PUBLIC HEALTH IN THE 21ST CENTURY SERIES
INFECTIOUS DISEASE MODELLING RESEARCH PROGRESS
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PUBLIC HEALTH IN THE 21ST CENTURY SERIES Health-Related Quality of Life Erik C. Hoffmann (Editor) 2009. ISBN: 978-1-60741-723-1 Cross Infections: Types, Causes and Prevention Jin Dong and Xun Liang (Editors) 2009. ISBN: 978-1-60741-467-4 Swine Flu and Pig Borne Diseases Viroj Wiwanitkit 2009. ISBN: 978-1-60876-291-0 Family History of Osteoporosis Afrooz Afghani (Editor) 2009. ISBN: 978-1-60876-190-6 Biological Clocks: Effects on Behavior, Health and Outlook Oktav Salvenmoser and Brigitta Meklau (Editors) 2010. ISBN: 978-1-60741-251-9 Infectious Disease Modelling Research Progress Jean Michel Tchuenche and C. Chiyaka (Editors) 2010. ISBN: 978-1-60741-347-9
PUBLIC HEALTH IN THE 21ST CENTURY SERIES
INFECTIOUS DISEASE MODELLING RESEARCH PROGRESS
JEAN MICHEL TCHUENCHE AND
CHRISTINAH CHIYAKA EDITORS
Nova Science Publishers, Inc. New York
Copyright © 2009 by Nova Science Publishers, Inc.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Infectious disease modelling research progress / [edited by] Jean Michel Tchuenche ... [et al.]. p. ; cm. Includes bibilographical references. ISBN 978-1-61470-090-6 (eBook) 1. Communicable diseases--Epidemiology. I. Tchuenche, Jean Michel. [DNLM: 1. Communicable Diseases--epidemiology. 2. Epidemiologic Research Design. 3. Models, Theoretical. WA 110 I433 2009] RA651.I54 2009 362.196'9--dc22 2009016556
Published by Nova Science Publishers, Inc. New York
Dedication In memory of C.O.A. Sowunmi (1934-2007)
CONTENTS Preface
ix
Chapter 1
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction: The SIS Case C.O.A. Sowunmi
1
Chapter 2
A Mathematical Analysis of Influenza with Treatment and Vaccination H. Rwezaura, E. Mtisi and J.M. Tchuenche
31
Chapter 3
A Theoretical Assessment of the Effects of Chemoprophylaxis, Treatment and Drug Resistance in TB Individuals Co-infected with HIV/AIDS C.P. Bhunu and W. Garira
85
Chapter 4
When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection P. Munz, I. Hudea, J. Imad and R.J. Smith
133
Chapter 5
A Review of Mathematical Modelling of the Epidemiology of Malaria C. Chiyaka, Z. Mukandavire, S. Dube, G. Musuka and J.M. Tchuenche
151
Chapter 6
A Mathematical Model of the Within – Vector Dynamics of the Plasmodium Falciparum Protozoan Parasite M. Teboh-Ewungkem, Thomas Yuster and Nathaniel H. Newman
177
Chapter 7
Mycobacterium Tuberculosis Treatment and the Emergence of a Multi-drug Resistant Strain in the Lungs G. Magombedze, W. Garira, E. Mwenje and C. P. Bhunu
197
Chapter 8
Mathematical Modeling for Tumor Growth and Control Strategies Sanjeev Kumar, Deepak Kumar and Rashmi Sharma
229
viii Chapter 9 Index
Contents With-in Host Modelling: Their Complexities and Limitations G. Magombedze
253 261
PREFACE As someone once said, publications in mathematical biology are so numerous that they are becoming themselves an epidemic. Previous collections in this field contain important materials as a basis for modern ground breaking research in disease dynamics, but their focus is either too narrow, or is a collections of conference papers, written for the advanced and experienced research students. This book attempts to complement the gap and is therefore intended to encourage the growing demand for interdisciplinary research which is still at a low level in the developing world where most of the diseases considered are prevalent. Training at the interface of mathematics and biology is increasing today, and virtually any advance in diseases dynamics requires a sophisticated mathematical approach in order to map out the parameters that drive them for control and containment of epidemic outbreaks. The goal of this book is two-fold: To expose students to the usefulness and applicability of mathematical knowledge in designing public health policies, and to introduce them to interdisciplinary research at the frontiers of mathematics and biology. They are meant to provide a glimpse into the diverse world of epidemiological modeling and to invite interested readers to experience, through a selection of epidemic diseases of global concern, the fascinating mathematical techniques at the interface between biology and mathematics. The materials presented herein describe, thoroughly analyze and interpret the dynamics of infectious diseases. The chapters are independent and can be used as basic case studies for any existing text material for upper undergraduate and graduate courses with a variety of audiences. The focus of this volume is essentially on the epidemiology of corruption, tuberculosis, influenza, tumor growth and malaria. The book is organized as follows: The first Chapter analyses the epidemiology of corruption and disease transmission as a saturable interaction (SIS case). Chapter 2 is a robust and in-depth study of an influenza model with treatment and vaccination. In Chapter 3, a theoretical assessment of the effects of chemoprophylaxis, treatment and drug resistance in TB infected individuals co-infected with HIV/AIDS is studied. Modelling an outbreak of zombies is considered in Chapter 4. Zombies are fictional, but it uses movies and popular culture to treat a zombie outbreak like a regular disease outbreak, while Chapter 5 is a review of previous studies on the epidemiology of malaria. A model of the sporogony cycle of Plasmodium falciparum, one of the agents responsible for malaria is studied in Chapter 6. Chapter 7, which is parallel to Chapter 3; deals with the treatment of mycobacterium tuberculosis and the emergence of a multi-drug resistant strain in the lungs. A model for tumor growth and its control is investigated in
x
Jean Michel Tchuenche and C. Chiyaka
Chapter 8. The complexity and limitation of within host dynamics models is briefly commented in Chapter 9. We would like to express our sincere appreciation to the reviewers and the members of the editorial board of Nova Science Publishers, Inc., involved in the oversight of this book. Finally, despite all the support in the production, at the end, the responsibility for the final product is entirely ours. Jean M. Tchuenche Christinah Chiyaka
(Dar es Salaam, TZ) (Bulawayo, Zim) December, 2008.
In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 1-30
Chapter 1
E PIDEMIOLOGY OF C ORRUPTION AND D ISEASE T RANSMISSION AS A S ATURABLE I NTERACTION : T HE SIS C ASE∗ C.O.A. Sowunmi† 6 Are Close, New Bodija, Ibadan 200221, Nigeria
Abstract The first section of this chapter explores corruption, modelled as a non-fatal transmissible infection which divides a community into 4 compartments: Susceptible, Infective, Removed and Resistant. The Removed class has age structure. The corruptionfree steady state, though unstable, can be stabilized by feedback control. The efficiency of the measure depends on a number of parameters, especially the rates at which susceptibles become resistant, and the infective are removed or become resistant. The second part analyses disease transmission as a saturable interaction. The use of the concept of saturable interactions as a framework for modelling disease transmission is explored with two SIS cases. In both cases, it is found that the disease-free steady state is always present whereas the endemic steady state may not be possible. Also, the instability of the disease-free steady state is part of the sufficient conditions for the existence of the endemic steady state. Finally, sufficient conditions for the local asymptotic stability and instability of the steady states are obtained.
Keywords: Corruption, non-fatal communicable disease, equilibria, stability, feedback control, Lyapunov, disease transmission, saturability, Age Structure.
1.
Epidemiology of Corruption
1.1.
Introduction
Official corruption, popularly called corruption, is here defined simply as giving or receiving a bribe. It is a habit that can be acquired and also lost. It is also possible to resist both ∗
Compiled by J.M. Tchuenche and published posthumously with permission from author’s wife, Prof. Bisi Sowunmi. † E-mail address:
[email protected],
[email protected] 2
C.O.A. Sowunmi
the giving and receiving of a bribe. It is thus similar in some respects to the communicable diseases which have been modelled in numerous publications, yet it differs in other respects which make it interesting to model. Corruption has been as much in the news as HIV/AIDS and should be accorded as much mathematical attention by way of modelling. Corruption divides a community in which it is active into four compartments, namely: (i) Susceptibles i.e., those who have neither taken nor offered a bribe but will take or offer it if sufficiently tempted, (ii) Infectives who have once taken or offered a bribe and will do so again. In particular, they tempt the susceptibles, thereby turning them into infectives, (iii) Resistants, who will no longer succumb to temptation, no matter what, and lastly, (iv) Removed who were once infective but are now rendered incapable of receiving or offering a bribe for a while (they could be in prison, on suspension or whatever). When they cease to be removed, they either go to the Infectives or the Resistant. The removed class has age structure with infinite life span. Everyone is born susceptible. Neither the birth rate nor the death rate is affected by the class to which an individual belongs.
1.2.
Model Equations
Let S(t), I(t), R1 (t), R2 (t) be the respective sizes at time t of the Susceptibles, Infectives, Removed and Resistant. Everyone is born into the susceptible class. The transition rates are non constant, but depend mostly on R1 (t). The rate of entering the removed class depends on I(t), R1 (t) and R2 (t). The endemic is not possible, but the corruption-free state is not likely to be stable, because the condition for stability is quite complicated. The population is assumed open and of total size N (t) at time t. Let the birth and death rates be b(N ) and γ(N ), respectively. The per capita rate a at which susceptibles flow into the resistant class is assumed to be a function of R1 alone. The reason is that the higher the likelihood of being removed, the more a susceptible is likely to become resistant. Thus, a is a monotone increasing function of R1 . However, it is possible that there are some who on their own will become resistant, hence, we assume a(0) > 0. For a similar reason, we assume that c = c(R1 ) and is monotone increasing but c(0) = 0. d is also a function of R1 alone, but is monotone decreasing, while e = e(R1 ) is monotone increasing. Since the removed Z class has age structure, we assume a density ρ, function of t and α ∞
ρ(t, α)dα. ρ(t, 0) is the rate at which infectives are removed. Let
such that R1 (t) =
0
∂h ∂h ∂h < 0, < 0 and > 0. The reason is as follows. ∂I ∂R1 ∂R2 The more infectives there are, the less the chances of their being sent to the R1 -class. The more infectives are sent to the R1 -class, the more wary those in the I-class will be. Finally, the more those in R2 -class, the more the infectives will be removed. ρ(t, 0) = h(I, R1 , R2 ), where
We set S(0) = S0 , I(0) = I0 , ρ(0, α) = ρ0 (α) and R2 (0) = R20 . We assume that the interaction between the S and I classes is saturable and is governed by a function F of S and I, and that F has the usual properties [1].
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction
3
We therefore have the following governing equations: dS dt
= −aR2 − F (S, I) + bN − γS,
dI dt
= F (S, I) + dR1 − cR2 − γI − h(I, R1 , R2 ),
∂ρ ∂ρ + ∂t ∂α
= −(d + e + γ)ρ,
dR2 dt
= eR1 − (γ − a − c)R2
(1.1)
1.3. Model Analysis The model consists of the system of equations (1.1) together with the initial conditions and the assumptions on the parameters stated in subsection 1.2 above. It is not difficult to determine a set of analytical conditions which will ensure global existence and uniqueness of a solution. We shall therefore proceed to investigate the steady states and their stability. Integrating the third equation of model system (1.1) w.r.t. α over [0, ∞) yields dR1 − ρ(t, 0) = −(d + e + γ)R1 . dt Thus,
dR1 = h(I, R1 , R2 ) − (d + e + γ)R1 . dt
(1.2)
Now, dN dt
= bN − γS − ρ(t, 0) + dR1 − γI + ρ(t, 0) − (d + e + γ)R1 + eR1 − γR2 , = mN,
(1.3) where m = b − γ. Rather than considering the mixed system (1.1) of ordinary and partial differential equations, we will consider the equivalent system made of the first, second and fourth equations of (1.1) of ordinary differential equations. It is easily seen that (S, I, R1 , R2 ) = (0, 0, 0, 0) is a solution of the system. This is the trivial steady state. A more interesting steady state, if it exists, would be the corruption-free steady state (S, I, R1 , R2 ) = (S0 , 0, 0, R20 ). Let I = 0 = R1 in the system of equations. Therefore, a(R1 ) = a(0) 6= 0 by assumption. Set a(0) = a0 , b = b(S0 + R20 ) = b0 , say. c(R1 ) = c(0) = 0. γ(S0 + R20 ) = γ0 , say. d(0) = d0 6= 0 since d is monotone decreasing and positive. e(R1 ) = e(0) = 0. Finally, h(0, 0, R20 ) = 0. Substituting (S, I, R1 , R2 ) = (S0 , 0, 0, R20 ) in the system, together with the parameter values yields −a0 R20 + b0 (S0 + R20 ) − γ0 S0 = 0,
(1.4)
(a0 − γ0 )R20 = 0
(1.5)
4
C.O.A. Sowunmi
From equation (1.4), we can have R20 6= 0 provided a0 = γ0 , while from equation (1.3), we also have S0 6= 0 if b0 = a0 . Thus, for arbitrary S0 , R20 6= 0 and provided a0 = b0 = γ0 , there is a corruption-free steady state (S0 , 0, 0, R20 ). Next, we investigate the possibility of an endemic steady state i.e., I0 , R10 6= 0 when S0 = R20 = 0. We see that the last equation of system (1.1) implies e(R10 )R20 = 0. Since e(R1 ) is monotone increasing and non-negative, it follows that R10 = 0, contrary to the assumption that R10 6= 0. Hence, an endemic steady state is not possible. We now investigate the stability of a corruption-free steady state by perturbing the system about the point (S0 , 0, 0, R20 ), given that S0 , R20 6= 0. Set S = S0 + s, I = 0 + i, R1 = 0 + r1 and R2 = R20 + r2 . Hence, ds dt
= (β − γ0 )s + (β − F2 )i + (β − a0 R20 )r1 + (β − a0 )r2 + M1 ,
di dt
= (F2 − γ0 − h1 )i + (d0 − c0 R20 )r1 + M2 ,
dr1 dt
= h1 + (γ0 + h1 − d0 )r1 + M3 ,
dr2 dt
= −γ0′ R20 s − γ0′ R20 i + {(a′0 + c′0 − γ0′ )R20 }r1 + (a0 − γ0′ R20 − γ0 )r2 + M4 ,
(1.6) where a′0 = a′ (0), c′0 = c′ (0), d′0 = d′ (0), γ0′ = γ ′ (S0 + R20 ), b′0 = b′ (S0 + R20 ), ∂h h1 = (0, 0, R20 ) and β = b(S0 + R20 ) + b′ (S0 + R20 )(S0 + R20 ) − γ ′ (S0 + R20 ). ∂I M1 , M2 , M3 , M4 are the nonlinear remainder terms: Written in matrix form, the perturbed system takes the form l11 l12 l13 l14 s s d i 0 l22 l23 0 i = dt r1 0 l32 l33 0 r1 l41 L42 43 l44 r2 r2
M1 M2 + M3 M4
If L denotes the community matrix in the last equation, then, its characteristic equation det(L − λI) when expanded becomes
λ2 − (l11 + l44 )λ + l11 l44 − l14 l41
2 λ − (l22 + l32 )λ + l22 l33 − l23 l32 = 0. (1.7)
The Routh-Hurwitz conditions for the quartic are easily obtained. Thus, l11 + l44 < 0 > l11 l44 − l14 l41 ,
(1.8)
l22 + l33 < 0 > l22 l33 − l23 l32 . In terms of the parameters of the system, necessary and sufficient conditions for local
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction
5
asymptotic stability of the corruption-free steady state are l11 + l44
= b′ N0 − r0′ (R20 + 1) < 0,
l11 l44 − l14 l41 = 0 > 0, l22 + l33
= F2 (s, 0) − 2γ0 − d0 < 0,
l22 l33 − l23 l32 = (F2 (S, 0) − γ0 )(γ0 + h1 − d0 ) − hi (γ0 + hi − c′0 R20 ) > 0. (1.9) Of the last four inequalities, only the first one is patently inconsistent. The remaining three can be logically consistent. From these, we infer that under suitable assumption, three out of the four characteristic roots have negative real parts while the odd one equals zero. This is a critical case of the type discussed by Lyapunov [2], where linearisation is inadequate for the resolution of the question of stability of a steady state. We will therefore apply the result of Lyapunov [2]. For this, it is convenient first to redesignate the dependent variables as s = y1 , i = y2 , r1 = y3 , r2 = y4 . Thus, the system of perturbed equations can be written as y = Ly˙ + M (y), (1.10) where y = (y1 , y2 , y3 , y4 )T , L = [lij ] and M (y) = (M1 (y), M2 (y), M3 (y), M4 (y))T . Next step is to rotate the coordinate frame in order to transform equation (1.10) into an appropriate form, such that one axis lies in the eigenspace of the zero root. We assume this subspace is 1-dimensional and the three non-zero roots are distinct. Let X0 be the kernel of L and X the orthogonal complement of X0 . Thus, any vector y can be written as y = x0 + x where x0 ∈ X0 and x ∈ X. Hence, y˙ = x˙ 0 + x˙ = L(x0 + x) + M(x0 + x) where L now denotes the linear operator which is represented by the matrix [lij ] in the original coordinate system. But, Lx0 = 0. Hence, x˙ 0 + x˙ = Lx + M(x0 + x). Obviously, Lx 6= 0, hence, it belongs to X. Let M(x0 + x) = M0 (x0 + x) + MX (x0 + x), where M0 ( ) ∈ X0 andMX ( ) ∈ X. Hence, equation (1.10) is equivalent to the pair x˙ 0 = M0 (x0 + x), (1.11) x˙
= Lx + MX (x0 + x)
In this latter form, we are almost ready for the final transformation preparatory to the application of the result of Lyapunov in [2]. The next step is to express the terms in system (1.11) in terms of the coordinates in the new basis. From equations (1.9), we see that we can take as basis for X0 , the vector T T −l14 l14 −l44 = , 0, 0, 1 , 0, 0, 1 . For convenience, we shall write ν for − . Hence, l11 l41 l11 the basis for X0 is (ν, 0, 0, 1)T denoted by a0 . As basis of the orthogonal complement X, we shall take the eigenvectors of the non-zero eigenvalues of L. From equation (1.7), since one root is zero, there must be another real root. The remaining pair may be real or complex
6
C.O.A. Sowunmi
conjugates. There is no loss of generality in assuming there are three real distinct roots. Denote these by λ1 , λ2 , λ3 and their corresponding eigenvectors by a1 , a2 , a3 . In terms of the basis {a0 , a1 , a2 , a3 } the first equation of (1.11) takes the form 3 X
x˙ 0 a0 = M0
xi ai
i=0
!
a0 .
Hence, x˙ 0 = M0 (x0 , x1 , x2 , x3 ).
(1.12)
Similarly, equation (1.12) takes the form 3 X i=1
x˙ i ai = L
3 X
xi ai
i=1
!
+
3 X i=1
But, L
3 X i=1
xi ai
!
=
Mi
3 X
3 X j=0
xj aj ai .
xi λi ai .
i=1
From the linear independence of {a1 , a2 , a3 }, it follows that x˙ i = λi xi + Mi (x0 , x1 , x2 , x3 ).
(1.13)
We now adopt Lyapunov’s technique in [2] to ensure that in the power series expansions of M1 , M2 , M3 there is no term in x0 having a lower degree than the x0 -term in M0 . Consider the homogeneous system λ1 x1 + M1 (x0 , x1 , x2 , x3 ) = 0, λ2 x2 + M2 (x0 , x1 , x2 , x3 ) = 0,
(1.14)
λ3 x3 + M3 (x0 , x1 , x2 , x3 ) = 0. (x0 , x1 , x2 , x3 ) = (0, 0, 0, 0) is a solution. Furthermore, λ1 , λ2 , λ3 6= 0. Assuming M1 , M2 , M3 are smooth enough, it follows that the Implicit Function Theorem applies and so we have x1 = X1 (x0 ), x2 = X2 (x0 ), x3 = X3 (x0 ) in an open neighbourhood of zero. Still continuing in the technique of Lyapunov [2], we substitute x1 = X1 + z1 , x2 = X2 + z2 , x3 = X3 + z3 in the equations (1.12 - 1.13) to obtain the system in x0 , z1 , z2 , z3 , viz: x˙ 0 = M0 {x0 , X1 + z1 , X2 + z2 , X3 + z3 ), (1.15) z˙i = λi (Xi + zi ) + Mi (x0 , X1 + z1 , X2 + z2 , X3 + z3 ) −
dX0 dt ,
= λi zi − Mi (x0 , X1 , X2 , X3 ) + Mi (x0 , X1 + z1 , X2 + z2 , X3 + z3 ), i − dX dx0 M0 (x0 , X1 + z1 , X2 + z2 , X3 + z3 ),
i = 1, 2, 3.
(1.16)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction
7
In equation (1.16), the terms in x0 alone are obtained by putting z1 = z2 = z3 = 0 in the dXi M0 (x0 , X1 , X2 , x3 ), from which it is clear nonlinear terms. The terms are given by − dx0 −dXi that the factor ensures that the terms in Xi cannot be of degrees less than that in M0 . dx0 We now have to compute the power series expansion of M0 (x0 , X1 +z2 , X2 +z2 , X3 + z3 ) about the point (0, 0, 0, 0). As a first step, we need to compute the remaining eigenvalues of L and their associated eigenvectors. −l44 −l14 = . We already have (ν, 0, 0, 1) associated with λ = 0, where ν = l11 l41 −l14 −l44 For λ = l11 + l44 = l11 (1 − ν) = λ1 we have (σ, 0, 0, 1), where σ = = . For l44 l41 p l22 + l33 + (l22 − l33 )2 + 4l23 l32 + λ2 = λ = , (1.17) 2 + + λ − l33 + , 1, w4 , where we have w1 , l32 w1+ =
(λ+ − l44 ){l13 l32 + l12 (λ+ − l33 )} + l14 {(λ+ − l33 )(l42 + l23 l32 )} , λ+ {λ+ − (l11 + l44 )}
and w4+ =
(λ+ − l11 ){l43 l32 + l42 (λ+ − l33 )} + l41 {(λ+ − l33 )l12 + l13 l32 } . l32 λ+ {λ+ − (l11 + l44 )}
(1.18)
Lastly, for −
λ =
(l22 + l33 ) −
p (l22 + l33 )2 + 4l23 l32 = λ3 , 2
(1.19)
− − − λ − l33 , 1, w4 , where w1− and w4− are obtained by replacing λ+ by λ− we have w1 , l32 in the expressions for w1+ and w4+ , respectively. Let {e1 , e2 , e3 , e4 } denote the standard basis for R I 4 while {a0 , a1 , a2 , a3 } denotes the basis made up of the eigenvectors which we have just obtained. The change of bases from standard to the newly obtained can be represented by the matrix ν 0 0 1 σ 0 0 1 + A= w+ λ − l33 1 w+ . 1 4 l32 − − λ − l33 − w1 1 w4 l32
Hence, AT is the matrix for the change of coordinates from (y1 , y2 , y3 , y4 ) to (x0 , x1 , x2 , x3 ). With reference to basis {e1 , e2 , e3 , e4 }, the nonlinear vector valued function M has components (M1 , M2 , M3 , M4 )T . Thus, with reference to basis
8
C.O.A. Sowunmi
{a0 , a1 , a2 , a3 }, the components (M0 , M1 , M2 , M3 )T are given by M0 ν σ w1+ w1− M 1 0 0 λ+ λ− M2 = 0 0 1 1 1 1 w4+ w4− M3
−1
M1 M2 M3 . M4
If B denotes the matrix in the above equation, then, B = (AT )−1 . We are interested in the coefficient of x20 . Each Mi consists of a homogeneous quadratic form in (y1 , y2 , y3 , y4 ) plus a remainder of higher degree terms. That is, each Mi consists of a term y T Qi y plus higher degree terms. 4 X
M0 =
B1s Ms = B1s yiT Qsij yj + higher degree terms
s=1 yiT B1s Qsij yj + higher degree terms 4 X yiT B1s Qsij yj + higher degrees i,j,s=1
= =
terms.
