control strategies via quarantine and isolation.
1
Introduction
Recently, many mathematical models have been used to investigate how to more effectively control emerging and reemerging infectious diseases such as SARS and smallpox via various disease control measures including vaccination, quarantine, and isolation (see, for example, Chowell, et al., 2003; Lipsitch, et al., 2003; McLean, et al., 2005; Riley, et al., 2003). Most of the studies have chosen to use simple deterministic ODE models of the SIR or S E I R type or their variations. While such simple models can very often provide many important insights into the disease transmission dynamics and have been used to address other important biological questions, the underlying assumptions used to obtain the simplicity of these model need to be examined carefully in order to make sure that the model predictions are reliable. One of the most commonly used simplifying assumptions is the exponential distribution assumption (EDA) on disease stages. In Feng et al. (2006), the appropriateness of using simple SEIR type of models to assess disease control strategies is investigated. They considered several epidemiological models with quarantine and isolation, but only a special case is considered b = 0). In this paper, we study a more general case in which early quarantine is included b> 0). The most basic and simple model is an extension of the standard SEIR compartmental model with the inclusion of quarantine (Q) and isolation classes, which is an ODE model (see (2.1)). This model uses the EDA for both the latent and infectious stages. Another ODE model with a gamma distribution for the disease stages is also considered (see (2.2)). The gamma distribution assumption (GDA) in general provides a more accurate description of the epidemiological process than the exponential distribution assumption. Hence, in many cases the gamma distribution model (GDM) is more realistic than the exponential distribution model (EDM). On the other hand, the GDM is more complex than the EDM and the mathematical analysis is more difficult. We demonstrate that the two models may produce conflict predictions, suggesting that the simple EDM may not be appropriate to use for assessing disease control programs. Following the approach of Feng et al. (2006) we also consider a general integral equation model by using arbitrarily distributed disease stages. It is shown that the general model reduces to the EDM or the GDM when the corresponding disease stage distribution is used. The results show that similar conclusions obtained in Feng et al. (2006) still hold for the case of"( > o.
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2
Zhilan Feng, Dashun Xu, Haiyun Zhao
Models with quarantine and isolation
In the standard S E I R model, the total population is divided into four epidemiological classes: the susceptible (S), exposed (E, individuals who are infected but not yet infectious), infectious (I, individuals who are capable of transmitting the disease), and removed (R). Under the assumption that the latent and infectious stages are exponential distributed the corresponding ODE model is
S' = p,N - {3Sf:t - p,S, E' = {3Sf:t - (a1 + p,)E, I' = alE - (51 + p,)I, R' = 511 - p,R.
(2.1)
" I " denotes the derivative with respect to time t. {3 is the transmission coefficient, a1 is the rate at which a latent individual becomes infectious, 51 is the recover rate and p, is the natural death rate. No disease-induced death is considered. All involved parameters are nonnegative constants, and all variables and parameters are listed in Table 1. Many extensions ofthe simple model (2.1) have been used to address important biological questions. One of such examples is obtained by incorporating quarantine and isolation as below:
S' = p,N - ).,(t)S - p,S + rSQ, So = (1 - b)).,(t)S - rSQ, E' = (1 - "()b).,(t)S - (X + a1 + p,)E, Q' = "(b).,(t)S + XE - (a2 + p,)Q, l' = alE - ( + 51 + p,)I, H' = a2Q + 1 - (52 + p,)H, R' = 511 + 52H - p,R, where
).,(t)
=
(3 [I(t)
+
(1; P)H(t)].
(2.2)
(2.3)
).,(t) denote the force of infection (a specific form is given below). In this model it is assumed that a fraction b of contacts (susceptible individuals who have had contacts with an infectious person) is actually infected, and that the other fraction (1- b) of contacts remains susceptible who will be quarantined (SQ) and will return to the S class at a rate r (see, e.g., Lipsitch, 2003). Among the infected individuals (b).,(t)S) a fraction "( will be quarantined (Q) at the early stage of infection (i.e., there is a rate, "(b).,(t)S, from S to Q directly). The fraction (1 - "()
The Uses of Epidemiological Models in the Study of . . .
153
of the exposed individuals who are not quarantined at the beginning of infection will be quarantined at a constant rate X throughout the latent period. The non-quarantined and quarantined (exposed) individuals will progress to the infectious stage at constant rates Q:1 and Q:2 respectively (the relationship between Q:1 and Q:2 will be discussed later). Infectious individuals will be isolated (H) at a rate ¢> and individuals in the H class will recover at a rate (h (the relationship between 81 and 82 will be discussed later). p E [0,1] is the fraction of reduction in the transmission rate of isolated individuals with p = 1, P = 0, and 0 < p < 1 representing a completely effective, completely ineffective, and partially effective isolation, respectively. It is known that the ordinary differential equation model (2.2) implicitly assumes the exponential distribution for the latent and infectious stages. More precisely, the exponential functions pE(S) = e- a1S and PJ(s) = e- O. This corresponds to a more aggressive control program which encourages early identification and quarantine of exposed individuals.
To incorporate early quarantine we modify the integral equations model in Feng et al. (2006) by allowing a fraction 'Y > 0 of infected individuals to be quarantined at their early exposure. The remaining fraction 1 = 1 - 'Y is moved into the latent class E and these individuals will either progress to enter the infectious class J or get quarantined at some later stage age s > O. The quarantined (infected) individuals will progress to the disease stage and will remain isolated. The effect of early quarantine will in general reduce the prevalence of the disease since the quarantined individuals will have a reduced transmission rate due to limited contacts with the general population. We adopt the same assumption as in Feng et al. (2006) to assume that b = 1. Then our model reads:
S(t)
=
lot p,Ne-/-L(t-s)ds -lo\(s)s(s)e-/-L(t-S)dS + Soe-/-L t ,
E(t)
=
1lot >..(s)S(S)PE(t - s)k(t - s)e-/-L(t-s)ds + E(t),
Q(t) = 11t
17
>..(s)S(s) [-PE(T - S)k(T - S)]PE(t - TIT -
xe-/-L(t-S)dsdT + 'Y
It
s)
>..(s)S(S)PE(t - s)e-/-L(t-S)ds + Q(t),
J(t) = 1lot lo1" >..(s)S(s) [-FE(T - S)k(T - s)]Pr(t - T)l(t - T) xe-/-L(t-S)dsdT + J(t), H(t)
=
1lot loU lo7 >..(s)S(s) [-FE(T - S)k(T - s)] X
[-Pr(u - T)i(u - T)]Pr (t - ulu - T)e-/-L(t-S)dsdTdu
t +11 lo7 >"(s)S(S)[-FE(T - S)k(T - s)]Pr(t - T)e-/-L(t-S)dsdT
+'Y R(t) =
r r>"(s)S(S)[-FE(T - s)]Pr(t - T)e-/-L(t-S)ds + H(t) '
io io
It17
>"(s)S(S)[-FE(T - s)][l - Pr(t - T)] xe-/-L(t-S)dsdT + R(t), (2.5)
The Uses of Epidemiological Models in the Study of . . . where >..(t)
I(t)
+ (1 -
155
p)H(t)
N . In this model, PE,P] : [0,00) -+ [0,1] describe the durations of the exposed (or latent) and infectious stages, respectively. More precisely, Pi(S) (i = E, I) gives the probability that the disease stage i lasts longer than S time units (or the probability of being still in the same stage at stage age s). Therefore, the derivative - Pi (8) (i = E, 1) gives the rate of removal from the stage i at stage age 8 by the natural progression of the disease. These duration functions have the following properties =
(3
It is assumed that latent individuals of stage age
8 are quarantined according to a given distribution described by k(8) : [0,00) -+ [0,1] with k(O) = 1 and k(oo) = o. That is, k(8) denotes the probability that exposed individuals have not been quarantined at stage age 8. Similarly, the function l(8) : [0,00) -+ [0,1] with l(O) = 1 and l(oo) = 0 describes the probability that infectious individuals have not been isolated at stage age 8. f.1 is the natural death rate and the disease-induced mortality is ignored. All variables and parameters are listed in Table 1. X(t) = Xo(t)e-/l t + Xo(t) (X = Q, I, H, R) which depends only on initial data (see Feng et al. 2006 for more detailed description of these functions). Obviously X(t) -+ 0 as t - 00. It can be shown that under standard assumptions on initial data and parameter functions the system (2.5) has a unique nonnegative solution defined for all positive time.
3
Reproductive numbers of the general model
We can calculate the effective (or control) reproductive number R c , and the basic reproductive number Ro in the absence of control. To see the biological meaning of the expression of Rc we introduce the following quantities:
(3.1)
156
Zhilan Feng, Dashun Xu, Haiyun Zhao Table 1: Definitions of frequently used symbols
Symbol S(t)
Definition Number of susceptible individuals at time t
SQ(t)
Number of susceptible individuals quarantined at time t
E(t)
Number of exposed (not yet infectious) individuals at time t
Q(t)
Number of quarantined (exposed) individuals at time t
I(t) H(t) R(t)
Number of susceptible individuals at time t
N
Total population size (constant)
Number of isolated (infectious) individuals at time t Number of recovered individuals at time t
C(t)
Number of cumulative new infections at time t
>.(t)
Force of infection at time t
f3
Transmission coefficient
ai, a
Rate at exposed individuals become infectious
8i , 8
Rate at which infectious individuals recover
f-t
Natural death rate
X, "*S*, H*
I*=;YTEkDIz>"*S*,
(3.9)
= (TEDI - TEkDIz +,TEkDIz)>"*S*, R* = J:...TE'h>"*S*,
where >..*
p,
= p,(Rc -
1). Obviously, U* exists only if Rc > 1. Similar
techniques used in Feng et al. (2006) can be applied to here to show that the reproductive number Rc determines the existence and stability properties of the equilibrium points of system (2.5).
4
Comparison of models with different sojourn distributions
It is easy to verify that, in the special case when PE(S) and PI(S) are exponential functions (i.e., e- as and e- 8s respectively), the system (2.5) reduces to the the following ODE model which we will refer to as the ED M (exponential distri bu tion model):
S'
= p,N - >..(t)S - p,S + rSQ,
+ a + p,)E, Q' = ,>"(t)S + XE - (a + p,)Q, I' = aE - (1) + 8 + p,)I, H' = aQ + ¢I - (8 + p,)H, R' = 8I + 8H - p,R.
E' = ;Y>"(t)S - (X
(4.1)
The Uses of Epidemiological Models in the Study of . . .
159
where A(t) is the same as given in (2.3). Note that this system has the same dynamic behavior as system (2.2) with b = I, a1 = a2 = a and 01 = 02 = o. If we assume that in the model (2.5) PE(S) and PI(S) are gamma distributed with parameters a and 0 respectively, i.e., m-1
k
(mas) PE (s ) =e -mQs" ~ l' k.
k=O
- -nos PI (s ) -e
n-1
L
k=O
k
(nos) k! '
then the model (2.5) simplifies also to an ODE model which we refer to as the GDM (gamma distribution model):
8' = f-LN - A(t)8 - f-L8, Ei
= ;YA(t)8 -
(X
+ ma + f-L)E1'
Ej=maEj-1-(x+ma+f-L)Ej, Q~ = 1'XE1 - (ma
j=2,···,m,
+ f-L)Q1,
= XEj + maQj-1 - (ma + f-L)Qj, j = 2,··· ,m, Ii = maEm - (¢ + nO + f-L)h, Ij = nOIj _ 1 - (¢ + nO + f-L)Ij, j = 2,··· ,n, Hi = maQm + ¢h - (no + f-L)H1, Hj = noHj _ 1 + ¢Ij - (no + f-L)Hj , j = 2,··· ,n, R' = nOin + noHn - f-LR, Qj
with I
=
2::7=1 I j ,
H
=
(4.2)
2::7=1 H j ,
where A(t) is the same as given in (2.3). We remark that the EDM is a special case of the GDM when m = n = 1. It was demonstrated in Feng et al. (2006) that the EDM and the GDM may provide contradictory predictions. Here we consider similar comparisons for the case l' > o. We first compare the model predictions using the reproductive number Re. From formulas (3.3), (3.4), and the exponential functions for PE and PI we get Rc for the EDM: (4.3) Similarly, using the gamma distribution function for PE and PI we get
Zhilan Feng, Dashun Xu, Haiyun Zhao
160 'Re for the GDM:
(ma)m (3 ~ (n8)j 'Re = (p, + ma)m p, + n8 (p, + n8)j
f;:o
X
[1- P ( 1
.
_ (1- / )(p, + ma)m p, + n8 (II. + ma + X)m p, + n8 + cP t'"
"n-l (no)' 6j=0 (f.I+noH)j ) ] .
~
"n-l 6)=0 (f.I+ n8 )J
(4.4) The plot of'Re for the EDM and GDM is shown in Figure 4.l. It is clear from Figure 4.1 that, similar to the case of 1 = 0 (see Figure 4.1(a) and (b)), the two models again produce inconsistent conclu1.5,,------------
1.5
~-----------
'Rc(~).X=O.l
-
-
'Rc(x). ~=0.1
'\:'. "''.
- - - - 'Rc(x.¢).x=¢
0.5
-
'Rc(¢). X= 0.1 'Rc<X). ~=0.1
\
- - - - 'Rc(X.~).X=~
0.5 (a)EDM
(b)GDM
0.1
0.2
OJ
X or
\ \
\
0.4
0.5
0.1
0.2
~
OJ
0.4
0.5
X or ¢
-
- 'Rc(~). X=r=0.2
-
'Rc(X). ~=r=0.2
-
-
'R c (¢)·x=r=O.2 'R c (X).¢=Y=0.2
-·-·'Rc(X.~).X=~,r=0.2
_._. 'Rc(X. ¢). X=~. Y= 0.2
...... 'Rc(n.X=~=0.2
...... 'Rc(r). X=~=0.2
\ \
~\
0.5
0.5
(c)EDM 0.1
(d)GDM 0.2 0.3 X. ~ or r
0.4
0.5
0.1
0.2 0.3 X. ~ or r
0.4
0.5
Figure 4.1: Comparison of the EDM (see (a) and (c)) and the GDM (see (b) and (d)) on the effect of various control measures in the reduction of the reproductive number 'Re. (a) and (b) are for the case of 1 = 0, and (c) and (d) are for the case of I> O. Both of the cases show that the two models are inconsistent regarding the effectiveness of control measures (represented by X, cP and ,) in the reduction of'Re.
The Uses of Epidemiological Models in the Study of ...
161
sions for 'Y > 0 (see Figure 4.1(c) and (d)). For example, in Figure 4.1(a) and (b), Rc for both models is plotted either as a function of ¢ for a fixed value of X = 0.1, or as a function of X for a fixed value ¢ = 0.1, or as a function of both X and ¢ with X = ¢. For any vertical line except the one at 0.1, the three curves intersect the vertical line at three points that represent three control strategies. The order of these points (from top to bottom) determines the order of effectiveness (from low to high) of the corresponding control strategies since a larger Rc value will most likely lead to a higher disease prevalence. The order of these three points (labeled by a circle, a triangle and a square) predicted by the EDM and the GDM is clearly different for the selected parameter sets, suggesting conflict assessments of interventions between the two models. Similar conclusions can be drawn for the case of 'Y > 0 by observing the curves of Rc as functions of X, ¢, or 'Y shown in Figure 4.1(c) and (d). Other parameter values used in Figure 4.1 are f3 = 0.2, P = 0.8, a = 1/7, and 8 = 1/10, corresponding to a disease with a latency period of l/a = 7 days and an infectious period of 1/8 = 10 days (e.g., SARS). We can also compare the two models by looking at the final epidemic size (or the cumulative number of infections during an epidemic), 0, which is defined by the equation
O'(t) = )..(t)S(t). Let 0(0) = 0 so that O(t) is the cumulative number of new infections at the end of an epidemic. As in Figure 4.1, the final size 0 (i.e., its value at the end of an epidemic) for both models is plotted as a function of one or two control parameters which are shown in Figure 4.2. The case 'Y = 0 is shown in Figure 4.2 (a) and (b)), and the case of'Y > 0 is shown in Figure 4.2 (c)-(f). Again, the EDM and GDM predict inconsistent outcomes for both cases. We also observe that the final size as a function of X and/or ¢ is not very sensitive to changes in 'Y (e.g., 'Y = 0.1 in Figure 4.2(c)(d) and 'Y = 0.5 in Figure 4.2(e)(f)). This is probably because of the value of p being small (or equivalently 1 - P being large), in which case the role of quarantine/isolation is limited. In fact, from the formulas (4.3) and (4.4) we can see that as a function of 'Y
Reb) for the EDM, and
~
a f3 - - - - ( 1 - p) /-l+a/-l+8
Zhilan Feng, Dashun Xu, Haiyun Zhao
162
200r--.....--~----;!::=:;:::;:::;;;::::::::;:::::::;-,
15
150
.~
.~
£
£
\~
\
"-
10
100
.. :
.......:... ..
.-.".
"-
\ ........ :._ ...
(0) EDM
(b)GDM
500L---~0~.2----70.74----~0.~6----~0.~8--~ 5~L---~0~.2----70.74----70.~6----~0.~8--~ r,¢ or X
y, $ or X
.~
\ "\
10 . . . . . . . . . . . . "'---"
---=-- ...
(c) EDM
500
......
. ........... '.
---..
:
(d) GDM
0.2
0.4
0.6
50 0
0.8
0.2
0.4
0.6
0.8
0.6
0.8
r,¢orx
r,¢orx ._. X-II.], ,all.1,varyt118 T
-x- O.2,y-O.S,\IM}'II>I' -,-0.2,,.-0.5 • ._. r~O.5,
.....,...,%
.
.~
.~
"ii
"ii
£
.5
'\
"-
""
'"
'""- ......
~
......
--
...:..
-
-
"- : .......
(e) EDM
500
0.2
:
(f) GDM
0.4
0.6
r,¢orx
0.8
50 0
0.2
0.4
r,¢orx
Figure 4.2: The effect of various control mearsures on the final size predicted by the EDM (see (a), (c) and (e)) and the GDM (see (b), (d) and (f)). (a) and (b) are for the case of"( = 0 while the others for the case of"( > O. For a convenient comparison, the curves for the final size C as a function of"( are plotted in all figures.
for the GDM. These inequalities also suggest that if 1 - P is not small enough, control via early quarantine b) may be impossible.
The Uses of Epidemiological Models in the Study of ...
163
To examine in more detail the difference between the two models, we can also look at Figure 4.3, where the cumulative number G(t) is plotted versus time t for various disease control measures. The EDM predicts that increasing ¢ from 0.1 to 0.2 leads to a reduction of 414 cases out of 662 of the final size, while increasing X from 0.1 to 0.2 leads to a reduction of 388 cases. This suggests that isolation might be more effective than quarantine (under the assumption that control effort is represented by the parameter values). On the other hand, the GDM predicts that the final size is reduced by 542 cases out of a total of 817 cases if ¢ is increased, and it is reduced by 592 if X is increased by the same value. This suggests that quarantine might be more effective than isolation. All parameters except those highlighted in the figures have the same values as in Figure 4.1 . 900
700 (a) EDM
800
600
(b) GDM
700 500
600
400
S
S
"
-
300
50
-- - - -
--
--
"
500 400 300
150 200 250 300 350 400 450 500 Time
100 150 200 250 300 350 400 450 500 Time
500r;:;r;;;::;:-~~-:::::::======t 450 (e) EDM
(d) GDM 500
400 350
S "
400
300
s
"
250
50 o
OL5"'0~1OC:-0--:1~50~20-:-0-;:2-:::50~3oo-:--:3~50=4~00~45§0~5oo Time
50
100 150 200 250 300 Time
Figure 4.3: The effect of various control mearsures on the final size predicted by the EDM (see (a) and (c)) and the GDM (see (b) and (d)). (a) and (b) are for the case of "I = O. (c) and (d) are for the case of "I = 0.1. We have also conducted stochastic simulations by looking at the final epidemic size based on the corresponding deterministic models. Some of the simulation results are shown in Figure 4.4. We observe that the
Zhilan Feng, Dashun XU, Haiyun Zhao
164
140.----~-_-~-~-~-~___,
140
(b)GDM
(a) EDM
120
120 100 ~80
e
60
0
0
40
20
60
100
80
120
140
160
100
20
.~
120
140
Time
Time -;=0.3 t-:~~·31
=41:0:3 ~
15
:l
.~
OJ
~
.5 "10
.
~
"
"100
"-
-
.....
(c)EDM
50 0
0.2
" ,,\ -
.....
~
-
(d)GDM
0.4
0.6 Xor
q,
0.8
50
0
0.2
0.6
0.4 Xor
q,
0.8
Figure 4.4: Three sample realizations are illustrated in the upper panel for the EDM and the GDM. The final epidemic sizes predicted by the EDM and the GDM under different control strategies are shown in the lower panel. For each fixed parameter value, the plotted final size is the mean of 200 realizations. The parameter values are (3 = 0.2, p = 0.8, c¥ = 1/7,0= 1/10, "y = O. In the upper panel, X = 0.3 and ¢ = 0.2. outcomes from stochastic simulations are consistent with that of the deterministic model. All parameter values have similar values as in other figures.
5
Conclusion
In this paper, we extended the results of Feng et al. (2006) by considering early quarantine of exposed individuals h > 0). The qualitative results seem unchanged regarding the difference between the simple and more complex models in predicting the effectiveness of disease control strategies. As we did in Feng et al. (2006), we formulated a general integral equations model (2.5) which assumes arbitrarily distributed dis-
The Uses of Epidemiological Models in the Study of· . .
165
ease stages. This general model reduces to the EDM-a simple ODE model of the SEI R type with quarantine and isolation-when the stage distributions are exponential. By demonstrating that the EDM may generate conflicting outcomes when compared with the GDM, another reduction of the general model when the stage distributions are gamma, we show that simple models (such as the EDM in this paper) may not be appropriate to use for assessing disease control strategies. A more detailed discussion on the drawbacks of the assumption of exponentially distributed disease stages is given in Feng et aI. (2006). Although the gamma distribution can provide in many cases a more reasonable description of the epidemiological processes, it may not solve the problem completely. Our general model allows us to consider other disease stage distributions within the same modeling structure, and hence makes it possible to compare outcomes from simple and more complex models.
References [1) Chowell, G., P. Fenimorea, M. Castillo-Garsowc and C. CastilloChavez. 2003. SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. J. Theor. BioI. 224: 1-8. [2) Feng, Z., W. Huang and C. Castillo-Chavez. 2001. On the role of variable latent periods in mathematical models for tuberculosis. Journal of Dynamics and Differential Equations. 13: 435-452. [3) Feng, Z. and H.R. Thieme. 2000a. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General Theory. SIAM J. Appl. Math. 61 (3): 803-833. [4J Feng, Z. and H.R. Thieme. 2000b. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages II: Fast disease dynamics and permanent recovery. SIAM J. Appl. Math. 61 (3): 983-1012. [5) Feng, Z., D. Xu, and H. Zhao. 2006. Epidemiological models with non-exponentially distributed disease stages and applications to disease control. Bulletin of Mathematical Biology. To appear. [6] Hethcote, H. and D. Thdor. 1980. Integral equation models for endemic infectious diseases. J. Math. BioI. 9: 37-47. [7J Hethcote, H., H.W. Stech and P. van den Driessche. 1981. Nonlinear Oscillations in Epidemic Models. SIAM J. Appl. Math. 40 (1): 1-9.
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[8] Lipsitch, M., T. Cohen, B. Copper et aI. 2003. Transmission dynamics and control of severe acute respiratory syndrome. Science. 300: 1966-70. [9] Lloyd, A. 200la. Realistic distributions of infectious periods in epidemic models. Theor. Pop. BioI. 60: 59-7l. [10] Lloyd, A. 2001b. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R Soc. Lond. B. 268: 985-993. [l1J MacDonald, N. 1978. Time lags in biological models. SpringerVerlag. New York. [12] McLean, A.R, M.M. Robert, J. Pattison and RA. Weiss. 2005. A case study in emerging infections. Springer-Verlag. Oxford University Press. New York. [13] Plant, RE. and L.T. Wilson. 1986. Models for age-structured populations with distributed maturation rates. J. Math. BioI. 23: 247262. [14J Riley, S., C. Fraser, C.A. Donnelly, A.C. Ghani et aI. 2003. Transmission dynamics of the etiological agent of SARS in Hong Kong impact of public health interventions. Science. 300: 1961-1966. [15] Taylor, H.M. and S. Karlin. 1998. An Introduction to Stochastic Modeling. Third ed. Academic Press. San Diego. [16J Thieme, H. 2003. Mathematics in Population Biology. Princeton University Press. Princeton.
167
Assessing the Burden of Congenital Rubella Syndrome and Ensuring Optimal Mitigation via Mathematical Modeling John W. Glasser Division of Viral Diseases National Center for Immunization and Respiratory Diseases Centers for Disease Control and Prevention Atlanta, Georgia 30333, USA E-mail: [email protected]
Maureen Birmingham Department of Communicable Diseases Surveillance and Response, South East Asia Regional Office World Health Organization c/o Ministry of Health N onthaburi 11000, Thailand E-mail: [email protected]
Abstract Congenital rubella syndrome (CRS) comprises a panoply of lesions, some fatal and others diagnosed too long after birth to be readily associated with first trimester maternal infections. Consequently, this syndrome is under-ascertained even by reviewing medical records for compatible conditions and confirming them serologically. Mathematical modeling provides an alternative to such studies, while also permitting health authorities to estimate the burden of this disease relatively easily and to evaluate means by which it might be mitigated. Our estimates of CRS from disease or serological surveillance and demographic information compare favorably to those from retrospective chart reviews in children's hospitals and clinics specializing on afflictions that may result from infection in utero. The CRS burden could be mitigated either by reducing susceptibility to infection on exposure among women of childbearing age (WCBA) or their risk of exposure. The first approach might involve vaccinating girls attending school or mothers delivering in hospital, neither of which however is universal. The second necessarily involves childhood vaccina-
168
John W. Glasser, Maureen Birmingham tion, via routine age-appropriate doses, mass campaigns, or both. But if sufficient coverage is not sustained, the immunization of some children only temporarily protects others, whose susceptibility may persist into adulthood. Unless vaccinated, their own children may be infected while playing with neighbors or attending school, and infect them. The females among these susceptible parents may be pregnant. The potential of childhood vaccination to exacerbate the future burden of CRS has been known since the early 1980s, but casually dismissed for lack of evidence that could not be apparent for some time. However, examples have recently begun appearing in the literature. Romania exemplifies a newly-recognized hazard: rubella occurs annually, but major outbreaks typically occur every 5 years. The period of these multiannual cycles doubled as falling birth rates decreased the supply of susceptible children, increasing the mean age of infection, and concomitantly CRS, a phenomenon that may be anticipated in China due to efforts to curb population growth (Gao and Hethcote 2006). We describe models with which possible strategies for mitigating the burden of CRS have been evaluated in these and other developing countries. Given estimated costs of feasible tactics, policymakers could consider whether to devote their scarce resources to this or other health problems. We encourage monitoring of disease or serology to ensure policy objectives are attained, even if modeling aided in vaccination program design.
1
The diseases
Rubella is a mild rash illness that generally afflicts children. But congenital rubella syndrome (CRS) - which includes spontaneous abortions, stillbirths, brain (microcephaly, mental retardation), cardiac, hearing, liver or spleen (hepatosplenomeglia), and visual (cataracts, glaucoma) deficits - may afflict the progeny of women infected during gestation, particularly their first trimesters. The nature of the lesion or combination presumably is determined by fetal ontogeny when mothers-to-be are infected (Cooper et al. 1969, Ueda et al. 1979, White et al. 1969).
2
Risk of infection
Cutts and Vynnycky (1999) compared a model in which the risk among susceptible people, termed force of infection, was constant (i.e., k in dp(a)/da = k[l- p(a)], where p denotes proportion immune, a age, and dp(a)/da is the rate of change with age), with a model in which this risk was allowed to vary between age intervals. They argued that models with constant risks were best. Because activity and disposition to interact
Assessing the Burden of Congenital Rubella Syndrome and···
169
with others roughly the same age peak during childhood, we estimate the coefficients of catalytic models (Muench 1959) from which age-specific forces of infection can be derived. Insofar as we choose models whose terms have significant likelihood ratios, our approach not only makes more sense biologically, but makes best use of available observations. Because some infections are asymptomatic and immunity is lifelong, serological surveys are preferred to disease surveillance for information about infection, not just of mothers as required for burden assessment, but children, as required to evaluate alternative means of mitigating the burden. New methods for assaying oral fluids (Ramsayet al. 1998) may increase parental willingness to involve children, and data from crosssectional surveys generally can be fitted by quadratic or cubic polynomials. Meanwhile, Farrington's (1990) model compensates for the most common limitations of observations from serological surveys.
3
Assessing the burden
The burden of CRS is under-ascertained via conventional surveillance partly because of spontaneous abortions, but also because infections may be asymptomatic. Even when symptomatic, the interval between maternal infections and stillbirths or diagnoses of birth defects limits appreciation of causality. Several investigators have attempted to ascertain more accurately the burden among living children by reviewing medical records for characteristic lesions and combinations in hospitals and outpatient facilities where care would be sought (Cutts et al. 1997), and determining if children with compatible conditions have antibodies to rubella. Because only living children can seek care, and not all do, these time- and other resource-consuming studies cannot estimate the true burden of CRS. As many as 5% of maternal infections result in spontaneous abortions and stillbirths (Table 1), while an unknown fraction result in mildly affected children who do not require medical care. The parents of still other children lack the resources required to care for their afflictions irrespective of severity. None of these children has a medical record. Because modeling estimates derive from infections among women of childbearing age (WCBA), they exceed those from chart reviews in hospitals and outpatient facilities caring for children with characteristic deficits. Modeling relies on (a) readily available demographic information, (b) estimates of gestational age-specific risks obtained from registries of infected pregnant women (e.g., Miller et al. 1982, Sever et al. 1969), and (c) surveillance for either incidence of rubella disease or prevalence of antibodies reflecting previous exposure to rubella virus. We will describe our methods for assessing the burden of CRS and
John W. Glasser, Maureen Birmingham
170
Table 1: Abortions, still and live births following infection during pregnancy (percent) 'lbtal
5 t14)
::;pontaneous abortions or stillbirths 3 (8)
105 (89)
8 (7)
5 (4)
18
United Kingdom Poland outbreak
258 (31)
523 (63)
45 (5)
826
28 (74)
7 (18)
3 (8)
38
Total ('10)
420 (41)
543 (53)
56 (5)
1019
lJountry and setting US outbreak Urban US
Live births
'l'herapeutlc abortions
29 (79)
37
lteterence Monif et al. 1966 Sever et al. 1969 Miller et al. 1982 Zg6niak -Nowosielska et al. 1996
provide two examples: Only WCBA were surveyed in Morocco, but in urban and rural settings. In Romania, a cross-section of the population by age was surveyed, and our estimate can be compared with passive surveillance.
3.1
Assessment methods
The number of women pregnant and infected in any given year can be approximated as the sum of products of age-specific numbers of women, birth rates, and risks of infection. Demographic information of this sort generally is available from government publications or, increasingly, web sites. Births underestimate pregnancies, but multiplying by 1.05 would correct for spontaneous abortions and stillbirths (Table 1), yielding a reasonable estimate of affected pregnancies. We estimate the risks of maternal infection (a) directly from case notifications or (b) indirectly via catalytic modeling of results from serological surveys. Surveys typically report numbers of sera tested and above thresholds deemed protective in each age group, from which the coefficients of suitable models may be estimated via maximum likelihood (Grenfell and Anderson 1985). We prefer the highest order polynomial observations warrant, but use Farrington's (1990) model when necessary. While catalytic modeling tacitly assumes equilibrium, annual outbreaks upon which multi-annual cycles may be superimposed - the quintessential feature of infectious diseases - affect these assessments minimally (Whittaker and Farrington 2004). Given a suitable catalytic model, one determines the risk of infection in each age interval by subtracting cumulative proportions at their beginnings, p(a), from those at their ends, p(a+n). Dividing by interval widths n, one obtains average annual risks, [p(a + n) - p(a)]jn. MUltiplying by the numbers in each interval and by age-specific live birth rates yields women experiencing term pregnancies and infections the same year by
Assessing the Burden of Congenital Rubella Syndrome and· . .
171
age. As gestation and years last roughly 40 and 52 weeks, respectively, 40/52 of these women experience these events simultaneously. During each week of gestation, we assume 1/40 of their developing fetuses risk infection. But if infected, the risk of being born with CRS varies with gestational age (Glasser 2007, Figure 3). We estimated those risks via logistic regression from published results of a large pregnancy registry (Miller et al. 1982).
3.2
Assessment examples
Our calculations for rural and urban Morocco during 2001 (Tables 2 and 3) are based on age-specific numbers of women, live birth rates, and risks of infection derived from catalytic modeling of observations from serological surveys (Figures 3.1), essentially cumulative proportions infected, differenced as described above. Assuming multiple births share the same fate, the product of women by age and age-specific birth rates are term pregnancies by age of mother. Their products with age-specific risks of infection are women who were pregnant and infected during a given year. Of these, 40/52 were infected while pregnant. Table 2: Estimated infants with CRS born to women in urban Morocco by age , 2001 Age group
J: O. fJ(N) - JL(N) is positive for N = 0 and negative for large N > O. It follows from these assumptions that there exists a unique number
K > 0 such that fJ(K) - JL(K)
=
O.
(2.2)
K is called the carrying capacity of the host population in absence of the disease, because N(t) -> K as t -> 00 provided N(O) > O. The disease divides the population into a susceptible part, with density 8(t), and an infective part, with density I(t),
N=8+I, 8' =fJ(N)(8 + q(l - p)I) - JL(N)8 I'
(JC(~)8I + qpfJ(N)I -
(JC~)8I,
(2.3)
JL(N)I - aI.
The infection is vertically transmitted at the probability p, p E [0,1]. Infected individuals reproduce at the reduced rate qfJ(N), q E [0,1]. a is the additional per capita rate of dying from the disease. The parameter (J is a compound parameter whose exact interpretation depends on the specific transmission mode of the parasite. In fungal plant diseases, (J factors in the average spore production of a typical infected plant and the conditional probability that an infection occurs once a spore has landed on a susceptible plant. In sexually transmitted diseases, (J combines the average sexual activity of a typical sexually active person and the conditional probability that a given sexual contact between a susceptible and an infective individual actually leads to an infection. The parameter (J will be of central importance in our analysis, and we call it the horizontal transmission coefficient. The contact function C(N) describes how the per capita amount or rate of contacts depend on the host population density N. These may be direct contacts as in sexually transmitted diseases or indirect contacts as through spores in fungal plant diseases. Again the precise interpretation depends on the type of disease. 1/N is the conditional probability that a given contact made by a susceptible individual actually occurs with an infective individual. In fungal plant diseases, C(N) is proportional to the probability at which a given spore lands on host plants rather than on the soil (or
Thanate Dhirasakdanon, Horst R. Thieme
190
somewhere else where it is wasted) provided that the host plant density is N. At low host plant densities, this probability should be roughly proportional to the plant density which suggests that C(O) = O. In sexually transmitted diseases, C(N) is proportional to the number of sexual contacts a typical sexually active person makes in a population with density N. Some models assume that C(N) is basically independent of N unless the population density is so low that a deterministic model like ours is not valid anyway. This assumption results in what is sometimes called standard (or frequency-dependent) incidence [5, 2.1] and is a special case of assuming C(O) > O. The studies in [6, 7] assume mass action (or density-dependent) incidence where C(N) is proportional to N such that C(N)jN does not depend on host density. Our analysis includes both standard and mass action incidence and all reasonable interpolations between these two extremes. A collection of contact functions that have been used in the literature can be found in [11, Sec.19.1]; another example, C(N) = (In(a + LIN), has been suggested for insect diseases [1, App.B]. We replace the equation for S by an equation for N,
N' =({3(N) - p,(N))N I'
o-G(N)t - 1)1
((1 - q){3(N) + ex) I,
+ qp{3(N)I -
p,(N)I - exI.
(2.4)
We introduce the fraction of infective individuals, I
(2.5)
f= N· ,
I'
N
By the quotient rule, f = N - f N ' following form in terms of Nand f,
=
N' f (TI' - N)· The model takes the
N' =N({3(N) - p,(N) - ((1- q){3(N)
f' = f ([aC(N)
+ ex) f) , (2.6)
- ex - (1 - q){3(N)] (1 - f) - q(1 - p){3(N)).
Assumption 2.2. All parameters are non-negative, q > 0 (the disease does not sterilize), p < 1 (vertical transmission typically is imperfect). C(N) is an increasing function of N ~ 0, C(N) > 0 for N > o. C(N) is continuously differentiable at N > O. .
3
The persistence equilibrium
The origin is an equilibrium where both the host and the parasite are extinct. Potentially there are equilibria of three other types: the parasite
Persistence of Vertically Transmitted Parasite Strains which· . .
191
extinction equilibrium (K,O) with the carrying host capacity K > 0, the host extinction equilibrium (0, f#) with f# > 0, where the host is extinct and the parasite persists (not in absolute density but in proportion), and the persistence equilibrium (N*, f*) where both host and parasite persist.
3.1
Uniqueness and existence
There is at most one persistence equilibrium [3]. We restrict our consideration to imperfect vertical transmission, p < 1. This excludes that all hosts are infective at equilibrium. The following equation can be derived for the persistence equilibrium, a* := aG(N*) = (
q(l - p)(3* + a - q(3*
p,*
+ 1) ((1 -
q
)(3*
+ a)
(3.1)
,
where (3* = (3(N*) is a decreasing and p,* = p,(N*) an increasing function of N* and (3(N*) - p,(N*) is a strictly decreasing function of N* (Assumption 2.1). For details see [3]. We define R(N)
=
aG(N) + qp(3(N) . p,(N) + a
(3.2)
R(N) is the basic replacement ratio of the parasite at host population density N, i.e., the average number of new infective hosts produced by one infective host in a completely susceptible population of density N. Notice that iL(rJHo: is the mean length of the infective period. aG(N) is the average rate at which a typical infective individual produces new infections by horizontal transmission if it is introduced into a completely susceptible population of density N. qp(3(N) is the average rate at which a typical infective individual produces new infections by vertical transmission if the population density is N. Recall the carrying host capacity Kin (2.2).
Theorem 3.1 ([3]). There exists at most one equilibrium (N*, f*) at which both host and parasite persist, N*, f* > O. If the persistence equilibrium exists, N* depends in a strictly decreasing wayan a while the fraction of infectives f* depends in a strictly increasing wayan a. The persistence equilibrium exists, with 0 < N* < K, if and only if the following two conditions are satisfied:
(a) R(K) > 1, (b) either
+ a - q(3(O) ~ 0 p,(O) + a - q(3(O) > 0
p,(0)
or { aG(O)
< (
and
q(l - p )(3(0) p,(0) + a - q(3(O)
+ 1) ((1 -
q)(3(O)
+ a).
192
Thanate Dhirasakdanon, Horst R. Thieme
Notice that condition (b) is satisfied if C(O) = 0 which includes the case of mass action incidence. Condition (a) guarantees that 1* > 0 while (b) guarantees that N* > O. If (a) holds but not (b), then the disease drives the host into extinction [14, 3].
3.2
Global stability of the persistence equilibrium
We state that the persistence equilibrium, when it exists, represents the long term behavior of the host-parasite dynamics.
Theorem 3.2. Let the assumptions (a) and (b) of Theorem 3.1 be satisfied. Then the persistence equilibrium (N* , 1*) is locally stable and all solutions N, f of (2.6) with N(O) > 0 and f(O) E (0,1] satisfy N(t) ----+ N* and f(t) ----+ 1* for t ----+ 00. Proof· The local stability of (N*, 1*) follows by linearization and a straightforward application of the Routh-Hurwitz criterion. The convergence of f and N is proved in [14, Sec.3].
o The proofs of the following global results can be found in [14, Sec.3]. The local results follow from standard linearized stability arguments in two dimensions.
Theorem 3.3. Assume that R(K) < 1. Then the equilibrium (K,O) is locally asymptotically stable and f(t) ----+ 0 and N(t) ----+ K for every solution of (2.6) with N(O) > 0, f(O) E [0,1]. Corollary 3.4. A completely vertically transmitted parasite (O' = 0) dies out, unless vertical transmission is perfect, p = 1, and the parasite is completely harmless, q = 1 and 0: = O. Proof. Let 0' = O. Recall that (3(K) = fL(K). So the sufficient condition for parasite extinction in Theorem 3.3 is satisfied if 0 < (l-pq)fL(K) +0:, i.e., if pq < 1 or 0: > O. 0
Theorem 3.5. Assume that R(K) > 1. Let 0'0 = O'C(O) , (3o and fLo = fL(O) and also assume fLo + 0: - q{3o > 0 and 0'0> (
Then N(t)
----+
q(l-p){3o +l)((l- q ){3o+o:). fLo + 0: - q{3o
0 as t
f(t)
--+
----+ 00
and
f# :=
0'0 0'0 -
(1 - qp){3o (1 - q){3o -
0: 0:
for every solution with N(O) ~ 0 and f(O) E (0,1].
>0
= (3(0)
Persistence of Vertically Transmitted Parasite Strains which· . .
4
193
The multiple strain model
We extend our model to allow for multiple strains of the parasite. We assume that there is cross-protection between the strains, i.e., a host that has been infected by one strain cannot be infected by another strain. We also assume that a host that has been infected one way (horizontally or vertically) cannot be again infected by the same strain the other way. In this section, we will consider arbitrarily many strains, let us say n, for some basic investigations. But soon we will restrict the consideration to two strains with the second strain only vertically transmitting. We will see that, if the horizontal transmission coefficient of the first strain is sufficiently high and the second strain is less virulent, the two strains can coexist. As before, N denotes the total density of hosts, S the density of susceptible, uninfected, hosts, I the total density of infected hosts, while I j denotes the density of hosts infected with strain j, n
n
S' = (S +
"£ qk(l - Pk)h ) (3(N) -
p,(N)S (4.1)
k=l
-
C(N)S ~ N
I
~ak k,
k=l
Ij
= I j (C(~)S aj + qjpj(3(N) - p,(N) - aj).
The parameters and parameter functions have the same meaning as before, but the epidemiologic parameters now carry an index which denotes the parasite strain. In the same way as for the one-strain model, we rewrite the system in terms of the total host density, n
k=l
Ij =Ij (a j C K, N OO :::;; K. 0 In the remainder of this paper we assume that no strain sterilizes the host and that every strain has imperfect vertical transmission, i.e.,
• qj > 0 and Pj < 1 for all j = 1, ... , n. Theorem 4.2. There exists some c > 0 such that n
lim sup LJj(t) :::;; 1 - c t---+oo
j=1
for all solutions of (4.5) with N(O) ~7=1 iJ(O) :::;; 1.
~
0, iJ(O)
~
0, j
= 1, ... , n, and
A formula for c > 0 is found in the subsequent proof, see (4.8).
Proof. Set ~ = minj=1 qj(1 - Pj) and a = maxj'=1 CTj. Then ~ the following inequality is obtained from (4.6),
l' :::;;aC(N)(1 -
f) -
> 0 and
~f3(N)f.
Let 100 limsuPHOO 1(t). By the fluctuation met~od ([4J ~r [11, Prop.A.22J), there exists a sequence tk --+ 00 such that f(tk) --+ foo and
Thanate Dhirasakdanon, Horst R. Thieme
196
/,(tk) ----> 0 as k ----> and (3 decreasing,
Since limsuPt_+<X) N(t) ~ K and C is increasing
00.
o ~ o-C(K)(1- 1
00
We solve this inequality for -00
f
1
00
)
-
~(3(K)l°o·
,
o-C(K)
1
~ o-C(K) + ~(3(K) < .
(4.8) D
Lemma 4.3. Let OjC(O) ~ (1- qjPj){3(O) + aj for j = 1, ... , n. Then, for all solutions with N(O) = 0, fJ (t) ----> 0 as t ----> 00, j = 1, ... ,n.
Proof. Let N(O) written as
=
O. Then N(t)
=
0 for all t ~ 0 and (4.6) can be
n
l' = L fJ (O"jC(O) (1 - 1) -
(1 - f){3(0) (1 - qjpj)
j=l
(4.9)
- lqj(1 - Pj )(3(0) - (1 - f)aj). By assumption,
n
l' ~ - L
fJqj (1 - Pj )(3(0)f.
j=l Set ~ = minj=l, ... ,n qj(1 - Pj). Then ~ > 0 and /' ~ -~(3(O)p. This implies that f(t) ----> 0 as t ----> 00. D Theorem 4.4. Let O"jC(O) ~ (1- qjPj )(3(0) + aj for j = 1, ... ,n. Then the host population is uniformly persistent: there exists some E: > 0 such that lim inft---> 00 N(t) ~ E: for all solutions with N(O) > O.
Proof. We use the language and the results in Section Appendix: Our state space is X = {(N,it, ... ,fn) E lR~+1,L:?=lfJ ~ 1}. We split up X as X = Xl l±I X 2 (disjoint union) with Xl = {N > O} n X and X 2 = {N = O} n X. Then X is closed and X 2 is compact. Both Xl and X 2 are forward invariant. By Lemma 4.3, (lJ .... , 0) is globally asymptotically stable for X 2 . By Theorem 4.1, every solution in X tends to the compact set X n {N ~ K}. System (4.5) has the form of Lemma A.7 with x = (N, it, ... , fn). By assumption, 91 (0, ... ,0) = (3(0) - jL(O) > O. By Lemma A.7, (0, ... ,0) is a uniform weak repeller for X n {Xl> O} = X n {N > O} = Xl. By Proposition A.6, the singleton set containing (0, ... ,0) is an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for Xl which is equivalent to the statement of the theorem. D
Persistence of Vertically Transmitted Parasite Strains which· . .
5
197
The two strain model with one strain only vertically transmitted
We restrict our consideration to two parasite strains. The second strain is only vertically transmitted while the first strain is transmitted horizontally and possibly vertically too. Somewhat imprecisely, we will speak about the first strain as the horizontally transmitted strain (HT strain) and about the second strain as the vertically transmitted strain (VT strain). System (4.2) specializes to 2
N' =N((3(N) - {l(N)) - L1k((1- qk)(3(N) k=l 1~
=h ( (JG(N)
N
-h - 12 N
+ O!k),
+ qlPl(3(N) - {l(N) -
)
O!l ,
(5.1)
1~ =h (q2P2(3(N) - {l(N) - 0!2)' and the system (4.5) specializes to 2
N' =N((3(N) - {l(N) -
L Jk((l- qk)(3(N) + O!k)) , k=l
J{ =h((JG(N)(l- h - h) - (l-QlPl)(3(N) - O!l 2
+ L h((l - qk)(3(N) + O!k)) ,
(5.2)
k=l
J~ =12(-(1- Q2P2)(3(N) - 0!2 2
+ :E Jk ((1 - qk)(3(N) + O!k)). k=l
(J is again called the coefficient of horizontal transmission. We assume that both strains do some harm to the host, O!j > 0 or qj < 1 for j = 1,2. However, neither strain sterilizes the host, i.e. qj > 0 for j = 1,2. Further vertical transmission is imperfect for both strains, pj < 1 for j = 1,2.
5.1
Coexistence equilibrium
By the last equation of (5.1), an equilibrium where both parasite strains and the host coexist satisfies
o =q2P2(3(N*) -
{l(N*) - 0!2·
(5.3)
198
Thanate Dhirasakdanon, Horst R. Thieme
Since the right hand side of this equation is strictly decreasing, we learn that N* is uniquely determined and does not depend on (1. We write (1* = (1C(N*), {3* = {3(N*), p,* = p,(N*). So {3* and p,* do not depend on (1 either, while (1* is proportional to (1. We define
(5.4) R2 (N) is the basic replacement number of the VT strain at population density N and is the special case of (3.2) for the VT strain ((1 = 0). The following is shown in [3J. Theorem 5.1. A (uniquely determined) coexistence equilibrium exists if and only if the following assumptions are satisfied:
(a) R2(0) > l. (b) The vertically transmitted second strain is less harmful than the horizontally transmitted first strain in the following way,
where N* is the unique solution ofR2(N*)
= l.
(c) The horizontal transmission coefficient is large enough,
Remarks 5.2. The equation R2(N*) = 1 in Theorem 5.1 (b) is equivalent to (5.3) by which the condition in Theorem 5.1 (b) is equivalent to
(5.5) The condition in Theorem 5.1 (c) is equivalent to
Notice that the right hand side of this inequality is strictly decreasing. Since N* ~ K, (5.6) also holds if N* is replaced by K. Since {3(K) = p,(K), the condition in Theorem 5.1 (c) implies that
(5.7) which is equivalent to condition (a) in Theorem 3.1 for q = ql, P = Pl. Let us assume that condition (b) in Theorem 3.1 also holds. Then we
Persistence of Vertically Transmitted Parasite Strains which· . . have a boundary equilibrium (N~, present. By (3.1),
ff, 0)
199
where only the HT strain is
Since the right hand side of this equation is decreasing, N~ < N*. We have the following stability results which will be proved and discussed elsewhere. Theorem 5.3. If the coexistence equilibrium exists, i.e. conditions (a), (b), (c) in Theorem 5.1 hold, it is locally asymptotically stable if
(i) the horizontal transmission coefficient (J" is sufficiently large, or (ii) the per capita birth rate f3 does not depend on the population density N. Under standard incidence, i.e. if the contact function C is constant, the coexistence equilibrium can be unstable for the following scenario: • P2 and q2 are close enough to 1, i.e.
the VT strain is almost perfectly vertically transmitted and causes almost no fertility reduction,
and • ql (I-pI) is close enough to 0, i.e. the HT strain is almost perfectly
vertically transmitted or sterilizes the host almost completely. Differently from the case of standard incidence, the coexistence equilibrium is locally asymptotically stable under mass action incidence (i.e. C(N)jN does not depend on N) if P2 is sufficiently close to 1. Whether or not the coexistence equilibrium is locally asymptotically stable whenever it exists is still an open question for mass action incidence.
5.2
Dynamic coexistence
The criteria for coexistence of both parasite strains and the host at equilibrium that were proved in Theorem 5.1 also guarantee dynamic coexistence. Theorem 5.4. The following are equivalent:
(i) There exists a coexistence equilibrium.
Thanate Dhirasakdanon, Horst R. Thieme
200
(ii) The horizontally and vertically transmitted strains coexist in the sense that there exists some c > 0 such that lim inf I j ( t) ;? c,
j
t~oo
= 1,2,
for all solutions of (5.1) with h(O) > 0,12(0) > 0, N(O) ;? h(O) + 12(0) . Existence of a coexistence equilibrium is necessary for the dynamic coexistence in (b) due to a general result [13, Thm.1.3.7]. The sufficiency is shown in Section 7. A global stability result can be shown if C and (3 do not depend on the population density N. Theorem 5.5. Assume that C and (3 are positive constants and that the coexistence equilibrium x* = (N*,Ii,In with N*,Ii,Ii, > 0 exists. Then all solutions of (5.1) with I 1 (0),h(0) > 0, N(O) ;? h(O) + h(O) converge towards the coexistence equilibrium.
The proof will be given in the next section.
6
Global stability for constant contact function and per capita birth rate
We consider the special case that C(N) and (3(N) do not depend on N. Notice that Assumption 2.1 implies that p'(N) > 0 for all N > O. In the following we show that whenever an endemic equilibrium exists where the VT strain is present (i.e. U > 0), then this equilibrium attracts all solutions with h (0) > 0 and 12(0) > O. Depending on a further threshold condition the host population goes extinct or converges to a positive limit. The equations for the strain fractions (5.2) become independent of the host equation,
((1 - h - h)aC + h,l + 12,2 - 1'1), f~ = 12 (f1l1 + 12,2 - 1'2), f{
=
h
(6.1)
where ,j =(1 - qj)(3 + aj,
i'j Notice that
=,j
i'j > 'j.
+ qj(l -
Pj)(3
= (1 -
qjPj)(3 + aj.
(6.2)
Persistence of Vertically Transmitted Parasite Strains which· . .
201
Lemma 6.1. The equilibrium ut, 0) with 0 < ft :::;; 1 exists if and only if !JC - 1'1 > 0, in which case
ft
=
!JC - 1'1 < 1. !JC - 1'1
(6.3)
Proof. ut,O) exists with 0 < ft if and only if (1- ft)!JC+ ftl'1 -1'1 = ~g::t > 0 and numerator and denominator have
o if and only if ft =
the same sign. Since 1'1 > 1'1, it :::;; 1 if and only if !JC - 1'1 > 0, in which case ft < 1. D
Lemma 6.2. An equilibrium 1 exists if and only if
Uf, U) with fi > 0, U > 0 and fi + U : :;
1. the equilibrium ut, 0) in Lemma 6.1 exists, and
2. ftl'1 - 1'2 > 0, where ft is given by (6.3). The equilibrium is unique. In case Uf, U) exists, we have 1'1 > 1'1 > 1'2 > 1'2, fhl + UI'2 = 1'2, and fi + U < 1. Proof. Suppose the equilibrium ut, 0) exists and ft :::;; 1, we have 1'1 > 1'1 > 1'2 > 1'2· Since
It 1'1 -
1'2 > O. Since
o
o. (II' f2) is an equilibrium "11
(1 - ff - fn(J"C + if'Yl + f~'Y2 - 1'1 = (1 - if - fn(J"C - (1'1 - 1'2) (by (6.5))
=
f~ )(J"C -
(1 - 1'2 -'Y:2'Y2 -
(1'1 - 1'2)
(by (6.5))
(ry1 - 1'2 - UbI - 'Y2) )(J"C - 'Yl (1'1 - 1'2) 'Yl = bl - 1'2)(J"C - (1'1 - 1'2) - 'Ylbl - 'Y2)(J"CU = 0 'Yl
(by (6.4)).
Since (1 - fi - f2)(J"C = 1'1 - 1'2 > 0, we have fi + 12 < 1. Conversely, suppose the equilibrium (If, 12) exists with fi > 0, 0, and fi + 12 ~ 1. Then (1 -
ff -
fn(J"C
if'Yl
+ if'Yl + f~'Y2 - 1'1 = 0,
+ f~'Y2 -
1'2
=
12 > (6.6) (6.7)
O.
We rewrite (6.7) as (1 - f2)bl - 1'2) = (1 - fi - f2hl + 12(1'2 -1'2) and see that 1'1 > 'Yl > 1'2 > 'Y2. We combine (6.6) and (6.7),
(1 - if - fn(J"C - (1'1 - 1'2)
= O.
(6.8)
We rearrange (6.7) as
(1 - if - f~h1 = - f~bl - 'Y2) + 'Yl - 1'2.
(6.9)
We combine (6.8) and (6.9),
(- f~bl - 'Y2)
+ 'Yl -
1'2)(J"C - 'Yl (1'1 - 1'2)
=
0,
(6.10)
which we rearrange as
Since 'Y1 > 1'2, we have
(J"C> 'Yl(1'l - 1'2) _ 1'lb1 - 1'2) + 1'2(1'1 - 'Yd
,
1'1 - 1'2 and so the equilibrium
-
,
'Y1 - 'Y2
'
> 'Yl,
(It, 0) in Lemma 6.1 exists. By (6.11),
Persistence of Vertically Transmitted Parasite Strains which· . .
203
Lemma 6.3. Every solution h(t), 12(t) with
> 0,
liminf h(t) t -H:X> lim sup J(t)
liminf 12(t) t----o,. ex;
lim sup (h(t)
=
t---+oo
> 0,
and
+ 12(t)) < 1,
t--+CX)
converges to the interior equilibrium (tl' U). Proof. We have
If =
F 1, 1~
= F 2, where
F 1(h,12) =h((1- h - 12)a-G + h/1
+ 12/2 -11 -
q1(1- pd,B)
((1 - h - h) (aC - Id - 12 b1 - 12) F2(h, h) =12 (in1 + 12/2 -/2 - q2(1 - P2),B) =h
=12( -(1- h -
12h2 + hb1 -/2) -
q1 (1 - pd ,B) ,
q2(1- P2),B).
From the assumption, the w-limit of the solution is contained in D
{(h h)
=
E]R2 :
Define p: D ~]R by p(h,12) aC -11 12
=
0 < h, 0 < 12, h
+ 12 < 1}.
hh(l~h-h)' Then
F p 1
=
11 -/2
pF2
= _ 12 +
11 -/2
h
1 - h - 12
1 - h - 12
q1(1- P1)f3 12(1 - h - h) ,
q2(1 - P2),B h (1 - h - h) ,
and so OpF1 oh OpF2 012
11 -/2 (1- h - 12)2 11 -/2 (1- h - 12)2
q1 (1 - P1),B 12(1- h - 12)2' q2(1 - P2),B
11(1 - 11 - 12)2'
Hence OpF1 oh
+
OpF2 q1(1- P1),B _ q2(1 - P2),B 0 and liminft-+oo 12(t) > 0 whenever h(O), 12(0) > O.
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Proof. The first assertion follows from Theorem 4.2. Now assume that the interior equilibrium (fl' N) exists. Let
= {(h,h) E]R2 : 0 ~ II, 0 ~ 12, h + h Xl = {(fl, h) EX: h > 0, h > O}, X 2 = {(fl, h) EX: h = 0 or h = O}. X
~ I},
There are two equilibria in X 2, (0,0) and (ft, 0). Every solution in X with h(O) = 0 converges (0,0) and every solution in X with h(O) = 0, h (0) > 0 converges to (ft, 0). In particular, the two equilibria form an acyclic set in X2. We rewrite system (6.1) in the form of Lemma A.7. From Lemma 6.1 and Lemma 6.2, we have gl(O,O)
= aD - 1'1 > 0
and
g2(ft,0) = ftn -
1'2> O.
By Lemma A.7, (0,0) is a uniform weak repeller for X n {II > O} and (ft,O) a uniform weak repeller for X n {h > O}. On X n {h = O} we have f~ = h((h -1){2 - q2f3(1- P2)), which is negative whenever h E (0,1]. Hence (0,0) is locally asymptotically stable for X n {II = O}. By Proposition A.6, (0,0) is an isolated invariant set for X. It is easy to see that (ft, 0) is locally asymptotically stable for X n {h = O}. Since (ft, 0) is a uniform weak repeller for X n {fz > O}, (ft,O) is an isolated invariant set for X by Proposition A.6. By Theorem A.4, X 2 is a uniform strong repeller for Xl, in particular lim inft-> = h(t) > 0 and liminft ---+= fz(t) > 0 whenever II(O) > 0 and h(O) > O. 0
Theorem 6.5. If the unique equilibrium (fl,Jn with f1 > 0, N > 0, f1 + N ~ 1 exists {i.e. if the conditions in Lemma 6.2 are satisfied}, then it is locally asymptotically stable and every solution of (6.1) with h(O) > 0 and fz(O) > 0 converges to this equilibrium. Proof. The local stability follows from a standard linearized stability analysis in two dimensions. The convergence results follow from Lemma 6.3 and Lemma 6.4. 0
We return to the full system (5.2) which, by (6.2), can be rewritten in this special case as N' = N(f3 - f.t(N) - f1"l1 - hY2),
f{ = h ((1- h - h)aG + II'YI + fz'Y2 - 1'1),
f~ = fz (f1"l1 + f2'Y2 - 1'2)'
(6.12)
Persistence of Vertically Transmitted Parasite Strains which· . .