Since yα = ... ...
yαT
=
M0 =
3 X
r=0 3 X
ATαr+1 xr xTr Aαr+1
r=0 4 X
3 X
(xTr Air+1 )B1s Qsij yj + higher degree terms
i,j,s=1 r=0
3 4 X X
=
(xTr Air+1 )B1s Qsij
=
ATjα+1 xα + higher degree terms
α=0
i,j,s=1 r=0 4 X
3 X
3 3 X X
xTr Air+1 B1s Qsij ATjα+1 xα + higher degree terms
i,j,s=1 r=0 α=0
... term in x20 is
4 X
i,j,s=1
... coefficient of x20 is 4 X
i,j,s=1
Ai1 B1s Qsij ATj1 x20
Ai1 B1s Qsij ATj1 =
4 X
i,j,s=1
Ai1 Qsij B1s A1j .
(1.20)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction
9
To obtain the r.h.s. of equation (1.20), we must compute Qsij for all s, i, j.
q1
q1
q1
q1
q1 q1 − F22 q1 q1 = q1 q1 q1 − a′′0 R20 q1 − a′0
Q1ij
q1
q1
q1 − a′0
q1
,
where q1 = b′′0 N0 + 2b′0 − γ0′′ S0
Q2ij
0
0
−γ0′ F22 − h11 − 2γ0′ −γ0′ −γ0′ − h13 = 0 −γ0′ −2d′0 − c′0 R20 − h22 −c′0 0
−γ0′ − h13
−γ0′
−c′0
0 0 0 0 h11 h12 − γ0′ h13 = −γ ′ h12 − γ ′ h22 − q2 h22 − γ ′ 0 0 0
Q3ij
−γ0′′
0
0
h13
h22 − γ0′
h33
a′0 − γ0′
,
where q0 = 2d′ (0) + e′ (0) + 2γ0′ . Finally, −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 4 Qij = −γ ′′ R20 −γ ′′ R20 q3 a′0 − γ0′ 0 0 −γ0′′ R20 −γ0′′ R20
−h33
−γ0′′ R20
,
where q3 = 2e′0 + (a′′0 + c′′0 + c′0 − γ0′′ )R20 , a′′0 = a′′ (0), b′′0 = b′′ (S0 + R20 ), γ ′′ = ∂2F ∂2h (S0 + R20 ), F22 = (S0 , 0), h12 = (0, 0, R20 ) and h11 , h22 , h13 , h33 are ∂I∂R1 ∂I∂R1 similarly defined.
10
C.O.A. Sowunmi
Hence, X
Ai1 Q1ij Aij
= q1 (ν + 1){ν + σ + w1+ + w1− } − w1+ a′0 ,
P2
Ai1 Q2ij A1j
= −{νσγ0′ + σ(γ0′ + h13 ) + w1+ c′′0 − wi− h13 },
Ai1 Q3ij A1j
= σh13 + w1+ {h22 − γ0′ (1 + ν)},
Ai1 Q4ij A1j
= −γ0′′ R20 {(ν + 1)(ν + σ + w1− ) + νw1+ w1− }.
i,j
ij
P
ij
P
ij
(1.21)
Thus, the coefficient of x20 = B11
X
Ai1 Q′ij A1j +B12
ij
X
Ai1 Q2ij A1j +B13
X
Ai1 Q3ij A1j +B14
X
Ai1 Q4ij A1j .
Now, B = (A−1 )T . Since A is known, A−1 is easily computed using MathCAD. Therefore, 1 (ν − σ) 1 (ν − σ) B= 0 0
(w4+ − w4− ) + w1− − w1+ (λ+ − λ− )(ν − σ)
σ(λ+ w4− − w4+ λ− ) + w1+ λ− − w1− λ+ (λ+ − λ− )(ν − σ)
ν(w4+ − w4− ) + w1− − w1+ (λ+ − λ− )(σ − ν)
ν(w4+ − w4+ ) + w1− − w1+ (λ+ − λ− )(σ − ν)
λ+
1 − λ−
1 + λ − λ−
λ−
λ− − λ+
λ+ λ+ − λ−
σ σ−ν ν ν−σ 0 0
Hence, the coefficient of x20
=
(ν − σ)−1 [q1 (ν + 1)(ν + σ + w1+ + w1− ) −a′0 w1+ + σγ0′′ R20 {(ν + 1)(ν + σ + w1− ) + νw1+ w−1 } +(λ− − λ+ )−1 {(σ(w4+ − w4− ) − (w1+ − w1− ))(νσγ0′ + σ(γ0′ + h13 ) − w1+ c′′0 + w1− h3 ) −λ+ (σw4− − w4+ ) − λ− (σw4+ − w1− )(σh13 + w1+ (h22 − γ0′ (1 − ν)))}].
(1.22)
If the r.h.s. of the above equation is not zero, then, by the result of Lyapunov [2] the corruption-free steady state is not stable, while if it vanishes, we will have to compute the coefficient of x30 . This is likely to be more cumbersome than the r.h.s. of this equation. Rather than get involved in this, we choose the case when the r.h.s. does not vanish and seek to stabilize the corruption-free steady state by means of a feedback control. Hence, we assume that the coefficient of x20 in M0 is not zero so that the corruption-free steady state is unstable.
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 11
1.4.
Stabilizing the Corruption-Free Steady State through Control
We begin with the linear parts of equations (2.24-2.25). Thus, z˙0 = 0, z˙1 = λ1 z1 , (1.23) z˙2 = λ2 z2 , z˙3 = λ3 z3 . The last three equations can be written in vector form as z˙ = Qz, where Q = diag[λ1 , λ2 , λ3 ]. Adopting the procedure in [3], we introduce a scalar valued function f of the real feedZ ±∞
back variable ρ, such that ρf (ρ) > 0 for ρ 6= 0, f (0) = 0 and
f (ρ)dρ is divergent.
0
f is called the characteristic of the servo motor or feedback mechanism. Let k, η be two constant 3-vectors, while u, v are real numbers. Finally, ρ is defined as a linear combination of z0 , z1 , z2 , z3 . Thus, ρ = vz0 + k · z. Our control problem therefore is to obtain sufficient conditions for the complete stability of the trivial solution of the system. z˙ = Qz + f (ρ)η, (1.24) z˙0 = f (ρ)u, where ρ = vz0 + k · z. We define the function T
V (z0 , z) = z Bz +
Z
(1.25)
ρ
f (τ )dτ,
(1.26)
0
where B is a positive definite, symmetric 3 × 3 matrix. V (z0 , z) is positive definite for all z0 , z. Reason: V (0, 0) implies ρ = 0 from equation (1.24). Hence, V (0, 0) = 0 by equation (1.26). On the other hand, V (z0 , z) = 0 implies z = 0 and ρ = 0 since the r.h.s. of (1.25) is the sum of positive terms. But, ρ = vz0 +k·z and v 6= 0, so that z0 = 0. Define C = −(QT B + BQ). Thus, C = C T . From equations (1.22 - 1.24), we have ρ˙ = v z˙0 + k · z˙ = vuf (ρ) + k · {Qz + f (ρ)η} = vuf (ρ) + K T Qz + f (ρ)k · η. Therefore, K T Qz + f (ρ)(k · η + vu) = ρ. ˙ Now, −V˙ = −(z˙ T Bz + z T B x˙ + f˙(ρ)ρ). ˙ Substituting for ρ from equation (1.25) above yields −V˙ = [z T Cz − f (ρ){η T Bz + z T Bη + k T Qz + f (ρ)(k · η + vu)}].
(1.27)
12
C.O.A. Sowunmi
Put K T Q = K ∗ and r = −(k · η + vu) in the last equation to obtain −V˙ = 4[z T Cz − f (η){η T Bz + z T Bη + k ∗ · z − rf (ρ)}].
(1.28)
For each t, equation (1.28) defines −V˙ as a monotone increasing function of r subject to the values of the remaining terms in the equation. Hence, increasing r increases the rate at which V decreases along a trajectory in (z0 , z)-space. As shown in [3], given any positive definite matrix C, there exists a unique symmetric matrix B also positive definite, which satisfies the equation C = −(QT B + BQ). Hence, let us choose C to be positive definite, and there will be a unique, symmetric positive matrix B which will fit into (1.26). Our goal is that −V˙ also be a positive definite function of (z0 , z). In matrix form, the r.h.s. of equation (1.28) is 1 C −(Bη + k ∗ ) z 2 . [z T , f (ρ)] 1 ∗ T f (ρ) −(Bη + k ) r 2
For the above to be a positive definite form in [z T , f (ρ)], it can be shown as in [3] that neces1 sary and sufficient conditions are that C be positive definite and r > (Bη + k ∗ )C −1 (Bη + 2 1 ∗ k ). Now, V˙ (0, 0) implies ρ = 0 = f (ρ), and conversely (z, ρ) = (0, 0) implies 2 (z0 , z) = (0, 0). Hence, −V˙ is a positive definite function of (z0 , z), provided 1 1 r > (Bη + k∗ )C −1 (Bη + k ∗ ), 2 2 since C is already chosen to be positive definite. Note also that r 6= −k · η.
(1.29)
(1.30)
(1.29 - 1.30) are the two fundamental control inequalities for the problem (cf. [3]. The next step is to apply (1.29) to our problem. Take C = diag(α1 , α2 , α3 ), αi > 0 for i = 1, 2, 3. Hence, B satisfies the equations bij = 0, i 6= j 2λi bii = −αi , ... bii =
i = 1, 2, 3
αi , i = 1, 2, 3 2λi
bij = 0, i 6= j. ...
B = diag
... Bη =
1 2
−α1 −α2 −α3 , , 2λ1 2λ2 2λ3
α1 η1 α2 η2 α3 η3 , , |λ1 | |λ2 | |λ3 |
T
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 13 Since C −1 = diag α11 , α12 , α13 , it follows that 3
1 1X 1 (Bη + k∗ )C −1 (Bη + k ∗ ) = 2 2 4 i=1
πi ηi ki∗ + |λi | πi
2
,
√ where πi = αi > 0. Hence, the fundamental control inequality (1.29) is 3
1X r> 4 i=1
πi ηi ki∗ + |λi | πi
2
(1.31)
We have seen that −V˙ is a monotone increasing function of r and when r satisfies (1.29), −V˙ becomes a positive definite function of (z0 , z). So far, our choice of (α1 , α2 , α3 ) has been arbitrary but for the positivity restriction. We now choose the α’s to minimize the r.h.s of (1.29) (cf [3]). This will give us the sharpest condition on r for −V˙ to be positive πi ηi k∗ definite, taking η, λi ’s and ki∗ ’s as fixed. + i is the sum of two numbers whose |λi | πi product is fixed. If they are both positive, then, their sum is minimum when they are equal. The case when both are negative can be converted to the preceding one. When they are of opposite signs there is no minimum, but since the sum is squared, the minimum is zero. Hence, 3 X πi ki∗ r> H(ηi ki∗ ), (1.32) |λi | i=1
where H is the Heaviside Unit Function. Next is the interpretation of (1.32) in terms of the parameters of the model.
1.5.
Discussion
We have already seen that the larger r is, the more rapid is the stabilization of the corruptionfree state. For a given r, efficiency of the feedback mechanism depends on how small we can make the r.h.s. of (1.32), i.e., how large we can make |λi | for all i = 2, 3. λ1 = l11 + l44 = β − γ0 + β − a0 = 2β − a0 − γ0 . Obviously, as a0 → ∞, λ1 → −∞. Now, 2λ2
= l22 + l33 +
2∂λ2 ∂h1
= −1 + 1 +
∂λ2 ∂h1
=
p (l22 + l33 )2 + 4l23 l32 ,
2(l22 +l33 )(−1+1)+4l23 +4l32 √ , (l22 +l33 )2 +4l23 l32 ′
0 R20 ) √2(h1 +d0 −c . 2
(l22 +l33 ) +4l23 l32
(1.33)
14
C.O.A. Sowunmi
Hence,
∂λ2 < 0, provided ∂h1
Correspondingly, for λ3 , we have
h1 + d0 − c′0 R20 < 0,
(1.34)
h1 + d0 − c′0 R20 > 0.
(1.35)
∂λ2 < 0, provided ∂h1
These are mutually exclusive. We can only have one or the other. This will be achieved by making use of the Heaviside function in (1.31). For (1.32), it would be enough to ensure that c′0 is large enough, whilst for (1.33), we would require that hi be large enough. The interpretation of the foregoing is as follows: If the corruption-free steady state is unstable, it is nonetheless possible to stabilize it absolutely through feedback control, by as much as possible encouraging direct movement of people from the Susceptible class to the Resistant class. In addition, those who are infective should be moved into the removed class. The efficiency of this feedback process depends on a number of factors, one of which is a0 : the rate at which people in the susceptible class become resistant purely on their volition, rather than through deterrence. Other factors are h1 and c′0 . If η2 k2∗ < 0, then, increasing c′0 so that (1.32) holds will increase the feedback efficiency. On the contrary, if the choice is that η3 k3∗ < 0, then, increasing h1 so that (1.33) holds, will increase the feedback efficiency.
2. 2.1.
Disease Transmission as a Saturable Interaction: The SIS Case Introduction
In the study of 2-sex population dynamics published in [4, 5], we began with the minimal analytical conditions on the interaction function; the function which encapsulates the dynamics of the interaction between the reproductive males and the reproductive non-gestating females. These minimal conditions were adequate for the proof of global existence and uniqueness of the solution of the model equations. One of the conditions is essentially a growth condition on the interaction function, and it is hardly surprising that one needed such a condition in order to prove global existence or uniqueness of solution. In paper [6], in order to prove boundedness and stability of steady states, additional conditions had to be imposed. Paper [7] was on stability, but of a time-discrete model, and the interaction function likewise needed additional conditions. The foregoing suggests the following approach to the modelling of any dynamical process which is driven by the interaction between some or all of its components, provided the interaction is saturable. We begin with what we consider as the minimal analytical conditions, and leave additional conditions on the interaction function to emerge from the qualitative study, other than existence and uniqueness of solution of the model equations. Disease transmission is a dynamical process driven by the interaction between the susceptibles and the infectives. It seems safe to assume that all such interactions are saturable. In a SIS model, there are just the two components, the susceptibles and the infectives. This
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 15 situation is analogous to that of 2-sex population dynamics or prey-predator interactions and therefore fits into the general formulation developed in [1]. We shall therefore proceed to the study of SIS models, which require a slight modification of the formulation in [1]. In the first SIS model, the function F (S, I) will be the incidence rate of new infectives when the susceptibles and infectives number S and I, respectively. Since F (S, 0) = F (0, I) = 0 ∀ S, I, it follows that F (S, I) = φ(S, I)SI for some function φ. The generalised law of the minimum [1] when applied to F yields the inequality F (S, I) ≤ min{k1 S, k2 I} for some positive constants k1 , k2 . k1 is therefore an upper estiF (S,I) mate of F (S,I) . Both are indicators of the ease or difficulty with S , whilst k2 is that of I which a disease is transmitted. In the second SIS model, the infective class has an age structure. Following the idea which was used in [4, 5] for the rate of new births, we assume the existence of a density Z ∞
π(α)F(S(t), J(t, α), α)dα,
function F such that the incidence rate at time t equals
0
where π(α) is the infectivity at class age α, and J(t, α) is the density of infectives aged α at time t. Obviously, F(0, x, α) = F(y, 0, α) = 0 for all positive x, y, α. We therefore have F(S(t), J(t, α), α) = Ψ(S(t), J(t, α), α)S(t)J(t, α) for some function Ψ. Thus, the incidence rate at time t also equals Z ∞ Ψ(S(t), J(t, α), α)π(α)J(t, α)dα. S(t) 0
Still following the pattern in [4, 5], we assume that the density function F satisfies the generalized law of the minimum in the form F(S(t), J(t, α), α) ≤ min{k1 S(t), k2 (α)J(t, α)}, where k1 , k2 (α) > 0. The function F is to a certain extent analogous to the interaction function in [4, 5]. Hence, by analogy, we expect that the functions such as min{k1 S(t), k2 (α)J(t, α)} and
k1 k2 (α)S(t)J(t, α) , k1 S(t) + k2 (α)]J(t, α)
which certainly satisfy all the conditions are admissible. In the next subsection, we shall formulate the governing equations of two SIS models, taken essentially from Brauer [8], but without his type of interaction functions.
2.2.
Formulation of Model Equations
Let Z ∞the total population at time t be N (t). Then, N (t) = S(t) + I(t) where I(t) = J(t, s)ds. We assume that there is a natural death rate D(N ), dependent on the popu0
lation size. There is also an additional mortality rate q among the infectives (a measure of the deadliness of the disease). P (s) is the Z fraction of infectives of class age s . P (0) = 1, ∞
and P is monotone decreasing such that
0
P (s)ds = τ < ∞ . P is also independent of
time. Hence, we are dealing with an autonomous case. Individuals are born into the S class
16
C.O.A. Sowunmi
0 J(t + s, s) ∂0 ∂ only. If we exclude death, then, P (s) = J is the derivative . If + J(t, 0) ∂t ∂s following a cohort, where change is due to recovery alone, then, 0 ∂0 d ∂ J = J(t, s) ln(P (s)). + ∂t ∂s ds Hence,
∂ ∂ + ∂t ∂s
d J(t, s) = − q + D(N ) − ln(P (s)) J(t, s). ds
(2.36)
When infectively is not age-dependent, J(t, 0) = F (S(t), I(t)).
(2.37)
When infectivity is age-dependent, J(t, 0)
R∞
=
0
π(α)F(S(t), J(t, α), α)dα, (2.38)
J(0, s) = J0 (s). Furthermore,
DN Dt
= (B(N ) − D(N )N − qI,
(2.39)
N (0) = N0 . The entire population is assumed to have a carrying capacity K, say. Hence, B(K)
= D(K),
B(N )
> D(N ), if N < K,
(2.40)
B ′ (N ) < D′ (N ), if N ≥ K. We further assume that B ′ (N ), D′ (N ) > 0. We can deduce the equation for S from the fact that S = N − I. However, we would rather focus on the subsystem of equations in N and I. Thus, equations (2.37) and the first equation of (2.38) will be written respectively as J(t, 0) = F (N (t) − I(t), I(t)), J(t, 0) =
2.3.
R∞ 0
(2.41)
π(α)F(N (t) − I(t), J(t, α), α)dα.
Age-independent Infectivity and Existence of Steady States
dN = First, we consider the case without age-dependent infectivity. At a steady state, dt dS ∂J = = 0 ∀ t ≥ 0. dt ∂t Let N (t) = N∞ , J(t, s) = J∞ (s), and I(t) = I∞ at a steady state. So, (B(N∞ ) − D(N∞ ))N∞ − qI∞ = 0,
(2.42)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 17 and
dI∞ d = − q + D(N∞ ) − ln(P (s)) J∞ (s). ds ds
(2.43)
Thus, from equation (2.43), we obtain J∞ (s) = exp {−(q + D(N∞ ))} P (s)J∞ (0). This can be written as J∞ (s) = J∞ (0)P (s)e−λs ,
(2.44)
λ = q + D(N∞ ).
(2.45)
where
Since I∞ =
Z
∞
J∞ (s)ds, equation (2.44) yields
0
I∞ = J∞ (0)Pˆ (λ), where Pˆ (λ) =
Z
∞
P (s)e−λs ds.
(2.46)
(2.47)
0
Therefore, from equation (2.36), we have
B(N∞ ) − D(N∞ ))N∞ = q Pˆ (λ)F (N∞ − I∞ , I∞ ).
(2.48)
From equation (2.37) at a steady state, J∞ (0) = F (N∞ − I∞ , I∞ ). Substituting this into equation (2.46), we have I∞ = Pˆ (λ)F (N∞ − I∞ , I∞ ).
(2.49)
qI∞ = (B(N∞ ) − D(N∞ ))N∞ .
(2.50)
From equation (2.42), Equations (2.49) and (2.50) define the steady states (N∞ , I∞ ). If I∞ = 0, equation (2.49) is satisfied, while (2.50) is satisfied provided N∞ = K. Hence, (K, 0) is a steady state. This is the non-trivial disease-free steady state and there is only one such since B(N ) = D(N ) only if N = K. N = 0 is trivial. At an endemic state I∞ > 0. If we can solve equation (2.50) for N∞ as a function of I∞ , then, we shall substitute for N∞ in equation (2.49) to obtain an equation in I∞ alone. Define f (N ) = (B(N ) − D(N ))N , then, f ′ (N ) = B(N ) − D(N ) + N (B ′ (N ) − D′ (N )), and f ′ (K) < 0, but, f ′ (N ) > 0 for sufficiently small values of N . Hence, f ′ (N ) = 0 for some N < K, equal to K0 say. We assume there are at least two branches of the inverse function f −1 . Specifically, let ′ f (N ) < 0 ∀ N ∈ (K0 , K). Let g denote the branch of f −1 defined on (0, I0 ) and extended by continuity to [0, I0 ] such that g(I0 ) = K0 . Thus, N∞ = g(qI∞ ) for I∞ ∈ [0, I0 ],
(2.51)
18
C.O.A. Sowunmi
where qI0 = K0 (B(K0 ) − D(K0 )). In equation (2.51), we note that λ is a function of N∞ . Hence, substituting for N∞ from equation (2.51) in equation (2.49) yields I∞ = Pˆ (q + Dog(qI∞ ))F (g(qI∞ ) − I∞ , I∞ ),
(2.52)
where I∞ ∈ [0, I0 ]. An endemic state will exist iff equation (2.52) has a non-trivial solution. This will happen iff the straight line Y = X and the curve Y = Pˆ (q + D o g)(qX))F (g(qX) − X, X)
(2.53)
dY (0) intersect over (0, I0 ). Suppose > 1, then, equation (2.52) has non-trivial solution dX if, but not only if, there is X1 , 0 < X1 < I0 such that Pˆ (q + Dog(qX1 )F (g(qX1 ) − X1 , X1 ) < X1 To obtain a sufficient condition with epidemiological meaning, we bring in the generalised law of the minimum. Thus, suppose Pˆ (q + D o g(qX1 ))k2 X1 < X1 . Therefore, k2 Pˆ (q + Dog(qX1 )) < 1.