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Corollary 6.6. The unique coexistence equilibrium (N*, J;, f2) of (5.2) with fi > 0, f2 > 0, fi + f2 ~ 1, and N* > 0 exists if and only if q2P2(3 > p,(0) + a2 and the unique equilibrium (ii, U) of (6.1) exists with fJ > 0 and Ii + U ~ 1. Actually fi = Ii and f2 = U. If this equilibrium exists, every solution of (5.2) with h (0) > 0, 12(0) > 0 and N(O) > 0 converges to this equilibrium as t -+ 00.
Proof· By (6.2) and Lemma 6.2, q2P2(3 > p,(0) (3 - p,(0) - i2 > 0 and to
+ a2
is equivalent to
(6.13) which is a necessary and sufficient condition for the existence of a solution N* > 0 to the equation (6.14) Cf. the first equation in (6.12). By Theorem 6.5, h (t) -+ Ii and 12(t) -+ U· By (6.13) there exist T ~ 0 and c > 0 such that (3 - p,(0) h (t)'"n - 12(th2 ~ c for t ~ T. Since p, is continuous, there is 0 > 0 such that (3 - p,(x) - h(t)'"n- 12(th2 ~ c/2 > 0 for x E [0,0] and t ~ T. Since N' = N((3 - p,(N) - hll - 1212) and since N(t) > 0 for all t, we have Noo ~ 0 > O. By the fluctuation method [4] [11, Prop.A.22] and (6.7) and (6.2), we have
o=N°O ((3 -
p,(N°O) - fill - f~'2)
=Noo ((3 - p,(Noo ) - fill -
f~'2)'
Since N°O ~ Noo > 0, (6.14) holds for both N°O and Noo in place of N*. Since N* is the unique solution of (6.14), we have N°O = Noo = N* which implies that N(t) -+ N* as t -+ 00. 0 Corollary 6.7. If q2P2(3 ~ p,(0) + a2 and the equilibrium (ii, U) of (6.1) with Ii > 0, U > 0 and Ii + U ~ 1 exists, then every solution of (5.2) with h(O) > 0, 12(0) > 0 and N(O) ~ 0 satisfies N(t) -+ 0, h(t) -+ Ii, 12(t) -+ f2 as t -+ 00.
Proof. Let h(t), 12 (t), N(t) be the solution with h(O) > 0, 12(0) > 0 and N(O) ~ O. From Theorem 6.5 we have (h, h) -+ (ii, U). By Lemma 6.2, we have fiTI + UI2 = i2' By the fluctuation method [4] [11, Prop.A.22], we have from Lemma 6.2 and (6.2) 0= N°O
((3 - p,(N°O) -
= N°O (_p,(N°O)
fill -
+ q2P2(3 -
f~'2) = N°O ((3 - p,(N°O) - i2) a 2)'
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Since f.1 is strictly increasing, we have q2P2(3 - f.1(x) - a < 0 for x > O. Hence NCO = 0 and so N(t) ----; O. D
7
Uniform strong coexistence
This section is devoted to the proof that the HT and VT strain and the host uniformly coexist, whenever the coexistence equilibrium exists. By assumption we always have three equilibria: the origin (0,0,0), (K, 0, 0) and the coexistence equilibrium (N*, fi, f2) where all components are positive. Another possible equilibrium is (N", ff, 0), at which the host and the HT strain persist, i.e., the first two components are positive. This equilibrium corresponds to the equilibrium (N*, j*) for the one strain model considered in Section 2 and Section 3. The dynamics of the twostrain model on the invariant set {!z = O} are the same as the dynamics of the one-strain model. If C(O) > 0, there are two more possible boundary equilibria: (0, It, 0) and (0, ii, 12). We will use the results of Section 6 for the invariant set {N = O}. The parameters 'Yj and ij, (3 and C in Section 6 must then be understood as being evaluated at N = 0, and the equilibrium coordinates ft and ii, 12 in Section 6 are the same as in the boundary equilibria just mentioned. The dynamics of the host-parasite model with two strains restricted to the invariant set {N = O} are the same as the dynamics of two dimensional model considered in Section 6 (with the exception of Corollary 6.6 and Corollary 6.7). Let us summarize some previous results (Corollary 3.4, Theorem 3.2 Theorem 3.5). Consult Definition A.5. Lemma 7.1. (a) Solutions with N(O) > 0, fICO) = 0, !z(0) ~ 0 converge to (K, 0, 0). (K, 0, 0) is locally asymptotically stable for {fI =
O}. (b) There exists at most one equilibrium (N", f", 0) with N", f" > O. If it exists, it attracts all solutions with N(O) > 0, fICO) > 0, !z(0) = 0 and is locally asymptotically stable for {!z = O}.
(c) There exists at most one equilibrium (0, ii, 12) with f'j > O. If it exists, it attracts all solutions with N(O) = 0, fICO) > 0, !z(0) > 0 and is locally asymptotically stable for {N = O}. (d)
There exists at most one equilibrium (0, ft, 0) with ft > O. If it exists, it attracts all solutions with N(O) = 0 = !z(0) and h(O) >
o.
If it exists and (N", ff, 0) does not exist, it attracts all solutions with !z (0) = 0 and fI (0) > 0 and is locally asymptotically stable for {!z = O}.
Persistence of Vertically Transmitted Parasite Strains which· . .
207
If it exists and (0, ii, U) does not exist, then it attracts all solutions with N (0) = and h (0) > and is locally asymptotically stable for {N = o}.
°
°
The last statement can be derived from the Poincare-Bendixson theory and a standard linearized stability analysis in the plane. For the next results consult Definition A.I.
Lemma 7.2. For the purpose of this lemma, let X = {(N,Jl, h) E ~t; h + 12 ~ 1}. If (N*, fi, fn exists, then: (a) (K, 0, 0) is a uniform weak repeller for X n {h > O}. (0,0,0) is a uniform weak repeller for X n {N > O}.
(b) Let (N~, ff, 0) exists. Then it is a uniform weak repeller for X n {12 > O}. (c) Let (0, fr, 0) exist. Then (0,0,0) is a uniform weak repeller for X n {h > O}. If (O,ff,U) does not exist, (O,fr,O) is also a uniform weak repeller for X n {N > O}.
(d) If (0, ii, U) exists, it is a uniform weak repeller for X n {N > O} and (0, fr, 0) is a uniform weak repeller for X n {12 > O}. Proof. The system (5.2) has the form in Lemma A.7 with x = (N, h, h). (a) As mentioned in Remark 5.2, the existence of the coexistence equilibrium implies that
By Lemma A.7, (K,O,O) is a uniform weak repeller for X n {X2 > O}
=
xn {h > O}. Let us turn to (0,0,0). By assumption, 91 (0,0,0) = 13(0) - JL(O) > O. By Lemma A.7, (0,0,0) is a uniform weak repeller for X n {Xl> O} =
Xn{N>O}. (b) As mentioned in Remark 5.2, N~ < N*. = 0, this implies
Since N* satisfies
q2P2!3(N*) -JL(N*) - a2
q2P2!3~ - JL~ - a2 > 0,
where !3~ = !3(N~) and JL~
93(N~, ff, 0)
= -
=
JL(N~). Then
(1 - q2P2)!3~ - a2
> -!3~ + JL~ + ff((1 -
+ ff((1 - qd!3~ + ad ql)!3~ + al) = 0,
with the last equality following from the first equation in (5.2) which is evaluated for N = N~ and h = ff, 12 = O. By Lemma A.7, (N~,ff,O) is a uniform weak repeller for X n {X3 > O} = X n {12 > O}.
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(c) Since (0, ff, 0) exists, o-C(O) - 1'1 second equation in (5.2),
> 0 by Lemma 6.1. By the
92(0,0,0) = o-C(O) - 1'1 > O. By Lemma A.7, (0,0,0) is a uniform weak repeller for X n {X2 > O} = X n {II> O}. If (0, ii, U) does not exist, then ff'Y1 :::;; 1'2 by Lemma 6.2. Since the coexistence equilibrium exists,R2(0) > 1 and q2P2f3(0) - p,(0) - 0:2 > 0 by Theorem 5.1. Then
gl (0, ff, 0) =13(0) + p,(0) - ff'Y1 ;;::: 13(0) - p,(0) - 1'2 =Q2P2f3(0) - p,(0) - 0:2 > O. By Lemma A.7,
(o,ff, 0) is a uniform weak repeller for
X n {Xl> O}
=
Xn{N>O}. (d) In this case, o-C(O) > 1'1 and ff'Y1 > 1'2. Then
g3(O, ff, 0) = -(1 - Q2P2)f3(O) - 0:2 + ff'Yl
=
-1'2 + ffil > O.
By Lemma A. 7, (0, ff, 0) is a uniform weak repeller for X X n {h > O}. Further
n {X3 > O} =
2
gl (0, if, fn = 13(0) - p,(0) -
L f~«l - Qk)f3(O) + O:k). k=l
Since N = 0, fj = ff solve the third equation in (5.2),
gl (0, if, f~)
=
Q2P2f3(0) - p,(0) - 0:2 > 0
by Theorem 5.1 (a). By Lemma A.7, (0, ff, U) is a uniform weak repeller for X n {Xl> O} = X n {N > O}. 0
Proposition 7.3. If (0, ff, 0) does not exist, there is some c
> 0 such
that (i) liminft-+oo N(t) ;;::: c for all solutions with N(O) > O.
> 0, h(O) > O. (iii) liminft-+oo h(t) ;;::: c for all solutions with N(O) > O,h(O) > 0, h(O) > O. (ii) liminft-+oo h(t) ;;::: c for all solutions with N(O)
Proof. Assume that (0, ff, 0) does not exist. By Lemma 6.1, o-C(O) :::;; (1 - QlPr)f3(O) + 0:1 and (i) follows from Theorem 4.4 with 0-2 = O. For (ii), we choose the state space X = {(N, h, h) E lR~; h + h :::;; 1; N > O}. By (i) and Theorem 4.1, all solutions in X are absorbed in a
Persistence of Vertically Transmitted Parasite Strains which· . .
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compact set which is contained in X. We split X = XIl±! X 2 with Xl = X n {h > O} and X 2 = X n {h = O}. The only equilibrium contained in X 2 is (K, 0, 0). (K, 0, 0) is globally stable for X 2 by Lemma 7.1 (a) and a uniform weak repeller for Xl by Lemma 7.2 (a). By Proposition A.6, it forms an isolated invariant set for X which is trivially acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii) . For (iii), we choose the state space X = {(N, h, h) E ~~; h + 12 ~ 1; N > 0, h > O}. By (i) and (ii) and Theorem 4.1, all solutions in X tend to a compact set which is contained in X. We split X = Xl l±J X 2 with Xl = X n {12 > O} and X 2 = X n {12 = O}. It follows from (i) and (ii) and [13, Thm.1.3.7] that the equilibrium (NH,ff,O) exists. It is the only equilibrium contained in X 2 and is globally asymptotically stable for X 2 by Lemma 7.1 (b). By Lemma 7.2 (b), it is a uniform weak repeller for X I and so forms an isolated invariant set by Proposition A.6 which is trivially acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl which implies (iii).
o
rtF,
Proposition 7.4. If (0, 0) exists, there is some c > 0 such that liminft-.oo h(t) ~ c for all solutions with h(O) > O.
Proof. We split up the state space X = {(N, h, h) E ~~, h + 12 ~ O} as X = XIl±! X 2 with Xl = {(N,h,12) E X;h > O} and X 2 = {(N, h, h) E X; h = O}. By Theorem 4.1, all solutions in X tend to the compact set X n {N ~ K}. X 2 only contains the equilibria (0,0,0) and (K, 0, 0). By Lemma 7.1 (a), all solutions in X 2 with N(O) > 0 converge towards (K, 0, 0) and (K, 0, 0) is locally asymptotically stable for X 2 . By Lemma 7.2 (a), (K, 0, 0) is a uniform weak repeller for Xl and thus an isolated invariant set for X by Proposition A.6. One readily checks that all solutions in X 2 with N(O) = 0 converge to (0,0,0), and (0,0,0) is locally asymptotically stable for X 2 n {N = O}. By Lemma 7.2 (a), (0,0,0) is a uniform weak repeller for {N > O}. Under the assumptions of this proposition, it is also a uniform weak repeller for {h > O} by Lemma 7.2 (c). So {(O,O,O)} is an isolated invariant set for X by Proposition A.6. Obviously M = {(O, 0, 0), (K, 0, O)} is acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl which implies the statement.
o
Proposition 7.5. If (0, some c > 0 such that
rtF, 0)
exists, but not (0, Ii, 12), then there is
(i) lim inft-+oo h (t) ~ c for all solutions with h (0)
> 0,
(ii) liminft-.oo N(t) ~ c for all solutions with h(O) > 0, N(O) > 0,
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(iii) lim inf t---+ 00 h(t) ;? c for all solutions with h(O) > 0, N(O) > 0, h(O) > O. Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N, h, h) E lR~; h + h ~ 1, h > O}. By Proposition 7.4 and Theorem 4.1, X contains a compact set to which all solutions in X tend. We split up X = Xl I±I X 2 with Xl = {(N, h, h) E X; N > O} and X 2 = {(N, h, h) E X; N = O}. All solutions in X 2 converge to (0, it, 0) and (0, it, 0) is locally asymptotically stable for {N = O}. Since (0, fl, 12) does not exist, (0, ft, 0) is a uniform weak repeller for {N > O} and in particular for Xl by Lemma 7.2 (c). By Proposition A.6, {(O, it, is an isolated invariant set and obviously acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies assertion (ii). For (iii), we take the state space X = {(N, h, h) E lR~; h + h ~ 1,N> O,h > O}. We split up X = X I I±IX2 with Xl = {(N,h,h) E X; h > O} and X 2 = {(N, h, h) E X; h = O}. By (i) and (ii) and Theorem 4.1, all solutions in X tend to a compact set contained in X. X 2 contains the equilibrium (N~,if,O) which exists by (ii) and [13, Thm.1.3.7] and is globally asymptotically stable for X 2 . By Lemma 7.2 (b), (N~,jf, 0) is a uniform weak repeller for X I and so an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for X I. This implies the assertion. 0
on
Proposition 7.6. Assume that the equilibria (0, it, 0) and (0, fl, i~) exist, but not (N~, if, 0). Then there exists some c > 0 such that (i) liminft---+oo h(t) ;? c for all solutions with h(O) > 0, (ii) liminft---+oo h(t) ;? c for all solutions with h(O) > 0, 12(0) > O. (iii) liminft---+oo N(t) ;? c for all solutions with h(O) > 0, h(O) > 0, N(O) > 0, Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N,h,h) E lR~;h + h ~ 1,h > O}. By Proposition 7.4 and Theorem 4.1, all solutions in X converge to a compact set which is contained in X. We split X = Xl I±I X 2 with Xl = X n {h > O} and X 2 = Xn{h = O}. Since (N~, itO) does not exist, (0, i#,O) is globally asymptotically stable for X 2 by Lemma 7.1 (d). By Lemma 7.2 (d), it is a uniform weak repeller for X 2 and so forms an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii). For (iii), we choose the state space X = {(N, h, h) E lR~; h + 12 ~ 1, h > 0,12 > O}. By (ii) and Theorem 4.1, all solutions in X tend to a compact set which is contained in X. We split X = Xl I±I X 2 with
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Xl = X n {N > O} and X 2 = X n {N = O}. By Lemma 7.1 (c), (0, ii, U) is globally asymptotically stable for X 2 . By Lemma 7.2 (d), it is a uniform weak repeller for Xl and so forms an isolated invariant set by Proposition A.5. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (iii).
o Proposition 7.7. Assume that (0, ff, 0), (0, ii, U) and (N~, ff, 0) exist. Then there exists some
(i) lim inft->oo h (t) ~
E
E
> 0 such that
for all solutions with
h (0) >
(ii) lim inft-+oo N (t) ~ E and lim inf t-+oo 12 (t) ~ h(O) > 0, 12(0) > 0 and N(O) > 0,
E
0,
for all solutions with
Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N, h h) E lR~; h + 12 ::::; 1, h > O}. By Proposition 7.4, all solutions in X converge to a compact set which is contained in X. We split X = X l I±JX2 with Xl = X n {N > 0,12 > O} and X 2 = X n {N = o or 12 = O}. By Lemma 7.1, (0, ii, U) is globally asymptotically stable for X n {N = 0, 12 > O}, (N~, ff, 0) is globally asymptotically stable for X n {N > 0, 12 = O} and (0, ff, 0) is globally asymptotically stable for X n {N = 0,12 = O}. This implies that (O,ff,O) is a uniform weak repeller for {N > O} and {h > O} and thus an isolated invariant set for X. By Lemma 7.2 (d), (0, ii, U) is a uniform weak repeller for {N > O} and forms an isolated invariant set for X. (N~, ff, 0) is locally asymptotically stable for X n {h = O} by Lemma 7.1 (b) and a uniform weak repeller for Xn{h > O} by Lemma 7.2 (b). By Proposition A.5, it forms an isolated invariant set for X. The set M consisting of these three equilibria is acyclic in X 2 as one can see from the dynamics described before. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii). 0
Appendix: Elements of persistence theory Let F : lR+. ~ lR n be locally Lipschitz and consider the ODE system x' = F(x). A set X c lR+. is called forward invariant, if all solutions with x(O) E X are defined for all t ~ 0 and x(t) E X for all t ~ O. X is called invariant, if all solutions with x(O) E X are defined for all t E lR and x(t) E X for all t E R The distance from a point x to a set Y is given by d(x, Y) = inf{llxYII;Y E Y}.
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Definition A.I. We assume that X is a forward invariant subset of ~+, X = Xl U X 2 , Xl n X2 = 0, with X 2 being a relatively closed subset of X and Xl forward invariant. Let Y2 ~ X2· Y2 is called a uniform weak repeller for X I if there exists some is > 0 such that limsupd(x(t), Y2 ) ~ 0 there exists some 8 > 0 with the following property: If x: lR+ --+ Y is a solution of x' = F(x), defined for all t ;? 0 with values in Y, and Ilx(O)-x*11 < 8, then Ilx(t)-x*11 < c for all t;? o. x* is called locally asymptotically stable for Y if is is locally stable and
(ii) There exists some 80 > 0 with the following property: If x : lR+ --+ Y is a solution of x' = F(x), defined for all t ;? 0 with values in Y, and Ilx(O) - x* II < 80 , then x(t) --+ x* for t --+ 00. x* is called globally asymptotically stable for Y if is is locally stable and
(iii) x(t) --+ x* for t --+ 00 for all solutions x : lR+ defined for all t ;? 0 with values in Y.
--+
Y of x'
=
F(x),
--+ IRn be locally Lipschitz and X ~ lR:;:' be forward invariant for the solutions of x' = F(x). Assume that X = Y1 I±J Y2 where Y1 is also forward invariant and X and Y2 are closed. Let x* E Y2 be an equilibrium, F(x*) = o. Assume that {x*} is a uniform weak repeller for Y1 and x* locally asymptotically stable for Y2. Then {x*} is an isolated invariant set for X.
Proposition A.6. Let F : IR:;:'
Proof. Let 81 > 0 be such that limsupt-+oollx(t) - x*11 > 81 for all solutions x: lR+ --+ Y 1 . Let 82 > 0 be such that limt-+oo x(t) = x* for all solutions x: lR+ --+ Y2 with Ilx(O) - x* II < 82 . Let 8 = ~ min(8 1 , 82 ). Let Me X n B8(X*) be an invariant set. We have M n Y1 = 0 because if Xo E M n Y1 and x : lR+ --+ Y1 is the solution with x(O) = Xo, then there is t E lR+ such that Ilx(t) - x* II > 81 > 8, and so x(t) (j. M, contradicting the invariance of M. Hence M C Y2 n B8(X*). Now suppose (to get a contradiction) that we can pick Zo E M ~ Y2 , Zo -I- x*. Let z : IR --+ M be the solution (defined on all lR) with z(O) = zoo We have z(t) E Y2 for all t E lR because if there is t' E lR such that z(t') E Y 1 , then there is t" > t' with Ilz(t") - x*11 > 81 > 8, again contradicting the invariance of M. Let c = ~llz(O) - x*11 and choose 8' > 0 such that Ilx(t) - x* II < c for every solution x : lR+ --+ Y2 with Ilx(O) - x*11 < 8'. Since M c B8(X*) and B8(X*) is compact, the a-limit set of z, a(z), is not empty and we have a(z) c M n Y2 c X n B8(X*). Since Y2 is closed, a(z) C Y2 n B8(X*).
214
Thanate Dhirasakdanon, Horst R. Thieme
Now pick Xo E a(z) C Y2 n Bt5(x*) and let x : 1R+ ---; a(z) be the solution with x(O) = Xo. Then limt_oo x(t) = x*, and since a(z) is closed, we have x* E a(z). Therefore there exists f < 0 such that Ilz(f) - x*11 < 6'. But then we have Ilz(O) - x*11 < c = ~llz(O) - x*ll, so we get a contradiction. Therefore M = {x*}. 0 The following easy lemma will be used again and again to show that a point is uniform weak repeller. Lemma A.7. Let gj : 1R+ ---; 1R be continuous, j = 1, ... , m. Consider the system of differential equations xj = Xjgj(x), j = 1, ... , m, x(t) = (Xl(t), ... xm(t)). Let X ~ 1R+ be forward invariant. Let x EX, j E {1, ... ,m}, Xj = 0, and gj(x) > O. Then x is a uniform weak repeller for X n {x j > O}.
Proof. Set c = gj(x)/2. By continuity, choose 6 > 0 such that Igj(x) gj(x)1 < c whenever Ix - xl < 6. So gj(x) ;;:: c whenever Ix - xl < 6. Suppose that x is not a uniform weak repeller for X n {Xj > O}. Then there exist a solution with Xj(t) > 0 for all t ;;:: 0 and limsuPt-oo Ix(t)xl < 6. Then x' (t) liminf ----L-() ;;:: liminfgj(x(t)) ;;:: t-oo Xj t t-oo
This implies Xj(t) ---;
00
as j ---;
00,
f
> O.
a contradiction.
o
Acknowledgement The authors thanks Stanley H. Faeth and Karl-Peter Hadeler for helpful discussions.
References [1] BRIGGS, C.J., H.C.J. GODFRAY, The dynamics of insect-pathogen interactions in stage-structured populations, The American Naturalist 145 (1995), 855-887 [2] BUSENBERG, S.N., K.L. COOKE, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin Heidelberg 1993 [3] FAETH, S.H., K.P. HADELER, H.R. THIEME, An apparent paradox of horizontal and vertical disease transmission, J. Bio!. Dyn. 1 (2007), 45-62 [4] HIRSCH, M.W., H. HANISCH, J.-P. GABRIEL, Differential equation models for some parasitic infections: methods for the study of
Persistence of Vertically Transmitted Parasite Strains which· . .
215
asymptotic behavior, Commun. Pure Appl. Math. 38 (1985), 733753 [5] HETHCOTE, H.W., The mathematics of infectious diseases, SIAM Review 42 (2000), 599-653 [6] LIPSITCH, M., M.A. NOWAK, D. EBERT, RM. MAY, The population dynamics of vertically and horizontally transmitted diseases, Proc. R. Soc. Lond. B 260 (1995),321-327 [7] LIPSITCH, M., S. SILLER, M.A. NOWAK, The evolution of virulence in pathogens with vertical and horizontal transmission, Evolution50 (1996), 1729-1741 [8] MEIJER, G., A. LEUCHTMANN, The effects of genetic and environmental factors on disease expression (stroma formation) and plant growth in Brachypodium sylvaticum infected by Epichloe sylvatica, GIKGS 91 (2000), 446-458 [9] SAIKKONEN, K., P. WALl, M. HELANDER, S.H. FAETH, Evolution of latency in foliar fungi, Trends in Plant Science 9 (2004),275-280 [10] THIEME, H.R., Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal. 24 (1993), 407-435 [11] THIEME, H.R, Mathematics in Population Biology, Princeton University Press, Princeton 2003 [12] THIEME, H.R, Pathogen competition and coexistence and the evolution of virulence, Mathematics for Life Sciences and Medicine (Y. Takeuchi, Y. Iwasa, K. Sato, eds.), Springer, Berlin Heidelberg 2007 [13] ZHAO, X.-Q., Dynamical Systems in Population Biology, Springer, New York 2003 [14] ZHOU, J., H.W. HETHCOTE, Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994), 809-834
216
Richards Model: A Simple Procedure for Real-time Prediction of Outbreak Severity* Ying-Hen Hsieh Department of Applied Mathematics, Chung Hsing University Taichung 401, Taiwan, China E-mail: [email protected]
Abstract We propose to use Richards model, a logistic-type ordinary differential equation, to fit the daily cumulative case data from the 2003 severe acute respiratory syndrome outbreaks in Taiwan, Beijing, Hong Kong, Toronto, and Singapore. This model enabled us to estimate turning points and case numbers during each phases of an outbreak. The 3 estimated turning points are March 25, April 27, and May 24. Our modeling procedure provides insights into ongoing outbreaks that may facilitate real-time public health responses when faced with infectious disease outbreak in the future.
1
Introduction
Prediction of the future is a risky but tantalizing endeavor in any discipline in the scientific studies of natural phenomena, be it that of climate change, seismic movement, or occurrence of deadly diseases, not to mention the ascertaining of social phenomena such as economic trends and market volatility. In recent decades, the utilization of mathematical models in the studies of infectious diseases (e.g., [2]) for the purpose of public health prevention and control has placed the predictive abilities of the models in high demanding, especially for newly emerging disease outbreaks where public health policy makers must decide on the best 'This research is supported by NSC of Taiwan under grant (95-125-M005-003). The Singapore part of the work was carried out while the author visited Institute of Mathematical Sciences, National Singapore University. The article was written while the author visited the School of Mathematics at University of New South Wales, Sydney, Australia, funded by Taiwan CDC grant (DOH95-DC-1407)
Richards Model: A Simple Procedure for Real-time Prediction··· 217 course of intervention measures as crucial scientific knowledge regarding the disease outbreak is being gathered and observations or theories can be tested as understanding of the phenomenon develops (e.g., [1, 30, 6, 23]). For novel infectious diseases such as the severe acute respiratory syndrome (SARS) outbreak of 2003, the importance of proper prediction of the disease severity at the early stages of the outbreak became even more evident [31, 4, 26, 10]. In a 1972 paper on predictions of future human populations, Keyfitz [16] made the distinction between two types of prediction. One is a "projection" which is a consequence of a set of assumptions; the other is a forecast, an unconditional statement of what will happen, albeit perhaps with a measure of the uncertainty. The two are related in the sense that, often, the methods for projection provide means with which forecasts are possible. In the aftermath of the SARS outbreak, for example, Massad et al. [23] attempted to analyze the distinction between forecasting and projection models as assessing tools for the estimation of the impact of intervention strategies, by providing a projection of what would have happened with the course of SARS epidemic if the universal procedures to reduce contact were not implemented in the affected areas. In an endeavor to assess the effectiveness of intervention measures during the SARS pandemic, Zhou and Yan [38] used Richards model, a logistic-type model [32], to fit the cumulative number of SARS cases reported daily in Singapore, Hong Kong, and Beijing. In that article, they obtained estimates for the cumulative case number and basic reproduction number for each affected area. However, only partial case data during the outbreak was used which influenced the accuracy of the result. More seriously, the inflection point of the logistic curve, which could provide vital information pertaining to the changing trends of the epidemic and possibly indicating changes in intervention and control, was not discussed. Hsieh et al. [12] proposed to use Richards model, along with the complete Taiwan SARS case data from the beginning of the outbreak to its end, to obtain an estimate of the accumulative case number. Moreover, the inflection point of the S-shaped epidemic curve was obtained which indicates the turning point of the outbreak in Taiwan when the daily number of infections starts to decrease. More recently, Hsieh and Cheng [14] use the SARS case data of Greater Toronto area (GTA) to demonstrate that even for a multi-staged epidemic, Richards model still can be used for real-time prediction of outbreak severity as well as real-time detection of turning points. In this work, we will give a complete overview of Richards model as a useful tool for public health purposes of instantaneous ascertaining of a short and ongoing disease outbreak. We will introduce some basics of Richards model in the next section. In Section 3, we will demonstrate
218
Ying-Hen Hsieh
the use of model in outbreaks where the cumulative case curve exhibits an S-shaped curve by using the SARS data of Taiwan, Beijing, and Hong. In Section 4 we will make use of the SARS data of GTA and Singapore to demonstrate that the same procedure can be used for realtime prediction of an outbreak with multiple waves. Finally, we give some remarks in Section 5.
2
Logistic and Richards models
The logistic model was first proposed by Verhulst [34] in 1838 to model population growth after reading Thomas Malthus' An Essay on the Principle of Population [22]. The model equation, also known as Verhulst equation, is as follows:
l'(t)
=
rl[l-
~],
(2.1)
where let) is the population size in question at time t, r is the intrinsic growth rate, and K is "scarrying capacity". In his 1995 book How Many People Can the Earth Support, Cohen [5] explained that Verhulst attempted to fit a logistic curve based on the logistic function to 3 separate censuses of the population of the United States of America in order to predict future growth. Interestingly, all 3 sets of predictions failed. This equation is also sometimes called the Verhulst-Pearl equation following its rediscovery by Pearl in 1920's (see, e.g., [28]). Pearl, together with Reed, used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930 [29]. A later attempt by Pearl and an associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion. This was passed in 1955. Alfred J. Lotka also derived the equation again in 1925, calling it the law of population growth [20]. In 1959, Richards [32] proposed the following modification of the logistic model to model growth of biological populations:
l'(t) = rl[l- (i)a]. K
(2.2)
The additional of the parameter a provide a measure of flexibility in the curvature of the S shape exhibited by the resulting solution curve. As a model for the growth of an epidemic outbreak, let) is the cumulative number of infected cases at time t in days, K is the carrying capacity or total case number of the outbreak, r is the per capita growth rate of the infected population, and a is the exponent of deviation from the standard logistic curve. Unlike models with several compartments commonly used to predict the spread of disease, the Richards model considers only the cumulative infective population size with saturation in growth as
Richards Model: A Simple Procedure for Real-time Prediction··· 219 the outbreak progresses, caused by decreases in recruitment because of attempts to avoid contacts (e.g., wearing facemask) and implementation of control measures. The basic premise of the Richards model is that the daily incidence curve consists of a single peak of high incidence, resulting in an S-shaped epidemic curve and a single turning point of the outbreak. These turning points, defined as times at which the rate of accumulation changes from increasing to decreasing or vice versa, can be easily located by finding the inflection point of the epidemic curve, the moment at which the trajectory begins to decline. This quantity has obvious epidemiologic importance, indicating either the beginning (i.e., moment of acceleration after deceleration) or end (i.e., moment of deceleration after acceleration) of a phase. The analytic solution of (2.2) is (2.3) It is trivial to show that ti is the only inflection point (or turning point denoting deceleration after acceleration) of the S-shaped epidemic curve obtained from this model. Moreover, tm = ti + (lna)jr in (2.3) is equal to the inflection point ti when a = 1, and approximates ti when a is close to 1.
3
Single wave out break
The Richards model fits the single-phase SARS outbreak in Taiwan [12] well. We give below the parameter estimation results and the theoretical epidemic curve for Taiwan SARS outbreak of February 23-June 12, 2003, using Richards model from [12] in Table 1 and Figure 3.1, respectively. The result indicated that the infection occurred on May 3, and the estimate for the maximum case number of K = 343.3 [95%CI: (340,347)] is merely 0.8% off the actual total case number of 346. Moreover, the case number data used was sorted by onset date. Given a mean SARS incubation of approximately 5 days [37], the inflection point for SARS in Taiwan could be traced back to 5 days before May 3, namely April 28. On April 26, the first SARS patient in Taiwan died. Starting April 28, the government implemented a series of strict intervention measures, including household quarantine of all travellers from affected areas [17]. In retrospect, April 28 was indeed the turning point of the SARS outbreak in Taiwan. It is also interesting to note that, using this method, relatively accurate estimates for the turning point of the epidemic and the final epidemic size can be obtained fairly early [14]. In this instance, estimate of turning point on May 3 can be obtained using case data of up to May 10, while CI interval for total case number of (298, 370) is obtained using
Ying-Hen Hsieh
220
Table 1: Estimates of parameters in Richards model using cumulative confirmed SARS case data in Taiwan (N=346) of selected time periods. t m =66.6 implies the turning point of epidemic is May 3. (Source: [12]) a 10.0632
875.8
95% C.L (0*,147247)
204.9
(185.2,224.6)
0.2737
4.7745 2.4169
253.1
(232.1,274.2)
67.508
0.1483
1.2326
334.2
(298.2,370.2)
67.432
0.1419
1.1694
342.1
(321.5,362.6)
66.6187
0.1359
1.0731
343.4
(339.7,347.1)
Time Period 2/25 - 4/28
tm
r
78.2193
1.1421
2/25 - 5/05 2/25 - 5/10
65.5108 66.9819
0.5343
2/25 - 5/15 2/25 - 5/20 2/25 - 6/15
K
*max(O, lower bound)
350F==::::::::::::::::================::;;iiiiii ,----------- . : • Real Data 300 --2/25-6/15.---------§-:..-----\ i 2/25-5/5 ~ 250 i • • • • 2/25-5/10--------Ir....,...-~~~~-"i ~:: . :--. _.--2/25-5/15 ---_ . ----.
.~ 200
1 u
150~----------_#:..-------~
100~---------~--------~ 50~--------~---------~
oL-__c=~~__________~ 2/25 3/7
3/17 3/27 4/6
4/16 4/26 5/6 Date
5/16 5/26 6/5
6/15
Figure 3.1: The theoretical epidemic curve for Taiwan SARS outbreak during February 23-June 12, 2003, using Richards model. Turning point is May 3. (Source: [12])
data up to May 15. This indicates that, if no deviation from the actual events had occurred, the authority could detect the turning point (for the better) of the outbreak about one week after its occurrence. Furthermore, a range for the final epidemic size could be estimated a month before the end of the outbreak. The real-time predictive potential of this procedure will be discussed in more details in the Conclusions section. We also note that although we did use the additional laboratory confirmed case data in Taiwan as detailed in [13], estimation studies have shown that the accuracy of the procedure will not be compromised even
Richards Model: A Simple Procedure for Real-time Prediction··· 221 if we did use the additional case data. For the purpose of illustration and comparison, we perform the same the 2003 procedure to the SARS data from other affected areas. epidemic, the largest outbreak of SARS occurred in Beijing in the of 2003. Multiple importations of SARS to Beijing initiated transmission in several healthcare facilities. The outbreak in Beijing March 5, and by late April daily hospital admissions for SARS exceeded 100 for several days. According to [18], total 2,521 cases of probable SARS occurred. We reconstruct the daily incidence data from epidemic curve given in Figure 1 of [18] and obtain the incidence data of 2380 cases with hospitalization dates between March 5 and 29 in 3.2. We also note that the official cumulative case number in is 2631 as published by World Health Organization (WHO) website (see or [36]). 180 150 120
'" "S"
i
90 60 30 0 3/5
3/15
3/25
4/4
4/14
4/24
5/4
5114
5/24
Date
Figure 3.2: The daily SARS incidence curve by hospitalization date for SARS outbreak during March 3-May 29, 2003. (Source: The data was used to estimate the parameters in Richards model. The parameter estimates and the resulting theoretical epidemic curve are in Table 2 and Figure 3.3, respectively. The estimate for total case number of J{ = 2352 [95%CI: 2369)] is somewhat less accurate than that of Taiwan SARS, due to the fact that not all probable cases (totaling 1521) were accounted for in the data used. Moreover, the epidemic curve data of was by hospitalization date, which was affected by variance in
Ying-Hen Hsieh
222
Table 2: Estimates of parameters in Richards model using cumulative confirmed SARS case data of 2380 cases in Beijing during March 5-May 29, 2003. ti=51.92 implies the turning point of epidemic is April 26. Period 3/5 - 4/25
a 7.62
tm
t;
K
0.846
78.14
75.74
25385.5
3/5 - 4/30
0.751
6.77
54.30
51.75
1798.1
(1707.7 - 1888.4)
3.53
55.41
52.24
2097.3
(2039.6 - 2155.0)
r
(0'
95% C.L 7012482.0)
3/5 - 5/05
0.398
3/5 - 5/10
0.321
2.80
55.47
52.27
2198.5
(2164.3 - 2232.7)
3/5 - 5/15
0.274
2.33
55.27
52.19
2264.7
(2238.2 - 2291.2)
3/5 - 5/20
0.242
1.98
54.90
52.07
2315.3
(2291.9 - 2338.6)
3/5 - 5/29
0.219
1.74
54.45
51.92
2351.8
(2334.8 - 2368.8)
*max(O, lower bound)
2500
j. 2000
- -
Real Data
I
/.
I
J J
t
§ 0 is the sum of infections due to direct contacts with the original 10 cases together with secondary contacts between susceptible individuals and individuals infected at any time (t-T) with 0 < T < t. Since the number of such infected individuals remaining at time t is B(T) i (t - T), i(t) satisfies the Volterra integral equation
i(t)
=
Set)
(fat "iJ!(N(t), T) i (t -
where
"iJ!(N(t) , T) =
T)dT + "iJ!(N(t), t)Io) ,
(2.1)
C~;~)) (3(T)B(T)
denotes the per capita infection rate at time t for an individual infected at time (t - T). This equation must be coupled with a demographic model for the total population. However, for many diseases, the time scale of the initial outbreak is much shorter than the demographic time scale, and births and non-disease deaths can be ignored. This leads to what is often called an epidemic model, where the change in the number of susceptible individuals is due to infection only, and the change in the total population is due to disease related deaths. Mortality due to disease is modelled by introducing M(T) as the fraction of the individuals infected at time (t - T) who are still alive at time t. The total population N (t) is the sum of the remaining susceptible individuals and the surviving infected individuals. Hence, N(t) and Set) satisfy the pair of equations
N(t) = Set)
+ fat M(T) i (t - T)dT + IoM(t),
:t
Set) = -i (t),
(2.2) (2.3)
together with the initial conditions
S(O) = So.
(2.4)
Note that M(T) ~ B(T), so that the integral in the above expression sums infected individuals as well as those recovered and immune to further infection. For simplicity, assume that M(O) = 1, and that N(O) = No = So + 10 . The model can be extended to include loss of immunity; however, this will not be covered in this chapter. Implicit in this model is a population, R(t), of individuals recovered from infection
James Watmough
240
and immune to further reinfection, which can be computed from the incidence, i (t - T), as follows:
R(t) = N(t) - S(t)
=
lt
-It
B(T) i (t - T)dT
(M(t) - B(t)) i (t - T)dT
+ Io(M(t) - B(t)).
Equations (2.1) through (2.4) constitute the general disease transmission model to be studied in the next sections. Kermack and McKendrick [15] studied the case with the per capita contact rate, C(N)jN, constant, and showed that epidemic-like solutions are possible only if the basic reproduction number defined in the next section is at least one, and that the epidemic will pass leaving a fraction of the population untouched by the infection. This special case, with the contact rate proportional to total population, has become known as mass action incidence.
3
The basic reproduction number
Consider an index case imported into a population composed entirely of susceptible individuals. The initial shape of the incidence curve is approximated by solutions to the linearization of (2.1) about the trivial solution i(t) = 0, S(t) = N(t) = SO. This equilibrium is referred to as the disease-free equilibrium. Note that there is a family of trivial solutions to Equations (2.1) through (2.4) parameterized by So. The linear model is the Volterra integral equation,
1
00
i(t)
=
i(t - T)A(T)dT
+ IoA(t),
(3.1)
where A(T) = SOW(SO,T) = C(So)(3(T)B(T) is the expected rate secondary infections are produced by an infected individual with infectionage T. The initial behaviour of solutions to Equation (2.1) is determined by the roots of its characteristic equation: (3.2) If Equation (3.2) has roots with real part greater than zero, then solutions to (4.1) with i (t) initially small will have i (t) increasing. That is, an outbreak will occur. In contrast, if all roots of (3.2) have negative real parts, then solutions to (3.1) with i (t) initially small will have i (t) remaining small and an outbreak does not occur.
The Basic Reproduction Number and···
241
It can be shown that (3.2) has a single real root, r*, and that all other roots have a real part less than that of r*. Further, this root is positive if and only if ¢(o) > 1 [9, 10, 16]. This root is referred to as the basic reproduction number, R o , and is given by
This number has a clear epidemiological interpretation. Since A(T) is the expected number of secondary infections caused by a person of disease age T, Ro is the expected number of secondary infections due to a single individual over the course of its infection. From the viewpoint of the epidemiologist, the basic reproduction number, R o , is defined as the expected number of secondary infections produced by an index case in a completely susceptible population [1, 7]. If Ro < 1, solutions to Equations (2.1) through (2.4) with 10 sufficiently small monotonically return to the disease-free equilibrium given by i(t) = 0. In contrast, if Ro > 1, then i(t) initially grows exponentially. Ro depends on the disease-free population size, So, through the contact rate C(So). Usually, this function is increasing with So. McNeil [18] discusses several historical epidemics and the effect of increasing population size and contact rates. There are several cautions to this interpretation of Ro. The assumption of homogeneity of contacts is likely invalid during the initial stage of the epidemic and is only valid after several secondary cases arise. The contact rate depends on many factors, such as age, setting (contacts in hospitals dominated the SARS epidemic in Toronto) or geographicallocation [19]. However, Ro remains a useful measure of the likelihood and severity of disease outbreaks.
4
The final size of a simple epidemic
For many diseases there is little disease induced death, and it is reasonable to assume the total population N (t) remains constant at the initial population size No = So + 10 . In this case, the model given by Equations (2.1) through (2.4) simplifies to the pair of equations studied by Kermack and McKendrick [14]:
i(t)
=
~S(t) = dt
Set)
(!at iJ!(No, T)i(t - T)dT +
-i(t),
iJ! (No , t)lo) ,
(4.1)
(4.2)
James Watmough
242
with S(O) = So. This simple model is also obtained if the per capita contact rate, C(N)jN, is constant, a case referred to as mass action incidence. Several results of this model are reviewed in the text of Radcliffe and Rass [21]. In particular, it is shown that Set) and i(t) are positive, and it follows that Set) is decreasing and must have some limit Soo = limt-+oo S (t). Dividing (4.1) through by S (t) and integrating with respect to t from 0 to 00 leads to
100 (I 1roo00 100
t
log ( : : )
=
=
a
=
'li(No, T)i(t - T)dT
'li(No, T)i(t - T)dt dT
T
io
roo
'li(No, T)dT io
(1
+ 'li(No, t)Io) + Rsa J,a
~~
=
~~ (So -
(4.4)
a
i(t)dt +
R I
~o a
00
=
dt (4.3)
(4.5)
i(t)dt + Io)
(4.6)
+ 10)'
(4.7)
Soo
Since the right hand side is finite, the limit SeX) is strictly positive, and the epidemic passes without infecting the entire population. Defining the attack rate, p, as the fraction of the population infected over the course of the epidemic, p
=~ S a
1
00
0
'(t)dt = So - Soo s a '
z
the final size equation can be written
1 ) log ( 1 _ p
=
pRo
Rolo
+ ----s;;'
(4.8)
Figure 4.1 illustrates that (4.8) has a single root between 0 and 1. Since Io « So, the last term involving 10 can be neglected. In this case, (4.8) has the single root p = 0 if Ro < 1, and a single root between 0 and 1 if Ro > 1. Thus, if Ro < 1, then there is no outbreak, and only a few cases arise from contacts with the index case. However, if Ro > 1, then an outbreak occurs, and secondary cases are expected to lead to tertiary cases and so on. The final size relation (4.8) was first derived by Kermack and McKendrick, who noted that Soo was not zero, or equivalently, that the attack rate was strictly less than one, and the epidemic passes without infecting every susceptible individual. The total number of people infected over the course of the epidemic is Kp = K - Soo.
The Basic Reproduction Number and···
243
log (_I ) I-p
p
Figure 4.1: The final size of a simple epidemic.
5
A final size inequality for a general model
The final size relation (4.8) applies if either N(t) is constant or if the per capita contact rate, C(N)/N, is constant. In this section, a final size inequality is derived for a more general model. It is reasonable to assume that C(N) is an increasing function of N with C(N)/N decreasing. The reasoning is that as population size increases a smaller fraction of the population is contacted by each infected individual. Further, assuming that any change in the total population is due to disease death implies that N(t) is decreasing with time t, and it follows that C(N(t))/N(t) is an increasing function of time t, so that
\J!(N(t), T)
~
\J!(No, T).
This bound can be used to estimate the final size of the epidemic in the general case. Let No be the initial population size, so that S(O) = N(O) = No. Repeating the steps preceding (4.7) leads to the following inequality:
1 (I ~ 1 (I 00
log ( : : )
t
\J!(N(t), T)i(t - T)dT + \J!(N(t) , t)Io) dt
=
00
=
t
\J! (No , T)i(t - T)dT + \J!(No, t)Io) dt
R o (So - Soo So
+ 10).
In terms of the attack rate, the final size inequality can be written as follows: log ~ pRo + RsoIo. (5.1)
(_1_) 1-p
a
As seen from the sketch in Figure 4.1, this final size inequality gives a lower bound for the attack rate p. If a lower bound can be placed on
James Watmough
244
N(t), giving an upper bound on 'J!(N(t), T), then this would lead to a lower bound on the attack rate. In practice however, this is not a simple estimate to obtain.
6
Examples
Several examples of simple disease transmission models are given in this section, beginning with the classic SIR and SEIR models and more general compartmental ODE models, and ending with a simple model with a discrete delay. The compartmental ode models are discussed in greater detail by Hethcote [12]. An interesting collection of recent modelling efforts is captured by Gumel [11]. An introduction to ODE models for biology in general can be found in many texts [8, 6,20].
6.1
The constant rate SIR epidemic model
The number of infected individuals at time t is a function of the incidence of infection i(t) and the fraction of infected individuals surviving to infection-age T as follows:
I(t) =
lot i(t - T)B(T)dt + IoB(T).
(6.1)
Three simplifying assumptions lead to a simple SIR epidemic model that is the core of most disease transmission models. First, suppose that B(T) = e-O: T • Substituting this form for B(T) in the expression for I(T) and dLerentiating leads to the di_erential equation
d
dt I(t) = i(t) - "(I(t). Second, suppose j3(T)
= 13 is
a constant. Then (2.1) simplifies to
i(t) = j3C(N(t))S(t)I(t) N(t) . .
. C(N(t)) . N(t) IS a constant, assumed to be unity, then Equations
Fmally, If
(2.1) and (2.3) simplify to the following system of ordinary differential equations.
dS = -j3S1 dt ' dI = j3S1 - aI dt '
-
(6.2) (6.3)
dR
-=aI dt
(6.4)
The Basic Reproduction Number and··· with S(O) = So and 1(0) simple model is
no
= 10 .
245
The basic reproduction number for this
= SO {'XJ {3e-
Jo
C' a and decreasing for (3S(t) < a (see Figure 6.1). Hence, if no > 1, let) initially increases until Set) < a/ (3 and then decreases to zero. If no < 1, then let) decreases to zero. The final size of the epidemic can be determined as follows. First divide (6.2) through by S and integrate to obtain
r
T
(S(T)) J(o S'(t) Set) dt = log S(O) = -(3 J l(t)dt. o Summing (6.2) and (6.3) and integrating leads to the relation
SeT)
+ leT) -
(So
+ 10)=
-a
loT l(t)dt.
Thus,
Set)
+ let) =
So
+ 10 + ~ In (~)
(6.5)
.
Setting limt---+oo let) = 0 gives the final size relation of Equation (4.8). Figure 6.1 shows a solution of (6.5) with So > a/ {3. For this simple example, the peak of the epidemic is
1 max = So
So
+ 10 + no (1 + In no)'
Further discussion can be found in the review of Hethcote [12]. I
a
f3 Figure 6.1: Solution of (6.5) with So > a/{3.
s
246
6.2
James Watmough
Simple compartmental models
The previous results can be applied directly to ordinary di_erential equation models with a single susceptible compartment where all infections arise in a single infective compartment. As a simple example of such a model, consider the SEIR model. dS = -{3S(EE + I), dt dE = {3S(EE + I) - K,E, dt dI dt =K,E-(a+6")I, -
dR = aI. dt
Here, E(t) is the number of latently infected individuals, and R(t) is the number of individuals recovered from infection with full immunity to further infection. Newly infected individuals first enter a latent stage of infection and progress to a fully infectious stage at a rate K,. Infectious individuals succumb to disease death at a rate 6" and recover from infection at a rate a. The parameter . E < < 1 is used to model a partially infectious latent stage. More generally, suppose there are n disease compartments and let x(t) E ~n denote the populations of these compartments at time t. Let b E ~n denote the relative infectiousness of each compartment. The model is given by the following system of ordinary dLerential equations. dS __ SC(N) bT dt N x, dx _ QSC(N)bT dt N x
(6.6) _
Vx.
(6.7)
The n x 1 matrix Q = (1,0"" ,O)T indicates that all new infections arise in the first disease compartment. The equations for R(t) and N(t) are left unspecified as they do not enter into the analysis of the simple case discussed here. For the SEIR model, bT = (E, 1), X=
V=
(6.8)
(:), (:K,
and
a:
6") .
(6.9)
(6.10)
The Basic Reproduction Number and···
247
Denoting the incidence of infection as i(t) as before, (6.7) can be written
!
(eVtx(t))
= eVtQ8(t)~~~(t)) bT x(t) = eVtQi(t).
(6.11)
The exponential of the matrix V is defined by the Taylor series (see, for example, [13]) e V =1 + V
V2
vn
V3
+ -+ -3! + ... + -+ ... 2 n!
Integrating (6.11) gives the population of each compartment, x(t) as follows: x(t) = e-Vtx(O) + i(t - r)e- VT dr.
1t
Since the introduced cases are assumed to be in the latent stage, x(O) Qlo, and
=
(6.12) Let u(r) = e-VTQ. the solution of
~~ =
-Vu with u(O) = Q. This
is interpreted as the distribution, after a time r, of a cohort of infected individuals initially all in the latent stage. It follows that B( r) is the sum of the entries of u(r) and {3(r)B(r) = bTcVTQ. Thus, from (3.3),
1
00
Ro = C(80 )
bT e-VTQdr
= C(80 )bT V- I Q,
Returning to the SEIR example, we find
V-I
= (:;:
Ro = E{3 /'\,
l~a)'
+!!... a
Notice that the (i,j) entry of V-I is the expected time an individual initially in compartment j spends in compartment i over the duration of its infection. Thus, the second term in Ro above is the product of l/a, the expected time in compartment I, and C(80 ), the rate an individual in compartment I produces secondary infections, giving the expected number of secondary infectious produced by an infected individual while in compartment 1.
248
6.3
James Watmough
The SLIAR model for influenza
It is not necessary that the disease progress through the compartments
in sequence as with the simple SE1R model. For example, the results also apply to the SL1AR model proposed by Arino et al. [2] illustrated in Figure 6.2.
Figure 6.2: The SL1AR model for influenza. Here the disease progresses from a latent stage (L) to an infectious stage which is either asymptomatic and mild (A) or symptomatic (I). The model assumes a fraction p of latently infected individuals develop symptoms and progress to I, whereas a fraction (1 - p) have an asymptomatic infectious period. The model is given by the following system of ordinary dLerential equations:
S' L' I' A' R' N'
= -i(t),
(6.13)
= i(t) - r;,L, = pr;,L - 0.1, = (1- p)r;,L - '1]A,
(6.14) (6.15)
= faI +'1]A,
(6.17)
= -(1 - f)al
(6.18)
(6.16)
with the incidence of infection i(t) given by
i(t)
=
~~l~~~;~~~ (tL(t) + (1 -
q)I(t)
+ 8A(t)),
and initial conditions
8(0) A(O)
= 8 0 , L(O) = la, 1(0) = 0, = 0, R(O) = 0, N(O) = No = So + 10 .
This model was proposed for influenza and studied by Arino et al.[2]. There they also extend the model to include treatment and vaccination. The model differs from the simple models of the previous sections by allowing the contact rate to depend on the number of infected individuals,
The Basic Reproduction Number and· ..
249
an issue also discussed by Brauer [4]. The parameter q is a reduction in the number of contacts made by individuals with symptomatic infections. Equations (6.14) through (6.16) can be written in the form
dx
dt
=
SC(N) QbT _ V N _ qI x x,
(6.19)
with (6.20) (6.21)
(6.22)
As before U(T) = e-VTQ, and from (6.12), I(t) = J; i(t - T)U2(T)dT, where U2(t) is the second component of u(t). Integrating (6.18) gives N(t) as follows:
N(T)
loT lot U2(t - T)i(T)dT dt (1- f)a loT iT U2(t - T)i(T)dtdT
=
No - (1 - f)a
=
No -
=
No - (1 - f)a
loT i(T - a) (l~Q U2(t + a - T)dt) da
= No - foT i(T - a)m(a)da
(6.23)
a
where m(a) = (1- f)aJo u2(S). The presence of q in the denominator of (6.19) does not effect R o , and
Ro=f3(~+ (l-q)p + 8(1- P)). K,
a
'f/
(6.24)
The final size of the epidemic can be estimated from (4.8). However, it is dLcult to put bounds on this estimate. A further discussion can be found in Arino et al. [3].
250
6.4
James Watmough
A discrete delay
As a final example, consider the model
i(t) = S(t) C(N(t)) ('JO i(t - T)B(T)dT N(t) iij with B(T) = e-aT and 8 > O. This model is similar to the SEIR model of Section 6.2, with f = 0, except that infected individuals now spend precisely 8 days in a latent stage before becoming infectious. In the SEIR model, individuals progress from E to I at a rate K, resulting in an exponential distribution for the waiting times in E (recall the first component of U(T) was e-I 1 then an orbit (the unstable manifold) connects the uninfected point (1,0) to the final state (800 ,0) whereby the number of remaining susceptible 8 00 < 1/ Ro can be obtained from the equation
(2.6) If Ro < 1 then the point (1,0) is (weakly) stable. There is no unstable manifold.
K.P. Hadeler
256
The case K> 0: Now the picture is very different. The function V as defined in (2.3) is no more an invariant of motion since .
V =
a
73K.
The point (8,1) = (0,0) is the only stationary point and this point is This claim follows immediately: The total number globally stable in of non-immune Q = 8 + 1 satisfies Q = -al. Hence 1 goes to zero and Q goes to a limit Qoo. Then the first equation in (2.2) implies Qoo = o. The Jacobian matrix at the point (0,0) is
JR.!.
(
-K K
0 ) . -a
Hence the point (0,0) is a node with incoming directions depending on the relative size of the parameters K and a. In applications we think of K « a. The eigenvectors are (0, If with eigenvalue -a and (a-K, Kf with eigenvalue -K. In the case K = 0 the time course of the epidemic is described by the unstable manifold of (1,0) which connects the points (1,0) and (800,0). For K > 0 the time course of the epidemic is described by the trajectory which passes through the point (1,0) at non-zero speed. Hence we cannot invoke continuous dependence on initial data to show that the two trajectories for K > 0 and K = 0 are in some sense close. However, we can use continuous dependence in the (8, I)-plane away from the line 1 = o. Consider any point (8, J) with 8 > 0 and J > 0 and the solution which passes through this point at time t = O. For the system with K = the trajectory lies on the curve given by the invariant of motion a - a (2.7) 1 = 73 log S - S + S - 73 log S + 1.
°
For t -> +00 and also for t -> -00 this trajectory approaches the line 1 = 0. If we introduce K> then for finite times, i.e., away from 1 = 0, nothing really changes. But along the line 1 = all trajectories pass this line with dl/ dS = -1 independently of K. Hence for K > the trajectory arrives at the line 1 = from S > > 0, 1 < < 0, passes through the line 1 = transversally and then converges towards the point (0,0). Now we follow, for K > 0, the trajectory through (8(0),1(0)) = (1, 0) forward in time. The function S (t) decreases from S (0) = 1 to for -> +00 while the function l(t) first increases to a maximum and then decreases to 0. Let Imax be the maximum of l(t) and let Smax be the corresponding value of 8. The point (Smax,lmax ) is located on the hyperbola
°
°
°
°
°
°
Epidemic Models with Reservoirs
1=
257
",S (3S
(2.8)
0'. -
which has a pole at S = 1/ Ro. We call this hyperbola the maximum prevalence curve. The qualitative behavior of the trajectory through (1,0) is the same for Ro ~ 1 and for Ro > 1. It appears that for this problem Ro has no qualitative meaning any more, except when considering the limiting case", -+ O. In Figure 2.1 the time course of the epidemic is depicted for", = 0 and for '" > 0, for Ro > 1 and for Ro < 1. Although the chosen value of '" is small, the effect on maximal prevalence is remarkably strong.
0.3
0.25 0.2
0.2
0.4
0.6
0.8 S
0.3
Figure 2.1: Time course of the epidemic for", = 0 (upper row) and for '" = 0.1 (lower row). 0'. = 1, left column: {3 = 2, right column: {3 = 0.5. If Ro > 1 then'" > 0 produces a higher prevalence, there are no remaining susceptible. If Ro < 1 there is no outbreak with", = 0 while there is an outbreak with", > o.
258
3
K.P. Hadeler
Singular perturbation
The system contains the small parameter /1,. For /1, = 0 the system has a one-dimensional manifold of stationary points (the line I = 0) which is hyperbolic (as a manifold) except at the point (8,1) = (1/ R o, 0). We expect that this manifold persists in some sense. Therefore we cast the problem into the standard form of a singular perturbation problem. Again we use the non-immune Q = 8 + I and Q = -aI. Then we put I = /1,J, use the equations for Q and for I, and 8 = Q - I, finally we divide the equation for J by /1,. The idea is that 1= 0(1) is small while J is not small. Then we get
Q= j
=
-/1,aJ, {3(Q - /1,J)J + (Q - /1,J) - aJ.
(3.1)
Then we scale time by T = /1,t and denote the derivative with respect to by'. When T runs from 0 to 1 then t runs from 0 to very large values. Afterwards we divide the Q' equation by /1,. Then we get
T
Q' = -aJ, /1,J'
= {3(Q - /1,J)J + (Q - /1,J) - aJ.
(3.2)
Now we have essentially a standard singular perturbation problem where the small parameter occurs also on the right hand side (which does not preclude the application of Fenichel's theory). If we put /1, = 0 then we get the slow manifold as J=
Q
(3.3)
a - (3Q which looks similar to (2.8). Indeed, it is the same expression if we identify I = /1,J and 8 = Q - /1,J = Q + 0(/1,). Hence we have found the following proposition. Proposition 1. For 8 < 1/ Ro and small /1, the hyperbola (2.8) is the slow manifold along which trajectories enter the stationary point (0,0). Of course convergence to the slow manifold is not uniform, and the manifold is disrupted at 8 = 1/ Ro because hyperbolicity is lost.
4
SIR model with demographic renewal
Here we endow the previous model (2.1) with demographic turnover but for the time being we exclude differential fertility or mortality,
S = p,- p,8 - /1,8 - {38I, j = -p,I + /1,8 + {381 - aI, R = -p,R+aI.
(4.1)
Epidemic Models with Reservoirs
259
R!
The positive orthant is positively invariant. All non-negative solutions are bounded in view of (djdt)(8+1 +R) = f.L(1-(8+1 +R)). Hence the existence problem is trivial. It suffices to study the two-dimensional system
s = f.L j
=
f.L8 - ",8 - (381,
-f.LI + ",8 + (381 - aI.
(4.2)
The Bendixson-Dulac criterion with weight function 1/81 excludes periodic orbits. For the discussion of stationary states it suffices to consider (4.2),
o = f.L -
f.L8 - ",8 - (381,
0= -f.LI + ",8 + (381 - aI.