(2.54)
Recall that λ = q + Dog(qX1 ). We therefore have the following result. dY (0) > 1 and there is a value of X = X1 > 0, such that k2 Pˆ (q + dX Dog(qX1 )) < 1, then, an endemic steady state X∞ < X1 exists. Lemma 1 If
For an interpretation of condition (2.54), we note that Z ∞ J(t + s, s) −λs ˆ e ds. P (λ) = J(t, 0) 0 The first factor in the integrand is the fraction of those infected at time t that stays infected till s units of time later, barring death due to natural causes or fatality of the infection, while the second is the probability of surviving that interval when the infective population equals X1 . The integrand is therefore the fraction that actually stays infective for s units of time. Pˆ (λ) has the dimension of TIME. It can be taken as a measure of the mean time spent in the infective class. The higher the fatality of an infectious disease, other factors being equal, the lower its Pˆ (λ) at a given population of infectives. k2 is an upper bound of the rate at which susceptibles are infected by an infective. Equation (2.54) is therefore the condition that at X1 , infectives do less than replace themselves. dY (0) It can be shown that = Pˆ (q + D(K)).F2 (K, 0) where F2 stands for the partial dX derivative of F w.r.t. its second variable. F1 is likewise defined. dY (0) < 1, there may be no endemic steady state. A sufficient but not necessary If dX condition for an endemic steady state is that there is X2 , 0 < X2 < I0 such that Pˆ (q + Dog(qX2 ))F (g(qX2 ) − X2 , X2 ) > X2 .
(2.55)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 19 But, unlike in the preceding case, we cannot make use of the generalised law of the minimum to obtain a sufficient condition. In fact, Pˆ (q + Dog(qX2 )k2 > 1,
(2.56)
does not guarantee (2.55).
2.4.
Stability of Equilibria
Let (N (t), I(t)) = (N∞ , I∞ ) be an equilibrium. Let its perturbation be N (t) = N∞ +n(t), I(t) = I∞ + i(t), J(t, Z J(0, s) = J∞ (s) + j(0, s) = J∞ (s) + j0 (s). Z s) = J∞ (s) + j(t, s), ∞
∞
j(t, s)ds = i(t). It can be shown, using
J∞ (s)ds = I∞ ,
By definition,
0
0
equations (2.36-2.39) that j satisfies the system ∂j ∂j + ∂t ∂s
= (−λ +
j(t, 0)
= {F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ )} i(t) + R2 ,
j(0, s)
= j0 (s),
∂ ∂s
ln(P (s)))j(t, s) − D′ (N∞ )J∞ (s)n(t) + R1 , (2.57)
where F1 and F2 denote the partial derivatives of F w.r.t. its first and second variables, respectively. R1 , R2 , . . . represent higher order remainder terms in the perturbations. By the usual method of characteristics, the system of equations (2.57) eventually yields Z t e−λs P (s)i(t − s)ds. i(t) = {F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ )} 0
−D′ (N∞ )F (N∞ − I∞ , I∞ ) +
Z
0
∞
Z
∞
e−λσ dσ
Z
0
0
t
e−λs P (s + σ)n(t − s)ds
P (s + t) −λs e j0 (s)ds + R3 . P (s)
(2.58)
Likewise from the pair dn dt
= {(B(N∞ − D(N∞ )) + N∞ (B ′ (N∞ ) − D′ (N∞ ))} n(t) − qi(t) + R4 ,
n(0) = n0 , (2.59) we obtain ρt
n(t) = n0 e − q where
Z
0
t
eρs i(t − s)ds + R5 ,
ρ = B(N∞ ) − D(N∞ ) + N∞ (B ′ (N∞ ) − D′ (N∞ )).
(2.60)
(2.61)
We recall from subsection 2.3 that ρ < 0. From the pair of equations (2.58) and (2.60), we will now obtain estimates of i(t) and n(t). Our interest is not in the rigorous proof of
20
C.O.A. Sowunmi
estimates of the asymptotic behaviour of i(t) and n(t), which can be achieved, but in the estimates themselves. For brevity in writing the terms, we introduce the following notations. a11
= F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ ),
a12
= D′ (N∞ )F (N∞ − I∞ , I∞ ),
b1 (t)
=
b2 (t)
= n0 eρt + R5 ,
R∞ 0
P (s+t) −λs j0 (s)ds P (s) e
+ R3 , (2.62)
P1 (s) = e−λs P (s), P2 (s) =
R∞ 0
e−λσ P (s + σ)σ,
P3 (s) = e−λs P2 (s). The system of equations in (2.58) and (2.60) can be viewed as a perturbed pair of linear Volterra integral equations. Taking its Laplace transform, with ξ as transform variable yields ˆi(ξ) = a11 Pˆ1 (ξ)ˆi(ξ) − a12 Pˆ3 (ξ)ˆ n(ξ) + ˆb1 (ξ), (2.63) q ˆ i(ξ) + ˆb2 (ξ). n ˆ (ξ) = ξ−ρ From system (2.63), we obtain the characteristic equation for the system as qa12 Pˆ3 (ξ) = 0. 1 − a11 Pˆ1 (ξ) + ξ−ρ
(2.64)
At the disease-free steady state a11 = F2 (K, 0) and a12 = 0. Hence, equation (2.64) reduces to 1 − F2 (K, 0)Pˆ1 (ξ) = 0. (2.65)
dY (0) Suppose > 1, i.e., Pˆ (q + D(K))F2 (K, 0) > 1, and condition (2.54) holds, so that dX an endemic steady state exists also. Set ξ = u + iv, so that equation (2.65) is equivalent to the pair R∞ F2 (K, 0) 0 e−us cos vs P1 (s)ds = 1, (2.66) R ∞ −us e sin vsP (s)ds = 0. 1 0 Let v = 0, then, the pair reduces to
F2 (K, 0) Z
∞
Z
∞
e−us P1 (s)ds = 1,
(2.67)
0
e−us P1 (s)ds is a monotone decreasing function of u. When u = 0, where 0 Z ∞ −us e P1 (s)ds reduces to Pˆ (q+D(K)). Having assumed that Pˆ (q+D(K))F2 (K, 0) > 0
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 21 1, it follows that equation (2.65) has a solution which is greater than zero. Hence, a positive characteristic root exists. If Pˆ (q + D(K))F2 (K, 0) > 1 and the condition (2.54) holds, then, an endemic steady state exists and the disease-free steady state is unstable. If Pˆ (q + D(K))F2 (K, 0) < 1, there may be no endemic steady state. Consider again the pair in (2.66). Since we are interested in u being positive or negative, we will consider the first equation of (2.66) alone, regarding v as a parameter of the equation. And here we ask: Can we find a sufficient condition on P1 that guarantees that all solutions of the second equation of (2.66) are negative for all v? We look for an equation in u which is independent of v such that if its solutions are negative, then, all solutions of the first equation of (2.66) are necessarily negative. Since | cos vs| Z ≤ 1, it follows that Z ∞
e−us cos vsP1 (s) ≤ e−us P1 (s). Hence, F2 (K, 0)
Z
0
∞
e−us cos vsP1 (s)ds ≤
∞
e−us P1 (s)ds.
0
e−us P1 (s)ds = 1,
(2.68)
0
is an equation in u such that any solution of the first equation in (2.66) for any v is less than Z ∞
or equal to its solution. F2 (K, 0)
e−us P1 (s)ds is a monotone decreasing function of
0
u. When u = 0, it takes the value Z ∞ P1 (s)ds = F2 (K, 0)Pˆ (q + D(K)) < 1, F2 (K, 0) 0
by our assumption. It follows that equation (2.67) has a negative solution which is also unique. Hence, all characteristic roots of equation (2.65) have negative real parts provided F2 (K, 0)Pˆ (q + D(K)) < 1. Thus, the disease-free steady state is asymptotically stable if F2 (K, 0)Pˆ (q + D(K)) < 1, and an endemic steady state may not exist. Next is to consider the stability of an endemic steady state. We assume that Pˆ (q + D(K))F2 (K, 0) > 1 and that condition (2.19) holds. The characteristic equation (2.64) for an endemic state can be written as Z s Z ∞ ρ(s−τ ) P3 (τ )e dτ e−ξs ds = 1. (2.69) a11 P1 (s) − qa12 0
0
Define W (s) = a11 P1 (s) − qa12 be written as
Z
∞
Z
0
s
P3 (τ )eρ(s−τ ) dτ,
s ≥ 0. Then, equation (2.69) can
W (s)e−(u+iv)s ds = 1 where ξ = u + iv.
0
Hence, equation (2.69) is equivalent to the pair R∞ −us cos vs ds = 1, 0 W (s)e R∞ 0
(2.70)
W (s)e−us sin vs ds = 0.
Let v = 0, then, the first equation in (2.70) is satisfied and the second equation reduces to Z ∞ W (s)e−us ds = 1. (2.71) 0
22
C.O.A. Sowunmi
Solving Z ∞equation (2.71) for u amounts to finding intersection of the curve W (s)e−us ds with the straight line z = 1. z= 0 Z ∞ Z ∞ −us W (s)e ds is a monotone decreasing function of u, thus, if W (s)ds > 1, there 0 0 Z ∞ W (s)ds > 1 an will be a unique intersection at a point (u, 1) where u > 0. Hence, if 0
endemic state if it exists will be unstable. For stability of a possible endemic state, we consider again the pair of equations in (2.70). We are interested in conditions which ensure that all solutions (u, v) have u < 0, regardless of v. Let us therefore consider only the first equation of (2.70) as an equation in u with v as parameter. We look for an equation in u which is independent of v and is such that if its solutions are negative, then, all solutions of the first equation of (2.70) as an equation in u, are necessarily negative. Since | cos vs| < 1, therefore, W (s) cos vs ≤ |W (s)|, Z ∞ |W (s)|e−us ds = 1, (2.72) 0
is such an equation. Since the left hand side of equation (2.72) is a monotone decreasing Z ∞ |W (s)|ds < 1 is a sufficient condition for the asymptotic function of u, it follows that 0
stability of an endemic state if it exists. We therefore have the following result.
Theorem 2 Given the model whose governing equations are (2.36 - 2.37), the second equation of (2.38) and (2.39) together with the conditions on the functions P , B and D, there always exists a disease-free steady state. An endemic steady state exists if Pˆ (q + D(K))F2 (K, 0) > 1 and condition (2.54) also holds. It may otherwise not exist. The disease-free steady state is stable if Pˆ (q + D(K))F2 (K, 0) < 1 and unstable if Pˆ (q + D(K))F2 (K, 0) > 1. Z s def P3 (τ )eρ(s−τ ) dτ, s ≥ 0, then, an endemic steady state if If W (s) = a11 P1 (s) − qa12 0 Z ∞ Z ∞ W (s)ds > 1. |W (s)|ds < 1, and unstable if it exists, is stable if 0
0
2.5.
Age-dependent Infectivity and Existence of Steady States
When infectivity is age-dependent, the governing equations are the same except that equation (2.37) is replaced by the second equations of (2.41) and (2.38), respectively. The analysis in subsection 2.3 remains the same up till equation (2.47). From equation the first equation of (2.38) at a steady state, Z ∞ π(α)F(N∞ − I∞ , J∞ (α), α)dα. (2.73) J∞ (0) = 0
Set x=
Z
∞ 0
π(α)F(N∞ − I∞ , J∞ (α), α)dα.
(2.74)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 23 Then, equations (2.44) and (2.49) can be written respectively as J∞ (s) = xP (s)e−λs , (2.75) I∞
= xPˆ (λ).
Hence, equation (2.73) becomes Z ∞ π(α)F(N∞ − xPˆ (λ), xP (α)e−λα , α)dα. x=
(2.76)
0
Equation (2.42) becomes (B(N∞ ) − D(N∞ ))N∞ = qxPˆ (λ).
(2.77)
Equations (2.76) and (2.77) are a simultaneous pair in x and N∞ . We observe that λ = q + D(N∞ ) which therefore depends on N∞ . At a disease-free equilibrium, I∞ = 0, whence J∞ (s) = 0. Thus, x = 0. When x = 0, the r.h.s. of equation (2.76) vanishes by a condition on F. Hence, equation (2.76) is satisfied by x = 0, for any N∞ . The l.h.s. of equation (2.77) vanishes when N∞ = 0 or N∞ = K. The first is trivial, hence, (N∞ , I∞ ) = (K, 0) is a solution of the pair (2.76) and (2.77). Therefore, the disease-free steady state always exists. An endemic steady state exists if the pair of equations has a solution (N∞ , x), where x > 0. As in subsection 2.3, we try to solve equation (2.42) for N∞ as a function of x, then, substitute for N∞ on the r.h.s. of equation (2.76) so that we have an equation in x alone. Equation (2.77) can be solved for N∞ as a function of x, provided the function f∗ (N∞ ) =
(B(N∞ ) − D(N∞ ))N∞ , Pˆ (q + D(N∞ ))
is invertible in some open interval. = Pˆ (q + D(N ))[{B(N ) − D(N ) + N (B ′ (N ) − D′ (N )}−
f∗′ (N )
−(B(N ) − D(N )N Pˆ ′ (q + D(N ))D′ (N )]/{Pˆ (q + D(N ))}2 , F∗′ (K) =
(2.78)
K{B ′ (K) − D′ (K)} < 0. Pˆ (q + D(K))
For sufficiently small N , f∗′ (N ) > 0. Hence, ∃ a maximal open interval (K00 , K) where f∗′ < 0, and f∗′ (K00 ) = 0. By the Inverse Function Theorem, f∗ is invertible over (K00 , K). Hence there exists an open interval (0, I00 ) such that f∗−1 maps (0, I00 ) onto (K00 , K) continuously. We now extend f∗−1 to [0, I00 ] by continuity and denote it by g∗ . Since K is the carrying capacity, N∞ ∈ (K00 , K). Equation (2.76) can now be written as Z ∞ π(α)F(g∗ (x) − xPˆ (q + D · g∗ (x), xP (α)e−α(q+Doh(α)) , α)dα. (2.79) x= 0
24
C.O.A. Sowunmi
Solving equation (2.77) amounts to finding the non-zero intersections of the line Y = X with the curve Z ∞ π(α)F{g∗ (X) − X Pˆ (q + Dog∗ (X)), XP (α)e−α(q+Dog∗ (X)) , α}dα. Y = 0
Suppose
dY (0) > 1, i.e., dX Z ∞
π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα > 1,
(2.80)
0
then, equation (2.77) has non-zero solution, if, but not only if, there is X2 , 0 < X2 such that Z ∞ π(α)F{g∗ (X2 ) − X2 Pˆ (q + Dog∗ (X2 )), X2 P (α)e−α(q+Dog∗ (X2 )) , α}dα < X2 . 0
(2.81) To obtain a sufficient condition with epidemiological significance, we involve the generalised law of the minimum which now takes the form: F(S(t), J(t, α), α) ≤ min{k1 S(t), k2 (α)J(t, α)},
(2.82)
for k1 , k2 (α) > 0, α ≥ 0. Set λ∗ = q + Dog∗ (X2 ). We assume that π(α) and k2 (α) are bounded and define K2 = sup k2 (α), Π = sup π(α). α
α
Suppose ΠK2
Z
∞
P (α)eαλ∗ dα < 1.
(2.83)
0
Then, by condition (2.82), Z ∞ π(α)J(g∗ (X2 ) − X2 Pˆ (q + Dog∗ (X2 )), X2 P (α)e−αλ∗ , α)dα 0
≤ X2 ΠK2
Z
∞
P (α)e−αλ∗ dα < X2 .
0
Thus, (2.83) implies (2.81). The interpretation of condition (2.83) is similar to that of (2.54). dY (0) Hence, if > 1 and the infectives numbering X2 do less than reproduce themselves, dX there will exist an endemic steady state. Otherwise, there may be no endemic steady state.
2.6.
Stability of Steady States
We proceed as in subsection 2.4, however, in place of the second equation of system (2.57), we shall have, from the second equation of (2.41) Z ∞ π(α)F2 (· · · , α)j(t, α)dα + FI 1 (n(t) − i(t)) + R6 , j(t, 0) = 0
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 25 where F2 (· · · , α) = F2 (N∞ − I∞ , J∞ (α, α), FI 1
R∞
=
0
π(α)F1 (N∞ − I∞ , J∞ (α), α)dα,
and the suffices attached to F denote partial derivatives of F as explained in subsection 2.4. For reasons which will soon be obvious, we prefer to write the boundary condition for j in terms of j as j(t, 0) =
Z
∞
{π(α)F2 (· · · , α) − FI 1 }j(t, α)dα + FI 1 n(t) + R6 .
0
(2.84)
We now have to obtain estimates of the solutions of the system of equations in j and n, namely, the first equation of (2.57), (2.84), the third equation of (2.57) and (2.59). From the first equation of (2.57), (2.84) and the last equation in (2.57) when s < t, we have, j(t, s) = P1 (s) − When s > t,
R ∞ 0
{π(α)F2 (· · · , α) − FI 1 }j(t − s, α)dα + FI 1 n(t − s) −
P1 (s) ′ 0 P1 (σ) D (N∞ )n(t
Rs
P1 (s) j(t, s) = j0 (s − t) − P1 (s − t)
(2.85)
− s + σ)J∞ (σ)dσ + R7 .
t
P1 (s) D′ (N∞ )n(σ)J∞ (s + σ − t)dσ + R8 , P (s − t + σ) 1 0 (2.86) and from system (2.59), we obtain as in subsection 2.4, Z
n(t) = n0 eρt − q
Z
∞
dσ
Z
t
0
0
eρs j(t − s, σ)dσ + R9 .
(2.87)
It can be shown that K0 ≤ K00 . We note that the value of N∞ for the endemic steady state, where infectivity is age-dependent, may not be the same as we had when infectivity was not age-dependent. Hence, the value of ρ in equation (2.87) may differ from its value in subsection 2.4. Since (K0 , K) ⊇ (K00 , K), it follows that ρ in equation (2.87) is negative. Recall that ρ = f ′ (N∞ ). The Laplace transform of equation (2.87), using ξ as the transform variable yields n ˆ (ξ) =
q n0 − ξ−ρ ξ−ρ
Z
∞
ˆ9. ˆj(ξ, σ)dσ + R
(2.88)
0
Likewise from equations (2.85) and (2.86), we obtain ˆj(ξ, s) = e−ξs P1 (s)
hR
+P1 (s)e−ξs
∞ 0 {π(α)F2 (· · ·
Rs 0
j0 (t)eξt P1 (t) dt
i , α) − FI 1 }ˆj(ξ, α)dα + FI 1 n ˆ (ξ) +
− P1 (s)D′ (N∞ )ˆ n(ξ)
Rs 0
(s−t) ˆ 10 . e−ξt JP∞1 (s−t) dt + R (2.89)
26
C.O.A. Sowunmi
We substitute for n ˆ from equation (2.88) in equation (2.89) to obtain an equation whose linear part can be regarded as a Fredholm integral equation in ˆj(ξ, ·). For brevity, we introduce the following notation before writing out the integral equation A1 (α)
= π(α)F2 (· · · , α) − FI 1 ,
C(ξ, s) = P1 (s)
Rs 0
e−ξt j0 (s−t) P1 (s−t) dt,
G(ξ, s) = D′ (N∞ )P1 (s) Thus, ∞
Rs 0
(2.90)
(s−t) e−ξt JP∞1 (s−t) dt.
q F I q 1 ˆ 11 . ˆj(ξ, s) = − G(ξ, s) ˆj(ξ, α)dα+C(ξ, s)+R e P1 (s)A1 (α) − e P1 (s) ξ−ρ ξ−ρ 0 (2.91) We further introduce the following notations: Z
−ξs
−ξs
A2 (α) = 1; B1 (s) = e−ξs P1 (s)., B2 (s) = −
{e−ξs P1 (s)I F1 q + qG(ξ, s)} . ξ−ρ
Hence, equation (2.91) can be written as Z ∞ ˆ 11 . ˆj(ξ, s) = {B1 (s)A1 (α) + B2 (s)A2 (α)}ˆj(ξ, α)dα + C(ξ, s) + R 0
Clearly, the linear part of the equation has a degenerate kernel, Z ∞ and applying a {B1 (s)A1 (s) + now familiar technique yields the following characteristic equation: 0
B2 (s)A2 (s)}ds = 1, i.e.,
L1 (ξ) − where L1 (ξ) =
Z
∞
L2 (ξ) = 1, ξ−ρ
(2.92)
e−ξs P1 (s)A1 (s)ds,
0
and L2 (ξ) = qI F1
Z
∞
e−ξs P1 (s)ds + qD′ (N∞ )
0
Z
∞
P1 (s) 0
Z
s 0
e−ξα J∞ (s − α) dα. P1 (s − α)
Equation (2.92) is of the same form as equation (2.64), and in order to apply the same method to it, we rewrite this equation in the form: Z
0
∞
−ξs
e
Z ′ P1 (s)A1 (s) − q FI 1 P1 (s) + D (N∞ )
We can now state the following Theorem.
s
ρτ
e dτ 0
Z
∞ 0
P1 (s + α − τ ) J∞ (α)dα ds. P1 (α) (2.93)
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 27 Theorem 3 Given the model whose governing equations are (2.36), the second equations of (2.38) and (2.41), and system (2.39) together with the conditions on the functions P , B and D, there always exists a disease-free steady state. An endemic steady state exists if Z ∞ π(α)F1 (K, 0, α)P (α)e−α(q+D(K)) dα > 1, 0
and there is X2 , 0 < X2 , such that ΠK2
Z
∞
P (α)e−αλ∗ dα < 1.
0
Otherwise, there may be no endemic steady state. The disease-free steady state is stable if Z ∞ π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα < 1, 0
and unstable if Z
∞
π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) da > 1.
0
If Z def W(s) = P1 (s)A1 (s)−q F1 P1 (s) + D′ (N∞ )
s 0
s ≥ 0, then, an endemic state if it exists, is stable if Z
eρ(s−τ ) dτ Z
Z
∞ 0
P1 (α + τ ) J∞ (α)dα , P1 (α)
∞
0
|W(s)|ds < 1, and unstable if
∞
W(s)ds > 1.
0
2.7.
Discussion
The idea of extending the framework of saturable interactions for the modelling of 2-sex population dynamics - to that of disease transmission, has worked for two SIS cases. It needs extending further to the cases with more than just the susceptible and infectives. The characteristic equation for the steady states becomes quite complicated owing to one single factor or parameter in particular, that is the fatality of infection. However, in the disease-free steady state, the characteristic equation (2.93) is Z ∞ e−ξs P1 (s)A1 (s)ds = 1, 0
where P1 (s) = e−(q+D(K))s P (s), is the only function of q. Let us write the equation as Z ∞ e−us cos vsP1 (s)A1 (s)ds = 1, X(u, v, q) = 0
28
C.O.A. Sowunmi
and Y (u, v, q) =
Z
∞
e−us sin vsP1 (s)A(s)ds = 0.
0
This pair defines u and v as implicit functions of q, provided the conditions of the Implicit Function Theorem are satisfied. In that case, if u = U (q) and v = V (q), it is easily shown dU dV that = −1, while = 0. Thus, near a disease-free steady state, increasing fatality dq dq tends to stabilize the steady state. If fatality is interpreted as removal of infectives, it is easy to see why this should be so.
2.8.
Summary of Results
2.8.1.
Age-independent Infectivity
The disease-free steady state exists always. It is stable if F2 (K, 0)Pˆ (q + D(K)) < 1, and unstable if F2 (K, 0)Pˆ (q + D(K)) > 1. The endemic steady state exists if F2 (K, 0)Pˆ (q+D(K)) > 1 and k2 Pˆ (q+Dog(qX1 )) < 1 for some X1 > 0. If F2 (K, 0)Pˆ (q + D(K)) < 1 and k2 Pˆ (q + Dog(qX2 )) > 1 for some X2Z > 0, an endemic steady state mayZnot exist. An existing endemic steady state is stable ∞
∞
if
0
|W (s)|ds < 1, and unstable if
W (s)ds > 1, where
0
W (s) = a11 P1 (s) − qa12
Rs 0
P3 (τ )eρ(s−τ ) dτ, s ≥ 0,
a11
= F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ ),
a12
= D′ (N∞ )F (N∞ − I∞ , I∞ ),
b1 (t)
=
b2 (t)
= n0 eρt + R5 ,
R∞ 0
P (s+t) −λs j0 (s)ds P (s) e
+ R3 ,
P1 (s) = e−λs P (s), P2 (s) =
R∞ 0
e−λσ P (s + σ)dσ,
P3 (s) = e−λs P2 (s). 2.8.2.
Age-dependent Infectivity
The disease-free steady state always exists. It is stable if Z ∞ π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα < 1 0
and unstable if
Z
∞ 0
π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα > 1.
Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 29 The endemic steady state exists if Z ∞ π(α)F2 (K, 0 < α)P (α)e−α(q+D(K)) dα > 1, 0
and ∃ X2 > 0 ∋ πK2
Z
∞
P (α)e−αλ∗ dα < 1 where K2 = sup k2 (α), π = sup π(α) and α
0
α
λ∗ = q + Dog∗ (X2 ). g∗ is defined as the extension of f∗−1 by continuity to [0, I00 ]. The endemic steady state may not otherwise exist. If Z ∞ Z s P1 (α + τ ) ρ(s−τ ) ′ e dτ W(s) = P1 (s)A1 (s) + q FI 1 P1 (s) + D (N∞ ) J∞ (α)dα , P1 (α) 0 0 Z ∞ |W(s)|ds < 1, and unstable s ≥ 0, then, an existing endemic steady state is stable if 0 Z ∞ W(s)ds > 1. if 0
P1 (s)
= e−λs P (s), where λ = q + D(K),
A1 (s)
= π(α)F2 (· · · , α) − FI 1 ,
FI 1
=
R∞ 0
π(α)F1 (N∞ − I∞ , J∞ (α), α)dα,
F2 (· · · , α) = F2 (N∞ − I∞ , J∞ (α), α).
Acknowledgments I am grateful to Prof. A.U. Afuwape of the Department of Mathematics, Obafemi Awolowo University, Ile-Ife, for a loan of the English translation of Lyapunov’s book.
References [1] C.O.A. Sowunmi, Saturation Processes, Math. Comput. Modelling 11 (1988), 250252. [2] A.M. Lyapunov, The General Problem of the Stability of Motion (Translated and edited by A.T. Fuller) Taylor and Francis (1992). [3] L. La Salle and S. Lefschetz, Stability by Liapunov’s Direct Method (with applications) Academic Press, N.Y., London (1961). [4] C.O.A. Sowunmi, A model of heterosexual population dynamics with age structure and gestation period. J. Math. Anal. Appl. 172(2), (1993), 390-411. [5] C.O.A. Sowunmi, An age-structured model of polygamy with density dependent birth and death moduli. J. Nig. Math. Soc. 11(3), (1992), 123-138.
30
C.O.A. Sowunmi
[6] C.O.A. Sowunmi, Stability of steady state and boundedness of a 2-sex population model. Nonlinear Analysis 39(2000), 693-709. Erratum 51, (2002), 903-920. [7] C.O.A. Sowunmi, Time-discrete 2-sex population model with gestation period. Mathematical modelling of population dynamics. Banach Center Publications vol. 63 (2004), 259-266. [8] F. Brauer, Models for the spread of universally fatal diseases. J. Math. Biol. 28 (1990), 451-462.
In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 31-84
Chapter 2
A M ATHEMATICAL A NALYSIS OF I NFLUENZA WITH T REATMENT AND VACCINATION 1
H. Rwezaura1,∗, E. Mtisi2 and J.M. Tchuenche1,† Department of Mathematics, University of Dar es Salaam, Box 35062, Dar es Salaam, Tanzania. 2 Dar es Salaam Institute of Technology, Tanzania.
Abstract The recent outbreaks of highly pathogenic avian influenza and associated human infections, arising primarily from direct contact with poultry in several regions of the world have highlighted the urgent need to prepare for the next influenza pandemic. Although measures such as closing schools, using face-masks, and keeping infected persons away from those susceptible (known as social distancing) may slow the effects of pandemic influenza, only vaccines and antiviral drugs are clearly efficacious in preventing infection or treating illness. Faced with the H5N1 pandemic threat and due to the lack of facilities for quarantine and isolation in some resource-poor countries (even though the availability of antiviral and the affordability of vaccine is also a challenging task), we construct a deterministic mathematical model with vaccination and treatment only in order to analyze their joint effect in curtailing an influenza epidemic. The results are interpreted in terms of the vaccination, treatment and vaccination and treatment-induced reproduction numbers, RV , RT , and RV T , respectively. We observe that vaccinating and treating individuals concurrently is more effective in slowing down the epidemic than concentrating on a cohort vaccination campaign or treatment campaign only. Population-level perversity cannot occur if the fitness ratio 0 < Hj < 1. Also, positivity and boundedness of solutions as well as persistence of the model are analyzed. We investigate the local and global stability of the steady states and observe that the treatment only sub-model likewise the vaccination-only sub-model undergoes the phenomenon of backward bifurcation, consequently, the full model also exhibits this phenomenon. Also, when RV T < 1, the model with inflow of infectives has a tri-stable equilibria where the disease-free equilibrium coexists with two stable endemic equilibria. Sensitivity analysis on the key parameters that drive the disease dynamics is performed in order to determine their relative importance to disease transmission. Numerical simulations are carried out to validate the model. ∗ †
E-mail address:
[email protected] E-mail address: jmt
[email protected] 32
H. Rwezaura, E. Mtisi and J.M. Tchuenche
MSC: 92B05, 92D30, 92C60, 93D05, 93D20. Keywords: Influenza, treatment, vaccination, basic reproduction number, stability.
1.
Introduction
Influenza, commonly known as flu, is an infectious disease of birds and mammals caused by Ribonucleic acid (RNA) virus of the family of Orthomyxoviridae. In humans, common symptoms of influenza infection are fever, sore throat, muscle pains, severe headache, coughing, weakness and fatigue. In more serious cases, influenza causes pneumonia, which can be fatal, particularly in young children and the elderly. It is transmitted from infected mammals through the air by coughs or sneezes, and from infected birds through their droppings (Wikipedia, 2007). The flu viruses are designated as types A, B, and C. Only influenza A and B cause major outbreaks and severe disease; influenza C is associated with common cold-like illness, principally in children. Influenza B virus usually causes a minor illness, but it does have the potential to cause more severe disease in older persons. Influenza A virus, however, causes pandemic (Parker et al., 2001). Historically, the number of deaths during a pandemic has varied greatly. Death rates are largely determined by four factors: the number of people who become infected, the virulence of the virus, the underlying characteristics and vulnerability of affected populations, and the effectiveness of preventive measures. Accurate predictions of mortality cannot be made before the pandemic virus emerges and begins to spread (URT, 2006). An influenza pandemic is a global epidemic caused by an especially virulent virus, newly infectious for humans, and for which there is no preexisting immunity (SPI, 2007). A pandemic occurs when a new influenza virus emerges and starts spreading as easily as normal influenza by coughing and sneezing. New influenza subtypes emerge as a result of a process called antigenic shift, which causes a sudden and major change in influenza A viruses. Because the virus is new, the human immune system will have no pre-existing immunity. This makes it likely that people who contract pandemic influenza will experience more serious disease than that caused by normal influenza and this is why pandemic strains have such potential to cause severe morbidity and mortality (SPI, 2007; WHOFS, 2007). Every year, influenza infects up to 20% of the US population, hospitalizes 200,000 people and kills 36,000 (CDC, 2007). Influenza pandemic is a rare but recurrent event. Three pandemic occurred in the previous century as shown in Table 1.1 (WHOFS, 2007). Table 1. Details of the previous three pandemics Name of Pandemic Spanish influenza Asian influenza Hong Kong influenza
Date 1918 1957 1968
Estimated deaths 40-50 million 2 million 1 million
Subtype involved H1N1 H2N2 H3N2
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The reason for the recurrent outbreaks is that the virus undergoes periodic antigenic shifts in its two outer membrane glycoproteins-hemagglutinin (H) and neuraminidase (N) for example, from H1N1 to H2N2 in 1957 and from H2N2 to H3N2 in 1968, thus, introducing a new virus into a population that has no protective serum antibody. No different subtypes of H and N have been identified for influenza B and C (Parker et al., 2001). Nowadays, the H5N1 subtype of influenza virus, also known as avian flu, has the potential to develop into a pandemic strain (ACP, 2006). The World Health Organization (WHO) has reported human cases of avian influenza A (H5N1), arising primarily from direct contact with sick or dead poultry or wild birds, or visiting a live poultry market, in Asia, Africa, the Pacific, Europe and the Near East. Infection in humans has occurred in three distinct waves of activity, since late December 2003. Overall, mortality in reported H5N1 cases is approximately 60% (CDCa, 2007). While the H5N1 virus does not yet infect people easily, infection in humans is very serious when it occurs, and so far, more than half of the people reported infected have died. Rare cases of limited human-to-human spread of H5N1 virus may have occurred, but there is no evidence of sustained human-to-human transmission. Nonetheless, because all influenza viruses have the ability to change, scientists are concerned that H5N1 viruses one day could be able to infect humans more easily and consequently spread from person-to-person. Since H5N1 viruses have not infected many humans worldwide, there is little or no immune protection against them in the human population and an influenza pandemic could begin if sustained H5N1 virus transmission occurs (CDCa, 2007). Prevention of influenza, particularly when it becomes pandemic, may be attempted by many measures, such as closing schools, using face masks, and keeping infected persons away from those susceptible. However, none of these measures are of clear value in preventing infection, even if they could be accomplished, but only vaccines and antiviral drugs are clearly efficacious in preventing infection or treating illness (Monto, 2006). They are the two most important medical interventions for reducing illness and deaths during a pandemic. However, until there is influenza pandemic, there is no evidence that vaccines or antiviral used in the treatment or prevention of such an outbreak would decrease morbidity and mortality. Since a pandemic vaccine is unlikely to be available during the first 4 to 6 months of the pandemic, appropriate use of antiviral may play an important role to limit mortality and morbidity, minimize social disruption, and reduce economic impact (HHS, 2007). The efficacy and effectiveness of influenza vaccines depend primarily on the age and immunocompetence of the vaccine recipient, the degree of similarity between the viruses in the vaccine and those in circulation, and the outcome being measured (CDCb, 2007). The impact of any vaccination strategy and antiviral drug use depends on how soon the pandemic starts. If it starts when there is no vaccine available and only limited supplies of antiviral drugs, it is more likely that targeted strategies for vaccine and antiviral drug use will be the only potential options. Flu viruses come in different strains that constantly mutate (making influenza quite unpredictable), until one that few people have immunity against emerges and is able to spread widely. Therefore, the flu vaccine must be reformulated every year to keep up with the fastevolving influenza virus. There is a potential danger, especially for developing countries where adequate health facilities are not generally available. Due to the lack of facilities in a country such as Tanzania, and the fact that most individuals live on less than a dollar a
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
day, affordability of vaccine in case of an epidemic is a challenging task. Consequently, the epidemic might spread and the only combating measure is the use of antiviral drugs. Therefore, we consider both vaccination and treatment in order to analyze their joint effect in curtailing an epidemic. This study is worth in its own right and we do hope the outcome will be useful to public health decision makers in the event of any epidemic/pandemic in the developing world.
1.1.
Motivation and Objectives
The motivation for writing this chapter comes from declared facts by the WHO that the world is as close as ever to the next pandemic (Democratis et al., 2006). Faced with the H5N1 pandemic threat, strategies designed to contain an emerging pandemic should be considered a public health priority. A number of mathematical models, using stochastic as well as deterministic formulations have been carried out to quantify the burden of a potential flu pandemic and assess various interventions (Alexander et al., 2007; Longini et al. 2004; Nu˜ no et al., 2006; Longini et al., 2005; Chowell et al., 2005 and Arino et al., 2008, to name but a few). Although findings in these studies seem reassuring, they assume the availability of an unlimited supply of antiviral drugs, vaccine, adequate health facilities for isolation and quarantine. These assumptions may of course not be realistic, especially in most resourcepoor countries in Sub-Saharan Africa. Given these limitations, whether a strategic use of vaccination and treatment can control the spread of influenza within a certain population is of great public health interest (Alexander et al., 2004). Since the failure of current influenza vaccines to protect all vaccine recipients warrants the determination of conditions necessary for a substantial reduction, approaching eradication of influenza infection in a population, this chapter therefore aims at exploring via mathematical modeling, the combined effects of both immunization (with a partially effective vaccine) and treatment on the transmission dynamics of influenza infection. Alexander et al., (2004) addressed the question of whether such a vaccine could ever completely stop the spread of infection and determines the minimal vaccine efficacy and vaccination rate required to control or eradicate infection in a population. Thus, the vaccination-only sub-model will not be analyzed herein. Understanding and analyzing the spread and control of an influenza epidemic in order to assess the ability to control the epidemic will be helpful in guiding intervention and policy decision. In addition to providing conceptual results such as the basic reproduction numbers, graphical representations of the model are illustrated.
1.2.
Methodology
We construct a deterministic mathematical model using a system of ordinary differential equations where the total population, denoted by N (t), is divided into a number of mutually exclusive sub-populations according to their epidemiological (or disease) status: susceptible (S), Infected and infectious (I), vaccinated (V ), Treated (T ), fully protected via vaccination (C) and recovered (R). We analyze the model qualitatively to determine the criteria for
A Mathematical Analysis of Influenza with Treatment and Vaccination
35
containing an epidemic/pandemic influenza in the presence of treatment and vaccination and use it to compute epidemic threshold numbers necessary for community-wide control of the disease. A sensitivity analysis on the key parameters that drive the disease dynamics is also performed. We evaluate the possibility that the disease may take-off in the absence of antiviral and vaccine via the classical basic reproduction number denoted by R0 . With this threshold, the qualitative mathematical properties of the model are studied, and the epidemiological consequences are discussed. If the parameter R0 is less than unity (R0 < 1), then the disease cannot spread in the population but, if R0 > 1, then, the spread of the disease in the population is always possible (Hethcote, 2000). R0 is calculated using the next generation operator method (Diekmann et al., 1990; van den Driessche and Watmough, 2002). This epidemiological quantity measures the average number of new cases generated by an infectious individual, for the duration of his/her infectiousness in a completely susceptible population. When vaccination and treatment strategies are applied, the aim is to reduce the threshold parameter to below unity in that strategy in order to prevent further spread of the disease. Alexander et al., (2004) analyzed a vaccination-only model with mass action incidence. We believe their results will hold in the standard incidence model considered herein, and for this reason, the vaccination-only sub-model will not be analyzed. The associated reproduction number for the vaccination-only sub-model denoted by RV will be computed for the purpose of analyzing the effects of the combination of vaccination and treatment on therapeutic measures taken one at a time. Similarly, for the antiviral-only scenario, the corresponding antiviral reproduction number is denoted by RT . For the case where vaccination and treatment are administered concurrently, the corresponding combined reproduction number is denoted by RV T . The epidemiological significance of the combined reproduction number, which represents the average number of new cases generated by a primary infectious individual in a population where the combined interventions are implemented, is that the pandemic may effectively be controlled (owing to the phenomenon of backward bifurcation) if the combined interventions can bring this threshold quantity to a value less than unity (the pandemic would persist otherwise). We use Descartes’ Rule of signs to analyze the local stability of the model and the the Korobeinikov-Maini’s (2004) type Lyapunov function for the persistence of the mass action incidence model. We investigate the possibility of periodic solutions by applying the classical Bendixon-Dulac criterion. A brief survey on previous works provides the context of the present study.
1.3.
Brief Review of Previous Studies
Mathematical models have long been recognized as useful tools in exploring complicated relationships underlying infectious disease transmission processes. Usually, the structure of the model is based on a set of causal hypotheses that describe current understanding of how different processes are interrelated (Spear, 2002). Unlike statistical models, their parameters generally have physical or biological meaning that allows their values to be estimated from literature as well as from field and experimental data. Like statistical mod-
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
els, mathematical models can be used to test competing hypotheses underlying research questions. Longini et al., (2004) used stochastic epidemic simulations to investigate the effectiveness of targeted antiviral prophylaxis to contain influenza and compare it with that of vaccination, if vaccine were available. They showed that although vaccination would be the best means for controlling influenza, 80% targeted antiviral prophylaxis for a possible 6 to 8 weeks is almost as effective as vaccinating 80% of the entire population, and thus targeted antiviral prophylaxis has potential as an effective measure for containing influenza until adequate quantities of vaccine are available. On the other hand, Lipsitch et al., (2007) designed and analyzed a deterministic compartmental model of the transmission of oseltamivir-sensitive and -resistant influenza infections during a pandemic and found that in the event of a future influenza pandemic for which antiviral drugs are used, there is a risk of resistance emerging and resistant strains causing illness in a substantial number of people. This would counteract the benefits of antiviral drugs but not eliminate those benefits entirely. Their model relies on realistic assumptions and it is hard to know how closely the model will mimic a real-life situation until the properties of an actual pandemic strain are known. They suggested that even if the benefits of antiviral drug use to control an influenza pandemic may be reduced, although not completely offset by drug resistance, the risk of resistance should be considered in pandemic planning and monitored closely during a pandemic. The use of vaccines to prevent human diseases is one of the major successes of modern medicine (Moghadas, 2004a), but it is well-known that, although vaccines can reduce or eliminate the incidence of infection, not all vaccines are 100% effective (McLean and Blower, 1993; Gandon et al., 2001). Some recent clinical studies have focused on the effect of partially effective (or imperfect) vaccines (waning and/or incomplete immunity) in controlling the transmission of infectious diseases. Alexander et al., (2004) evaluated the impact of a partially effective preventive vaccine on the control of influenza infection, using a new deterministic mathematical model. Their results showed that the disease can be controlled if infected individuals are not continuously introduced into the population and the eradication of the disease may not be feasible if infected individuals are continuously recruited. Therefore, increasing the level of vaccination will always reduce the level of epidemicity of the disease and vaccination can still be used to prevent a severe epidemic. However, since influenza can also be introduced into the population by recruitment of infected individuals this poses a challenge to the developing countries as they might have no access to vaccines throughout the duration of a pandemic. The use of partially effective preventive vaccine to prevent influenza infections in case of a pandemic will still be a challenging task for many developing countries like Tanzania. Longini et al., (2005) assessed the combined role of targeted prophylaxis, quarantine and pre-vaccination in containing an emerging influenza strain at the source and found that combinations of targeted antiviral prophylaxis, pre-vaccination, and quarantine could contain the strains. Nu˜ no et al., (2006) analyzed a more complex compartmental model to study the role of hospital and community control measures, antiviral and vaccination in combating a potential flu pandemic in a population of high and low-risk individuals. Their results suggested that countries with limited antiviral stockpiles should emphasize their use therapeutically (rather than prophylactically).However, countries with large antiviral stockpiles can achieve
A Mathematical Analysis of Influenza with Treatment and Vaccination
37
greater reductions in disease burden by implementing them both prophylactically and therapeutically. Their study showed that the prospect of combating the next flu pandemic is promising, provided a number of control measures (especially the use of a combined intervention strategies) are put in place in an efficient manner. Likewise Chowell et al., (2005) used a compartmental epidemic model to describe the transmission dynamics of pandemic influenza. Their results indicated that containment of the next influenza pandemic could require the simultaneous implementation of multiple component interventions that include effective isolation of hospitalized cases and reductions in the susceptibility of the general population through, increasing hygiene, using protective devices (e.g., face masks), prophylactic antiviral use and vaccination. The impact of targeted influenza vaccination and antiviral prophylaxis/treatment (oseltamivir) in high risk groups (elderly, chronic diseases), priority (essential professionals), and total populations was compared by Doyle et al., (2006) who suggested that if available initially, vaccination of the total population is preferred but if not, for priority populations, seasonal prophylaxis seems the best strategy. For high risk groups, antiviral treatment, although less effective, seems more feasible and cost effective than prophylaxis and should be chosen, especially if the availability of drugs is limited. Their results suggested a strong role for antiviral in an influenza pandemic. Alexander et al., (2007) developed a mathematical model to evaluate the effect of delay (which is not considered herein) in initiating a course of antiviral treatment on containing a pandemic. Their model assumed the availability of an unlimited supply of antiviral drugs for the entire course of a pandemic. Their findings showed that, due to uncertainty in the attack rate of a pandemic strain, initiating therapy immediately upon diagnosis can significantly increase the likelihood of disease control and substantially reduce the required community-level of treatment. This suggests that reliable diagnostic methods for influenza cases should be rapidly implemented within an antiviral treatment strategy. Recently, Arino et al., (2008) analyzed a similar model to ours. This study was almost completed before the authors learned about their work. Even though the titles have the same keywords, there are significant differences in the approach and content. Nevertheless, the common results agree on the point. They considered treated -susceptible, -latent infective and -asymptomatic classes, respectively, derived analytic expression for the final size of the epidemic and compared their numerical computations with those of recent stochastic simulation influenza models, which have great potential for predictions of outcomes and design of control strategies. However, some of the stochastic model parameters have considerable uncertainty and are not very amenable to sensitivity analysis. Nevertheless, compartmental models effectively allow a complete analysis of parameter space. For more details, see Arino et al., (2008) and the references therein. While we investigate local and global stability of steady states and observe that the treatment only sub-model exhibits the phenomenon of backward bifurcation which has important public health implications, they do not. An economic analysis of influenza vaccination and anti-retroviral treatment for healthy working adults has been carried out by Lee et al., (2002), and it was found that vaccination in a variety of settings is cost beneficial in most influenza seasons. The model is robust enough to provide great insights in curtailing an outbreak by analysing the impact of both vaccination and treatment in combating the spread of the disease. The rest of this chapter is organised as follows: In the next Section, we formulate the
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
model and carry out its complete analysis. Section 3 presents the sensitivity analysis and numerical simulations, while in Section 4, we briefly discuss and conclude the chapter.