We write these two equations as 1=
f.L-f.L8 -",8 (38 '
1=
",8 a+f.L-(38'
(4.3)
equate these two expressions for I and obtain a quadratic equation for S,
¢(8) == (3f.L8 2
-
f.L((3
We define
+ a + f.L)8 -
"'(f.L
+ a)8 + f.L(a + f.L) = O.
(4.4)
a+f.L
S= -(3-' The quadratic equation has always two real positive roots 8 1 < 8 2 , We find that
Hence S1
< min(l, S) ~ max(l, S) < S2·
(4.6)
From the second equation (4.3) it follows that at the stationary point (8 2 , I) the component I is negative. At the stationary point (81 ,!) the I component is positive. Hence we get, with 8 = S1, the following proposition. Proposition 2. For all choices of the parameters with", > 0 there is a unique feasible stationary point (s,1). The component S is the smaller solution of the quadratic equation (4.4), the component I is given by any of the two expressions (4.3) with S = S. The point (8,1) is globally stable in
lRt.
K.P. Hadeler
260
Proof: The system is dissipative, there are no periodic orbits and there is a single stationary point. By the Poincare-Bendixson theorem this point is a global attractor in IR? D Now we explore the local character of the point (8, J). Proposition 3. The point (8, J) is linearly stable.
Proof The Jacobian at the stationary point is (4.7) and hence, using (4.3), /1 tr J
+ 0: -
{38
> 0,
= -(/1 + '" + (3J) - (/1 + 0: -
(38)
< 0,
det J = (/1 + '" + (3J)(/1 + 0: - (38) + (38(", + J) >
o.
0
Proposition 4. The trajectory through (1, 0) starts with dI/ d8 = -1..
Either the function I(t) increases to maximum prevalence J at the stationary point or it first increases to maximal prevalence and then approaches the equilibrium prevalence 1. Proof The stationary point is the intersection of the isoclines given by (4.3). Either the trajectory through (1,0) enters the stationary point before crossing the j = 0 isocline or it crosses it a first time before eventually converging to the stationary point. This intersection of the trajectory and the isocline gives the maximal prevalence along the trajectory. D Now we discuss how the case with small '" > 0 is related to the classical case", = o. We introduce the basic reproduction number as
Ro =
_(3_.
(4.8)
0:+/1
If '" = 0 then (1,0) is a stationary point which describes the uninfected population. If Ro =J- 1 then there is a second stationary point A
/1
1
1= - ( 1 - -). /1+0: Ro
(4.9)
This point is feasible if and only if Ro > 1. Now we assume Ro =J- 1 is fixed and '" > 0 is small. Then again we have two scenarios. i) If Ro < 1 then there are a feasible stationary point (8,1) close to (8,1) = (1,0) and a non-feasible stationary point close to (8,1), away from (1,0).
Epidemic Models with Reservoirs
261
ii) If Ro > 1 then there are a feasible stationary point (8, I) close to (8,1) and a non-feasible stationary point close to (1,0). Hence, as long as 0:, {3 are kept fixed and I'>, is sufficiently small, the cases Ro < 1 and Ro > 1 are well distinguished. The differences disappear if I'>, gets large. Next we further explore the effect of the reservoir. Following an idea of Thieme we distinguish infected individuals by the source of their infection. Let h denote those which became infected by the reservoir, and let 12 be those who became infected by contact with another infected individual. Then the system becomes A
s = p, - p,8 jl
j2
1'>,8 - {381,
+ 1'>,8 - o:h, = -p,12 + {381 - o:h
=
-p,h
R = -p,R+o:l with 1 = h + h of the infected.
At the equilibrium
(4.10)
(8, I) we find the following partition
Proposition 5. At equilibrium let II be the number of infected by contact with the reservoir and 12 the number of infected by contact with other infected. Then -
12
{3J2
= ----. I'>,
The proportion v
=
•
v
+ {31
(4.11)
hi 1 satisfies +- {31 = 8I'>, 1
(I'>, I'>,
+ (31
)
- v .
(4.12)
Hence the proportion of infected by the reservoir is initially 1 and then decreases to the equilibrium proportion v = 1'>,/(1'>, + (3I).
5
Homogeneous demographic model
In models with demographic replacement and differential mortality it makes sense to assume that the transmission term is homogeneous of degree 1. Since recovery, death and birth are modeled by linear terms, the resulting models are homogeneous systems of differential equations [3]. Such models typically do not have stationary point (except the origin). Like in linear systems the solutions of primary interest are exponential solutions. For the stability theory of exponential solutions see
K.P. Hadeler
262
[7]. Here we introduce a reservoir into a model from [3], .
I
S = b1S + b2I + b3R - I-£ I S - ",S - (3S p' .
I
1= -1-£21 + ",S + (3S p - 0'.1, (5.1)
R= -1-£3R+aI with P
=
S
+ I + R. We assume
and we introduce The basic reproduction number for the homogenous problem is (3
Rhomog _
o
- b1 - 1-£1
(5.2)
+ 1-£2 + 0'.
As usual the numerator contains those rates which favor infection (only (3) and the denominator contains those rates which work against infection, mortality of infected 1-£2 (not just differential mortality), recovery 0'., and washout by demographic growth b1 - 1-£1. Following [3] we can look at this problem in two ways, we can either discuss the homogeneous system (5.1) or we can project it to a twodimensional set. The projection
S x = p'
Y=
carries the system (5.1) to the set {x, Y ~
I
P a: x + Y ~
1},
b1x + b2y + b3(1 - x - y) - (3xy - ",x - X(blX + b2y + b3(1- x - y)), iJ = -1-£2Y + (3xy + ",x - ay - y(b 1x + b2y + b3(1- x - y)) (5.3)
i; =
while I
7] = -
R
carries it to the set {(~, 7]) : ~, 7] ~ a},
(5.4)
Epidemic Models with Reservoirs
263
In the first approach we must discuss the non-linear eigenvalue problem
= blS + b21 + b3R - J.lIS - /'i,S - j3SI/ P, AI = -J.l21 + /'i,S + j3SI/P - aI, )"R = -J.l3R + aI. )"S
(5.5)
It seems that the approach of [3] does not work for /'i, > O. Therefore we base our analysis entirely on the projected systems. In [3] the following has been shown for what in the present context is a limiting case, /'i, = O.
Proposition 6. Let /'i, = O. If R~omog ::;; 1 then the only feasible exponential solution of (5.1) is the uninfected solution with exponent bl - J.lI and the eigenvector (1,0, O)T. If R~omog > 1 then there is a second feasible exponential solution with exponent 5. ::;; bl - J.lI and eigenvector (S, 1, Ii) with 1 > O. This infected solution attracts all solutions with I > 0 in the sense of homogeneous systems. For the case
/'i,
> 0 we find the following.
Proposition 7. Let /'i, > O. Then: i) The system (5.3) has no limit sets which intersect the boundary part defined by x + y = 1. ii) All trajectories of the system (5.4) stay bounded. iii) The system (5.4) has no periodic orbits and no closed chains of saddle-saddle connections. The system has a feasible stationary point. iv) Every trajectory of the system (5.4) converges to a stationary point. v) The system (5.1) has at least one feasible exponential solution. All solutions converge to exponential solutions in the sense of homogeneous systems. Proof i) Denote z
i
=
=
1 - x - y. Then
-(bIx + b2 y + (b 3
-
b3 (x + y)))z + ay.
Hence, along x + y = 1 the vector field points strictly inward, except at the point (1,0). At (1,0) we have x = -/'i, < O. ii) The line x + y = 1 corresponds to R = O. The systems (5.3) and (5.4) are equivalent except for the line x + y = 1 which corresponds to (~, 7]) at infinity. Since limit sets of (5.3) stay away from that line, trajectories of (5.4) stay eventually away from 00 and have (compact) limit sets. iii) To (5.4) we use the multiplier 1/(~7]) and apply the criterion of Dulac. The divergence is b
b3
~2
~27]
- -2 - -
/'i,
a
7]2
~
- - - - < O.
264
K.P. Hadeler
Hence there are no periodic orbits and no closed chains. Then the theorem of Poincare-Bendixson yields the existence of at least one stationary point. iv) There are only finitely many stationary points since the right hand side is rational and non-degenerate. v) The statement is just a reformulation of the previous statements in terms of homogeneous systems. 0
6
Special homogeneous system
We consider the special case with constant rates and differential mortality b1 = b2 = b3 = b, /11 = /13 = /1, /12 = /1 + 8. This system is best suited to be compared to a case fatality model [10], [11],
S = bP -
/1S - ",S - j3SI/P,
j = -/11 - M - aI + ",S + j3SI/ P,
R = -/1R+aI.
(6.1)
The projected system becomes
(6.2) The reproduction number (5.2) becomes j3 -b+a+8
Rhomog _
o
(6.3)
= o. If R~omog ~ 1 then there is only the uninfected exponential solution. If R~omog > 1 then there is, in addition, a unique infected exponential solution. ii) Let", > o. Then the system (6.2) has a unique stationary point and the system (6.1) has, up to a positive factor, a unique exponential solution.
Proposition 8. i) Let",
Proof i) This statement has been proven in [3]. ii) For the stationary states we have
o = b(1 + ~ + 1]) - "'~ o=
"'~
+
j3~1] 1+~+1]
-
j3~1]
1+~+1]
a1](1
-
a~1]
+ 1]) - 81] .
, (6.4)
Epidemic Models with Reservoirs
265
Adding these equations leads to
(b - a1])(1
+ ~ + 1]) = 81].
(6.5)
We solve for ~ and insert the expression for ~ into the second equation of (6.4). The resulting equation can be written as follows, (6.6) The left hand side of (6.6) is positive for small 1] and increases to +00 when 1] goes to the pole b/a. The right hand side of equation (6.6) has a unique pole at p E (0, b/a), and it is negative for 1] < p. If 1] runs from p to infinity then the right hand side decreases from +00 to 1. Since p < b/ a there is a unique intersection. D
7
Case fatality
Traditionally in epidemic modeling additional mortality caused by the infectious disease is modeled by differential mortality as in (6.1). If one looks closely at the stochastic interpretation then infected stay infectious all the time and eventually recover with the rate a or die with the rate f-L + 8. Hence death and recovery are considered as competing risks. In [5] (see also [1]) the original idea of Daniel Bernoulli has been followed according to which infected exit from the infected state with some rate , > 0 and then either die with some "case fatality" c or recover with some probability 1 - c. Then the system assumes the form
S=
bN - f-LS - ""S - (lSI/P,
j
""S + (lSI/ P - f-LI - ,I,
=
R=
(1 - chI - f-LR.
(7.1)
Although the parameters a,8 and "c are connected by some simple equations,
8 = C/, a = (1 - ch, , = a + 8, c = 8/ (a + 8),
(7.2)
and hence the systems (6.1) and (7.1) are equivalent in the sense that each case fatality system corresponds to one differential mortality system and conversely (except for c = I), the interpretations are quite different because a,8 are rates while c is a probability. The system (7.1), for varying parameters, in particular for the limiting case c = 1 (where
KP. Radeler
266
the correspondence (7.2) breaks down), has been studied in [10], [I1J for Ii = O. The tools developed there can be used to study the relation between (7.1) and (6.1) also for the case Ii > O.
Acknowledgement The author thanks Christina Kuttler and Johannes Muller for useful suggestions.
References [lJ Andreasen, V., Disease regulation in age-structured host populations. Theor. Population BioI. 36, 214-239 (1989)
[2] Brauer, F., The Kermack-McKendrick epidemic model revisited. Math. Biosc. 198, 119-131 (2005) [3J Busenberg, S., Radeler, KP., Demography and epidemics. Math. Biosci. 101,63-74 (1990) [4J Diekmann, 0., Reesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley 2000 [5J Dietz, K, Reesterbeek, J.A.P., Daniel Bernoulli's epidemiological model revisited. Math. Biosci. 180, 1-21 (2002)
[6] Esteva, L., Radeler, KP., Maximal prevalence and the basic reproduction number in simple epidemics. In: C. Castillo-Chavez et al. (eds), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Springer-Verlag, IMA Vol. Math. Appl. 126, 31-44 (2002) [7J Radeler, K.P., Periodic solution of homogeneous equations. J. Diff. Equ. 95, 183-202 (1992) [8] Rethcote, R.W., The mathematics of infectious diseases. SIAM Review 42, 599-653 (2000) [9] Rethcote, R.W., Wang, W., Li, Y., Species coexistence and periodicity in host-host-pathogen models. J. Math. BioI. 51, 629-660 (2005) [10J Safan, M., Spread of infectious diseases: Impact on demography, and the eradication effort in models with backward bifurcation. Dissertation Dept. Math., University of Tubingen (2006) [11] Safan, M., Radeler, KP., Dietz, K, Effects of case fatality on demography. submitted.
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[12] Singh, S., Chandra, P., Shukla, J.B., Modelling and analysis of the spread of carrier dependent diseases with environmental effects. J. Biological Systems 11, 325-325 (2003) [13] Sengar, Shikha, Modelling and analysis of the spread of infectious diseases: Effects of environmental and ecological factors. Ph.D. Thesis. Department of Mathematics, Indian Institute of Technology Kanpur (2005) [14] Thieme, H.R., Mathematics in Population Biology. Princeton University Press (2003)
268
Global Stability in Multigroup Epidemic Models Hongbin Guo, Michael Y. Li, Zhisheng Shuai Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta, T6G 2Gl, Canada Email: [email protected]@math.ualberta.ca. zshuai@math. ualberta. ca
Abstract The question of the uniqueness and global stability of endemic equilibria of multigroup epidemic models of SEIR type is revisited. After a brief review of the literature, we prove that, for a general class of n-group models with bilinear incidence, the endemic equilibrium is unique and globally stable when the basic reproduction number is greater than 1. Our proof utilizes a global Lyapunov function well-known in the literature. The key to our proof is a complete description of the complex patterns exhibited in the derivatives if the Lyapunov function uses results from graph theory.
1
Introduction
Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations. Heterogeneity can result from many factors. Individual hosts can be divided into groups according to different contact patterns such as those among children and adults for Measles and Mumps, or to distinct number of sexual partners for sexually transmitted diseases including HIV / AIDS. Groups can be geographical such as communities, cities, and countries, or epidemiological to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Multigroup models can also be used to investigate infectious diseases with multiple hosts such as West-Nile virus and vector borne diseases such as Malaria. For a recent survey of multigroup models, we refer the reader to [27]. A multigroup model is, in general, formulated by dividing the population of size N(t) into n distinct groups. For 1 ~ k ~ n, the k-th group
Global Stability in Multigroup Epidemic Models
269
is further partitioned into four compartments: the susceptibles, the exposed (latent), infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), h(t), and Rk(t), respectively. For 1 ~ i, j ~ n, the disease transmission coefficient between compartments Si and Ij is denoted by fJij ? 0, and fJij = 0 if there is no transmission of the disease between the two compartments. The new infection occurs in the k-th group is given by n
LfJkj SkIj. j=1
(1.1 )
The form of incidence in (1.1) is bilinear. Other incidence forms have been used in the literature, depending on the assumptions made about the mixing among different groups. Within the k-th group, it is assumed that death occurs in Sk, E k , h, and Rk compartments with rate constants d~ , df, di" and df, respectively. These rates may include death due to natural causes and due to the disease. The influx of individuals into the k-th group is given by a constant A k , which are assumed to be susceptible. After infection, an individual remains in the latent class before becoming infectious. We assume that the transfer rate from Ek to Ik is fk so that l/fk is the mean latent period for the k-th group. We assume that individuals in h recover with a rate constant "(k, so that l/"(k is the mean infectious period for the k-th group, and once recovered they remain permanently immune for the disease. Based on these assumptions, the following system of differential equations can be derived n
S~ = Ak - d~Sk - LfJkjSk1j, j=1 n
E~
=
I~ =
R~
LfJkjSkIj - (df + fk)Ek, j=1 fkEk - (di, + "(k)h,
k = 1,2, ... , n.
(1.2)
= "(h - dfRk,
All parameters are non-negative constants, and we assume fk, d~, df, di" df > 0 for all k. For each k, adding all equations in (1.2) gives
+ Ek + h + Rk)' =
(Sk
Ak - d~ Sk - df Ek - di,h + df Rk + Ek + Ik + Rk),
~ Ak - d'k(Sk
where d*
Rk)
k
Hence limsuPt---+oo(Sk + Ek + h Similarly, from the Sk equation we obtain lim SUPt---> 00 Sk
= min{d~, df, di" df} > O.
~ ~.
+ ~
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
270
~. Observe that the variable Rk does not appear in the first three
e~uations of (1.2). This allows us to consider first the following reduced system for Sk, Ek and h n
S~ = Ak - d~Sk - 'Lf3kj S k1j, j=1 n
E~ = 'Lf3kjSk1j -(d~ + Ek)Ek , j=1 I~ = EkEk - (d{ + "Ik)Ik,
(1.3)
in the following region of the non-negative cone of ]R.3n
(1.4)
The behaviours of Rk can then be determined from the last equation in (1.2). It can be verified that region r is positively invariant. System (1.3) always has the trivial equilibrium Po = (S~, 0, 0, ... , S~, 0, 0), where = ~, 1 ~ k ~ n, is the equilibrium of the Sk popula-
s2
k
tion in the absence of disease (Ek = Ik = 0, 1 ~ k ~ n). For this reason, Po is called the disease-free equilibrium. An equilibrium P* = o
(Si, Ei ,Ii , ... , S~, E~,I~) in the interior r of r, namely, St" Et,,Tk > 0, 1 ~ k ~ n, is called an endemic equilibrium. The matrix B = (f3ij) encodes the patterns of contacts and transmission among groups that are built into the model. Throughout the paper, we assume that B is irreducible (see Section 2 for definition of irreducibility). Biologically, this is equivalent to assuming that any two groups have a direct or indirect route of transmission. Set
Ro
=
p(Mo),
(1.5)
where
f3ij Ei S P
M _ ( 0-
(df
)
+ Ei)(d{ + "Ii) 1~i,j~n'
(1.6)
and p denotes the spectral radius. The parameter Ro is a key threshold parameter for (1.3), and is referred to as the basic reproduction number (see, e.g., [28]). Its biological significance is that if Ro < 1 then the disease dies out, and that if Ro > 1 then the disease becomes endemic (see, e.g., [27, 28]). A long-standing open question in mathematical epidemiology is that whether a mUlti-group model such as system (1.3)
Global Stability in Multigroup Epidemic Models
271
has a unique endemic equilibrium P* when Ro > 1, and that whether P* is globally asymptotically stable when it is unique. We prove the following theorem, which settles this open problem for system (1.3), and any other model that can be converted to the same form.
Theorem 1.1. Assume that B = (/3ij) is irreducible. If Ro > 1, then system (1.3) has a unique endemic equilibrium P*, and P* is globally o
asymptotically stable in
r.
One of the earliest work on multi group models is the seminal paper by Lajmanovich and Yorke [17] on a class of SIS multigroup models for the transmission dynamics of Gonorrhea. The global stability of the unique endemic equilibrium is proved using a quadratic global Lyapunov function. Subsequently, more complicated multigroup models have been proposed and analyzed, see e.g., [1, 9, 10, 11, 12, 13, 22, 24, 26, 27] and references therein. Hethcote [10] proved global stability of the endemic equilibrium for multigroup SIR model without vital dynamics. Beretta and Capasso [1] derived sufficient conditions for global stability of the endemic equilibrium for multigroup SIR model with constant population in each group. Thieme [26J proved global stability of the endemic equilibrium of multigroup SEIRS models under certain restrictions. The most recent result on global stability is Lin and So [22] for a class of SIRS models with constant group sizes, in which they proved that the endemic equilibrium is globally asymptotically stable if the cross-group contact rates /3ij, i -=I- j, are small or if the recovery rates in each group are small. On the other hand, results in the opposite direction also exist in the literature. For a class of n-group SIR models with proportionate incidence, uniqueness of endemic equilibria may not hold when Ro > 1 (see [13, 27]). In light of these results, complete determination of the global dynamics of these models is essential for their application and further development. In the case when n = 1, system (1.2) is the single population SEIR model, whose global dynamics have been completely determined [15, 20, 21]. The basic reproduction number in (1.5) becomes
/3EA
Ro where
/3
=
/3u
=
(E + d) b
+ d) d '
and all subindices are suppressed. When Ro :::;; 1, the o
disease-free equilibrium is globally stable in the feasible region r, while if Ro > 1, Po becomes unstable, and a unique endemic equilibrium P* s globally stable in r. The global stability of P* was first established in [20] (also see [19, 21]) using a geometric approach to global stability of Li and Muldowney. Another proof was given in [15J using a global Lyapunov function.
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
272
Our proof of Theorem 1.1 uses the following global Lyapunov function n
V
=
L Vk [(Sk -
Si" lnSk)
+ (Ek -
k=l
~
Ei" lnEk) + k f+ Ek (Ik - Ii" lnlk)]. k
(1.7) We note that V is a linear combination of functions of form
which were used in [15] for the single group SEIR models. The form of Lyapunov functions in (1.7) has long been known in the literature of ecological models. Its introduction can be traced to a paper of Goh [7] in which basic properties of this class of functions are discussed. Applications to other ecological and epidemic models can be found in [5, 22] and references therein. The key to our analysis is finding the right choice of coefficients Vk in (1.7) such that the derivative V'is nonnegative. This is made possible by a complete description of the complex patterns exhibited in V', using graph theory. As structured models are being used to describe more and more complicated biological problems, we expect this form of Lyapunov functions and our graph theoretical analysis will have much wider applicability. In the next section, we recall some definitions and results from graph theory, and prove a preliminary result regarding system (1.3). In Section 3, we present the proof of our main result, Theorem 1.1.
2
Preliminaries
Let E = (eij)nXn, F = (fij)nxn be nonnegative matrices, namely, all of their entries are nonnegative. We write E ~ F if ekj ~ Aj for all k, j, and E > F if E ~ F and E =I- F. For n > 1, an n x n matrix F is reducible if, for some permutation matrix Q,
and F l , F3 are square matrices. Otherwise, F is irreducible. The following properties of nonnegative matrices are standard (e.g., see [3]). PI. If F is nonnegative, then the spectral radius p(F) of F is an eigenvalue, and F has a nonnegative eigenvector corresponding to p(F). P2. If F is nonnegative and irreducible, then p(F) is a simple eigenvalue, and F has a positive eigenvector x corresponding to p(F). P3. If 0 ~ E ~ F, then peE) ~ p(F). Moreover, if 0 ~ E E + F is irreducible, then peE) < p(F).
< F and
Global Stability in Multigroup Epidemic Models
273
P4. If F is nonnegative and irreducible, and E is diagonal and positive (namely, all of its entries are positive), then FE is irreducible. Irreducibility of matrices can be easily tested using the associated directed graphs. A directed graph G n is a set of n vertices and a set of directed arcs joining two vertices. It is strongly connected if any two distinct vertices are joined by an oriented path. The directed graph G(F) associated with an n x n nonnegative matrix F is a directed graph of n vertices, 1,2"" ,n, such that, there exists an arc (j, k) leading from j to k if and only if Aj i= 0. We have the following property. P5. Matrix F is irreducible if and only if G(F) is strongly connected. An oriented cycle in a directed graph is a simple oriented path from a vertex to itself. A directed tree is a connected directed graph with no cycles. A directed tree is said to be 'rOoted at a vertex, called root, if every path between a non-root vertex and the root is oriented towards the root. We refer the reader to [23] for more details. Consider the linear system (2.1)
Bv =0, where 2:1#1/311 - /312
-/321 2:1#2 /321 ...
(2.2)
and /3ij ;;: 0, 1 ~ i, j ~ n. (.
Lemma 2.1. Assume that the matrix (/3ij)nxn is irreducible and n ;;: 2. Then the followings hold.
(1) The solution space of system (2.1) has dimension 1. (2) A basis of the solution space is given by (V1,V2,'"
,vn ) = (0 11 ,022 ,'" ,Onn),
where Okk denotes the cofactor of the k-th diagonal entry of B, 1 ~ k ~ n.
(3) For 1 ~ k
~
n,
Okk =
L
II
/3jh > 0,
(2.3)
TE'lrk (j,h)EE(T)
where 'lI'k is the set of all directed trees of n vertices rooted at the k-th vertex, and E(T) denotes the set of directed arcs in a directed tree T.
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
274
Proof. Since the sum of each column in B equals zero, we have
(2.4) where C jk denotes the cofactor of the (j, k) entry of B. Since B is singular, we know that (C11 , C I2 ,··· ,C1n ) is a solution of system (2.1). Therefore, by (2.4), (C 11 , C 22 ,··· ,Cnn ) is also a solution of system (2.1). For 1 ~ k ~ n, in the k-th column of B, the diagonal entry, Ll# 13kl, equals the negative of the sum of nondiagonal entries. By a result on directed graphs in [23, p. 47, Theorem 5.5], we obtain Ckk
L
=
II
13j h.
TElI'k (j,h)EE(T)
Since (/3ij) is irreducible, its associated directed graph is strongly conI1 /3jh nected, by P5. Thus, for each k, at least one term in L TElI'k (j,h)EE(T)
is positive. Therefore, Ckk > 0 for k = 1,2,··· ,n. Since C 11 is a (n -1) minor of B, we know rank(B) = n - 1, and the solution space of (2.1) has dimension 1, completing the proof of Lemma 2.1. 0 As an illustration of (2.3), let n = 3 and '][\ be the set of all directed trees rooted at the first vertex. Then, as shown in Figure 1, '][\ = {Tl, Tn, and E(Tn = {(3, 2), (2, I)}, E(Tf) = {(2, 1), (3, I)}, E(Tt) = {(2,3), (3, I)}. Therefore,
Tr,
C11
=
L
II
T{E1I'l (j,h)EE(T{)
3
2
2
3
,
T.'
Figure 1. All directed trees with three vertices and rooted at 1. A unicyclic graph is a directed graph that is obtained from a collection of directed rooted trees by joining their roots and contains a unique
Global Stability in Multigroup Epidemic Models
275
oriented cycle. For 1 :::; l :::; n, let D(n, l) denote the number of unicyclic graphs with n vertices whose cycle has length l. Then (2.5) and
n
I: D(n, l)l.
nn =
(2.6)
l=l
For proofs of these relations, we refer the reader to [2, Chapter 2]). In a directed rooted tree, if we add a directed arc from the root to any non-root vertex, we obtain a unicyclic graph, see Figure 2.
.i
"-
"- ......
k
Figure 2. A unicyclic graph is formed from a directed tree rooted at vertex k by adding a directed arc from k to j. Let Ro be defined in (1.5). We first establish the following result for system (1.3). Results like Proposition 2.2 are known in the literature, at least for some special classes of model (1.3) (see [10, 22, 27]). We provide a proof for completeness and to demonstrate our derivation of
Ro· Proposition 2.2. Assume B hold.
=
((3ij)
is irreducible.
Then followings
(1) If Ro :::; 1, then Po is the unique equilibrium and it is globally stable in
r.
(2) If Ro > 1, then Po is unstable and system (1.3) is uniformly pero
sistent in
r.
Proof. Let 8 = (81 ,'" ,8n ), 8 0 = (8~,··· ,8~), and
M(8)
= (
E
(d i
jE
(3i i8;
+ Ei)(di + 'Yi)
)
l";i,j~n
.
For 1 :::; k :::; n, since 0 :::; 8k :::; 8Z, we have 0:::; M(8) :::; M(80 ) = M o, and if 8 =I- So, then M(S) < Mo. Since B is irreducible, we know M(8)
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
276
is irreducible for 0 < 8 ~ 8 0 . Therefore, by P3, p(M(8)) < p(Mo) Ro ~ 1 provided 8 i= 8 0 . This implies that
=
M(8) I = I has only the trivial solution I = (h,'" ,In)t = 0, and that Po is the only equilibrium of system (1.3) in r if Ro ~ 1. Let (Wl,W2,'" ,wn ) be a left eigenvector of Mo corresponding to p(Mo), i.e., (Wl,W2,'"
,wn)p(Mo) =
(Wl,W2,'"
Since Mo is irreducible, we know, by P2,
Wk
,wn)Mo.
> 0 for k = 1,2",' ,n. Set
Then n
L'
=
~ (df + fk~(dr + ')'k) [fkE~ + (df + fk)I~] ,wn ) [M(8) I - I]
= (Wl,W2,'"
~ (Wl,W2,'" =
,wn ) [Mo I - I] [p(Mo) -1](Wl,W2,'" ,wn ) I ~ 0,
Furthermore, if Ro L' = 0 implies
= p(Mo) < 1, then L' = 0 {=} I = O.
(Wl,W2,'"
If 8
i= 8
0
,
if Ro
,wn)M(8)I =
~
1.
If Ro
(Wl,W2,'"
,wn)I.
,wn)Mo =
(Wl,W2,'"
= 1, then (2.7)
then
,wn)M(8)
0 in a neighborhood of
Po in f, by continuity. This implies Po is unstable. Using a uniform persistence result from [6] and a similar argument as in the proof of Proposition 3.3 of [19], we can show that, when Ro > 1, the instability of Po implies the uniform persistence of (1.3). This completes the proof of Proposition 2.2. 0
Global Stability in Multigroup Epidemic Models
277
Uniform persistence of (1.3), together with uniform boundedness of o
solutions in r, implies the existence of an equilibrium of (1.3) in Theorem D.3 in [25] or Theorem 2.8.6 in [4]).
0
r
(see
Corollary 2.3. Assume B = ((3ij) is irreducible. If Ro > 1, then (1.3) has at least one endemic equilibrium.