2. 2.1.
Model Framework and Analysis Model Framework
The model consists of six ordinary differential equations which specify the rate of change of six categories of individuals in the population. The total population, denoted by N (t) consisting of a class of susceptible individuals (S), a class of vaccinated individuals (V ), a class of infected and infectious individuals (I), a class of treated individual (T ), a class of individuals who recover with temporary immunity (R) and a class of fully protected individuals via vaccination (C). The susceptible population is increased by recruitment of individuals (either by birth or immigration), and by loss of immunity. In this class, individuals can incur the disease but are not yet infected. Individuals in this class are reduced through vaccination, infection and by natural death. The population of vaccinated is increased by vaccination of susceptible. Vaccinated individuals may become fully protected or may become infected at lower rate than unvaccinated (those in class S) because the vaccine does not confer immunity to some vaccine recipients. The vaccinated class is thus diminished by this infection (moving to class I), by moving to a class of fully protected individuals (C) and further decreased by waning of vaccine-based immunity (moving to class S), and by natural death. G E U/
1 U /
S P
K
Z
1 H E
V
P
N
I P
J
T
WS
P
OD
C P
Figure 1. Model flowchart.
R P
1 O D
A Mathematical Analysis of Influenza with Treatment and Vaccination
2.2.
39
Descriptions of Variables and Parameters
The population of infected individuals is increased by recruitment of infected individuals from outside the population, as well as by infection of susceptible individuals including those who remain susceptible despite being vaccinated. It is diminished by natural death, by disease-induced death and by treatment (moving to class T ). The population of treated individuals is increased by treating the infected individuals and is decreased by individuals recovering from their infection (moving to class R), by disease-induced death and by natural death. Since the immunity acquired by infection wanes with time, the recovered individuals become susceptible to the disease again. Thus, recovered class is increased by individuals recovering from their infection through treatment and is decreased as the natural immunity wanes (moving back to class S) and by natural death. The transfer diagram for these processes is shown in Figure 1. The notations used in the flow diagram are described below. S(t) V(t) I(t) T(t)
: : : :
C(t) R(t) η ω γ κ τ ǫ π λ
: : : : : : : : : :
β Λ
: :
δ ρ µ
: : :
α
:
Susceptible individuals at time t Vaccinated group at time t Infected/infectious class at time t Individuals receiving treatment at time t (may also represent the hospitalized class) Protected class at time t Recovered class at time t Rate at which susceptible individuals are vaccinated Rate at which the vaccine-based immunity wanes Rate of acquiring protective antibodies Treatment rate Recovery rate Vaccine efficacy (ǫ ∈ [0, 1]) Measures the drug efficacy in increasing the recovery rate (π ≥ 1) Effectiveness of the drug as a reduction factor in disease-induced death (0 < λ ≤ 1) Contact rate and βI N the force of infection Recruitment rate of individuals into the population, a fraction ρ of which are infective Rate at which the immunity from previous infections wanes Fraction of recruited individuals who are already infected Natural death (or emigration) rate which is assumed to be the same for all sub-populations Disease induced death rate.
Note that if ǫ = 0, then, the vaccine is useless and if ǫ = 1, the vaccine is 100% effective, while if 0 < ǫ < 1, the vaccine is leaky. Also, in the I-class, λ(≥ 1) may
40
H. Rwezaura, E. Mtisi and J.M. Tchuenche
represent compliance (those who refuse to seek medical support).
2.3. The Model βSI dS = (1 − ρ)Λ − + δR − (µ + η)S + ωV, dt N dV βV I = ηS − (1 − ǫ) − (γ + ω + µ)V, dt N dI βSI βV I = ρΛ + + (1 − ǫ) − (κ + µ + λα)I, dt N N
(2.1)
dT = κI − (µ + τ π)T − (1 − λ)αT, dt dR = τ πT − (µ + δ)R, dt dC = γV − µC, dt with initial conditions S(0) = S0 , V (0) = V0 , I(0) = I0 , T (0) = T0 , R(0) = R0 , C(0) = C0 and N (t) = S(t)+V (t)+I(t)+T (t)+R(t)+C(t), where N (t) is the total population at time t.
2.4.
Model Analysis
Lemma 1 The feasible set of system (2.1) is given by 6 : S+V +I +T +R+C ≤ Ω = {(S, V, I, T, R, C) ∈ R I+
Λ µ }.
Proof. Adding the differential equations in the model system (2.1) gives dN = Λ − µN − λαI − (1 − λ)αT ≤ Λ − µN. dt
(2.2)
The feasible region for (2.1) is 6 Ω = {(S, V, I, T, R, C) ∈ R I+ : S+V +I +T +R+C ≤
Λ }. µ
(2.3)
Since from (2.2) lim sup N (t) ≤ t→∞
Λ , µ
(2.4)
then, the global attractor of (2.1) is contained in Ω. Thus, the dynamics of the model will be considered in Ω. Lemma 2 The set Ω is positively invariant.
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Proof. On the side of Ω, we have S=0⇒
dS = (1 − ρ)Λ + δR + ωV, dt
V =0⇒
dV = ηS, dt
I=0⇒
dI = ρΛ, dt
T =0⇒
dT = κI, dt
R=0⇒
dR = τ πT, dt
C=0⇒
dC = γV. dt
(2.5)
Since all the parameters are positive and any vector field on ∂Ω, the boundary of Ω, is tangent or has inward direction, the region Ω of biological interest is positively invariant, 0
and any solution of system (2.1) with initial point on ∂Ω enters Ω, the interior of Ω and remains there. Lemma 3 Existence: A solution of model system (2.1) is feasible in Ω. Proof. Since system (2.1) is dissipative, that is, all feasible solutions are uniformly bounded 6 , then, any solution with initial value in Ω attains its maximum in a proper subset Ω ⊂ R I+ and remain in there. Hence, Ω is compact and positively invariant.
2.5.
Positivity of Solutions
Model system (2.1) describes the dynamics of a human population, therefore, it is important to prove that the susceptible, vaccinated, infected (infectious), treated and recovered individuals are non-negative for all time. In other words, we want to prove that all solutions of the system with positive initial data will remain positive for all t > 0. Theorem 4 Positivity: Let the initial data be S(0) > 0, V (0) ≥ 0, I(0) ≥ 0, T (0) ≥ 0, R(0), C(0) ≥ 0 ∈ Ω. Then, solutions of S(t), V (t), I(t), T (t), R(t), C(t) of system (2.1) are positive for all t > 0. dR Proof. For t ∈ [0, T¯] , = τ πT − (µ + δ)R ≥ −(µ + δ)R, and by freshman integration, dt we obtain R(t) ≥ R(0)e−(µ+δ)t ≥ 0, µ + δ < +∞. (2.6) A similar argument on the remaining variables yields: T (t) ≥ T (0)e−[µ+τ π+(1−λ)α]t ≥ 0, −(µ+γ+ω)t−(1−ǫ)β
V (t) ≥ V (0)e
C(t) ≥ C(0)e−µt ≥ 0
I(s) 0 N (s) ds
Rt
I(t) ≥ I(0)e−(µ+κ+λα)t ≥ 0, ≥ 0,
−(µ+η)t−β
S(t) ≥ S(0)e
I(ζ) 0 N (ζ) dζ
Rt
≥ 0,
.
(2.7) This shows that the solution of (2.1) is such that min{S(t), V (t), I(t), T (t), R(t), C(t)} ≥ 0 in its interval of existence Ω.
42
H. Rwezaura, E. Mtisi and J.M. Tchuenche 2000 1800 1600
Susceptible
1400 1200 1000 800 600 400 200 0
0
1000
2000
3000 4000 Treated
5000
6000
7000
Figure 2. Evolution of S and T classes towards equilibrium. Lemma 5 The ω-limit set of any orbit of the system (2.1) with initial point in Ω is a rest point. Proof. Since the vector field related to (2.1) is inward on ∂Ω and Ω is compact, then, the 0
omega-limit set of each orbit with initial point in Ω is a non-empty subset of Ω. By the Poincar´ e-Bendixon Theorem (Perko, 2000), this set must be an equilibrium point. Lemma 5 is graphically illustrated in Figure 2. For simplicity, we show the phase portrait of S and T only, which converges to a positive equilibrium.
2.6.
The Model in the Absence of Inflow of Infectives (ρ = 0)
At an equilibrium point, the RHS of system (2.1) equals zero and substituting ρ = 0, we have βS ∗ I ∗ + δR∗ − (µ + η)S ∗ + ωV ∗ = 0, Λ− N∗ ηS ∗ − (1 − ǫ)
βV ∗ I ∗ − (γ + ω + µ)V ∗ = 0, N∗
βS ∗ I ∗ βV ∗ I ∗ + (1 − ǫ) − (κ + µ + λα)I ∗ = 0, N∗ N∗ κI ∗ − (µ + τ π)T ∗ − (1 − λ)αT ∗ = 0, τ πT ∗ − (µ + δ)R∗ = 0, γV ∗ − µC ∗ = 0.
(2.8)
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Thus, R∗ =
τ πT ∗ , µ+δ
C∗ =
T∗ =
κI ∗ , µ + τ π + (1 − λ)α
V∗ =
ηN ∗ S ∗ , (1 − ǫ)βI ∗ + (µ + ω + γ)N ∗
γV ∗ µ (2.9)
S∗ =
[Λ + δR∗ + ωV ∗ ]N ∗ βI ∗ + N ∗ (µ + η)
2.7. Non-existence of the Trivial Equilibrium Communicable diseases may be introduces into a community by the arrival of new infective individuals from outside the population. For example, travelers may return home from a foreign trip with an infection acquired abroad (Brauer and van den Driessche, 2001). For as long as the recruitment term Λ is not zero, the population will not be extinct. This implies that there is no trivial equilibrium point, i.e., (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ) 6= (0, 0, 0, 0, 0, 0).
2.8.
Disease-Free Equilibrium (E0 )
At the disease-free equilibrium, we have I0 = 0 = T0 = R0 , therefore, Λη Ληγ Λ(µ + ω + γ) , , 0, 0, 0, ). (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω µ(µ + γ)(µ + η) + µ2 ω (2.10) Remark: If β = 0 so that the only infectives are those who have entered the population from outside, this reduces system to a linear non-homogeneous system for which every ρΛ solution approaches a certain equilibrium E e , with I e = . We would expect a κ + µ + λα steady state with β > 0 to satisfy I ∗ ≥ I e . E0 = (
2.9.
Computation of the Reproduction Numbers R0 , RV , RT and RV T
Applying van den Driessche and Watmough technique (2002), the treatment and vaccination-induced reproduction number RV T is given by: RV T =
βµ[µ + γ + ω + (1 − ǫ)η] . [(µ + γ)(µ + η) + µω](µ + κ + λα)
(2.11)
The threshold quantity RV T can be interpreted as the number of infected people generated by one infected individual introduced into the population in the presence of vaccination and treatment. In the absence of treatment κ = 0, we have the vaccination-induced reproduction number given by: βµ[µ + γ + ω + (1 − ǫ)η] . (2.12) RV = [(µ + γ)(µ + η) + µω](µ + λα) In absence of vaccination η = 0 = γ = ω = ǫ, we have the treatment-induced reproduction number given by β . (2.13) RT = µ + κ + λα
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
In the absence of any intervention strategy (i.e., without vaccination and treatment) η = 0 = γ = ω = ǫ = κ, we have the basic reproduction number given by β . µ + λα
R0 =
(2.14)
We note that in the I-class, λα ≃ α, that is λ = 1 in this case, since those who are administered drugs are in the T -class.
2.10.
Local Stability of the Disease-Free Equilibrium E0
One of the most important concerns in the analysis of epidemiological models is the determination of the asymptotic behaviour of their solutions which is usually based on the stability of the associated equilibria (Moghadas, 2004a). These models typically consist of a disease-free equilibrium and at least one endemic equilibrium. The local stability of the disease-free equilibrium is determined based on a threshold parameter RV T , known as the vaccination and treatment-induced reproductive number. The Jacobian matrix at E0 is given by
JE0
=
−K11 ω −βS0 0 δ η −K22 −(1 − ǫ)βV0 0 0 0 0 K33 0 0 0 0 κ −K44 0 0 0 0 τπ −K55
(2.15)
where
βS0 + (1 − ǫ)βV0 − (µ + κ + λα), N0 K44 = (µ + τ π) + (1 − λ)α, K55 = (µ + δ), and Λ(µ + ω + γ) Λη S0 = , V0 = . (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω K11 = (µ + η), K22 = (µ + ω + γ), K33 =
The eigenvalues of JE0 are ς1 = −(µ + δ), ς2 =
βS0 +(1−ǫ)βV0 N0
− (µ + κ + λα),
ς3 = −(µ + τ π + (1 − λ)α), ς4 = − 12 {2µ + η + γ + ω + ς5 = − 21 {2µ + η + γ + ω −
p (η + ω)2 + γ(γ + 2ω − 2η)}, p (η + ω)2 + γ(γ + 2ω − 2η)}.
(2.16)
A Mathematical Analysis of Influenza with Treatment and Vaccination
45
The following eigenvalues ς1 , ς3 , ς4 , ς5 are negative. Thus, the local stability of E0 depends on the sign of ς2 . That is, ς2 =
βS0 + (1 − ǫ)βV0 − (µ + κ + λα), N0
= −(µ + κ + λα) + = (µ + κ + λα)[
βµ[µ + γ + ω + (1 − ǫ)η] [(µ + γ)(µ + η) + µω]
(2.17)
βµ[µ + γ + ω + (1 − ǫ)η] − 1] [(µ + γ)(µ + η) + µω](µ + κ + λα)
= (µ + κ + λα)(RV T − 1). Thus we have established the following result: Lemma 6 The DFE of the model system 2.1 (when ρ = 0) is locally asymptotically stable if RV T < 1 and unstable if RV T > 1. Proof. An equilibrium is asymptotically stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part. Since the eigenvalues of JE0 are negative except ς2 , it follows that the DFE is locally asymptotically stable if ς2 < 0 and unstable if ς2 > 0. Note that ς2 < 0 if and only if RV T < 1 (cf. 2.17). Since under conditions (a)(i) and (c)(i) of Lemma 10 the model system 2.1 has a unique endemic equilibrium, therefore, eliminating any possibility of backward bifurcation, we can show that under those assumptions, the DFE of system 2.1 will be GAS when RV T < 1.
2.11.
Global Stability of the Disease-Free Equilibrium E0
Lemma 7 The DFE of the model system 2.1 (when ρ = 0) is globally asymptotically stable (GAS) whenever RV T < 1. Proof. The proof is based on using a comparison Theorem (cf. Lakshmikantham et al., 1989; p.31). Note first of all that we can write the equations of (I, T, R) for the system 2.1 as follows; dI dt I I dT = ϕ(t) T ≤ ϕ(0) T , (2.18) dt R R dR dt
46
H. Rwezaura, E. Mtisi and J.M. Tchuenche
where βS (1 − ǫ)βV + − (µ + κ + λα) 0 0 N ϕ(t) = N . κ −(µ + τ π + (1 − λ)α) 0 0 τπ −(µ + δ) (2.19) and ϕ(0) is the value of ϕ(t) evaluated for the variables S(t), V (t), and N (t) when t = 0. The eigenvalues of ϕ(0) are
ς1 = −(µ + δ), ς2 =
βS0 +(1−ǫ)βV0 N0
− (µ + κ + λα) = (µ + κ + λα)(RV T − 1),
ς3 = −(µ + τ π + (1 − λ)α), Using the fact that the eigenvalues of the matrix ϕ(0) all have negative real parts if ς2 < 0 i.e.,if RV T < 1, it follows that the differential inequality system (2.18) is stable whenever RV T < 1. Consequently, (I(t); T (t); R(t)) → (0; 0; 0) as t → ∞. Thus, by a comparison Theorem (cf. Lakshmikantham et al., 1989; p.31), (I(t); T (t); R(t)) → (0; 0; 0) as t → ∞. Substituting I = 0 = T = R in the first two equation of (2.1) gives S(t) → S ∗ , V (t) → V ∗ and C(t) → C ∗ as t → ∞. Thus, (S(t), V (t), I(t), T (t), R(t), C(t)) → (S ∗ , V ∗ , 0, 0, 0, C ∗ ) as t → ∞ for RV T < 1 and hence, E0 is GAS if RV T < 1 (for ρ = 0).
2.12.
Effects of Public Health Measures (Treatment and Vaccination)
In order to study the effects of public health measures in slowing down the spread of the influenza epidemic in a community, we investigate the role of the basic reproductive number (which is a measure of the power of a disease to invade a population under conditions that facilitates maximal growth) on influenza eradication. We rewrite the treatment-induced reproductive number, the vaccination-induced reproductive number and the vaccination and treatment-induced reproductive number as follows: The quantity RT can be rewritten as 1
βR0 , β + κR0
(2.20)
µ[µ + γ + ω + (1 − ǫ)η] , [(µ + γ)(µ + η) + µω]
(2.21)
RT = R0 [
1+
κ µ+λα
]=
while RV can be rewritten as RV = R0
A Mathematical Analysis of Influenza with Treatment and Vaccination
47
and finally, RV T can be rewritten as RV T = R0
µ[µ + γ + ω + (1 − ǫ)η] = HV T R0 , κ [(µ + γ)(µ + η) + µω][1 + µ+λα ]
(2.22)
β µ + λα RV T µ[µ + γ + ω + (1 − ǫ)η] . := = κ ] R0 [(µ + γ)(µ + η) + µω][1 + µ+λα
where R0 = and HV T
Since µ[µ + γ + ω + (1 − ǫ)η] < [(µ + γ)(µ + η) + µω][1 + HV T =
κ ] , then, µ + λα
µ[µ + γ + ω + (1 − ǫ)η] < 1. κ ] [(µ + γ)(µ + η) + µω][1 + µ+λα
(2.23)
HV T is the factor by which public health measures (treatment and vaccination) reduce the number of secondary infections. It is also referred to as fitness ratio (Smith? and Blower, 2004). Since 0 < HV T < 1, then, population level perversity is not possible and influenza vaccine combined with treatment will always have a beneficial impact. 1 We also note that in the presence of treatment only, HT = < 1, HV = κ 1 + µ+λα 1 − ǫηµ M γη < 1, where M = µ(µ + γ + ω + η). Therefore, vaccination and treatment 1+ M taken separately are always beneficial, but with minimal impact compared to the combined strategy as shown below (cf. equations 2.24 and 2.25). If R0 < 1, it is possible that influenza will not spread into an epidemic (and therapeutic measures may not be necessary) and for R0 > 1, we now determine the necessary condition for slowing down the spread of influenza. Following Hsu Schmitz (2000), we have ∆V T := R0 − RV T = R0 [1 − HV T ] for which ∆V T > 0 is expected to slow down the spread of the influenza epidemic in a community. This condition is satisfied for all 0 < η, ǫ, κ < 1. We note that under this condition, the factor H multiplying R0 is less than unity (HV T < 1), indicating that public health measures have the capability of reducing the number of secondary infections. Now, we determine the threshold level of vaccination and treatment coverage that guarantee disease eradication. It is well-known that, although vaccines can reduce or eliminate the incidence of infection, not all vaccines are 100% effective. Also, treatment efficacy is of paramount importance in curtailing an epidemics. This section addresses the following questions: What proportion of susceptible people must be immunized and/or treated in order to prevent an endemic spread of influenza? By addressing this question, one would expect to explore the impact of three major parameters associated with a vaccination program on disease transmission (Moghadas, 2004a): - vaccination coverage level (η); - efficacy of
48
H. Rwezaura, E. Mtisi and J.M. Tchuenche
vaccine (ǫ); - and treatment rate (κ). Differentiating RV T partially with respect to ǫ, η and κ, we obtain, ∂RV T µηR0 =− ∂ǫ [(µ + γ)(µ + η) + µω][1 +
κ µ+λα ]
< 0,
µR0 (µ + γ + ω)(ǫµ + γ) ∂RV T =− < 0, κ ∂η [(µ + γ)(µ + η) + µω]2 [1 + µ+λα ]
(2.24)
βµ[(µ + γ + ω) + (1 − ǫ)η] ∂RV T =− < 0. ∂κ [(µ + γ)(µ + η) + µω](µ + κ + λα)2 From (2.24), the necessary conditions ∆V T > 0,
∂RV T ∂RV T ∂RV T ηc , κ > κc , η > η¯c , κ > κ ¯ c and ǫ > ǫ¯c . In conclusion, if the vaccine efficacy is low, influenza may not be controlled using vaccination alone because the corresponding value of ηc required is large (Alexander et al., 2004) and perhaps not feasible in rural and poor communities settings.
A Mathematical Analysis of Influenza with Treatment and Vaccination
49
Now, comparing RV T and RV , RV T and RT , we have RV T = R0
µ[µ + γ + ω + (1 − ǫ)η] βRV , = κ [(µ + γ)(µ + η) + µω][1 + µ+λα ] β + κR0
(2.27)
which implies that RV T < RV . Also RV T = R0
since 0
RV T (1, 1), the (1 + µ + λα)[(µ + γ)(µ + 1) + µω] threshold for disease eradication is given by RV T (1, 1) < RV T < RV T (0, 0) = R0 .
2.14.
(2.29)
Endemic Equilibrium and Its Stability
The model system (2.1) has an endemic equilibrium point (EE), where the infected components I is non-zero, given by Ee = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ), with R∗ =
τ πκI ∗ , (µ + δ)[µ + τ π + (1 − λ)α]
T∗ =
κI ∗ , µ + τ π + (1 − λ)α
V∗ =
ηN ∗ S ∗ , (1 − ǫ)βI ∗ + (µ + ω + γ)N ∗
S∗ =
[(1 − ρ)Λ + δR∗ + ωV ∗ ]N ∗ , βI ∗ + N ∗ (µ + η)
C∗ =
γV ∗ µ
I∗ = −
ρΛN ∗ βS ∗ + (1 − ǫ)βV ∗ − (µ + κ + λα)N ∗
(2.30)
50
2.15.
H. Rwezaura, E. Mtisi and J.M. Tchuenche
Stability Analysis when RV T > 1
Theorem 8 The model system (2.1) has no periodic orbits and its unique EE is GAS when RV T > 1.