3
Proof of Theorem 1.1
Denote an endemic equilibrium, whose existence is established in Corollary 2.3, by P* = (S;, E;,1;,'" ,S~, E~,1~),
Sk' E k, Ik > 0 for k = 1, 2, ... ,n. In this section, we prove that P* is globally asymptotically stable when Ro > 1. In particular, this implies o
that the endemic equilibrium is unique in the region r when it exists. Set (3.1) i3ij = (3ij S; Ij , 1 ~ i,j ~ n, n ~ 2, and
2::1# 131/ -/321 -1312 2::1#21321 '"
-i3nl -i3n2
B=
(3.2)
-i31n
-i32n
2::1#n i3nl
Then, by Lemma 2.1, a basis for the solution space of the linear system (3.3)
Bv =0 can be written as
(3.4) where C kk denotes the cofactor of the k-th diagonal entry of B, 1 ~ k ~ n. By the irreducibility of B, we know that (i3ij) is irreducible and Vk = C kk > 0, k = 1"" ,n, by Lemma 2.1. For n = 1, i.e., the case of single group SEIR model, Theorem 1.1 is well know (e.g. see (15]). We only consider the case n ~ 2. Let VI,'" ,vn be chosen as in (3.4). Set n
V
=
L Vk [(Sk k=1
Si., lnSk)
+ (Ek -
Ei., lnEk)
+
~
k E: Ek (h - Ii., lnh) ]. (3.5)
278
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
Differentiating V and using the equilibrium equations n
Ak = drS'k
+ L(3kj S 'k 1j,
(3.6)
j=l n
(df + Ek)E'k = L(3kj S 'k 1j,
(3.7)
j=l
(3.8) and
(3.9) which follows from (3.7) and (3.8), we obtain
(3.10)
Global Stability in Multigroup Epidemic Models
279
since
S'k Sk
Sk
+ S'k
- 2;;:: 0,
and the last equal sign holds if and only if Sk = S'k. In the above derivation, we have substituted the two incidences of Ak using (3.6). Next, for V1,'" ,Vn as in (3.4), we claim
for all (h,'" ,In) E lR+.. To see this, we note that n
n
n
LVk L{3kj S 'k I j k=1 j=1
n
n
n
= LVj L{3jk S ;h = L (L{3jk S ;Vj)h. (3.12) j=1
k=l
k=1
j=1
It suffices to show k = 1,2"" ,n.
(3.13)
In fact, by (3.1)-(3.3) and (3.9), we have
[
{3ll~i Ii
.... :
{3nl~~Ii
{31nSi I~ .. ,
{3nnS~I~
1
and this leads to (3.13). Using inequality (3.10), notation 13kj in (3.1), and identity (3.11), we have
Denote
(3.14)
280
Hongbin Guo, Michael Y. Li, Zhisheng Shuai o
In the following we show Hn ~ 0 for all (81, E 1, h,'" ,8n , EnIn) E r. Since the proof for the general case uses a graph theoretic approach and is quite abstract, we choose to illustrate the main ideas first with detailed proofs for cases n = 2 and n = 3. Then major steps of the proof for arbitrary n ~ 2 will be given. Case n = 2: In this case, we obtain from (3.3) V1 = /321 and V2 Substituting V1, V2 into (3.28) and expanding H2 we have
- - (
8i -
h81Ei Ii 8 i E 1 Iz8 1Ei I;8iE1 -
3 - 82
-
I182E; E2I;) Ii 8:;' E2 - E:;' Iz
8; 3 - 82
-
Iz 8 2 E; E2I; ) I;8;E2 - E:;'I2 .
H2 = i321i3n 3 - 8 1
- - ( 8i + i32d312 3 - 8 1
- - (
+ i312i321 -
-
+ i312i322
(
-
8;
= /312,
E1Ii) Eih EIIi) Eih (3.15)
Note that the middle two terms have the same coefficients /321/312' We show that this is not a coincidence and can be seen as follows: write the subindices of /3ij'S in each coefficient in the form (3.16) respectively. Each expression in (3.16) defines a transformation from row 1 to row 2 and possesses a unique cycle of length 1 or 2. Moreover, the two coefficients in (3.15) corresponding to the transformations with a 2cycle (a cycle with length 2) are the same, since they arise from rotations of the same 2-cycle. Therefore, the terms in H2 can be naturally grouped according to the length of cycles appearing in their coefficients.
Note that
Global Stability in Multigroup Epidemic Models
281
and
/32Ii312 (6 -
Si _
I 2S 1Ei _ E1Ii _ S2 _ hS2E2 _ E2I2) , :::: 0 I 2 S i E 1 Eih S2 Ii S2 E2 E2I2 '" .
Sl
We thus obtain H 2(Sl, E 1, h, S2, E 2, 12) ~ 0, and thus V' ~ 0, for all o
(Sl,E1,h,S2,E2,I2) E r. From (3.10), we know V' = 0 if and only if Sk = Sic, k = 1,2, and H2 = O. Moreover, irreducibility of matrix B, or equiv~le~tly, the strong connectedness of the directed graph G(B), implies (321(312 > O. Consequently, we obtain from (3.17) V'=O
Sk=Sic, Ek=aE'k, h=aI'k,
{=:::>
k=1,2,
where a is an arbitrary positive number. Case n - 3:
In this case H 3
= ~ 6
k,j=l
-(3 _ S'kS _
v (3k kJ
k
SkIjE'k _ EkI'k) S* 1* E E* I . k J k k k
(3.18)
From (3.4) we obtain V1
= /332/321 + /331/321 + /323/331,
V2 = V3 =
+ /313/332 + /312/332, /312/323 + /321/313 + /313/323.
/331/312
(3.19)
Substituting these expressions of Vk into H 3 , we observe that H3 is the sum of 33 = 27 terms of forms
(3.20) or - - - ( Sk SkIjEk EkIk) (3rk(31k(3kj 3 - -S - S* 1* E - E* I ' k kj k kk
(3.21 )
where {r, l, k} is a permutation of {I, 2, 3}, and 1 ~ j ~ 3. Write the subindices of /3ij'S in (3.20) and (3.21) in the form of transformations
l k}
r { l k j
and
rlk} {k kj ,
(3.22)
respectively. When j = k, lor r, both transformations in (3.22) possesses cycles of length 1, 2 or 3. The terms in H3 will be grouped together according to the length of cycles appearing in (3.22).
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
282 When j
= k,
both transformations in (3.22) have a I-cycle {: :
~},
and accordingly, the terms in (3.20) and (3.21) satisfy
- - - ( Sk SklkEk Eklk) /3rl/3lk/3kk 3 - Sk - Sk1k Ek - Ekh :(: 0, and
- - - ( Sk Sk1jEk Eklk) /3rk/3lk/3kk 3 - Sk - SkI; Ek - Ekh :(: O. When j
= r,
the first transformation in (3.22) produces two distinct
3-cycle patterns
!
{~ ~}
{~ 7;}.
and
There are 6 terms in H3 of a
3-cycle form, three of them correspond to each cycle pattern, and thus have the same coefficients i3rli3lki3kr or i3rki3kli3lr' These six terms can be divided into two groups and each has a sum of form si., SkIrEi., Ek I ;) + /3- /3- /3- (3 /3-rl /3-lk /3-kr (3 - Sk - S*k1* Ek - E*k Ik lk kr rl r /3/3(3 §L SlIkEi ElIi) +/3kr rl lk - Sl - Si I; El - Ei Il
s;
SrItE;
S r - S*r1* Er l
-
ErI; )
E*r T I
(3.23) When j = r, the second transformation in (3.22) has a 2-cycle
~ : ; }. Also, when j = l, both transformations in (3.22) have a 2-cycle * k l . There are altogether 12 terms in H3 corresponding to 2-cycle { *lk}
{
patterns. Each 2-cycle pattern corresponds to 2 terms in H3 with the same coefficients. These 12 terms can be grouped into 6 pairs and each has a sum of form
/3-rk /3-lk /3-kr (3 -
§1. - SkIrE'k - Ek I ;) Sk
-- /3-rk /3-lk /3-kr (6 -
S*k1*rEk
E*kIk
+ /3-kr /3-lk /3-rk (3 - !E:.. S -
§1. - SkIrE'k - EkI; Sk
S*k1* Ek r
E*k Ik -
r
!E:.. S r
-
SrIkE; S*r1* E k r
SrhE; S*I*E rkT
ErI;)
E*r T I
0
:(:,
(3.24)
or
= /3-rl /3-lk /3-kl (6 -
Err) =-::..z:.E*I rT
S'k _ SkIlE'k _ EkI'k _ Sk S*I*E E*I kl k kk
§I _ SlhEi _ ElIi) Sl
S*I*E l k l
E*I II
:(:
O. (3.25)
Global Stability in Multigroup Epidemic Models
283
In summary, each coefficient in H3 corresponds to a transformation in (3.22) which possesses a unique cycle of length 1,2, or 3. By property (3) of Lemma 2.1, the number of transformations in (3.22) with an 1cycle is given by D(n, I) x I, I = 1,2,3. In particular, by (2.5), the number of I-cycles in (3.22) is D(3, 1) x 1 = 9, the number of 2-cycles is D(3,2) x 2 = 12, and the number of 3-cycles is D(3,3) x 3 = 6. Therefore, by (2.6), 33
= 27 = D(3, 1) x 1 + D(3, 2) x 2 + D(3, 3) x 3.
This shows that all terms in H3 are accounted for in our grouping according to cycle patterns and lengths in (3.22). Therefore, we have shown o
0 for all (Sl,E 1 ,h,S2,E2,I2,S3,E3,h) E f, and thus V' ~ O. From (3.10), we know V' = 0 if and only if Sk = S'k, k = 1,2,3, and H3 = O. We claim that if Sk = S'k, k = 1,2,3, then
H3
~
(3.26) where a is an arbitrary positive number. It suffices to show that H3 = 0 implies 1
~
k,r
~
3.
(3.27)
If f3kr = 0, for some k #- r. Then, by the irreducibility of B = (f3ij) , or equivalently, the strong connectedness of the G(B), we necessarily have - f3klf3lr
#- 0, for I #- k, r. Therefore, either a 3-cycle {klr} Irk exists or both
2-cycles {
7~ :} and {~ ~ :} exist. In either case, (3.27) follows from
H3 = 0, and from relations (3.23), (3.24), and (3.25). If all f3ij #- 0, i #- j, then i3kli3lr #- 0, for I #- k, r, and the same argument shows that (3.27) holds. We thus obtain Vi = 0
{==:}
Sk = S'k, Ek = aE'k, h = aI'k,
k = 1,2,3,
where a is an arbitrary positive number. Case n ;;::: 2:
We have
(3.28) Here Vk = Ckk as given in (2.3) is a sum of nn-2 terms, each of which is a product of (n - 1) i3i/S whose subindices can be represented by
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
284
all arcs in a directed tree T rooted at the k-th vertex, by Lemma 2.1(3). Therefore, subindices of Vk 13kj, which is a product of n 13i/S, are represented by all directed arcs of a unicyclic graph Q obtained by adding an oriented arc (k, j) from the root k to the vertex j to the directed tree T, as shown in Figure 2. Unicyclic graph Q has a unique cycle CQ of length 1 ~ I ~ n. Furthermore, all l rotations of the I-cycle CQ give rise to l terms in Hn with the same coefficients, and these I terms are naturally grouped together. We can show, as in the cases of n = 2,3, the sum ofthese I terms is nonpositive. More specifically, using (2.3), we can first group all terms in Hn according to the cycle length present in their coefficients, then further group the terms of the same cycle length according to their cycle patterns represented in the unicyclic graph Q, as shown in the following H
n
=
=
~ v
{3- .
L k kJ k,j=1
t
1=1
(3 _ 8"k _ 8k 1j E "k 8 8* 1* E k
k
J
_ EkI;') E* I k k k
[L L ( II !3jh) QED(n,l) (1',m)EE(CQ) (j,h)EE(Q)
( 3 _ 8; _ 8 1' 1m E ; _ E1'I;)] 81' 8;I;"E1' E;I1'
(3.29)
n
=L
1=1
[QED(n,l) L ((j,h)EE(Q) II !3jh)
L
(3 _ 8; _ S1'1m E ; 81'
(1',m)EE(CQ)
8;I;"E1'
_ E1'I;)] E;I1' '
where V(n, l) presents the set of all unicyclic graphs of n vertices with an oriented cycle of length l, CQ is the oriented cycle of length l in a unicyclic graph Q E V(n, l), and E(CQ), E(Q) represent the sets of arcs in CQ, Q, respectively. Since the cardinality of E(Q) is n, the coefficient of each term in (3.29), TI(j,h)EE(Q) 13jh, is a product of n 13i/S. The cardinality of the set V(n, l) is (3.30) and the cardinality of E(CQ) is the length l of the cycle CQ. By the identity (2.6) n
nn =
L D(n, l)l, l=1
(3.31)
Global Stability in Multigroup Epidemic Models
285
we see that all terms in Hn are accounted for in our grouping (3.29). For the oriented cycle CQ in any Q E D(n, l), we have
(3.32) By (3.29) and (3.32), we know H n (Sl, E 1 , h,··· ,Sn, En'!n) ~ 0 for all o
(Sl, E 1 , h,··· ,Sn, Enln) E r. Therefore, we have V' ~ we claim that if Sk = S'k, 1 ~ k ~ n, then
o. Furthermore, (3.33)
where a is an arbitrary positive number. It suffices to show that
when i3kr =f. By the irreducibility of (fJij), there exist 1 ~ m1, m2, ... , ms ~ n, 0 ~ s ~ n - 2 such that k, r, m1,··· ,m s are distinct, and
o.
the product i3kri3rm li3mlm2 ... i3m s k =f. o. Furthermore, there exists a unicyclic graph Q E D(n, l) such that CQ = {(k, r), (r, md,··· ,(m s , k)} and IT i3j ,h =f. O. Therefore, from (3.29) and (3.32), we know (j,h)EE(Q)
Ek - b.. E; and lJ.. Ik -- Ix. I; 1·f H n -- 0 . From (3.10) and (3.33), we know that V' ~ 0 for all (Sl, E,I,· .. ,Sn,
§Jr.. -
o
r,
and V' = 0 {o} Sk = S'k,Ek = aE'k,h = al'k,k = 1,2,··· ,no Substituting Sk = S'k, Ek = aE'k, and h = alk into the first equation of system (1.3), we obtain
En'!,) E
n
0= Ak - d~S'k - a LfJkjS'klj. j=l
(3.34)
Since the right-hand-side of (3.34) is strictly decreasing in a, we know, by (3.6), that (3.34) holds if and only if a = 1, namely at P*. Therefore, the only compact invariant subset of the set where V' = 0 is the singleton {P*}. By LaSalle's Invariance Principle, P* is global asymptotically o
stable in
r
if Ro
> 1. This completes the proof of Theorem 1.1.
Acknowledgments This research is supported in part by grants from the Natural Science and Engineering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI). Both HG and ZS acknowledge the support
286
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
of the Josephine Mitchell Graduate Scholarships from the Department of Mathematical and Statistical Sciences at the University of Alberta. The authors also acknowledge the financial support from NCE-MITACS Project "Transmission Dynamics and Spatial Spread of Infectious Diseases: Modelling, Prediction and Control".
References [1] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, and S.A. Levin (Eds.), Mathematical Ecology, Singapore World Scientific, Teaneck, NJ, 1986,317-342. [2] F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge University Press, Cambridge, 1998. [3] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [4] N. P. Bhatia and G. P. Szego, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, Vol. 35, Springer, Berlin, 1967. [5] H.I. Freedman and J.W.-H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985), 69-86. [6] H.I. Freedman, M.X. Tang, and S.G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat. 6 (1994), 583-600. [7] B.S.Goh, Global stability in many-species systems, Am. Nat. 111 (1977), 135-143. [8] H. Guo and M.Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng. 3 (2006), 513525. [9] K.P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci. 146 (1997), 15-35. [10] H.W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol. 14 (1978), 338-349. [11] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000), 559-653. [12] H.W. Hethcote and H.R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985), 205-227.
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[13] W. Huang, K.L. Cooke, and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV / AIDS transmission, SIAM J. Appl. Math. 52 (1992), 835854. [14] A. Korobeinikov, A Lyapunov function for Leslie-Gower predatorprey models, Appl. Math. Lett. 14 (2001), 697-699. [15] A. Korobinikov and P.K. Maini, A Lyapunov function and some properties for SEIR, SIS Epidemic models, Math. Biosci. Eng. 1 (2004), 157-160. [16] A. Korobeinikov and G.C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett. 15 (2002), 955-960. (17] A. Lajmanovich and J.A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), 221-236. (18] J.P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [19) M.Y. Li, J.R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999), 191-213. [20) M.Y. Li and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci. 125 (1995), 155-164. [21] M.Y. Li and L. Wang, Global stability in some SEIR epidemic models, in: C. Castillo-Chavez et. al. (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, The IMA Volumes in Mathematics and Its Applications, Vol. 126, Springer, New York, 2002, 295-312. [22) X. Lin and J.W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations, J. Austral. Math. Soc. Ser. B 34 (1993), 282-295. [23) J.W. Moon, Counting Labelled Trees, William Clowes and Sons, London, 1970. [24] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. and Comp. Sci. 10 (2000), 63-79. [25] H.L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
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[26] H.R. Thieme, Local stability in epidemic models for heterogeneous populations, in: V. Capasso, E. Grosso, and S.L. PaveriFontana (Eds.), Mathematics in Biology and Medicine, Lecture Notes in Biomathematics 57, Springer 1985, 185-189. [27] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [28] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29-48.
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Epidemic Models with Time Delays* Wendi Wang School of Mathematics and Statistics, Southwest University Chongqing 400715, China E-mail: [email protected]
Abstract This article overviews mathematical approaches for analysis of epidemic models with time delays. We start by introducing the methodology of modeling epidemic diseases with time delays and illustrating the D-subdivision method for characteristic equations. Then we show the application of the Liapunov functional method. Lastly, we present the persistence techniques to determine conditions under which diseases are permanent in the population.
1
Introd uction
Since the pioneer work of Kermack and McKendrick [35], mathematical models of differential equations have become important tools in analyzing the spread and control of infectious diseases [1, 6, 7, 15, 16, 17, 21, 19,30,31,33,37,48,49]. Although models of ordinary differential equations play fundamental roles, time delays are common and important in modeling epidemic diseases. First, the incubation periods of infectious diseases, the infection periods of infective members and the periods of recovered individuals with immunity can be represented by time delays. More importantly, these delays can not be neglected in most of cases. This is not only because the lengthes of delays may be long, for example, the incubation time for HIV could reach 10 years, but also because quite different dynamical behaviors of mathematical models can be induced by time delays. The objective of present work is to survey recent advances of delayed epidemic models. This paper is organized as the follows: L Mathematical models with time delays 2. The methods of stability analysis 3. Persistence of diseases 4. Summary *The author is partly supported by NSF of China (No. 10571143).
290
2
Wendi Wang
Mathematical modeling
As any other mathematical modeling, we must begin from selecting model's structures to construct epidemic models. Basically, we follow the methodology of compartments from Kermack and McKendrick [35]. Then we branch according to the durations of infectious diseases. If the disease spreads quickly and is not fatal, we can ignore the birth and death of populations. On the other hand, we should consider demographic structures if an infectious disease persists a long time or the disease-induced mortality rate is high. For illustration purpose, we present several typical epidemic models to show ingredients of models ignoring birth and death processes or models with demographic structures, and ways of introducing time delays. We always adopt the following nomenclature: t: time;
N: population density or population number;
S: the number or density of susceptible individuals; E: the number or density of exposed individuals;
I: the number or density of infectious individuals; R: the number or density of recovered individuals.
2.1
Models without demographic structure
For an SIR type of disease, if r is the latent period of the disease, we have
S' (t) = -(3S(t)I(t), / (t)
=
(3S(t - r)I(t - r) - "(I(t),
R' (t) = "(I(t),
where {3 is the contact coefficient and "( is the recovery rate (Ma et al., [33]). Cooke and Yorke [12] proposed the following model for gonorrhea epidemics:
dS dt = -(3S(t)I(t)
+ (3S(t -
r)I(t - r),
dI dt = (3S(t)I(t) - (3S(t - r)I(t - r), where r is the infection period of the disease.
Epidemic Models with Time Delays
2.2
291
Model with vital dynamics
General demographic structure of a population in the absence of disease takes the form:
~ = B(N)N -
D(N)N,
(2.1)
where N is the population size, B(N) is the per capita birth rate and D(N) is the per capita death rate. Some structures frequently used in literature are listed below:
(1) B(N) = D(N) = /-1, where /-1 is a constant. This means that the birth and death are balanced so that the population size is a constant, which simplifies mathematical analysis (see, for example, Ma et al., [33], Wang [39]); (2) The net birth rate of population is a constant A, and the per capita death rate of population is a constant d. Then vital dynamics is given by dN (2.2) ill =A-dN. One advantage of this population dynamics is that the population size is variant and there is a saturation effect for population growth. Further, mathematical analysis for epidemic models from this demographic structure may be much easier. But this seems reasonable only when population size is approximately a constant or there is recruitment from outside; (3) The population growth is simulated by the logistic equation. A typical differentiation of the logistic growth into birth and death process is given by B(N) = b - arNjK and D(N) = d + (1a)r N j K with r = b - d (see Gao and Hethcote [18]).
(4) B(N) = be-aN,D(N) = /-1 with a > O,b > 0,/-1 > 0. This means that population grows according to the Ricker law [13]; ~,D(N) = /-1 with p > O,q > O,m > 0,/-1 > 0. This type of vital dynamics was proposed by Mackey and Glass [34] and used by Jin and Wang [24];
(5) B(N) =
= ~ + L,D(N) = /-1 with A > O,L > 0,/-1 > 0. This structure represents a constant immigration rate A together with a linear birth term LN, which is used by Cooke, van den Driessche and Zou [13], Wang and Zhao [45, 46, 47].
(6) B(N)
On the basis of the population dynamics (2.2) and the standard incidence, Hethcote and van den Driessche [22] proposed the following SIS
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292
epidemic model:
l
(t) = 'x(l- ~\tl))l(t) - ,Xa(l - ~~t-=-~))l(t - w) - (d + f)l(t),
N' (t)
=
(2.3)
A - dN(t) - EI(t),
where w is the infectious period and a = exp( - (d + f)W), which gives the survival fraction of infectives through the infection period. Beretta and Kuang [2] proposed the following model for bacteriophage infection:
~~
=
as(t)(l- S(t)
~ l(t)) _ KS(t)P(t),
dl = KS(t)P(t) _ Ke-Jl-i TS(t - T)l(t - T) - /-Lil(t), dt dP = (3 - KS(t)P(t) dt
(2.4)
+ bKe-Jl-iTS(t - T)l(t - T) - /-LpP(t),
where S is the density of susceptible bacteria, I is the density of infected bacteria, P is the density of viruses or bacteriophages, and T is the survival fraction of infected bacteria through the infection period. In this model, only susceptible bacteria S are capable of reproducing by cellular division according to the logistic growth, whereas the infected bacteria, under the genetic control of virulent phages, replicate phages inside themselves up to the death by lysis after a infection period. However the infected bacteria I still compete with susceptible bacteria S for common resources. Cooke and van den Driessche [9] studied an SEl RS model with two time delays: latency period and the period of temporary immunity:
dS dt
=
bN( ) _ bS( ) _ 'xS(t)l(t) t t N(t)
l( _ ) -bT
+ "( t T e
E(t) = it 'xS(u)l(u) e-b(t-U)du t-w N(u) , dl dt
=
'xS(t - w)l(t - w) _ (b )l( ) N(t-w) +"( t,
R(t) =
,
(2.5)
l~T "(l(u)e-b(t-U)du,
where w is the incubation period, T is the period of temporary immunity, b is the per capita birth rate and the per capita death rate, ,X is the valid contact rate. In the study of HIV models, it is interesting to introducing distributive delays. Culshaw, Ruan and Webb in [14J incorporated continuous
Epidemic Models with Time Delays
293
delays into a model of ceU-to-cell spread of HIV-1:
~~
=
reG(t)
dI dt = kr I
jt
-00
(1 - G(t~:I(t)) - krG(t)I(t), (2.6)
G(u)J(u)F(t - u)du - f.LrI(t),
where G(t) represent the concentration of healthy cells, J(t) is the concentration of infected cells, re is the effective reproductive rate of healthy cells, GM is the effective carrying capacity of the system, kr represents the infection of healthy cells by the infected cells in a well-mixed system k~/kr is the fraction of cells surviving the incubation period, f.LI is th~ death rate of the infected cells, and F is the kernel function. In paper [44], another time delay that mimics the density regulation effects of cells is also incorporated:
4ft- = reC(t)(l - f~oo f(t-~~(s)+I(s)) )ds { dft = k~ J~oo G(u)I(u)F(t - u)du - f.LrI(t),
krG(t)J(t), (2.7)
where f is the kernel function. If a population can be split into juvenile group and adult group, and an epidemic disease propagates only in the adult community, then a maturation delay can be introduced into the epidemic model. In [13], Cooke, van den Driessche and Zou investigated the following model:
~~ = dN dt
=
)..(N(t) - J(t)) ~~l)
-
(d + E + 'Y)J(t) ,
B(N(t _ T))N(t _ T)e- d1T
-
dN(t) - El(t),
where S is the density of susceptible adults, J is the density of infected adults, N = S + J, T is the maturation time of juveniles, d 1 is the per capita death rate of juveniles, A is the valid contact rate, and E is the disease-induced death rate. Summarizing above discussions, we have models that can be split into two types: one with coefficients independent of time delays (type I), the other one with delay-dependent coefficients (type II). The stability analysis of models is distinguished according to these types.
3
Analysis of local stability
The basic approach for the local stability analysis of a delayed system is to linearize it at an equilibrium and then consider the characteristic
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294
equation of the linearized system. For a linear system (3.1) where L is a linear operator from C([ -T, 0], Rn) into Rn. If we write
(3.2) where ry((}) is an n x n matrix whose elements are of bounded variation on [-T,O], then the characteristic equation of (3.1) is
where I is an identity matrix. By the stability theory of delayed differential equations [20, 27, 43], we have Theorem 3.1. The equilibrium is stable if all characteristic roots of (3.3) have negative real real parts, and is unstable if there is one characteristic roots has the positive real part. If there is only one delay, the characteristic equation takes the form: ~(>') := P(>.)
where
+ Q(>.)e- AT
n
P(>.)
= 2:::: ak>.k, k=O
=
(3.4)
0,
m
Q(>.)
= 2:::: bk>.k,
(3.5)
k=O
where ak and bj are constants. For a system of type I, ak and bj are independent of the delay, while some of them are the functions of the delay for the system of type II.
3.1
Models with constant coefficients
Typical characteristic equations of type I systems in epidemiology are
(>. + a) + be-AT = 0, (>.2 + al>' + ao) + (b l >' + bo)e- AT
=
O.
(3.6) (3.7)
The basic approach of stability analysis for such equations is the Dsubdivision method (see, for example, the book of Kolmanovskii and Myshkis [26]). For a fixed delay T, we consider the equation ~(iy) = 0 with a variable y. As y varies in IR = (-00,00), the graphs of the equation split the parameter space of ai and bj into a number of domains
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295
in each of which, the stability is unchanged. The exact stability information can be drawn with the aid of the direction of the real parts of characteristic roots as a point crosses the curves. We illustrate this by (3.6). In the a-b plane, the curves of A(iy) = 0 are given by
a = -ycot(yr),
b = y/ sin(yr)
(3.8)
and a
+ b = o.
(3.9)
These curves divide the plane into two parts: domain U that lies in the left of (3.9) and above (3.8), and the domain V that lies in the right of (3.9) and below (3.8). U is an unstable domain because a point in the negative a-axis corresponds to an characteristic root A = -a > O. Similarly, V is a stable region. Furthermore, since the curve of (3.8) lies above the line of b = a with a ~ 0 for any r > 0, the domain of Ibl < a is absolutely stable, i.e., a system with the characteristic equation (3.8) is stable irrespective of the length of the delay. Advanced techniques for the applications of the D-subdivision method can be found in Ma's book [32J.
b
a
Figure 3.1: The solid curve is the graph of (3.8), the dashed line is the graph of b = a with a ~ 0 and the solid line is the graph of a + b = o. Generally speaking, an equilibrium of epidemic models is stable when r = O. Basically, this means that all the characteristic roots have negative real parts. As the time delay r increases, these roots continuously move in the complex plane. The equilibrium becomes unstable if one root crosses the pure imaginary root from the left to the right. If this root stays in the right of the pure imaginary root afterwards, the equilibrium remains unstable. if it goes back to the left and all the other roots always stay in the left, the equilibrium becomes stable again. This phe-
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296
nomenon of stability transitions is called as stability switches. Multiple stability switches can occur in epidemic models. For (3.4) with ak and bj independent of the delay, Cooke and van den Driessche [10] proved that there may be only finite stability switches and eventually unstable state may occur as the delay increases. Criteria for stability switches are given in [27, page 83]: Theorem 3.2. Suppose that P(>.) and Q(>.) are analytic functions in Re>. > 0 and satisfy the following conditions:
(i) P(>.) and Q(>.) have no common imaginary root; (ii) P(-iy) = P(iy),Q(-iy) = Q(iy) for real y; (iii) P(O)
+ Q(O) i- 0;
(iv) limsup{IQ(>')/ P(>')I : 1>'1 ~ := IP(iy)1 2
(v) F(y) of real zeros.
00,
Re>. ?: O}
< 1,-
-IQ(iy)12 for real y has at most a finite number
Then the following statements are true:
(a) If F(y) = 0 has no positive roots, then no stability switch occur; (b) If F(y) = 0 has at least one positive root and each of them is simple, then a finite number of stability switches may occur and eventually, the equation becomes unstable.