Proof. The proof is based on reducing the model (2.1) into a 2-dimensional one by consecutively eliminating C, T, R and S respectively, to obtain: dV Λ βµV I = η( − V − I) − (1 − ǫ) − (γ + ω + µ)V, dt µ Λ (2.31) βµI Λ βµV I dI = ρΛ + ( − V − I) + (1 − ǫ) − (κ + µ + λα)I. dt Λ µ Λ Let X and Y denote the right hand sides of the first and second equations of (2.31), respec1 tively. Consider the Dulac function D = for V > 0, I > 0. Then, VI ∂(DX) ∂(DY ) η Λ 1 Λρ µβ + = −{ 2 ( − 1) + ( 2 + )} < 0, ∂V ∂I V µI V I Λ
(2.32)
Λ > 1. Thus, by Dulac’s criterion, there are no periodic orbits in Ω|Ω∗ where µI Ω∗ = {(S, V, I, T, R, C) ∈ Ω : I = 0 = T = R}. Since Ω is positively invariant, and the endemic equilibrium exists whenever RV T > 1, then it follows from the Poincar´ e-Bendixon Theorem (Perko, 2000), that all solutions of the limiting system originating in Ω remain in Ω for all t. Further, the absence of periodic orbits in Ω implies that the unique endemic equilibrium of the special case of the model system (2.1) is GAS whenever RV T > 1. since
2.16.
Endemic Equilibria when ρ > 0
Conditions under which none or multiple endemic equilibria exists are explored below (see alternative proof in Appendix D(2) when it is assumed that the C-class plays no further part in the disease transmission or when γ = 0). In order to find these conditions, we use the first equation of (2.31) to express the variables V in terms of the variable I when I 6= 0. This gives (at an arbitrary equilibrium)
V∗ =
Λ η(Λ − µI) · . µ Λ(µ + γ + ω + η) + (1 − ǫ)µβI
(2.33)
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51
Substitution of equation (2.33) into the second equation of (3.47) gives the following polynomial P (I, ρ) = −
(1 − ǫ)β 2 µ2 3 I − µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)]I 2 Λ
−Λ[(µ + γ + ω + η)(µ + κ + λα − β) + β(ǫη − (1 − ǫ)µρ)]I +Λ2 ρ(µ + γ + ω + η) = 0, =
(1 − ǫ)β 2 µ2 3 I + µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)]I 2 Λ +Λ[(µ + γ + ω + η)(µ + κ + λα − β) + β(ǫη − (1 − ǫ)µρ)]I −Λ2 ρ(µ + γ + ω + η) = 0,
= AI 3 + BI 2 + [C0 − (1 − ǫ)Λµρ]I + Λ2 ρ(µ + γ + ω + η) = 0, = IQ(I) − ρ{(1 − ǫ)ΛµI − Λ2 (µ + γ + ω + η)} = 0, = AI 3 + BI 2 + CI + D, (2.34) where
Q(I) = AI 2 + BI + C0 , A=
(1 − ǫ)β 2 µ2 , Λ
B = µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)],
(2.35)
C0 = Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη], C = C0 − Λβµρ(1 − ǫ), D = −Λ2 ρ(µ + γ + ω + η). Since all the model parameters are nonnegative, it follows from (2.34) that A ≥ 0 and D ≤ 0. Furthermore, if RV T < 1 or RT < 1, then C0 > 0. That is, C0 = Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη], = Λ[(µ + γ + ω + η)(µ + κ + λα) − β(µ + γ + ω + (1 − ǫ)η)], γη Λ [µ(µ + γ + ω + η + )(µ + κ + λα) − µβ(µ + γ + ω + (1 − ǫ)η)] µ µ Λγη (µ + κ + λα), − µ =
52
H. Rwezaura, E. Mtisi and J.M. Tchuenche = Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)[1 − − Λγη µ (µ + κ + λα), = Λ(µ + γ + ω + η(1 +
βµ[µ+γ+ω+(1−ǫ)η] [(µ+γ)(µ+η)+µω](µ+κ+λα) ]
Λγη γ ))(µ + κ + λα)(1 − RV T ) − (µ + κ + λα), µ µ
Λγη γ (µ + κ + λα) + Λ(µ + γ + ω + η(1 + ))(µ + κ + λα)(RV T − 1)]. µ µ (2.36) Alternatively, C0 can be written in terms of RT as follows = −[
C0 = Λ(µ + κ + λα)[(µ + γ + ω + η) − RT (µ + γ + ω + (1 − ǫ)η)], = Λ(µ + κ + λα)(µ + γ + ω + η)[1 − RT (1 −
ǫη (µ+γ+ω+η) )]
(2.37)
= Λ(µ + κ + λα)(µ + γ + ω + η)(1 − ϑRT ). where ϑ = 1 −
ǫη < 1. We now consider the following three case: (µ + γ + ω + η)
Case 1: C0 < 0 (RV T > 1 or RT > 1), this is equivalent to η < ηc < η¯c , κ < κc < κ ¯ c . In this case, by Descartes Rule of Signs, IQ(I) = P (I, 0) = 0 has at most one positive root regardless of the sign of B, and one zero root. In this case, Q(I) has a unique positive root, denoted by I ∗ . Since P (I, ρ) is a decreasing function of ρ for positive I, it follows that P (I ∗ , ρ) < 0 for ρ > 0. Furthermore, P (I, ρ) → ∞ as I → ∞. Thus, P (I, ρ) has a unique positive root for all ρ ≥ 0, and this unique positive root must be at I > I ∗ . That is, when the model has a unique endemic equilibrium with ρ = 0, recruitment of infected individuals introduces no new equilibria but serves to shift the existing (unique) equilibrium to a higher disease state. Case 2: C0 > 0 (RV T < 1 or RT < 1) this is equivalent to η > ηc > η¯c , κ > κc > κ ¯c. If B < 0 and B 2 − 4AC0 > 0, then, IQ(I) = P (I, 0) = 0 has two positive and one zero roots. Since A ≥ 0, C > 0 and D < 0, then, by Descartes Rule of Signs, P (I, ρ) = 0 has at most three positive roots (when the aforementioned threshold is less than unity). If B > 0 then, by Descartes Rule of Signs, Q(I) = 0 has no positive roots and P (I, ρ) = 0 has at most one positive root. Case 3: C0 = 0 (RV T = 1), or RT = 1 provided ǫ = 0. This is equivalent to κ = κc , and Q(I) reduces to (AI + B)I = 0. In this case, the model has a unique endemic equilibrium if B < 0 and no endemic equilibrium if B > 0 orA = 0. Since A ≥ 0, C < 0 and D < 0, then, by Descartes Rule of Signs, P (I, ρ) = 0 has at most one positive root regardless of the sign of B. We note here that Case 2 confirms that the model exhibits backward bifurcation (for which its existence is shown in Section 2.18). The results of this section are summarized in the following Theorem. Theorem 9 Suppose ρ > 0 in (2.1).
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53
(a) If C0 ≤ 0, then (i) the model has a unique endemic equilibrium if η ≤ ηc ≤ η¯c , κ ≤ κc ≤ κ ¯c. (b) If C0 > 0, then (i) the model has unique endemic equilibria if B > 0, (ii) the model has three equilibria if B < 0 (where the disease-free equilibrium coexists with two stable endemic equilibria).
2.17.
Equilibria when ρ = 0
Notice that when ρ = 0, P (I, ρ) = 0 is reduced to IQ(I) = 0 for which I ∗ = 0 is a solution corresponding to the the DFE of (2.1), and P (I, ρ) < IQ(I) for ρ > 0 and I > 0. From a similar argument as above, the following result which we state without proof can be established (see alternative proof in Appendix D(1) with same assumption as in D(2) above). Lemma 10 Suppose ρ = 0 in (2.1). (a) If C0 < 0, then (i) the model has a unique endemic equilibrium if η < ηc < η¯c , κ < κc < κ ¯c. (b) If C0 > 0, then (i) the model has no endemic equilibria if B > 0, (ii) the model has two endemic equilibria if B < 0 and B 2 − 4AC0 > 0. (c) If C0 = 0, then (i) the model has a unique endemic equilibrium if B < 0 (ii) the model has no endemic equilibrium if B > 0 orA = 0.
2.18.
Existence of Backward Bifurcation
It has been shown (Alexander et al., 2004) that the vaccination only sub-model undergoes the phenomenon of backward bifurcation (BB), that is, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for the disease control and eradication. We note that I ∗ in system (2.30) exists provided RV T < 1, consequently, the full model will also exhibit this phenomenon (see Figure 3 for the co-existence of a DFE and an EE when RV T < 1). Hence, the following result: Lemma 11 The DFE of the model system (2.1) co-exists with a stable EE when RV T < 1.
54
H. Rwezaura, E. Mtisi and J.M. Tchuenche 1
0.8
Stable EE I
0.6
0.4
Unstable EE
0.2
Stable DFE 0 0.8
0.85
0.9
0.95
1
Reproduction number
Figure 3. Backward bifurcation diagram for the full model.
Proof. It is sufficient to show that I ∗ above is epidemiologically relevant if RV T < 1 as shown below.
βS ∗ + (1 − ǫ)βV ∗ − (µ + κ + λα)N ∗ < βS0 + (1 − ǫ)βV0 − (µ + κ + λα)N0 = =
βΛ(µ + ω + γ) (1 − ǫ)βΛη Λ + − (µ + κ + λα), (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω µ
=
βµ[µ + γ + ω + (1 − ǫ)η] Λ (µ + κ + λα)[ − 1], µ [(µ + γ)(µ + η) + µω](µ + κ + λα)
=
Λ (µ + κ + λα)[RV T − 1], µ (2.38)
The last expression is positive provided RV T < 1. Hence the proof.
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2.19.
55
Local Stability of the Endemic Equilibrium E1
Consider the following reduced model dS βSI = (1 − ρ)Λ − + δ(N − S − V − I − T ) − (µ + η)S + ωV, dt N dV βV I = ηS − (1 − ǫ) − (γ + ω + µ)V, dt N
(2.39)
dI βSI βV I = ρΛ + + (1 − ǫ) − (κ + µ + λα)I, dt N N dT = κI − (µ + τ π)T − (1 − λ)αT, dt defined in 4 Φ = {(S, V, I, T ) ∈ R I+ } ⊂ Ω.
For existence and uniqueness of endemic equilibrium E1 = (S ∗ , V ∗ , I ∗ , T ∗ ), its coordinates should satisfy the condition S ∗ > 0, V ∗ > 0, I ∗ > 0, T ∗ > 0. From (2.39) at equilibrium, we have (1 − ρ)Λ −
βS ∗ I ∗ + δ(N ∗ − S ∗ − V ∗ − I ∗ − T ∗ ) − (µ + η)S ∗ + ωV ∗ = 0, N∗
ηS ∗ − (1 − ǫ) ρΛ +
βS ∗ I ∗ N∗
βV ∗ I ∗ − (γ + ω + µ)V ∗ = 0, N∗
+ (1 − ǫ)
∗I ∗
βV N∗
(2.40)
− (κ + µ + λα)I ∗ = 0,
κI ∗ − (µ + τ π)T ∗ − (1 − λ)αT ∗ = 0 At the steady states of the reduced model (2.39), the Jacobian matrix J(E1 ) is given by
J(E1 )
βI ∗ ω−δ − N ∗ − X11 (1−ǫ)βI ∗ η − N ∗ − X22 = (1−ǫ)βI ∗ βI ∗ N∗ N∗ 0 0
∗ − βS N∗ − δ ∗ − (1−ǫ)βV N ∗∗ ∗ βS +(1−ǫ)βV − N∗
κ
−δ X33
0 0 −X44
X11 = (µ + δ + η),
X22 = (µ + γ + ω),
X33 = (µ + κ + λα),
X44 = (µ + τ π + (1 − λ)α).
, (2.41) (2.42)
We use the additive compound matrices (or geometric) approach of Li and Moldowney (1996) to analyze the stability of the endemic equilibrium. From the Jacobian matrix J(E1 ) ,
56
H. Rwezaura, E. Mtisi and J.M. Tchuenche
the second additive compound matrix is given by
[2]
J(E1 )
=
K11
(1−ǫ)βI ∗ N∗
0
∗
− βI N∗ 0 0
− (1−ǫ)βV N∗ K22 κ η 0 0
∗
0 0 K33 0 η
βS ∗ N∗
βI ∗ N∗
+δ ω−δ 0 K44 κ 0
δ 0 ω−δ 0 K55
(1−ǫ)βI ∗ N∗
0 δ
βS ∗ − N∗ − δ , 0 (1−ǫ)βV ∗ − N∗ K66
(2.43)
where
∗
− (2µ + δ + η + γ + ω), K11 = − (2−ǫ)βI N∗ K22 =
βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗
− (2µ + δ + η + κ + λα),
∗
K33 = − βI N ∗ − (2µ + δ + η + τ π + (1 − λ)α), K44 =
βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗
(2.44)
− (2µ + γ + ω + κ + λα),
∗
K55 = − (1−ǫ)βI − (2µ + γ + ω + τ π + (1 − λ)α), N∗ K66 =
βS ∗ +(1−ǫ)βV ∗ N∗
− (2µ + κ + α + τ π).
The Local stability of the endemic equilibrium point E1 is demonstrated by the following Theorem of Li and Wang (1998). Theorem 12 An n × n real matrix A is stable if and only if A[2] is stable and (−1)n det(A) > 0. Proof. In order to determine det(J(E1 ) ), the following simplified equations of system (2.40) are used. (1 − ρ)Λ + δ(N ∗ − V ∗ − I ∗ − T ∗ ) + ωV ∗ βI ∗ = + µ + δ + η, S∗ N∗ ηS ∗ βI ∗ = (1 − ǫ) + (γ + ω + µ), V∗ N∗ ρΛ βS ∗ βV ∗ − ∗ = ∗ + (1 − ǫ) ∗ − (κ + µ + λα), I N N κI ∗ = µ + τ π + (1 − λ)α. T∗
(2.45)
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57
Then, from the Jacobian J(E1 ) and the simplified expressions (2.45), we have βI ∗ − ω−δ N ∗ − X11 ∗ − X22 η − (1−ǫ)βI ∗ det(J(E1 ) ) = N (1−ǫ)βI ∗ βI ∗ N∗ N∗ 0 0 (1−ρ)Λ+δ(N ∗ −V ∗ −I ∗ −T ∗ )+ωV ∗ − S∗ η = βI ∗ N∗ 0
=
κ
S ∗ V ∗ T ∗ (N ∗ )2
∗
− βS N∗ − δ
− (1−ǫ)βV N∗
βS ∗ +(1−ǫ)βV ∗ N∗
κ
ω−δ ∗ − ηS V∗
∗
− X33 ∗
(1−ǫ)βI ∗ N∗
0
−δ
− βS N∗ − δ (1−ǫ)βV ∗ − N∗ − ρΛ I∗ κ
0 0 −X44 −δ 0 0 ∗ − κI T∗
{[βI ∗ V ∗ (1 − ǫ)]2 [(1 − ρ)Λ + δ(N ∗ − V ∗ − I ∗ − T ∗ ) + ωV ∗ ]
+βηδI ∗ S ∗ N ∗ (I ∗ + T ∗ )[S ∗ + (1 − ǫ)V ∗ ] + (βI ∗ )2 S ∗ V ∗ (1 − ǫ)[(ω − δ)V ∗ + ηS ∗ ] +β 2 ηS ∗ 3 I ∗ 2 + ηρΛS ∗ N ∗ 2 [(1 − ρ)Λ + δ(N ∗ − I ∗ − T ∗ )]} > 0, = (−1)4 det(J(E1 ) ) > 0. (2.46) is
[2] J(E1 )
[2] J(E1 )
From the second additive compound matrix in (2.43), the stability of demonstrated as follows: For the endemic equilibrium point E1 = (S ∗ , V ∗ , I ∗ , T ∗ ), let P = [2] diag(1, −1, 1, −1, 1, −1) be the diagonal matrix. Then, the matrix J(E1 ) is similar to the matrix given by
[2]
P J(E1 ) P −1
=
K11
(1−ǫ)βI ∗ N∗
0 ∗ − βI N∗ 0 0
− (1−ǫ)βV N∗ K22 κ η 0 0
∗
0 0 K33 0 η βI ∗ N∗
βS ∗ N∗
+δ ω−δ 0 K44 κ 0
δ 0 ω−δ 0 K55
(1−ǫ)βI ∗ N∗
0 δ
βS ∗ − N∗ − δ , 0 ∗ − (1−ǫ)βV N∗ K66 (2.47)
where K11 , K22 , K33 , K44 , K55 , K66 are as defined above. [2] Since similarity preserves the eigenvalues, then matrix J(E1 ) is stable if and only if the [2]
matrix P J(E1 ) P −1 is stable. Therefore, we apply the following Theorem due to McKenzie (1960) which establishes that if A has a negative dominant diagonal, then, A is stable. Theorem 13 (Sufficiency) If an n × n matrix A is dominant diagonally and the diagonal is composed of negative elements (aii < 0 ∀ i = 1, .., n), then, the real parts of all its eigenvalues are negative, i.e., A is stable.
58
H. Rwezaura, E. Mtisi and J.M. Tchuenche [2]
This can be done by examining if the matrix P J(E1 ) P −1 is diagonally dominant in rows, since its diagonal elements are negative ∗
h1 = − (2−ǫ)βI − (2µ + δ + η + γ + ω) + N∗
(1−ǫ)βV ∗ N∗
+0−
βS ∗ N∗
− δ + δ + 0,
= − Nβ∗ [S ∗ + (2 − ǫ)I ∗ − (1 − ǫ)V ∗ ] − (2µ + δ + η + γ + ω) < 0.
Using have,
βS ∗ +(1−ǫ)βV ∗ N∗
∗
βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗
h2 = − (1−ǫ)βI + N∗ ∗
= − 2(1−ǫ)βI + (κ + µ + λα) − N∗ ∗
+ = −[ 2(1−ǫ)βI N∗ h3 = −κ −
βI ∗ N∗
ρΛ I∗
= (κ + µ + λα) −
ρΛ I∗
from the third equation of (2.45) we
− (2µ + δ + η + κ + λα) + 0 + ω − δ + 0 + δ,
ρΛ I∗
− (2µ + δ + η + κ + λα) + ω,
+ µ + δ + η − ω] < 0.
− (2µ + δ + η + τ π + (1 − λ)α) + ω − δ +
βS ∗ N∗
+ δ,
= −[ Nβ∗ (I ∗ − S ∗ ) + 2µ + δ + η + κ + τ π + (1 − λ)α − ω] < 0.
Using have,
βS ∗ +(1−ǫ)βV ∗ N∗
h4 = =
βI ∗ N∗ ǫβI ∗ N∗
βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗
+ η + (κ + µ + λα) −
= −[ ρΛ I∗ + µ + γ + ω − For
ρΛ I∗
+µ+γ+ω >
h5 = η − κ − =
ǫβI ∗ N∗
ǫβI ∗ N∗
(2.49)
from the third equation of (2.45) we
− (2µ + γ + ω + κ + λα),
ρΛ I∗
− (2µ + γ + ω + κ + λα),
(2.50)
− η] < 0.
+ η,
(1−ǫ)βI ∗ N∗
∗ −V ∗ ) −[ (1−ǫ)β(I N∗
Finally, using
ρΛ I∗
= (κ + µ + λα) − +η+
(2.48)
βS ∗ +(1−ǫ)βV ∗ N∗
− (2µ + γ + ω + τ π + (1 − λ)α) +
(1−ǫ)βV ∗ , N∗
(2.51)
+ 2µ + γ + ω + κ + τ π + (1 − λ)α − η] < 0.
= (κ + µ + λα) −
ρΛ I∗
from the third equation of
A Mathematical Analysis of Influenza with Treatment and Vaccination
59
(2.45) we have, ∗
h6 = − βI N∗ −
(1−ǫ)βI ∗ N∗
+
βS ∗ +(1−ǫ)βV ∗ N∗
∗
= − (2−ǫ)βI + (κ + µ + λα) − N∗ ∗
= −[ (2−ǫ)βI + N∗
ρΛ I∗
ρΛ I∗
− (2µ + κ + α + τ π), (2.52)
− (2µ + κ + α + τ π),
+ µ + τ π + (1 − λ)α] < 0.
We have all values h1 , h2 , h3 , h4 , h5 , h6 < 0, therefore, all the diagonals are negative. Thus, from Theorems 12 and 13, the system has a local stability at the endemic equilibrium point.
2.20.
Global Stability of the EE E1 when RV T > 1
The global stability of endemic equilibrium point E1 is demonstrated by the following Lemma of Li and Muldowney (1993). ∂g Lemma 14 Let Ω ⊆ R I n be a simply connected region and ( ∂u (u(t, x0 )))[2] is the second additive compound matrix of the Jacobian matrix of vector function g(u) at u(t, x0 ). Assume that the family of linear systems
z(t) ˙ =
[2] ∂g z(t), x ∈ Ω, (u(t, x0 )) ∂u
(2.53)
is equi-uniformly asymptotically stable. Then, (i) Ω contains no simple closed invariant curves including periodic orbits, homoclinic, heteroclinic cycles, (ii) each semi-orbit in Ω converges to a single equilibrium. In particular, if Ω is positively invariant and contains a unique equilibrium x ¯, then x ¯ is globally asymptotically stable in Ω. Proof. The proof is based on the approach of Yan and Li (2005). From the second additive [2] compound matrix J(E1 ) (2.43) above which we reproduce here for convenience
[2]
J(E1 )
=
K11 (1−ǫ)βI N
0 − βI N 0 0
− (1−ǫ)βV N K22 κ η 0 0
0 0 K33 0 η βI N
βS N
+δ ω−δ 0 K44 κ 0
δ 0 ω−δ 0 K55 (1−ǫ)βI N
0 δ
βS −N −δ , 0 − (1−ǫ)βV N K66
(2.54)
60
H. Rwezaura, E. Mtisi and J.M. Tchuenche
where − (2µ + δ + η + γ + ω), K11 = − (2−ǫ)βI N K22 =
βS+(1−ǫ)β(V −I) N
− (2µ + δ + η + κ + λα),
K33 = − βI N − (2µ + δ + η + τ π + (1 − λ)α), K44 =
βS+(1−ǫ)β(V −I) N
(2.55)
− (2µ + γ + ω + κ + λα),
− (2µ + γ + ω + τ π + (1 − λ)α), K55 = − (1−ǫ)βI N K66 =
βS+(1−ǫ)βV N
− (2µ + κ + α + τ π),
we have the linear system with respect to the solutions (S(t), V (t), I(t), T (t)) written as ˙ 1 (t) = K11 W1 (t) − (1 − ǫ)βV W2 (t) + ( βS + δ)W4 (t) + δW5 (t), W N N ˙ 2 (t) = (1 − ǫ)βI W1 (t) + K22 W2 (t) + (ω − δ)W4 (t) + δW6 (t), W N ˙ 3 (t) = κW2 (t) + K33 W3 (t) + (ω − δ)W5 (t) − ( βS + δ)W6 (t), W N
(2.56)
˙ 4 (t) = − βI W1 (t) + ηW2 (t) + K44 W4 (t), W N ˙ 5 (t) = ηW3 (t) + κW4 (t) + K55 W5 (t) − (1 − ǫ)βV W6 (t), W N ˙ 6 (t) = βI W3 (t) + (1 − ǫ)βI W5 (t) + K66 W6 (t). W N N Set V (W ) = max{Xl |Wl | : l = 1, 2, ..., 6}, where Xl > 0 (l = 1, 2, ..., 6) are constants. Then, direct calculation leads to the following inequalities: X1 (1 − ǫ)βV d+ X1 |W1 (t)| ≤ K11 X1 |W1 (t)| + | |X2 |W2 (t)| dt X2 N X1 βS 1 |( N + δ)|X4 |W4 (t)| + X +X X5 |δ|X5 |W5 (t)|, 4 d+ X2 (1 − ǫ)βI X2 |W2 (t)| ≤ K22 X2 |W2 (t)| + | |X1 |W1 (t)| dt X1 N X2 X2 + X4 |ω − δ|X4 |W4 (t)| + X6 |δ|X6 |W6 (t)|,
A Mathematical Analysis of Influenza with Treatment and Vaccination
d+ dt X3 |W3 (t)|
61
3 ≤ K33 X3 |W3 (t)| + X X2 |κ|X2 |W2 (t)| βS X3 3 + X5 |ω − δ|X5 |W5 (t)| + X X6 |( N + δ)|X6 |W6 (t)|,
X4 βI X4 d+ X4 |W4 (t)| ≤ K44 X4 |W4 (t)| + | |X1 |W1 (t| + |η|X2 |W2 (t)|, dt X1 N X2 (2.57)
d+ X5 X5 |W5 (t)| ≤ K55 X5 |W5 (t)| + |η|X3 |W3 (t)| dt X3 X5 X5 (1−ǫ)βV + X4 |κ|X4 |W4 (t)| + X6 | N |X6 |W6 (t)|, X6 βI X6 (1 − ǫ)βI d+ X6 |W6 (t)| ≤ K66 X6 |W6 (t)| + |X5 |W5 (t)|. | |X3 |W3 (t)| + | dt X3 N X5 N
where
d+ denotes the upper right-hand derivative. Consequently, we have dt d+ V (W (t)) ≤ φ(t)V (W (t)), dt
(2.58)
with φ(t) = max{K11 +
X1 (1−ǫ)βV X2 | N
K22 +
X2 (1−ǫ)βI | X1 | N
K33 +
X3 X2 |κ|
K44 +
X4 βI X1 | N |
K55 +
X5 X3 |η|
K66 +
X6 βI X3 | N |
+
X3 X5 |ω
+
+
+
X2 X4 |ω
X1 βS X4 |( N
− δ| +
− δ| +
+ δ)| +
X1 X5 |δ|,
X2 X6 |δ|,
X3 βS X6 |( N
+ δ)|, (2.59)
X4 X2 |η|,
X5 X4 |κ|
+
|+
+
X5 (1−ǫ)βV X6 | N
|,
X6 (1−ǫ)βI |}. X5 | N
For convenience, we further assume the following hypothesis: [2] Since the matrix J(E1 ) is diagonally dominant in rows and its diagonal elements K11 , K22 , K33 , K44 , K55 , K66 are negative, then, there exist constants Xl > 0 (l = 1, 2, ..., 6) such that sup{K11 +
X1 (1−ǫ)βV X2 | N
K22 +
X2 (1−ǫ)βI | X1 | N
K33 +
X3 X2 |κ|
+
+
X3 X5 |ω
|+
X1 βS X4 |( N
+ δ)| +
X2 X4 |ω
− δ| +
X2 X6 |δ|,
− δ| +
X3 βS X6 |( N
+ δ)|,
X1 X5 |δ|,
62
H. Rwezaura, E. Mtisi and J.M. Tchuenche K44 +
X4 βI X1 | N |
K55 +
X5 X3 |η|
K66 +
X6 βI X3 | N |
+
+
X4 X2 |η|,
X5 X4 |κ|
+
+
X5 (1−ǫ)βV X6 | N
X6 (1−ǫ)βI |} X5 | N
|,
(2.60)
< 0.
Therefore, by the boundedness of solution of (2.39), there exists ̺ > 0 such that φ(t) = −̺ < 0, and thus V (W (t)) ≤ V (W (s))e−̺(t−s) , t ≥ s ≥ 0. This shows that the second compound system (2.56) is equi-uniform asymptotically stable, and from Lemma 14, the system (2.39) has no non-constant periodic solution and the unique endemic equilibrium E1 is globally asymptotically stable in R I 4. For mathematical convenience, the analysis of the persistence of the model is carried out below using the mass-action incidence model. Mathematically speaking, if inflow of infected individuals is allowed, then, the epidemiological implication is that the DFE will not exist and eradication of the disease may not be feasible. In this case the public health objective is to minimize the level of epidemicity.
2.21.
The Model with Mass-Action Incidence dS = (1 − ρ)Λ − βSI + δR − (µ + η)S + ωV, dt dV = ηS − (1 − ǫ)βV I − (γ + ω + µ)V, dt dI = ρΛ + βSI + (1 − ǫ)βV I − (κ + µ + λα)I, dt dT = κI − (µ + τ π)T − (1 − λ)αT, dt
(2.61)
dR = τ πT − (µ + δ)R, dt dC = γV − µC, dt
2.22.
Persistence of Solutions of the Model with Mass-Action Incidence (ρ = 0)
Permanent co-existence of a dynamical system for which none of the variables is zero at steady states is biologically meaningful for survival of competing species, or the persistence of a disease, or existence of the endemic steady state, even though for disease dynamics, the disease-free equilibrium is the goal of any intervention strategy. The permanence of
A Mathematical Analysis of Influenza with Treatment and Vaccination
63
the disease destabilizes the disease-free equilibrium and since RV T > 1, the interior or endemic equilibrium will exist.
Lemma 15 The solution of model system (2.61) is uniformly persistent.
Proof. The proof is based on the approach by McCluskey (2006). Consider the following Lyapunov function (Korobeinikov and Maini, 2004) which is defined and continuous for all S, V, I, T, R, C ≥ 0, VL = c1 (S − S ∗ lnS) + c2(V − V ∗ lnV ) + c3 (I − I ∗ lnI) +
+c4 (T − T ∗ lnT ) + c5 (R − R∗ lnR) + c6 (C − C ∗ lnC),
where c1 , c2 , c3 , c4 , c5 , c6 are constants. Differentiating VL with respect to time t gives S∗ ˙ V∗ ˙ I∗ T∗ ˙ V˙ L = c1 (1 − )S + c2 (1 − )V + c3 (1 − )I˙ + c4 (1 − )T + S V I T ∗ ∗ C ˙ R ˙ )R + c6 (1 − )C, +c5 (1 − R C S∗ )[Λ − βSI + δR − (µ + η)S + ωV ] + = c1 (1 − S V∗ )[ηS − (1 − ǫ)βV I − (γ + ω + µ)V ] + +c2 (1 − V ∗ I +c3 (1 − )[βSI + (1 − ǫ)βV I I T∗ )[κI − (µ + τ π)T − (1 − λ)αT ] + −(κ + µ + λα)I] + c4 (1 − T R∗ C∗ +c5 (1 − )[τ πT − (µ + δ)R] + c6 (1 − )[γV − µC], R C S∗ )[βS ∗ I ∗ − δR∗ + (µ + η)S − ωV − βSI + δR] − = c1 (1 − S S∗ ) − [(µ + η)S + ωV ] + −c1 (1 − S ∗ V )[ηS − (1 − ǫ)βV I − (γ + ω + µ)V ] + +c2 (1 − V I∗ +c3 (1 − )[βSI + (1 − ǫ)βV I I T∗ )[κI − (µ + τ π)T − (1 − λ)αT ] + −(κ + µ + λα)I] + c4 (1 − T ∗ R C∗ +c5 (1 − )[τ πT − (µ + δ)R] + c6 (1 − )[γV − µC], R C
64
H. Rwezaura, E. Mtisi and J.M. Tchuenche
S∗ )[β(S ∗ I ∗ − SI) + δ(R − R∗ ) + S S∗ )[(µ + η)(S ∗ − S) + ω(V − V ∗ )] +c1 (1 − S VI I V∗ )[η(S − S ∗ ∗ ∗ ) + (µ + γ + ω)( ∗ − 1)V ] +c2 (1 − V V I I V I∗ ∗ V +c3 (1 − )[βI(S − S ∗ ) + (µ + κ + λα)( ∗ − 1)I] I V V T∗ I +c4 (1 − )[(µ + τ π + (1 − λ)α)(T ∗ ∗ − T )] T I C∗ V R∗ ∗ T )[(µ + δ)(R ∗ − R)] + c6 (1 − )[µ(C ∗ ∗ − C)]. +c5 (1 − R T C V Without loss of generality, let c1 = 1 = c2 = c3 = c3 = c4 = c5 = c6 , and = c1 (1 −
(
S V I T R C , , , , , ) = (x1 , x2 , x3 , x4 , x5 , x6 ). S ∗ V ∗ I ∗ T ∗ R∗ C ∗
Then, ∗ )2 V˙ L = −(µ + η)( (S−S ) + (1 − S
S∗ ∗ ∗ S )[β(S I
− SI)+
+δ(R − R∗ ) + ω(V − V ∗ )] +(1 −
V∗ V )[η(S
+(1 −
I∗ I )[βI(S
+(1 −
T∗ T )[(µ
+ τ π + (1 − λ)α)(T ∗ II∗ − T )]
+(1 −
R∗ R )[(µ
+ δ)(R∗ TT∗ − R)] + (1 −
= −(µ + η)( (S−S S
− S ∗ VV∗ II ∗ ) + (µ + γ + ω)( II∗ − 1)V ] − S ∗ VV∗ ) + (µ + κ + λα)( VV∗ − 1)I]
∗ )2
C∗ ∗ V C )[µ(C V ∗
(2.62)
− C)]
) + f (x1 , x2 , x3 , x4 , x5 , x6 ),
where f (x1 , . . . , x6 ) = (1 −
1 ∗ ∗ x1 )[βS I (1
+(1 −
− x1 x3 ) + δR∗ (x5 − 1) + ωV ∗ (x2 − 1)]
1 ∗ x2 )[ηS (x1
− x2 x3 ) + (µ + γ + ω)(x3 − 1)x2 ]
+(x3 − 1)[βS ∗ I ∗ (x1 − x2 ) + I ∗ (µ + κ + λα)(x2 − 1)] +(1 −
1 ∗ x4 )[T (µ
+ τ π + (1 − λ)α)(x3 − x4 )]
+(1 −
1 ∗ x5 )[R (µ
+ δ)(x4 − x5 )] + (1 −
1 ∗ x6 )[µC (x2
− x6 )]. (2.63)
Collecting the like terms in βS ∗ I ∗ we have; f (x1 , · · · , x6 ) = βS ∗ I ∗ [(1 −
1 − x1 + x2 (1 − x3 ) + x3 )] + F (x1 , · · · , x6 ), x1
(2.64)
A Mathematical Analysis of Influenza with Treatment and Vaccination
65
where F (x1 , x2 , x3 , x4 , x5 , x6 ) are the remaining terms. Without loss of reality and for convenience, we assume that F = 0, that is, x2 = 1 = x3 = x4 = x5 = x6 , and we have, (S − S ∗ )2 1 V˙ L = −(µ + η) + βS ∗ I ∗ (2 − − x1 ), S x1
(2.65)
1 − x1 ≤ 0 by the arithmetic mean-geometric mean inequality (McCluskey, x1 2006), with equality if and only if x1 = 1. Hence, V˙ L ≤ 0 with equality iff S = S ∗ and consequently VL is a Lyapunov function. Let P (t) := (S(t), V (t), I(t), T (t), R(t), C(t)), then, rewriting V˙ L as follows, we can determine the constant ξ such that where 2 −
lim inf P (t) > ξ.
(2.66)
t→∞
That is, ˙ ˙ ˙ V˙ L = c1 (S − S ∗ ) SS + c2 (V − V ∗ ) VV + c3 (I − I ∗ ) II ˙
˙
˙
R +c4 (T − T ∗ ) TT + c5 (R − R∗ ) R + c6 (C − C ∗ ) C C, R V Λ ∗ = c1 (S − S )[ S − βI + δ S − (µ + η) + ω S ]
+c2 (V − V ∗ )[η VS − (1 − ǫ)βI − (γ + ω + µ)]
(2.67)
+c3 (I − I ∗ )[βS + (1 − ǫ)βV − (κ + µ + λα)] +c4 (T − T ∗ )[κ TI − (µ + τ π) − (1 − λ)α] T +c5 (R − R∗ )[τ π R − (µ + δ)] + c6 (C − C ∗ )[γ VC − µ].
Since V˙ L ≤ 0, constants c1 , c2 , c3 , c4 , c5 , c6 can be found such that V˙ L ≤ −(µ + η)(S − S ∗ ) − (γ + ω + µ)(V − V ∗ ) − (κ + µ + λα)(I − I ∗ ) −[µ + τ π + (1 − λ)α](T − T ∗ ) − (µ + δ)(R − R∗ ) − µ(C − C ∗ )] ≤ −ξ[(S − S ∗ ) + (V − V ∗ ) + (I − I ∗ ) + · · · + (C − C ∗ )],
(2.68)
where ξ = min{µ, µ + η, µ + δ, γ + ω + µ, κ + µ + λα, µ + τ π + (1 − λ)α}, ∴ 0 < ξ ≤ lim inf P (t) ≤ lim sup P (t) = t→∞
t→∞
Λ . µ
(2.69) (2.70)
Since all the variables in P (t) are continuous and bounded with derivatives in L∞ , then, [0,T¯] applying a slightly modified version of Barbalat’s Lemma (1959) which is stated without proof in Appendix B on equation (2.68) above yields
66
H. Rwezaura, E. Mtisi and J.M. Tchuenche
(S − S ∗ ) + (V − V ∗ ) + (I − I ∗ ) + (T − T ∗ ) + (R − R∗ ) + (C − C ∗ ) → 0 as t → ∞, and the only largest invariant subset in the set Ω1 ⊂ Ω given by Ω1 = {(S, V, I, T, R, C) : S = S ∗ , V = V ∗ , I = I ∗ , T = T ∗ , R = R∗ , C = C ∗ } (2.71) is E ∗ = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ). Therefore, by Lyapunov-LaSalle’s invariance properties, system (2.61) is uniformly persistent.
2.23.
Treatment-Only Submodel (with Mass-Action Incidence)
We assume that treatment is the only intervention adopted. In this case η = 0 = ω and the model reduces to the following system dS = (1 − ρ)Λ − βSI + δR − µS, dt dI = ρΛ + βSI − (κ + µ + λα)I, dt
(2.72)
dT = κI − (µ + τ π)T − (1 − λ)αT, dt dR = τ πT − (µ + δ)R, dt For simplicity of the mathematical analysis, we normalize the treatment-only submodel (2.72) by defining the new variables µ µ µ µ s = S, i = I, tˆ = T, r = R, and the parameters Λ Λ Λ Λ δ κ τ α Λ ˜ ˜ ˜ ˜ = , τ˜ = , α ˜ = , to obtain β = 2 β, t = µt, δ = , κ µ µ µ µ µ ds ˜ + δr ˜ − s, = (1 − ρ) − βsi dt˜ di ˜ − (˜ = ρ + βsi κ + 1 + λα ˜ )i, dt˜ (2.73) dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ, dt˜ dr ˜ = τ˜π tˆ − (1 + δ)r, dt˜ with invariant region given by 4 Ω2 = {(s, i, tˆ, r) ∈ R I+ : s + i + tˆ + r ≤ 1}.
(2.74)
System (2.73) can be decoupled and reduced to a 3-dimensional system with variables s, i and tˆ. Since the use of the dimensionless variables s = Λµ S,
A Mathematical Analysis of Influenza with Treatment and Vaccination µ i = Λ I, tˆ = replaced by
µ Λ T,
and r =
µ ΛR
67
leads to further simplifications, system (2.73) is now
ds ˜ + δ(1 ˜ − s − i − tˆ) − s, = (1 − ρ) − βsi dt˜ di ˜ − (˜ = ρ + βsi κ + 1 + λα ˜ )i, dt˜
(2.75)
dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ, dt˜ for which the disease-free equilibrium when (ρ = 0) is given by ET0 = (1, 0, 0). To determine the local stability of ET0 , the Jacobian of the normalized reduced (NR) model is evaluated at the DFE to yield ˜ −(δ˜ + 1) −(β˜ + δ) −δ˜ JE 0 = (2.76) 0 β˜ − (1 + κ ˜ + λα ˜) 0 T 0 κ ˜ −(1 + τ˜π + (1 − λ)α) ˜
The eigenvalues of JE 0 are ς1 = −(δ˜ + 1), ς2 = β˜ − (1 + κ ˜ + λα ˜ ), and ς3 = −(1 + τ˜π + T (1 − λ)α). ˜ The eigenvalues ς1 and ς3 are negative. Thus, the local stability of ET0 depends on the sign of ς2 = β˜ −(1+ κ ˜ +λα ˜ ) = (1+ κ ˜ +λα ˜ )(RT −1). Therefore, the DFE of the treatment-only submodel (2.73) is locally asymptotically stable if RT < 1 and unstable if RT > 1. Lemma 16 The treatment only submodel (2.73) has a unique endemic equilibrium (EE) ET∗ = (s∗ , i∗ , tˆ∗ ) ∈ Ω2 when ρ = 0 iff RT > 1.
Proof. Equating the RHS of the NR model to zero and solving for the variables, we obtain ˜ ˜ (RT − 1)(1 + δ) 1 ˜ (RT − 1)(1 + δ) ˆ∗ = κ , i∗ = s∗ = , where , t ˜ T (1 + κ˜ ) ˜ T (A + κ RT β˜ + δR Aβ˜ + δR ˜) A β˜ A = 1 + τ˜π + (1 − λ)α, ˜ RT = . Thus, ET∗ exists provided RT > 1. Hence 1+κ ˜ + λα ˜ the proof. From (2.75) we can further reduced the 3-dimensional system to a 2-dimensional system by replacing s by 1 − i − tˆ. That is, di ˜ − i − tˆ)i − (˜ = ρ + β(1 κ + 1 + λα ˜ )i, dt˜ dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ. dt˜
(2.77)
Lemma 17 The treatment-only sub-model (2.73) has no periodic solutions in Ω2 |Ω0 when RT > 1. Proof. The EE exists only when RT > 1 (see Lemma 16) and i + tˆ + r → 0 as t˜ → ∞. Since Ω0 = {(s, i, tˆ, r) ∈ Ω2 : i = 0 = tˆ = r; s = 1}, let G be the sum of the RHS of 1 system (2.77) and ψ = be a candidate Dulac’s function. It follows that itˆ 1 ρ div(ψG) = − ( 2 + β + κ) < 0. tˆ i
(2.78)
68
H. Rwezaura, E. Mtisi and J.M. Tchuenche 0.9 0.25
0.8 0.7
0.2
0.15
0.5 Treated
Infected
0.6
0.4
0.1
0.3 0.2 0.05
0.1 0
0
0.05
0.1
0.15 0.2 Treated
0.25
0.3
0.35
0
0
(a)
0.1
0.2
0.3 Infected
0.4
0.5
0.6
(b)
Figure 4. Phase portraits of i and tˆ showing the effects of treatment on infectives.
Hence, by Dulac’s criterion, there are no periodic orbits in Ω2 |Ω0 . Since Ω2 is positively invariant and the EE exists whenever RT > 1, then by the Poincar´ e-Bendixon Theorem (Perko, 2000), all solutions of the limiting system originating in Ω2 remain there for t˜. Further, the absence of periodic orbits in Ω2 implies that the unique EE of the special case of the treatment-only submodel (when ρ = 0) is GAS whenever RT > 1. Now, suppose ρ = 0, then, the following result holds.
Theorem 18 If the treatment only submodel has a unique endemic equilibrium, then, it is GAS and the disease persists within the population.
Proof. The two-dimensional simplex Ω2 is bounded and the submodel has no periodic orbits, homoclinic orbits, or polygons (see Lemma 17) and by the Poincar´ e-Bendixon Theorem (Perko, 2000), the omega-limit set of every solution in Ω2 is an equilibrium point. Since Ω2 is positively invariant, it follows from Lemma 17 that the omega-limit set in Ω2 |Ω0 must be the EE. Figure 4 illustrates the effect of treatment on the infective class. Treatment reduces the number of infectives to a lower lever, but does not eradicate the disease due to the flux of infectives. The parameter values used are (a): ˜ α ˜ = 0.01, δ = 0.15, κ ˜ = 0 : 0.1 : 1, λ = 0.5, π = 0.6, ρ = 0.1, τ˜ = 0.25, β˜ = 4.5, with initial values s0 = 0.9; i0 = 0.1; tˆ0 = 0; r0 = 0. (b): α ˜ = 0.01, δ˜ = 0.15, κ ˜ = 0.5, λ = 0.5, π = 0 : 0.1 : 1, ρ = 0.1, τ˜ = 0.25, β˜ = 4.5, ˆ and s0 = 0.9; i0 = 0.1; t0 = 0; r0 = 0 In Figure (a), κ ˜ varies from 0-1 with π constant, while in (b), π varies from 0-1 with κ ˜ constant.
A Mathematical Analysis of Influenza with Treatment and Vaccination
2.24.
69
Existence of Backward Bifurcation in the Treatment-Only Model
We investigate the existence of backward bifurcation (or subcritical bifurcation, i.e., positive endemic equilibria exist for RT < 1 near the bifurcation point) in the treatment-only model using the normalized reduced system (2.77) for which at steady state we have, ˜ − i∗ − tˆ∗ )i − (˜ ρ + β(1 κ + 1 + λα ˜ )i∗ = 0, ∗ ∗ ∗ κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ = 0.