3.2
Models with delay-dependent coefficients
Stability analysis for models with delay-dependent coefficients may be much more complicated because components of an endemic equilibrium could be the functions of the delay, and therefore, the endemic equilibrium does not exist when the delay is larger. This means that the delay should be confined to an interval to discuss its stability analysis. As a consequence, stability switches of models with delay-dependent coefficients may be quite different from those of models with delayindependent coefficients. Beretta and Kuang [3] proposed a geometric procedure to determine stability switches for models with delay-independent coefficients, illustrated that an equilibrium can be eventually stable after transitions of several stability switches. We present the method by the following characteristic equation: (3.10)
Here, we assume that the coefficients a, band c are real smooth functions of T, have continuous derivatives in T and satisfy
(3.11)
Epidemic Models with Time Delays
297
°
(3.11) means that>. = is not a root of (3.10). Then, we want to find conditions such that pure imaginary roots occur. Set>. = ±iw with w > 0. Without loss of generality, we consider only>. = iw. Substituting it into (3.11), we have
P(iw, r)
+ Q(iw, r)e-
iWT
= 0,
(3.12)
where
P(iw, r) = b(r) + iwa(r),
Q(iw, r) = c(r).
It follows from (3.12) that
b(r) cos(wr) = - c(r) ,
. (
) _ w(r)a(r) c(r) .
(3.13)
+ b2 -
(3.14)
sm wr -
As a consequence, we obtain
F(w, r)
:=
IP(iw, rW - IQ(iw, r)1
=
w2 a2
c2 = 0.
Therefore, the necessary conditions for the occurrence of pure imaginary roots are Ic(r)1 > Ib(r)l,
a(r)=;fO, {
(3.15)
b2
2
w(r) = (~)1/2. 2 a
Let (3.15) hold. In order to obtain sufficient conditions for the occurrence of pure imaginary roots, we set O(r) = w(r)r. Then O(r) satisfies cos(O(r))
b(r)
= - c(r) ,
. (O( )) r
sm
=
w(r)a(r) c(r).
(3.16)
Evidently, if O(r) is a solution of (3.16), so is O(r) +2mr. Hence, we have
rw(r) = O(r) + 2mr, which leads to r=
If
rn :=
O(r) + 2mr w(r) O(r) + 2mr w(r)
and (3.15) holds, we obtain the sufficient condition for the occurrence of pure imaginary roots: (3.17)
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298
By Beretta and Kuang [3], the direction of the pure imaginary roots crosses the imaginary axis as T increases can be computed by the formula:
dReA
sgn ~ I.>-=iw = sgnR(T), where
R(T)
:=
a2(T)w(T)W' (T)(a(T)b(T)
(3.18)
+ C2T)
+w 2(T)a 2(T)(a' (T)b(T) - a(T)b' (T)
(3.19)
+ C2(T)).
Example 1. We consider an extended Bass model with an evaluation stage:
dA(t) dt
-- =
p('y + AA(t - T))(O - A(t - T)) - aA(t),
(3.20)
where A(t) is the number of adopters of a new product, A is the valid contact rate of adopters of the product with potential adopters, 0 is the capacity of potential adopters of the product, T is the average time for an individual to evaluate the product, a = 8 + ZJ and p = e-(Hp)T. Here, 8 is the birth rate and the death rate of a population, "( is the intensity of advertisement of product, ZJ is the discontinuance rate of adopters of the product, and p is the rate that individuals leave the evaluation class since they have decided not to buy the product. See paper [40] for more details. This model admits a unique positive equilibrium
A*
= -
P"( + PA 0 - a
+ J (p"( -
PA 0
+ a)2 + 4 p2 A "( 0
2PA
.
Its characteristic equation is:
where
q=p(AO-2AA*-"(). By (3.15), we obtain
F(W,T) := w 2 If a 2
+ a 2 -l.
< q2, this equation admits a positive solution (3.21 )
Thus, (3.22)
Epidemic Models with Time Delays
299
Assume that 8(T) E (0,27l") satisfies (3.22). Then we have
Tn =
8(T) +2n7l" W(T) ,
Sn(T) = T - Tn·
(3.23)
Since it is difficult to obtain the zeros of Sn(T) analytically, we fix the parameters and use numerical calculations. We take C = 20, a = 0.2,.A = 0.1, I = O.l,p = e- 0 .2T . Then numerical calculations show that W(T) exists when 0 :( T :( 6.5047 and
W(T) = p' (1.9 _ 0.1 19.0 P - 2.0 + J 44~.0 p' - 76.0 p + 4.0) _ 0.04 2
}1/2
{
Furthermore, q(T)
< 0 when T varies in the interval. Thus, we define 8(T) = 7l" + arctan(- W(T)), a
_ 8(T) +2n7l" Tn (T) W(T) . By numerical calculations, we see that only So (T) = 0 has two roots TOl = 1.50164 and T02 = 4.41394. Furthermore, by (3.19) we obtain R(Tod = 1 and R( T02) = -1. Hence, the equilibrium is stable when 0 :( T < 1.50164 and 4.41394 < T, is unstable if 1.50164 < T < 4.41394. Further analysis
Figure 3.2: two roots of SO(T)
Wendi Wang
300
shows that there is a stable periodic solution in 1.50164 < r < 4.41394 (see [40]). Li and Ma extended the geometric method of paper [3] in ([28, 29]). We consider D()", r) := P()", r) + Q().., r)e- AT = 0, (3.24) where
P()", T)
=)..2
+ a(T) .. + e(T), Q().., T) = beT)>' + d(T)
with T ~ O. We make the following assumptions: (AI) a( T), b( r), e( T) and d( r) are continuously differentiable in R+o
=
[0, +(0);
i= 0 for any T E R+o; P(iy, T) + Q(iy, T) i= 0 for any T E R+o;
(A2) e( T) + d( r) (A3)
(A4) Any pure imaginary root of (3.24) is simple; (A5) All roots of (3.24) have negative real parts when
T
= O.
(AI) ensures that the solutions of (3.24) are continuous in T. (A2) means that>. = 0 is not a root of (3.24). (A3) implies that P(>., T) = 0 and Q(>', T) = 0 do not have a common pure imaginary root. (A4) guarantees the transection of pure imaginary roots with the imaginary axis. (A5) indicates that (3.24) is stable when T = O. We begin from conditions for the occurrence of pure imaginary roots of (3.24). Substituting).. = iy into (3.24) and separating its real part and imaginary part, we obtain
b(T)ysinyr + d(r) COSyT = -[e(r) _ y2], { d(T) sin yT - b(T)y cos yT = a(T)y,
(3.25)
It follows that
F(y,T) where
:=
y4 - hCT)y2
+ hCr) = 0,
h(T) = b2(T) + 2c(r) - a2(r), h(T) = C2(T) - d2(T), h(T) = N(r) - 412(T).
To determine positive solutions of (3.26) easily, we assume
C3.26)
Epidemic Models with Time Delays (A6) Each of !i(T) any T E R+.
= 0 (j = 1,2,3)
When h(T) > 0, if F(y,T)
y! =
~ [!leT) + Jh(T)]
has at most one positive root for
= 0 has roots y~ '
301
and y~, we have
y~ = ~ [!leT) -
Jh(T)].
(3.27)
Assume that yeT) is a positive root of (3.26). In order to make iY(T) a pure imaginary root of (3.24), Y(7) must satisfy (3.25). From (3.25), we have
. _ -b(T)Y(C(T) - y2) + a(T)d(T)Y smYT b2(T)y2 + d2(T) , d(T)(C(T) _ y2) + a(T)b(T)y2 { COSyT = b2(T)y2 + d2(T) .
(3.28)
If T = 7* satisfies (3.28), then iY(T*) is a pure imaginary root of (3.24). Now, we replace y(T)T in (3.28) by OCT) to obtain
. {)( ) _ -b(T)y(C(T) - y2) + a(T)d(T)y ~ ( ) T b2(T)y2 + d2(T) - 'P y, T ,
sm
{
cos O( T)
= _
d(T)(C(T) - y2) + a(T)b(T)y2 ~ .1,( ) b2(T)y2+d2(T) 'l-'Y,T.
(3.29)
(3.29) determine a function: 'P arctan -;j' 7r
if sin{} > 0, cos{} > 0; if sin{}
= 1, cosO = 0;
'P 7r + arctan -;j ,
ifcosO
< 0;
37r 2 '
ifsinO
=
2'
{)( T) =
'P 27r + arctan -;j'
(3.30)
-1, cosO
ifsin{} < 0, cos{}
= 0;
> O.
If T = T* satisfies (3.28), it follows from (3.28) and (3.29) that there exists kENo = {O, 1,2,3, ... } such that
y(T*)T*
=
O(T*)
+ 2k7r.
(3.31 )
Now, we define
8(T) ~ y(T)T - {}(T). 27r For convenience, we assume also
(3.32)
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302
(A7) There exist at most finite solutions in equation S(7) = k for any kENo" and every solution of it is simple. Let (AI )-(A 7) hold. By the above discussions, we see that the necessary and sufficient condition that ±iy(7*) are roots of (3.24) is that 7* is a root of y(7)7-0(7) = k that is, S(7) = k, 21f
'
where 0(7) is defined by (3.30), and y(7)(7 E (0:, fJ)) is a positive root of (3.26). Now, we consider the direction that a pair of pure imaginary roots crosses the imaginary axis as 7 varies [28]. Theorem 3.3. Let (Al)-(A7) hold. If 7* E (0:, fJ) such that A = iY(7*) is a root of (3.24), the direction of A( 7), when 7 increases in a neighborhood of 7 = 7*, depends upon
v = sgn {tl (7*)} sgn { d~~7) IT=TJ . (1) If V = 1, when 7 passes through 7 = 7*, then A = A(7) passes through the imaginary axis from the left to the right;
(2) If V = -1, when 7 passes through 7 = 7*, then A = A( 7) passes through the imaginary axis from the right to the left.
The following two theorems [28, 29] give criteria for the stability and ultimate stability of delay-dependent equations. Theorem 3.4. Let (Al)-(A7) hold. If there is no positive solution in equation F(y,7) = 0 for any 7 E (0,00), or there is no positive solution in equation S(7) = k for any kENo, then (3.24) is stable for any
7 E [0, +00). Theorem 3.5. Let (Al)-(A7) hold and (3.24) have pure imaginary roots for some 7 E (0, +00). Suppose that the equation
has a unique positive root y+(7). Then we have
(i) (3.24) is ultimately stable if Y+(7) is defined on a finite interval; (ii) Assume that Y+(7) is defined in an infinite interval. Then (3.24) is ultimately stable ijlimsupS+(7) if lim sup S+(T)
T---?+oo
> O.
T---?+oo
< 0, and is ultimately unstable
Epidemic Models with Time Delays
Example 2.
Let us consider an SElS Model [28]:
dS dt = (b - d)S(t) - (3S(t)I(t) dE
dt
=
303
+ I'I(t),
(3S(t)I(t) - (3S(t - T)I(t - T)e- dy - dE(t),
(3.33)
dI dy dt = {3S(t - T)I(t - T)e- - (d + CY + I')I(t), where b is the birth rate of the population, d is the death rate of the population, {3 is the disease transmission coefficient, T is the incubation period, CY is the disease-induced death rate and I' is the recovery rate of infected individuals. When 9 = b - d > 0 and w := d + a + I' > I'e- dY , (3.33) has a unique endemic equilibrium P*. Its characteristic equation is:
(oX + 9 + f(T))(oX + w) - w(oX + g(l(T) - l))e->'Y = 0, where
(3.34)
I'e- dY f(T) = w-I'e- dY ·
It is easy to see that P* is stable when
F(y, T)
:=
T
= O. Set
y4 + g2 f2(T)y2 - w2g2[1 - 2f(T)]
(3.35)
= O.
When w ~ 31', (3.35) has a unique positive root y(T), which is defined in T E (0, +(0). If I' < w < 31', (3.35) has a unique positive root y(T), defined in T E ((In ~)/d, +(0). Thus, the existence interval for Y(T) is infinite in each case. Since limy--->oo f(T) = 0, we have limy--->oo Y(T) = vwg. Note that
It follows that lim S(T)
=
y--->oo
+00.
Consequently, Theorem 3.5 shows that P* is ultimately unstable. For another type of characteristic equation:
D(oX, T)
:=
P(oX, T)
+ Q(oX, T)e->'y + R(oX, T)e- 2>.y
where n
P(oX,T) = I>k(T)oX k , k=O m
k=O I
R(oX,T)
=
LTk(T)oXk, k=O
n > max{m,l},
= 0,
304
Wendi Wang
which is important in the analysis of epidemic models, Li and Ma [29] present excellent criteria for stability and stability switches.
4
Liapunov direct methods
The basic method for stability analysis of an equilibrium is to consider its characteristic roots. The advantage of this approach is that sharp conditions may be found in some cases. However, it is difficulty to apply when the number of equations is more than two or when there are several delays. However, Liapunov direct method could be a good option in these cases. We introduce the Liapunov functional method for the case of finite delay. Let G([-T, 0], Rn) denote the set of continuous functions mapping [-T, 0] into Rn. With linear operations and the norm defined by Ilepll = max lep(e)1 for any ep E G([-T, 0], R n), G([-T, 0], Rn) becomes (lE[-r,O]
a Banach space. If x(t) is continuous in [to - T, to + A) with A > 0, for to ~ t < A, we define Xt = x(t + e), e E [-T, 0]. Then a system of an autonomous delay-differential equations can be written as (4.1) Let R+ = [0,00). Suppose that x = 0 is an equilibrium of (4.1). Then we have the following fundamental stability theorems [20, 27]. Theorem 4.1. Suppose f : G -+ R n takes bounded sets of G into bounded sets of Rn, and WI, W2 : R+ -+ R+ are continuous nondecreasing functions, satisfying WI (0) = W2 (0) = 0, lim WI (r) = +00 and r-+oo
WI(r) > 0'W2(r) > 0 for r > O. If there is a continuous functional V : G -+ R such that
Then x = 0 is stable, and every solution is bounded. If we also have w2(r) > 0 for r > 0, then x = 0 is globally stable. We say V : G -+ R is a Liapunov function on a set G in G if V is continuous on G, the closure of G, and V ~ O. Let
S = {¢ E G: V(¢) = O}, M = largest invariant set in S which is invariant with respect to (4.1). Theorem 4.2. If V is a Liapunov function on G and Xt(¢) is a bounded solution that remains in G, then Xt(¢) tends to M as t -+ 00.
Epidemic Models with Time Delays
305
We now apply the Liapunov method to consider the stability of an
SEIRS model with two delays [38]. s'
= b - As(t)i(t) + j3i(t - r) - bs(t)
i'
= Aas(t - w)i(t - w) - b + b)i(t),
(4.2)
where s is the fraction of susceptible individuals, i is the fraction of infectious individuals, b is the birth rate and death rate of the population, A is the average number of adequate contacts of an infectious individual per unit time, w is the latent period of the disease, r is the immune period of the population, 'Y is the recovery rate of infectious individuals, a = exp(-bw) and (3 = 'Ye-bT. (4.2) is a simplification of (2.5). Indeed, the population N in (2.5) is a constant. If e(t) is the fraction of exposed individuals and ret) is the fraction of recovered individuals, wee obtain (4.2) and set) + e(t) + i(t) + ret) = 1. The model has two time delays and delay-dependent coefficients. Set T = max{r,w}. Due to our background, we will consider (4.2) in the set
It is easy to show that D is positively invariant for (4.2). It is clear that (1,0) is the disease free equilibrium of (4.2).
Theorem 4.3. If Ro (1,0) is globally stable.
=
Aa/ ('Y
+ b)
~ 1,
the disease free equilibrium
Proof. Note that the second equation can be rewritten as
i' = Aas(t)i(t) - b + b)i(t) + Aa[s(t -
w)i(t - w) - s(t)i(t)]
t
=
i(t) [Aas(t) -
b + b)]- Aa
!J
(s(u)i(u))du.
t-w
If
Xt = (s(t + e), e(t + e), i(t + e), ret + e)), e E [-T,O],
we define
t
Va(Xt) = i(t)
+ Aa
J
s(e)i(e)de.
t-w
The derivative of Va along solutions is V~ (Xt)
= i(t) [Aas(t) - b + b)] = b + b)i(t) [Ros(t) - 1].
Wendi Wang
306
Let S = {
. + C)U2, h
{
u3 = >'U2 - P,U3' (4.4) We consider a Liapunov functional defined by
1 1 2 + -2W3U3 1 2 1 * V(Ut) = -WI(UI + U2) 2 + -U + -/3IS 2 2 2 2 where Wi > 0 (i Note that
lh it f(s)
0
t-s
2 u2(v)dvds,
= 1,2,3) are to be determined later. 1 2 1 2 1 2 V(ut) ~ "2WI(UI + U2) + "2U2 + "2W3U3'
Calculating the derivative of V along the solutions of (4.4), we obtain
V'(Ut) = WI(UI + U2)[-p,UI - (>. + P, + C)U2] + (/31 + (32)1*UIU2
-(>. + P, + C- /32S*)U~ + /3IS*U2 Jo f(S)U2(t - s)ds h
+W3U3U; + ~/3IS*U~ - ~/3IS* Jo f(s)u~(t - s)ds. h
By simplifications, we obtain
V' (Ut) = -wIP,uI - [WI (>. + P, + c) + (>. + p, + C- /32S*)]U~ +[-WIP, - WI (>. + P, + c) + (/31 + (32)I*]UIU2
+ W3>'U2 U3 - W3p,U~
+/31S*U2 Joh f(S)U2(t - s)ds + ~/3IS*U~ - ~/3IS* J: f(s)u~(t - s)ds. (4.5)
Wendi Wang
308 Choose
Wl
> 0 such that
Note that For any U2, we have
/3S*U2
t
io
U2(t - s)ds
~ ~/3S*u~ + ~/3S* t 2 2 io
f(s)u 2(t - s)ds.
It follows from (4.5) that
V'(ut} ~ -wlp,ui -
Wl().
+ P, + c)u~ -
W3P,U~
+ W3).U2 U3·
(4.6)
If we choose W3 such that
i.e.,
4(). + p, + C)p,(/31 ).2().
+ /32)1* + 2p, + c)
it follows that the right hand side of (4.6) is negative definite. Conse0 quently, the equilibrium is stable. For the application of Liapunov functional method to the global stability of an endemic equilibrium, readers can refer to [41].
5
Conditions of disease persistence
For epidemic models, it is important to know conditions for the stability of endemic equilibria. That the endemic equilibrium is stable means the spread of infectious diseases. However, it could be very hard to perform mathematical analysis for endemic equilibria for many models. In such cases, we turn to the persistence theory of dynamical systems. A good way for this is to analyze the limit set of flows on the diseasefree subspace to see if it repels positive solutions (see paper [36] for the general theory, and papers [47, 50] for the applications). Another alternative is to adopt persistence functionals [23] Let us illustrate the second one by SE1S model (4.2) [38].
Ro = ).a/b + b) > 1. Then there is a positive constant E such that each positive solution (s(t),i(t)) of (4.2) satisfies i(t) ;? E ift is large. Theorem 5.1. Suppose
Epidemic Models with Time Delays
309
Proof. Let us consider a positive solution (s(t), i(t)) of (4.2). According to this solution, we define t
V1 (t)
=
i(t)
J
+ AC¥
s(())i(())d().
t-w
Then we have
v; (t) = i(t) [AC¥S(t) -
b + b)] = b + b)i(t)(Ros(t) -
1).
(5.1)
J
Since Ro > 1, we have h := ~(1- o ) > o. Claim: For any to > 0, it is impossible that i(t) :::;; h/2 for all t ;? to. Suppose the contrary. Then there is a to > 0 such that i(t) :::;; h/2 for all t ;? to. The first equation of (4.2) is:
s' = b - As(t)i(t)
+ (3i(t -
Then, for t ;? to,
s' (t) > b - (Ah/2
T) - bs(t).
+ b)s(t)
which implies
J t
s(t) > e-(AI1 /2+b)(t-t o)[s(to)
+ b e(Ah/2+b)(O-to)d()] (5.2)
to
> where 0
b (1 _ e-(AIl/2+b)(t-to)) Ah/2+b '
< s(to) is used. Since AI)2+b =
2
Ro +1' we have
s(t) > _2_(1 _ e-(Ah/2+b)(t-t o)). Ro + 1
(5.3)
Choose T1 > 0 such that
~(1 - ~) = e-(Ah/2+bm. 4 Ro
(5.4)
Then (5.3) implies
3Ro + 1 s(t) > 2Ro(Ro + 1) It is easy to see
R>
Jo.
6.
=
R,
£ or t
>t T 7 0 + 1·
(5.5)
Then, by (5.1) we have
v; (t) > b + b)i(t)(RRo -
1),
for t;? to
+ T1 ·
(5.6)
Wendi Wang
310 Set
i=
min i(to
liE[-w,O]
+ TI + w + B).
Statement: i( t) ): i for all t ): to + TI. Suppose the contrary. Then there is a T2 ): 0 such that
i(t) ): i, for to + TI ~ t i(to + TI + w + T2) = i,
~
to
+ TI + w + T2,
i' (to + TI + W + T2) ~ o. However, the second equation of the model is:
i' = Aas(t (5.5) implies that for t
=
to
w)i(t - w) - (-y + b)i(t).
+ TI + w + T 2, we have
i' (t) ): [Aas(t - w) -
(-y + b)]i. > (-y + b) [RoR - 1]i. > O.
This is a contradiction. Thus, i(t) ): i for all t ): to quence, (5.6) leads to
v; (t) > (-y + b)i.(RoR -
1)
for t): to
+ TI .
As a conse-
+ TI,
which implies that as t -+ 00, VI (t) -+ 00. This contradicts VI (t) ~ 1 + Aaw. Hence, the claim is proved. By the claim, we are left to consider two possibilities. First, i(t) ): h/2 for all large t. Secondly, i(t) oscillates about h/2 for all large t. Define
(5.7) We hope to show that i(t) ): 12 for all large t. The conclusion is evident in the first case. For the second case, let hand t2 satisfy
i(tI) = i(t2) = h/2
i(t) < h/2
for tl < t < t2'
If t2 - tl ~ TI + w, since i' (t) > -(-y + b)i(t) and i(tI) = h/2, it is obvious that i(t) ): 12 for h < t < t2' If t2 - tl ): TI + w, by the second equation of the model i = Aas(t - w)i(t - w) - (-y
we obtain i'
> -(-y + b)i(t).
+ b)i(t),
Epidemic Models with Time Delays
311
It leads to i(t) ~ 12 for t E [h,tl +Tl +wJ. For tl +Tl +w::;;; t::;;; t2, we have
Set i*
= 8E[-w,O] min i(h + Tl + w + e)
~ [2.
Proceeding exactly as the proof for above claim, we see that i(t) ~ i* ~ ~ 12 for t E [h, t2J. Since this kind of interval [h, t2J is chosen in an arbitrary way (we only need tl and t2 are large), we conclude that i(t) ~ h for all large t in the second case. In view of our above discussions, the choices of Tl and 12 are independent of the positive solution, we have actually proved that any positive solution of (4.2) satisfies i(t) ~ 12 for all large t. The proof is complete. D
h for tl + Tl + w ::;;; t ::;;; t 2. Consequently, i(t)
6
Summary
In this paper, we have presented the approaches of mathematical modeling for epidemic models with time delays. It has been illustrated that the incubation period, infection period and immune-lasting period could be represented by delays. We have shown that the stability of some equations can be analyzed by the D-subdivision method. Then criteria to test stability switches for systems of delay-independent coefficients and for systems of delay-dependent coefficients are introduced, respectively. The Liapunov functional method is also given to show its power when there are more than two equations or more than two delays. Finally, we have shown the way to prove epidemic disease is persistent under suitable conditions. In all, our aim is to present basic mathematical techniques for analysis of epidemic models with time delays.
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[19] S.A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread. Nonlinear dynamics and evolution equations. 137-200. Fields Inst. Commun. 48. Amer. Math. Soc. Providence. RI. 2006. [20] J.K. Hale and S.M.V. Lunel, Introduction to functional differential equations. Springer-Verlag. New York. 1993. [21] H.W. Hethcote, The mathematics of infectious diseases. SIAM Review 42 (2000), 599-653. [22] H.W. Hethcote and P. van den Driessche, Two SIS epidemiologic models with delays. J. Math. Biol. 40 (2000), 3-26. [23] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics. Cambridge University Press. Cambridge. 1998. [24] Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl. 308 (2005), 343-364. [25] Z. Jin and Z. Ma, The stability of an SIR epidemic model with delays. Math. Biosc. Engi. 3 (2006), 101-109. [26] V. Kolmanovskii and A. Myshkis, Applied Theory of Differential Equations. Kluwer Academic Publishers. 1992. [27] Y. Kuang, Delay differential equations with applications in population dynamics. Academic Press. Inc. Boston. 1993. [28] J. Li, Studies of Epidemic models. PhD Thesis of Xian Jiaotong University. 2002. [29] J. Li and Z. Ma, Stability switches in a class of characteristic equations with delay-dependent parameters. Nonlinear Anal. Real World Appl. 5 (2004),389-408. [30] J. Li, Y. Zhou, Z. Ma, and J.M. Hyman, Epidemiological models for mutating pathogens. SIAM J. Appl. Math. 65 (2004), 1-23. [31] M.Y. Li, H. L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission. SIAM J. Appl. Math. 62 (2001), 58-69. [32] Z. Ma, Mathematical modelling and studies of population ecology. Anhui Education Press. Hefei. 1996. [33] Z. Ma, Y. Zhou, W. Wang and J. Zhen, Epidemic Dynamics and its Mathematical Modelling. Chinese Academic Press. Beijing. 2004. [34] M.C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287-289. [35] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics. part 1, Proc. Roy. Soc. London Ser. A 115 (1927), 700-721.
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[36] H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J. Math. Anal. 24 (1993), 407-435. [37] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002), 29-48. [38] W. Wang, Global behavior of an SEIRS epidemic model with time delay Applied Mathematics Letters. 15 (2002), 423-428. [39] W. Wang, Population dispersal and disease spread. Discrete Contino Dyn. Syst. Ser. B. 4 (2004), 797-804. [40] W. Wang, P. Fergola, S. Lombardo and G. Mulone, Mathematical models of innovation diffusion with stage structure. Applied Mathematical Modelling 30 (2006), 129-146. [41] W. Wang and Z. Ma, Global dynamics of an epidemic model with time delay. Nonlinear Analysis: Real World 3 (2002), 365-373. [42] Wendi Wang and G. Mulone, Threshold of disease transmission on a patch environment. J. Math. Anal. Appl. 285 (2003), 321-335. [43] J. Wu, Theory and applications of partial functional-differential equations. Applied Mathematical Sciences. 119. Springer-Verlag. New York, 1996. [44] Z. Wang and W. Wang, Bifurcation analysis of a model of cellto-cell spread of HIV-I with two distributed delays, J. Southwest University 32 (2007), 22-27. [45] W. Wang and X. Zhao, An epidemic model in a patchy environment. Math. Biosci. 190 (2004), 97-112. [46] W. Wang and X. Zhao, An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math. 65 (2005), 1597-1614. [47] W. Wang and X. Zhao, An epidemic model with population dispersal and infection period. SIAM J. Appl. Math. 66 (2006), 1454-1472. [48] G.F. Webb, M.J. Blaser, H. Zhu, S. Ardal and J. Wu, Critical role of nosocomial transmission in the Toronto SARS outbreak. Math. Biosci. Eng. 1 (2004), 1-13. [49] Y. Zhou, Z. Ma and F. Brauer, A discrete epidemic model for SARS transmission and control in China. Math. Comput. Modelling 40 (2004), 1491-1506. [50] X. Zhao and X. Zou, Threshold dynamics in a delayed SIS epidemic model. J. Math. Anal. Appl. 257 (2001), 282-291.
315
A Simulation Approach to Analysis of Antiviral Stockpile Sizes for Influenza Pandemic* Shenghai Zhang Centre for Infectious Disease Prevention and Control Public Health Agency of Canada E-mail: [email protected]
Abstract The model described here is developed for simulating an influenza pandemic. It intends to explain the manner in which the pandemic develops in a specific community. It shows how the longitudinal cases occurring pattern is built from inception and how this pattern is affected by various sizes of stockpiles of antiviral agents combining some preventing strategies. It assumes that potential inter-household and household-school contacts play very important role in the disease transmission. It uses statistical data about characteristics of households in the community and the parameters of disease transmission to generate repeated random trials of possible outcomes following the introduction of infective individuals into that community, then to track the statistically determined pathway of the infection and average the outcome results. It aims to describe the dynamics of pandemic itself, then the affection of antiviral treatment based on the available sizes of stockpile of the drugs can be obtained.
1
Introduction
Avian flu is unprecedented in its scope as an animal disease and it poses a greater challenge to the world than any previous infectious disease [1]. The virus has spread to birds in many countries in Africa, Asia, Europe *The author is grateful to Dr. Ping Van and his colleagues for their valuable suggestions. This is only a personal point of view and it does not represent any official views of the organization. The research was done before the middle of May, 2006 and presented in the Canada-China Workshop of Infectious Disease Modeling" at Xi'an, China.
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Shenghai Zhang
and the Middle East [2]. The cases of human infection with the H5N1 avian influenza virus are increasing. The transmissibility of the virus among humans could lead to a global influenza pandemic, although the effect of the transmissibility of the virus, either from birds to humans or from one person to another, is not fully understood. At the beginning of a pandemic, the possible public health measures we have so far is to give antiviral medicine to those infected with flu, to quarantine areas and to isolate people. Even if a pandemic cannot be stopped, it is said that such measures can buy time for health authorities to improve their response strategies and stave off the disease until a pandemic vaccine can be produced. Therefore, one of the pressing public health questions is whether and how we stockpile antiviral drugs which can be used for treatment in the early phases of a flu pandemic. A discrete-time stochastic simulation model of influenza spread is used to estimate the distribution of infective people, the number of deaths and the duration of the pandemic of using the different antiviral stockpile sizes to treat infection. The first case of an influenza pandemic could occur in any of a variety of locations and households. The subsequent spread of the pandemic might be very different according to the availability of anti-flu drugs, for instance, for susceptible population. And the spread of the epidemic might also be different according to the presence or absence of school-age children, for instance, in the households first affected [5]. It is assumed that the schools and households are most important in evolution of the pandemic. The model described here was developed for simulating an influenza pandemic. It intends to explain the manner in which the pandemic develops in a specific community. It shows how the longitudinal cases occurring pattern is built from inception and how this pattern is affected by various sizes of stockpiles of antiviral agents combining some preventing strategies. The information about possible social contacts in the community and the parameters possibly describing transmission of the disease are used to determine the probability that a susceptible person will be infected in a day (also see [6], [7] and [5]). It uses statistical data about characteristics of households in the community and the probabilities of disease transmission to generate repeated random trials of possible outcomes following the introduction of infective individuals into that community, to track the statistically determined pathway of the infection and then to average the outcome results. It aims to describe the dynamics of pandemic itself, then the affection of antiviral treatment based on the available sizes of stockpile of the drugs can be obtained. The paper is organized as follows. In section 2 the network of potential inter-household and household-school contacts by which the disease can be transmitted is described for a community of 3200 people. Assumptions about transmission of the disease and transmission probabilities between contacts in the social network are given in section 3.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 317 Following the main results summarized in section 4, we will discuss the potential application of the methodology provided in this article and its limitations.
2
Methods
An outbreak of influenza pandemic in 1918 generated more than 400 million cases worldwide and the number of deaths was more than 20 million, based on the estimation by Cunha [4]. The schools as well as households were important in the evolution of the pandemic. So, the simulation will be based on information from data about the size, composition and immune status of households and the details of school classes and other components. Two separate groups of parameters are used: The first group of parameters defines the components involved in community (such as sizes of households and schools) and the structure of their inter-connections. These parameters define the social and related knowledge about the community. This knowledge, then used with a second group of parameters which describe how the infection moves between individuals and between households and the other sites. Motivated by the idea in the paper by Longini et al. [6], a community of population of 3200 people is stochastically generated by the age distribution and approximated household size published by Statistics Canada (see Table 1). To generate typical families for the simulation,
Table 1· Proportion of the population by sex and age group Age group U. .4 5 .. 9 10 .. 14 15 .. 19 20 .. 24 25 .. 29 30 .. 34 35 .. 39 40 . .44 45 . .49 50 .. 54 55 .. 59 60 .. 64 65 .. 69 70 .. 74 75 .. 79 80 .. 84 85 .. 89 90 and over 'lotal
Ganada (%)
rate %)
Female (%)
5~
5.b
b.l
6.0 6.6 6.7 7.0 6.8 7.0 7.5 8.6 8.0 7.0 6.0 4.6 3.7 3.3 2.7 1.9 1.0 0.5 lOU.U
6.2 6.9 6.9 7.2 6.9 7.1 7.7 8.7 8.1 7.0 6.0 4.5 3.6 3.1 2.3 1.5 0.6 0.3 1uu.u
5.8 6.4 6.4 6.7 6.6 6.8 7.4 8.5 8.0 7.0 6.0 4.6 3.8 3.4 3.0 2.3 1.3 0.7 100,0
Note: Population as of July 1, 2004. Source: Statistics Canada, CANSIM, table 051-0001. Last modified: 2005-02-10.
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318
information on household composition in Canada is needed. In the following, a child means a person with age less than 19 years old; a young child is a child who is less than 5 years old. Values of parameters about families structure can be obtained from Canada 2001 Census population data released from Statistics Canada [3]. Among all families, 36.55% of them are ones without any child at home; 63.45% of them are families with one child or more children. The 43.02% of families with at least one child have only one child; 39.30% of them have two children; 17.68% of them are ones with three or more children at home. Among the people who are 65 or older, 25.4% of them are living alone; 48.4% of them are living with their spouses or partners; 26.2% of them are living with their children or living with other arrangements. The proportion of lone parents who are 65 or older among the families which have 65 or older people is 5.43%. The 75.32% of families with at least one child are couple families. Among these couple families, 37.08% of them have only one child; 42.90% of them have two children; and the families with three or more children at home are 20.02%. The 36.55% of all families are couple families without any child at home. The 24.68% of families with one child or children at home are lone-parent families. The 61.14% of lone parent families are ones with one child; 28.30% of them have two children; The percentage of ones with three or more children is 10.56%. These parameters determine the hierarchy of decomposition for households. For example, the households with three people are generated, based on the hierarchy of decomposition shown in the Figure 2.1.
~ g, g-
0
'" '< 0
~
...,o
~ 0 8
8
g-
"@
8
i
@
u. I
.g
e:+
i
@
'"
e:+
Figure 2.1: An example of household decomposition
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 319 Table 2 shows the living arrangement for senior people in Canada. In general, the household sizes as shown in Table 3. Table 2: Living arrangements of seniors aged 65 and over by sex and age group, Canada, 2001 Sex
Living alone
Males Females
(%) 16.0 34.8
Living with spouse or partner (%) 61.4 35.4
Living with children
Living in health care into (%) 4.9 9.2
(%) 13.3 12.1
Other
(%) 4.4 8.4
Table 3: Household size
#
Persons % of households
The community with a population of 3200 is arranged by four neighborhoods with the age structure and the families introduced above. Each neighborhood will have two small day-care centres and a large day-care centre. There are two elementary schools and one high school in the community. The community decomposition is shown in Figure 2.2. The "S", community (3200)
neighbourhood I
/ t \
family···
~ +
1+\
family
neighbourhood 2
neighbourhood 3
neighbourhood 4
/ t \
/ t \ ""tl
family···
family...
family family
~+I
I
I children I I \
I
family
/ t \ ~ +I
I children I \
I
m11\
family
ay care
/
J~ \
[88[8[8 G [8 1------1-1
High School
1--1-------'
Figure 2.2: The structure of the community
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320
"L" and "E" in the figure are "small day-care centre", "large day-care centre" and "Elementary school" , respectively.
3
Model
As an infectious disease, influenza follows SIR time line shown in Figure 3.1. It has incubation period, latent period, infectious period and recovery period. The onsets of symptoms are fever, respiratory symptoms, nasal discharges, cough, headache and sore throat. However, there is a time period between the start of infection and the onset of symptoms, which is incubation period. The length of incubation period is roughly the same as the time between being infected and becoming infectious that is, capable of passing on the infection to others, which is latent period. Then, infectious period follows. Once the infectious period ends, the recovery period starts, during which the infected individual is no more transmitting the infection to others. Based on the research done by Stiver [11], the influenza antiviral agents can modify the severity of illness (relative reduction in influenza complication rate) about 50%, and reduce the duration of illness about 1.5 to 2.5 days. The reduction of time to resume normal activity is 1.5 day to 3 days. However, the treatment is most effective when given within 48 hours after the onset of illness; the earlier the antiviral is started after the onset of symptoms of influenza-like illness, the better the treatments. The work here aims at statements about the distribution of infective people, the number of deaths and the duration of a pandemic in a typical community in Canada at certain sizes of stockpile for antiviral drugs under given scenarios. Symptoms
Viral infection
No infectious
Recovering
Incubation
Latent E
:>
~
Infectious E
~
Figure 3.1: SIR time line Transmission probabilities are important parameters for the simulation model. It is obvious that there is no such parameters for an unknown
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 321 pandemic. However, statistical models have been used for the analysis of infectious disease data from family studies in a community (see [9] and [10]), and the household transmission probabilities have been provided by analyzing data from influenza A(H3N2) [6]. The transmission probabilities within small groups, large day-care centers, elementary schools, middle schools, and high schools are 0.04, 0.015, 0.0145, 0.0125 and 0.0105, respectively. The probabilities within a family are 0.08 (between a child to a child), 0.03 (between a child to an adult (vise versa)), and 0.04 (between an adult to an adult (vise versa)). The transmission probability that a preschool child is infected from a infective person in his neighborhood (other than his/her group or his/her family) is 0.00004; the probability for a school child can be 0.00012; the probability for an adult is increased to 0.00016. The probability that a pre-school child is infected from an infective person in the community (other than his/her neighborhood) is 0.00001; the probability for a school child is 0.00003; and the probability for an adult is 0.00004. However, to describe the transmission accurately, the process may be decomposed into contact process and infection process. Let Pcontact is the probability that a contact is a sufficient contact for transmission of influenza. The Pcontact depends on the age groups and is given by Longini et al. [12]. The calculation of P, the probability that a susceptible is infected for possible contacts with infectives, is shown as the following. P = min(1)PcontactNcontact, 1),
where 1> is determined by baseline attack rate; the Ncontact is a number of contacts in a day. The 1> for an unknown pandemic has to be determined by a simulation study. The 1> used in this article is determined by the baseline age specific attack rates given by Longini et al. (see Table S4 in "Supporting Online Material" for the paper by Longini et al. [12]). For example, the 1> = 0.1 corresponds to the situation that the overall illness attack rate is 33%. It is assumed that the efficacy of the antiviral drug for symptomatic disease given infection is variable as the time for giving the drug, based on the discussion by Stiver [11]. If a person becomes ill and takes antiviral drugs within 48 hours, the duration of illness will be reduced by 1.52.5 days (uniform distribution) [11]. In this paper, it is assumed that antiviral drugs are only for therapeutic use, excepting for high risk group people. The antiviral efficacy for infectiousness is 0.80 ; and the antiviral efficacy for symptomatic disease given infection is 0.60, which are the same as ones in the article by Longini et al. [12]. The antiviral efficacy for symptomatic disease given infection is how much an antiviral agent will reduce the probability that an infected person will develop influenza symptoms compared with an infected person who is not using an antiviral
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Shenghai Zhang
agent. The antiviral efficacy for infectiousness is how much an antiviral agent will reduce the probability that an infected person will transmit influenza to others compared with an infected person who is not using an antiviral agent. An asymptomatic infection is assumed to be 50% as infectious as a symptomatic infection, based on some studies. It means that the clinical attack rate is the half of infection rate (serologic attack rate). The distribution of incubating days was estimated by Longini et. al. [12J: 30% of infected persons have an incubation of one day; 52% infected persons have two days and 18% of infected persons have three days. The distribution of infectious days is as follows: 30% for three days; 40% for four days; 20% for five days and 10% for six days. Now, the calculation of the probability that a susceptible person is infected on a day can be shown as the following example. Considering an elementary school child, suppose he is a susceptible and he is exposed to a number of infective children and adults in his household, his neighborhood cluster, his school and his social groups. Because infective people may have different infectious forces for infecting a susceptible person, the" numbers" of infective people in each group is given by the following models: suppose the number of new infective people in a group is y(t) on the day t, a total of Y(t) infective people in this group is based on the new infective people in the past five days and Y(t) = L~~~(5,t) y(t- j)'I1 j. Here, \[J j describes the force of an infected person who is still infectious j days after initially contracting the diseases. Table 4 gives the force of an infective person who takes antiviral drugs in time, based on the results provided by Daley and Gani [13J. Let Ync(t), Yna(t), Yes (t) , YsI(t) and Y s2 (t) are the total of infective people in each group, respectively; and let Pnce, Pnae, Pes, PsI, and Ps2 are transmission probabilities from a infective person in a corresponding group to the child. Then, the probability P(t) that the child becomes a new infective person on day t is as the following. P(t) = 1- (1- Pnce)Ync (t)(1- Pnac)Yna(t)(l- Pes)Yes(t)
x (1 - PsdYsdt) (1 - Ps2)Y 2(t). S
Figure 3.2 shows how the probabilities that a susceptible child is infected on a day is calculated.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes ... 323 A susceptible child
Infected? Yes Prob.~ 1- II., (I-p.l'
Infected
)~E_ _Ye_s_---11
Use antiviral drugs
No
It-_NO__~ ~su~sc;ep~tib§ley----.J
~~ Figure 3.2: Calculation of the probability
It is also interesting to know the probability, Pn(t), that there are certain number of infective people in the community at time t, where Pn (t) =Prob{ there are n infective people in a certain population at time t}, where t = 0.1, ... , refers to time in day. The calculation of this probability is based on the dynamics of disease in population level. In this case, it is assumed that the community is open, it means that patients are allowed to arrive from outside of the community and spread the disease to other people. Four forces are considered: the first is the immigration force, bring new cases; the second force is the infectious force, spreading the disease; the third force is the death force or immunity force and the removing patient force: emigration force. This scenario is described by the Immigration-infectious- cure (immunity)-Emigration model studied by Daley and Gani [13]. The important assumption in this model is that each of these four forces acts independently of the other three. An arrival patient occurs with probability >"6t; a patient could leave the system with probability f-t6t; given n - 1 patients in the system at time t, the probability of a new case could occur in the time interval (t, t + 6t) is (n -1)1I6t; given n + 1 patients in the system at time t, the probability
Shenghai Zhang
324
of a patient being recovered or death is (n the following equation:
+1)w6t.
The Pn(t) satisfies
Through the generating function: 00
l: sn Pn(t)
Gt(s) =
n=O
the solution can be obtained (if there are no patients with illness in the time 0):
Pn(t)
= [exp(-(v -
w)t)(v -
min{n+.tc,no+.tc} ~
W
~
] (no+e+j-l) [v-wexp(-(v-W)t) -(v - w)t) j
J=O
no + ~
w)]~-~ [ v-wexp / - (v-w )t)]no+~
.)
( n+~-J
v(exp(
1)
[w(l - exp( -(v - w)t))C+:-
[vexp(-(v-w)t)-w]
0-
j
J
+)
The parameters in the formula can be obtained from doing the simulation on the individual level.
4
Results
In the simulations, it is assumed that an initial person infected with the newly emergent influenza strain is randomly introduced to the community of 3200 people. Then, there are three possible outcomes: there is a large epidemic (more than four cases in the community); there is a small epidemic (more than one but less than or equal to four cases in the community); no further people are infected except for the original case. Also, it is assumed that the first case is a symptomatic infection. It is known that the transmissibility of the new strain depends on the illness attack rates, which are unknown. The only situation of the overall illness attack rate is of 33% is discussed in this paper. The effect on the number of infections of using different antiviral stockpile sizes to treat infection is estimated. It is estimated that the probability of controlling the infection to at most four cases in the community is 0.77, when the stockpiles cover 30% of the population.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes . " 325 Suppose there is no antiviral stockpile, the possible intervention is household quarantine and the probabilities of withdrawing ill people to their homes are 0.8, 0.75 and 0.50 for preschool children, school child and adults, respectively. The chance that there is no large epidemic exists in this scenario. However, this chance is small, the probability of the outbreak of larger epidemic is greater. The simulations are run to calculate the number of infection for each day. Figure 4.1 shows the average numbers of infective adults and children respectively on each day, when there is no antiviral stockpile. In Figure 4.1, horizontal axis represents time (in day); the vertical axis represents the numbers of infective adults (upper curve) and children (lower curve). 140 120 100 80 60 40 20 0
0
20
40
60
80
100
120
Figure 4.1: Numbers of infective adults and children in each day without antiviral stockpile Suppose the antiviral stockpile covers 30% of the population. It means that we have enough antiviral stockpile to be dispensed to 960 ill persons in the community of 3200 people. Based on the simulations, it is estimated the probability that the number of cases can be controlled under four in the community is 0.77. This is based on the scenario that the original case can not be identified in a very early day of the infection and the only prevention is the limited quarantine: withdrawing ill people to their homes. However, there is a 23% chance for a large epidemic, that is, the infected rate will be bigger than 15/10000. It is interesting that once a large epidemic happen, the majority of people in the community will be infected under this scenario. The maximum averaged number
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326
of infective people in a day is 74 when the worst cases is considered. It will be about 30th day after the initial case was introduced into the community. The dynamic change of infective people can be seen in Figure 4.2. In Figure 4.2, horizontal axis represents time (in day); the vertical axis represents the numbers of infected adults (upper curve) and children (lower curve). Here, it is assumed that patients who are given antiviral drugs can get the drugs in time, that is, the treatment can be obtained within 48 hours without delay. 45 40 35 30 25 20 15 10 5 0
0
20
60
80
100
120
Figure 4.2: Numbers of infective adults and children in each day with antiviral drug treatment in time If antiviral stock pile is available for 30% population, but only 80% of patients who get antiviral drugs can get treatment in time, then the dynamic is shown by Figure 4.3. It is interesting that there will be a strong chance of curing all infected people in about 20 days after the initial case is introduced in the community. However, once it becomes a larger epidemic, it needs more than two or three months to cure all infected people and extinguish the infection sources in the same community. The point here is that the epidemic will be controlled within 20 days or it will need more that 60 days. The frequency of the duration of the epidemic is shown in Figure 4.4, based on 100 simulations. The vertical axis represents the numbers of simulations corresponds to the duration (in days represented by horizontal axis) of the epidemic per 100 simulations. The death could happen. The death rate for the worst case is shown in Figure 4.5, which is an accumulative death. For example, the accu-
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 327 45 40 35 30 25 20 15 10 5 20
40
60
80
100
120
140
Figure 4.3: Numbers of infective adults and children in each day with 80% of antiviral drug treatment in time
70 60 50
o
7
17
27
37
47
57
67
77
87
97
Figure 4.4: The frequency of duration of the pandemic among 100 simulations
mulative death reaches the highest in the 60th day, which is 17/100000 with antiviral stockpile covering 30% population.
Shenghai Zhang
328 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
10
20
30
40
50
60
70
Figure 4.5: The accumulative death rate per 2000 people
5
Conclusion and discussion
The goal of this article is to build a simulation framework to answer the following question: What is the probability that we can control the number of infected people in a community under certain level if there is only limited antiviral drug stockpile. It is estimated that the probability of controlling the infection to at most 4 cases in the community of 3200 people is 0.77 if the stockpiles can cover 30% of the population and patients who get medical treatment can get anti-viral drugs in time. The results were obtained based on some naive scenarios. In practice, the situation may be more complex. For example, although diagnostic tests for influenza are available for all patients, the tests may take extra time for some patients because office-based tests may be not available for all clinicians. Also, it was estimated by Stiver [ll])that once an influenza outbreak in a community, the 30%-50% individuals with ILl (influenza like illness) who may not have influenza will accept antiviral drugs treatments, if anti-influenza drug therapy is prescribed to all people with ILL Considering this situation, the size of stockpile for antiviral drugs should be inflated. The discussion in this paper is under the assumption of the use of antiviral drugs for treatment of patients. The potential use of antiviral agents for prophylaxis has been investigated in many articles (see, for example, [14] and [6]) and the use of antiviral agents for containing an emerging influenza pandemic was studied by Ferguson [15] and Longini et al. [12].
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 329 The future extension of this work could be into more complex situation, where, for example, the disease progression considering the status of vaccination of the population and the use of antiviral drugs for prophylaxis could be important components of the model. It could be modeling to compare the effectiveness of a combination of antiviral treatment and the quarantine intervention strategies against a new strain of influenza.
References [1] http:j jwww.who.intjcsrjdiseasejinfluenzajenjindex.html.onMarch 28,2006. [2] http:j jwww.MercuryNews.com on March 6, 2006.
[3] http:j jwwwI2.statcan.cajenglishjcensusOl on November 6, 2005. [4] B.A. Cunha, Influenza: historical aspects of epidemics and pandemics. Infect. Dis. Clin. North Am. 18 (2004), 141-155. [5] B.M. Sayers and J.J. Angulo, A new explanatory model of an SIR disease epidemic: A knowledge-based, probabilistic approach to epidemic analysis. Scandinavian Journal of Infectious Disease 37 (2005), 55-60. [6] I.M. Longini Jr., E.M. Halloran, A. Nizam, and Y. Yang, Containing pandemic influenza with antiviral agents, American Jouranl of Epidemiology, 159 (2004), 623-633. [7] J.C. Hudson, Geographical diffusion theory: Studies in Geography, Vo1.19, Northwestern University, Evanston, Illinosis 1972. [8] F.T. Cadham, The use of a vaccine in the recent epidemic of influenza, Canadian Medical Association Journal, 11 (1919), 519-527. [9] I.M. Longini, J.S. Koopman, M. Haber, and G.A. Cotsonis, Statistical inference for infectious diseases, American Journal of Epidemiology, 128 (1988), 845-859. [10] D.L. Addy, I.M. Longini, and M.S. Haber, A generalized stochastic model for the analysis of infectious disease final size data, Biometrics, 47 (1991), 961-974. [11] G. Stiver, The treatment of influenza with antiviral drugs, Canadian Medical Assocation Journal, 168 (2003), 49-57. [12] I.M. Longini Jr., A. Nizam, and S. Xu, et al., Containing Pandemic Influenza at the Source, Science, 309 (2005), 1083-1087. [13] D.J. Daley, and J. Gani, Epidemic modelling: an introduction, Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, United Kingdom 1999.
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[14] R.D. Balicer, M. Huerta, and I. Grotto, Trackling the next influenza pandemic, BMJ, 328 (2004), 1391-1392. [15] N.M. Ferguson, D.A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437 (2005), 209-214.
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Modeling and Simulation Studies of West Nile Virus in Southern Ontario Canada Peter BuckA , Rongsong Liu B , Jiangping Shuai A , Jianhong Wu c , Huaiping Zhu C A Foodborne,
Waterborne and Zoonotic Infections Division Centre for Infectious Disease Prevention and Control Public Health Agency of Canada 255 Woodlawn Road West Unit 120 Guelph, Ontario NiH 8Jl, Canada B Department of Mathematics, Purdue University 150 N. University Street, West Lafayette, IN 47907-2067, USA C Department of Mathematics and Statistics, York University Toronto, Ontario, M3J lP3, Canada
Abstract
We carry out a preliminary mathematical modelling study of the West Nile Virus in the Peel region, Ontario Canada. The surveillance data from this region is used to show that the noncrow family birds which are susceptible to West Nile Virus are one of the key factors responsible for the endemic of the virus in the region. The density of dead crow is no longer an accurate indicator for the virus after the virus has sustained in the region, while the density of mosquitoes can give us better quantification of the risk level of the West Nile virus.
1
Introduction
Although West Nile Virus (WNV) was isolated in the West Nile district of Uganda in 1937[21J, and WNV in the Eastern Hemisphere has been maintained in an enzootic cycle involving culicine mosquitoes and birds[1l,12J, WNV activities in North America were not recorded until August of 1999 in the borough of Queens, New York City[7,16 J. Despite this long delay of invasion into North America, the virus has rapidly expanded spatially within the subsequently several years and evidences seem to indicate that WNV becomes a permanent fixture of the North America medical landscape[18J. It is this endemic nature of WNV that motives this research.
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Peter Buck, Rongsong Liu, Jiangping Shuai, ...
Through past successful WNV surveillance program in Southern Ontario, it is confirmed that the virus has caused serious problems in certain bird populations in North America. For the last several years it has been seen the drop of the number of avian species (such as American crows and blue jays) in southern Ontario. In 1999, in the New York area, the crow population crashed by about 90 per cent in a few months. In Canada, American Crow population also crashed in 2002. Since birds are the host and reservoir of WNV, the decline of some bird populations will likely affect both the virus spread pattern and risk levels, and the effectiveness of the existing surveillance systems. For example, it was previously believed that Canada would face a high-risk WNV year in 2005. However, WNV risk was reported relatively low in the year. This might be explained by the decline of American Crow population in the region. It is therefore important for us to explore how the virus might affect the population of major bird species. This study can then give us guidelines for surveillance program, surveillance focuses, and dead bird collecting and testing strategies. There have been some modeling studies on the transmission dynamics of West Nile Virus, and most of the models are autonomous system of ordinary differential equations or discrete systems. The work[14J summarized the available models and made comparative studies of two continuous and discrete-time West Nile Virus models. In the previous modeling studies, the avian species was considered as one family. In the paper, we will develop a mathematical model by classifying the bird species into crow-family and non-crow family. The density of mosquitoes is modeled by time-dependent Gaussian functions. The focus is to study the impact of WNV on the ecology of birds. By using the surveillance data of the major avian species in Southern Ontario, numerical simulations allow us to investigate if the WNV is responsible for the decline of the number of certain species of birds in the region, and to study how surveillance program can be made more effective to alert the possible outbreak and recurrence of the WNV.
2
The model formulation
WNV is transmitted from birds to birds by mosquitoes. When a female mosquito feeds on an infected bird, it picks up the virus and transmit it to other uninfected susceptible [l,6,19J. Occasionally, infected mosquitoes will feed on mammals such as horses, dogs, cats, and humans, and transmit the virus to them. Mammals are the dead-end hosts, however, and they do not contribute to the transmission cycle. Most birds do not become ill when they got infected with this virus, but crow family birds (including crows, ravens, magpies, blue jays, gray jays and Stellar's jays[6J)
Modeling and Simulation Studies of West Nile Virus in . . .
333
are susceptible to the WNV and often die when they are infected, due to inflammation of many organs including the brain (encephalitis) caused by the virus[81. For this reason, crow family birds have been chosen as an indicator species for the presence of WNV. The crow family birds may not contribute significantly to the spread of the virus because of the higher mortality rate, and they die from the infection in a short period. Therefore, we will divide all birds susceptible to the virus into two classes: crow family and non-crow family, based on whether they show symptoms of illness or not and we will denote their populations sizes by B 1 (t) (crow family birds) and B 2 (t) (the remaining birds under consideration). We shall investigate the impact of the second class of birds for the ongoing outbreak and spread of the virus. For non crow family birds, we divide them into susceptible, infected and recovered, denoted by B 2s , B 2i , B 2r , respectively. While for crow family birds, we divide them into susceptible and infected, denoted by B 1s , B 1i . Due to the higher mortality rate, we ignore the recovered class for crow family birds since most infected crow family birds will die due to the disease. And we will abuse notations and use, for example, B 1s to denote the number of susceptible crow family birds at time t. Let Nb = N 1b + N 2b, where N 1b = B 1s + B1i and N 2b = B 2s + B2i + B 2r are the total number of crow family birds and non crow family birds. Figure 1 gives the flow chart of the model. . Based on the assumptions and the flow chat, we have the following equations to model the avian species: dB 1s
&
dBli
dt
= h 1(B1s ) =
Bls
b1{31 Nb Mi - d 1i
dB2s & = h2(N2b ) dB2i
B1s
b1{3l Nb Mi - dlsB ls , B 1i,
B 2s
b2{32 Nb Mi -
B 2s Nb
- - = b2{32--Mi - d 2B 2i -
dt
r
d B 2 2s, B
(1)
2i,
dB2r
- - = r B2i - d 2B2n
dt
where h 1 (.) and h2(.) are birth functions of the Bl and B2 class, bi , i = 1, 2 are the biting rate of a mosquito on ith kinds of birds, Mi(t) is the number of infected adult mosquitoes. Cross-infection between birds and mosquitoes is modeled using mass-action normalized by the bird density, {31 and {32 are the probability of transmission from infected mosquitoes to crow family and non crow family susceptible birds, respectively, d1s and d 1i are the mortality rates of susceptible and infected crow birds respectively, r is the recover rate of infected non crow family birds, and d 2 is the mortality rate of the non-crow family birds B 2 ·
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Peter Buck, Rongsong Liu, Jiangping Shuai, ...
~
~dUCeddeath
~ irth
B2s
- .......- - - t L-_,--_...J
death
Figure 1: Flow chart of the model. Let Nm(t) be the total number of (adult female) mosquitoes, divided into infected mosquitoes Mi (t) and susceptible mosquitoes N m (t)Mi(t). Death and reproduction of mosquitoes are assumed not to depend on their infection status, and so the number of infected adult mosquitoes Mi(t) are assumed to obey
dMi _ -d M. dt -
m,
+
(N _ M.)b1a1B 1i + b2a 2B 2i m,
Nb
'
(2)
where al and a2 are the probability of transmission from susceptible mosquitoes to crow family and non-crow family infected birds, respectively, dm is the natural death rate of mosquitoes which varies over time.
3
Surveillance data and numerical simulations
Since WNV is particularly virulent in crow family birds, the dead American crows and blue jays were usually used as the indicator of the arrival of WNV in a geographic area. In [4], for example, the data of the number of WNV human disease cases and the density of dead crows in New York State from 2001 to 2003 are used to develop a threshold value of 0.1 dead crows per square mile (0.04 dead crows per square km) as a risk indicator for WNV in New York State. Our purpose in this paper is to validate this threshold value as the indicator and a corresponding similar threshold for the establishment phase of the virus. We will carry out the
Modeling and Simulation Studies of West Nile Virus in . . .
335
study for the Peel Region of Southern Ontario, and use the surveillance data from this area and model based simulations of the transmission dynamics.
3.1
Birds
CDC[6j listed 284 species of bird which have been reported to CDC's West Nile Virus avian mortality database from 1999-present. Using the information for North American birds[lOj, we assume that the region under consideration has two crow family birds (American crows and blue jays) and 41 other species which are susceptible to WNV (these include American Robin, Red-winged Blackbird among others). By the breeding bird census (BBC)[lOj, we estimate the density (number of birds/lOO ha (1 square km)) of crow family birds in the range [35,110] and the density of non crow family birds in the range [465,1024]. Based on those estimation, in our simulations we will assume that the initial value for the susceptible crow family birds B ls = 50 and susceptible non crow family birds B2s = 650.
3.2
Mosquitoes
From the adult mosquito surveillance data of the Peel region, we note that the distribution of mosquitoes is quite heterogeneous. We concentrate on the trap of mosquitoes, we call it TX (trap x), mainly because this tra~ has relatively complete data over the mosquito season, and we will consider this trap as a representative sample in the surrounding region by ignoring the spatial heterogeneity. We recall that there is a standard method[17j to calculate mosquito population density: take the total number of captured mosquitoes, by species, and divide it by the number of trap nights. Figure 2 gives the density of mosquitoes in the trap TX from the year 2002 to 2004. Observation of the density of mosquitoes for the year 2002 to 2004 in Figure 2 suggests that a) The shape of the density of mosquitoes for each year has binormal distribution; b) The density of mosquitoes achieves its first peak around July 15, and then reaches its second peak at the end of August. Therefore, we use the following combination of Gaussian functions to simulate the total number of mosquitoes n
Nm(t)
=
L
2 [Ail exp( -(t.- (i - 1) * 365 - aid / bil)
i=l
+Ai2 exp( -(t - (i - 1) * 365 - ai2)2/bi2)] ,
(3)
r