(2.79)
˜ At steady state, system (2.79) can be expressed The force of infection is given by χ = βi. in terms of χ as follows: χ ˆ∗ χ − t ) − (˜ = 0, κ + 1 + λα ˜) ˜ β β˜ χ χ ρ + χ(1 − − tˆ∗ ) − = 0, ˜ R β T χ ˜ tˆ∗ = 0. κ ˜ − [1 + τ˜π + (1 − λ)α] ˜ β ρ + χ(1 −
(2.80)
Expressing the variables tˆ∗ in terms of the variable χ gives tˆ∗ =
κ ˜χ . ˜ β[1 + τ˜π + (1 − λ)α] ˜
(2.81)
Substituting (2.81) into the first equation of (2.80) gives the following quadratic equation: ˜ + τ˜π + (1 − λ)α)]( Q(i, ρ) = [1 + κ ˜ + τ˜π + (1 − λ)α)]χ ˜ 2 + β[1 ˜ R1T − 1)χ ˜ + τ˜π + (1 − λ)α)] −ρβ[1 ˜ = 0, 2 = aχ + bχ + c = 0. (2.82) ˜ + τ˜π + (1 − λ)α)]( where a = [1 + κ ˜ + τ˜π + (1 − λ)α)], ˜ b = β[1 ˜ R1T − 1) ˜ + τ˜π + (1 − λ)α)] and c = ρβ[1 ˜ = 0. Since all the model parameters are nonnegative, it follows from (2.82) that a > 0 and c ≤ 0. Furthermore, if 0 < RT ≤ 1, then b ≥ 0 and if RT > 1, then b < 0. Notice that when ρ = 0, the quadratic equation Q(χ, ρ) is reduced to a linear equation b aχ + b = 0, so that χ = − . If b ≥ 0, that is, 0 < RT ≤ 1, then, χ ≤ 0. Therefore, a no endemic equilibrium exists whenever 0 < RT ≤ 1. But for b < 0, i.e., RT > 1 then, χ > 0,and in this case, the treatment-only submodel has a unique endemic equilibrium if and only if b < 0, i.e., RT > 1, ruling out backward bifurcation in this case. If ρ > 0, then, since a > 0 and c < 0, the quadratic equation Q(χ, ρ) has at most one positive root regardless of the sign of b, i.e., RT ≤ 1 or RT > 1. Hence, the treatment only submodel has a unique endemic equilibrium regardless of RT ≤ 1 or RT > 1. This result indicates the possibility of backward bifurcation due to the existance of the endemic equilibrium when RT ≤ 1. If b = 0 (i.e., RT = q1) then the quadratic equation Q(χ, ρ) has two roots with opposite signs, namely, χ = ± −c a since a > 0 and c ≤ 0. In this case, the treatment-only submodel has a unique endemic if c < 0, and ρ > 0. Hence, the following result is established (an alternative proof of this Theorem is given in Appendix C):
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
Theorem 19 The treatment only submodel (2.73) has (i) a unique endemic equilibrium if c = 0 (for ρ = 0) and b < 0(i.e., RT > 1), (ii) tow equilibria if c < 0 for ρ > 0, RT ≤ 1, (iii) no endemic equilibrium otherwise. Case (ii) indicates the possibility of backward bifurcation in the the treatment only submodel (2.80) when RT ≤ 1. To check for this, the discriminant b2 − 4ac is set to zero and the result solved for the critical value of RcT , from which it can be shown that backward bifurcation occurs for values of RT such that RcT < RT < 1. But, since a > 0 and c ≤ 0, then, b2 −4ac = 0 when b2 = 0 = 4ac i.e., when (c = 0 ⇒ ρ = 0) and (b = 0 ⇒ RT = 1).
3. 3.1.
Sensitivity Analysis and Numerical Simulations Sensitivity Analysis
Sensitivity indices allow us to measure the relative change in a state variable when a parameter changes, while its analysis is commonly used to determine the robustness of model predictions to parameter values (since there are usually errors in data collection and presumed parameter estimations). The sensitivity indices to the parameters in the model are calculated in order to determine parameters that have a high impact on RV , RT , RV T , and that should be targeted by intervention strategies (Chitnis et al., 2008). These indices tell us how crucial each parameter is to disease transmission and prevalence. In order to avoid repetition, we omit the calculations of the sensitivity indices of the endemic equilibrium E ∗ . Nevertheless, it is noted that disease prevalence is directly related to the endemic equilibrium point, specifically to the magnitudes of I ∗ . The fraction of infectious humans, I, is especially important because it represents the people who may be clinically ill, and is directly related to the total number of influenza deaths. For a detail description of these concept, see (Chitnis et al., 2008). The most important parameter for equilibrium disease prevalence is the human-mosquito contact rate β. Other important parameters are the treatment rate κ, the vaccine efficacy ǫ and the fraction of recruited individuals who are already infected ρ. In determining how best to reduce human mortality and morbidity due to influenza, it is necessary therefore to know the relative importance of the different factors responsible for its transmission and prevalence. Initial disease transmission in the absence of any intervention strategy is directly related to R0 , and disease prevalence is directly related to the endemic equilibrium point, specifically to the magnitudes of I and R. In computing the sensitivity analysis, we use method described by Chitnis et al., (2008). The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives. Therefore, the normalized forward sensitivity index of a variable, u, that depends differentially on a parameter, p, is defined as: Υup =
∂u p × . ∂p u
(3.83)
A Mathematical Analysis of Influenza with Treatment and Vaccination
3.2.
71
Sensitivity Indices of RV T
The sensitivity indices of RV T =
βµR0 [µ + γ + ω + (1 − ǫ)η] with respect to its (κR0 + β)[(µ + γ)(µ + η) + µω]
(eight) parameters are given by VT ΥR µ
=
2µ + γ + ω + (1 − ǫ)η µ(2µ + γ + ω + η) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω
VT ΥR γ
=
γ γ(µ + η) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω
VT ΥR ω
=
ωµ ω − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω
VT ΥR η
=
η(µ + γ) η(1 − ǫ) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω
VT = − ΥR ǫ
VT ΥR = β
ǫη , µ + γ + ω + (1 − ǫ)η
κR0 , κR0 + β
VT ΥR R0 =
(3.84)
β , κR0 + β
VT = − ΥR κ
κR0 . κR0 + β
Similarly, we can derive the expression for R0 , RV and RT . When RV T is expressed in µ[µ + γ + ω + (1 − ǫ)η] with respect to terms of RT , the sensitivity index of RV T = RT [(µ + γ)(µ + η) + µω] VT RT is given by ΥR RT = 1. βRV β + κR0 VT = 1. Here, we note that R can also be expressed with respect to RV is given by ΥR V T RV RV RT in terms of the three reproduction numbers RV , RT , and R0 as RV T = . R0 When RV T is expressed in terms of RV , the sensitivity index of RV T =
By evaluating the sensitivity indices of R0 , RV , RT and RV T using the parameter values in the Table 3.1, it can be seen (Table 3.2) that the sensitivity indices of R0 , RV , RT and RV T with respect to β does not depend on any parameter values. The most sensitive V = −1.6254 = ΥRV T , parameter for RV and RV T is the vaccine efficacy ǫ, since ΥR ǫ ǫ then, increasing (or decreasing) ǫ by 10% decreases (or increases) RV and RV T by 16%. Also the vaccination rate η is an important parameter for RV and RV T since by increasing (or decreasing) η by 10% decreases (or increases) RV and RV T by 5.5%. For R0 as well as RV the most sensitive parameter is the disease-induced death rate α, since by increasing the disease induced death rate α decreases the life expectancy which tends to reduce R0 and RV , consequently, treatment is absolutely necessary (since combined strategy applied concurrently is more beneficial) in order to curtail the epidemic. The rate of loss of immunity δ may be responsible for RV to be very sensitive
72
H. Rwezaura, E. Mtisi and J.M. Tchuenche Table 2. Parameters definition and values
η ω γ κ τ ǫ π λ β Λ ρ δ µ α
Definition vaccination rate (days−1 ) waning rate of vaccine-based immunity (days−1 ) rate of acquiring protective antibodies (days−1 )
treatment rate (days−1 ) recovery rate (days−1 ) vaccine efficacy drug efficacy effectiveness of the drug contact rate (days−1 /person) recruitment rate (days−1 ) fraction of recruited infectives (days−1 ) rate of loss of immunity (days−1 ) outflow rate of individuals (days−1 ) the disease induced death rate (days−1 )
Range 0.3 − 0.7 0.003 0.143 0.25 − 0.7 0.14 − 0.25 0.3 − 0.9 0.3 − 0.9 0 0. Thus, the public health objective in this case is to minimize the level of epidemicity as shown on Figure 8 (for the effects of treatment on infectives and vaccination on susceptibles). Figure 6 shows that vaccination and treatment applied concurrently quickly lower the level of epidemicity. Figure 7: The parameter values used here are the same as those in Figure 5, but with the following initial conditions S0 = 1000; V0 = 500; I0 = 0; T0 = 0; R0 = 0; C0 = 70. A similar trend to that in Figure 5 is observed due to the combine effects of vaccination and treatment. Figures 8 (a) and (b): These depict the effects of vaccination and treatment on susceptible and infected individuals, respectively.
4. 4.1.
Discussion and Conclusion Discussion
This work is based on the construction and use of a mathematical model for the transmission dynamics of influenza with treatment and vaccination in a human population. The main innovation of this work with respect to previous influenza models is that we explicitly consider both vaccination with an imperfect vaccine and treatment (with treatment efficacy), as well as disease induced death rate, where a fraction of susceptible individuals is vaccinated per unit time with inflow of infectives. Indeed, this model which includes only vaccination and treatment is more appropriate for developing countries where adequate health facilities are not generally available for mass hospitalization, isolation and quarantine. Our proposed model, likewise others, incorporates some essential parameters (such as vaccine and antiviral drugs efficacy, rate at which immunity wanes from previous infection, and
74
H. Rwezaura, E. Mtisi and J.M. Tchuenche Susceptible
Vaccinated
1000
2000
500
1000
0
0
50
100
0
0
50
Infected
Treated
1000
2000
500
1000
0
0
50
100
0
2
1000
1
0
0
50 4
Recovered 2000
0
100
50
100
0
x 10
0
100
Protected
50
100
Figure 5. Phase portrait of the evolution of each class against time.
5000 S V I T R C
4500 4000 3500 3000 2500 2000 1500 1000 500 0
0
50
100
150
Figure 6. Evolution of classes against time on the same phase plane.
A Mathematical Analysis of Influenza with Treatment and Vaccination Susceptible
Vaccinated
1000
2000
500
1000
0
0
50
0
100
0
50
Infected 2000
500
1000
0
50
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100
2000
2
1000
1
0
0
50 4
Recovered
0
100
Treated
1000
0
75
50
0
100
100
Protected
x 10
0
50
100
Figure 7. Evolution of each class against time with different initial values. 1000
16000
900 14000
800 12000 Infected/infectious
Susceptible
700 600 500 400
10000 8000 6000
300 4000
200 2000
100 0
0
200
400
600
800 1000 1200 Vaccinated
(a)
1400
1600
1800
2000
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3000 4000 Treated
5000
6000
7000
(b)
Figure 8. Effects of vaccination (a) and treatment (b).
the disease-induced death rate) of influenza transmission, which enable the assessment of various anti-influenza preventive strategies, and their epidemiological consequences. The model is given in the form of a non-linear ODE, and our analytical results show that:
76
H. Rwezaura, E. Mtisi and J.M. Tchuenche (i) In the absence of recruitment of infected humans into the community, the DFE exists and consequently, influenza can be controlled using treatment and vaccination.
(ii) The basic reproduction number R0 is reduced whenever treatment or vaccination is introduced as a therapeutic measure (i.e., RV < R0 and RT < R0 ). (iii) Also, the vaccination and treatment-induced reproduction number RV T reduces both RV and RT . Thus, concurrent administration of vaccination and treatment is more adequate in curtailing the epidemic. This is consistent with the fact that control measures are necessary to reduced the value of the basic reproduction number R0 , and if this quantity is less than unity, the disease can be eradicated. Thus, a combined control program is more effective in curtailing the epidemic. Since it is well-known that infected persons do migrate from one region to another, this is incorporated into our model by assuming that a proportion (ρ) of recruited individuals are infected. That is, a proportion (1 − ρ)Λ of recruited individuals are susceptible, while the remaining fraction ρΛ is infected. The consequence of this inflow of infectives is that the DFE equilibrium does not exist any longer. Thus, no amount of preventive measures can eradicate influenza from the population, but can lead to significant reductions in influenza infections. For ρ > 0, the disease remains endemic (persists) in the community, and the number of infectives can only be reduced by reducing ρ or Λ, or by increasing the treatment and vaccination rate κ and η (as well as the vaccine and drug efficacy ǫ, and π), respectively. Nevertheless, the drug efficacy π does not appear explicitly in the expression of RV T , and this could indicate that vaccination is more important in preventing an influenza outbreak (i.e., prevention is better than cure). We use a continuous vaccination program (where a fraction of susceptible individuals is vaccinated per unit time), and do hope that the result is independent of the type of vaccination program adopted (i.e., using cohort vaccination or a combination of both cohort and continuous vaccination). Therefore, the present model can be refined in various ways: For instance, it is instructive to determine whether or not the results obtained using continuous vaccination or a combined continuous and cohort vaccination (where a fraction of the newly-recruited members of the community are vaccinated) will hold. It is also worthwhile to investigate the effect of co-circulating influenza strains and the effect of seasonality (using time-dependent transmission coefficients) on the transmission dynamics of this disease. It is important to note that our model is relaxed by considering a model with mass action incidence and without inflow of infectives in order to study the persistence of solutions of the system. In general, results are given regarding invariant regions, existence, positivity and stability of equilibria. Basically, we have established the following results: (a) Local and global stability of the DFE of the model without inflow of infectives (ρ = 0). (b) Local and tri-stability stability analysis of the EE when ρ > 0. (c) Persistence of the model system (2.1) with ρ = 0.
A Mathematical Analysis of Influenza with Treatment and Vaccination
77
(d) Public health measures significantly reduce influenza infections. (e) The treatment-only sub-model with mass action incidence has no periodic orbits, and its EE if its exists, is GAS and the disease will persist. (f) The treatment-only sub-model with mass action incidence exhibits the phenomenon of backward bifurcation, likewise the vaccination-only sub-model (cf. Alexander et al., 2004), consequently, the full model also undergoes the same phenomenon when it reproductive threshold RV T < 1 as shown in Theorem 9, Lemma 10 and Section 2.18. Also, it is algebraically shown that the model with inflow of infectives has a tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity - a dynamical feature that has been observed in TB dynamics (Gumel and Song, 2008). (g) Sensitivity of the model parameters to determine their relative importance in disease transmission. (h) Low fitness ratio is beneficial and reduce the probability of population-level perversity. We have performed sensitivity analysis on a mathematical model of influenza transmission to determine the relative importance of model parameters to disease transmission and prevalence by computing sensitivity indices of the reproductive numbers, which measures initial disease transmission (Chitnis et al., 2008). Mathematical modeling of influenza can play a unique role in comparing the effects of control strategies. We begin such a comparison by determining the relative importance of model parameters in influenza transmission. The model is based on a continuous vaccination program . The same conclusion will hold if a continuous vaccination program only (where a fraction of susceptible individuals is vaccinated per unit time) is considered (Sharomi et al., 2007). The threshold fitness ratios Hj = 1, j = T, V indicates that above this value, infected treated, vaccinated, respectively, generate more secondary infections than infected untreated, and unvaccinated individuals (Smith? and Blower, 2004). Numerical simulations help examine the dynamics and suggest some properties of these models that we were unable to prove mathematically. One of them shows that our model may have a backward bifurcation where the DFE and EE co-exist.
4.2.
Conclusion
In summary, mathematical models are potentially useful tools to aid in the design of control programs for infectious diseases. We developed an epidemiological model of human influenza with vaccination and treatment and used it to predict trends in infection as well as possible control measures. The model incorporates several realistic features including vaccination of susceptible and drug treatment for infected individuals. The qualitative and quantitative mathematical properties of the models are studied, their biological consequences and some control strategies are discussed, and the results of the models are compared with previous ones. It is shown that the full model exhibits the phenomenon
78
H. Rwezaura, E. Mtisi and J.M. Tchuenche
of backward bifurcation whereby two equilibria, namely the disease-free and the endemic equilibrium co-exist. Explicit thresholds of vaccination and treatment rates are established for which the infection will be controlled under certain levels. Realistic model parameters are used to validate the model.
Acknowledgments HR acknowledges with thanks the Eastern Africa Universities Mathematics Program for partial support and the Center for International Mobility for student exchange program at Lappeenranta University of Technology (Finland) within the framework of North-SouthSouth Network Program.
Appendix A C0
=
Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη],
=
Λ[(µ + γ + ω + η)(µ + κ + λα) − β(µ + γ + ω + (1 − ǫ)η)],
=
Λ γη [µ(µ + γ + ω + η + )(µ + κ + λα) − µβ(µ + γ + ω + (1 − ǫ)η)]− µ µ Λγη (µ + κ + λα), − µ
= Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)[1 − −
Λγη (µ + κ + λα), µ
βµ[µ + γ + ω + (1 − ǫ)η] ] [(µ + γ)(µ + η) + µω](µ + κ + λα)
= Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)(1 − RV T ) − = −[
Λγη µ (µ
+ κ + λα),
Λγη γ (µ + κ + λα) + Λ(µ + γ + ω + η(1 + ))(µ + κ + λα)(RV T − 1)]. µ µ
Appendix B Barbalat Lemma (1959) Lemma 20 Let x 7→ F (t) be a differentiable function with a finite limit as t → ∞. If F˙ is uniformly continuous, then F˙ → 0 as t → ∞.
Appendix C Alternative Proof of Theorem 19 In order to find the conditions for the existence of multiple equilibria, we use the second equation of (2.79) to express the variables tˆ∗ in terms of the variable i∗ when i∗ 6= 0. This
A Mathematical Analysis of Influenza with Treatment and Vaccination
79
gives (at equilibrium) tˆ∗ =
κ ˜ i∗ . (1 + τ˜π) + (1 − λ)α ˜
Substituting it into the first equation of (2.79) gives the following quadratic equation: ˜ + τ˜π + (1 − λ)α)]i ˜ + τ˜π + (1 − λ)α)]i Q(i, ρ) = [˜ κ + β(1 ˜ 2 + (1 + κ ˜ + λα ˜ − β)[1 ˜ −ρ[1 + τ˜π + (1 − λ)α)] ˜ = 0, = ai2 + bi + c = 0, ˜ + τ˜π + (1 − λ)α)], ˜ + τ˜π + (1 − λ)α)] where a = [˜ κ + β(1 ˜ b = (1 + κ ˜ + λα ˜ − β)[1 ˜ and c = ρ[1 + τ˜π + (1 − λ)α)]. ˜ Since all the model parameters are nonnegative, it follows that a > 0 and c ≤ 0. Furthermore, if RT < 1, then b > 0, that is, ˜ + τ˜π + (1 − λ)α)], b = (1 + κ ˜ + λα ˜ − β)[1 ˜ = (1 + κ ˜ + λα ˜ )[1 + τ˜π + (1 − λ)α)](1 ˜ −
β˜ 1+˜ κ+λα ˜ ),
= (1 + κ ˜ + λα ˜ )[1 + τ˜π + (1 − λ)α)][1 ˜ − RT ]. Notice that when ρ = 0, the quadratic equation Q(i, ρ) is reduced to a linear equation b ai + b = 0, so that i = − . If b ≥ 0, i.e., RT ≤ 1, then, i ≤ 0. Therefore, no endemic a equilibrium exists whenever RT ≤ 1. But, for b < 0, i.e., RT > 1, i > 0. Therefore, there exists a unique endemic equilibrium. If ρ > 0, and since a > 0 and c < 0, then, the quadratic equation Q(i, ρ) has two positive roots for b > 0 (RT ≤ 1). Hence, the treatment only sub-model has an endemic equilibrium regardless of RT ≤ 1 or RT > 1. This result indicates the possibility of backward bifurcation due to the existence of the endemic equilibrium when RT ≤ 1. Case (ii) of Theorem 19 thus implies that the model exhibits the phenomenon of backward bifurcation in the the treatment only sub-model (2.72) when RT ≤ 1, for b > 0. To check for this, we solve for β˜c when b = 0 which implies RT = 1) and obtain β˜c = 1 + κ ˜ + λα ˜, ˜ from which it can be shown that backward bifurcation occurs for values of β such that β˜c < β˜ < 1 + κ ˜ + λα ˜ and ρ > 0.
Appendix D (1) Endemic Equilibria when ρ = 0 The endemic equilibria of model system (2.1) with ρ = 0 (if they exist) cannot be cleanly expressed in closed form. In order to find the existence of these equilibria, we use (2.1) to express the variables S, V, T, R in terms of I when I 6= 0, with the assumption that the Cclass plays no further part in the disease transmission or γ = 0. Let the endemic equilibrium of system (2.1) without the fully protected class be denoted by E ∗ = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ ).
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H. Rwezaura, E. Mtisi and J.M. Tchuenche
Then, expressing E ∗ in terms of I ∗ , we have (−a11 − a12 I ∗ /(a7 − a5 I ∗ ))(a8 + a13 I ∗ /a4 ) ∗ = S ηω − (a9 + βa6 I ∗ /(a7 − a5 I ∗ ))(a11 + a12 I ∗ /(a7 − a5 I ∗ )) −η(a8 + a13 I ∗ /a4 ) V∗ = ηω − (a9 + βa6 I ∗ /(a7 − a5 I ∗ ))(a11 + a12 I ∗ /(a7 − a5 I ∗ ))
(4.85)
κI ∗ T∗ = µ + πτ + (1 − λ)α κπτ I ∗ , R∗ = (µδ )(µ + πτ + (1 − λ)α)
where
a1 = µ + πτ a2 = (1 − λ)α a3 = µ + δ a4 = a3 (a1 + a2 ) a5 = a2 κ + (a1 + a2 )αλ a6 = µ(a1 + a2 ) a7 = Λ(a1 + a2 ) a8 = (1 − ρ)Λ a9 = (µ + η) a10 = (1 − ǫ) a11 = γ + ω + µ a12 = βa6 a10 a13 = δκρτ a14 = κ + µ + λα a15 = a9 (γ + µ) + µω. Substituting (4.85) into the third equation of system (2.1) gives (at equilibrium) P (I ∗ ) ≡ I ∗ (b3 I ∗2 + b2 I ∗ + b1 ) = 0,
(4.86)
where b3 = a12 (a4 a14 (βa6 − a5 a9 ) + a13 (a5 η − a6 )) +a5 (a4 a5 a14 a15 + β6 a11 (a13 − a4 a14 )),
b2 = a4 a7 a9 a14 (a12 − 2a5 a11 ) + βa6 a7 a11 (a4 a14 − a13 ) + βΛa4 a6 (a5 a11 − a12 ) +a12 η(a4 a5 Λ − a7 a13 ) + 2a4 a5 a7 a14 ηω,
b1 = a27 a214 µa15 (1 − RV T ).
(4.87)
Clearly I ∗ = 0 is a solution to the equation (4.86) which corresponds to the disease free equilibrium of system (2.1) with ρ = 0. The remaining quadratic equation will be analysed for the existence of a positive solution which corresponds to the endemic equilibrium. Since all parameters are positive it follows from (4.87) that b3 > 0 and the sign b1 corresponds to that of 1 − RV T . The existence of the equilibria are summarized in the following theorem. Theorem 21 Suppose ρ = 0 in (2.1). Then (i) a unique endemic equilibrium if b1 < 0 ⇐⇒ RV T > 1, (ii) a unique endemic equilibrium if b2 < 0 or b22 − 4b3 b1 = 0, (iii) two endemic equilibria if b1 > 0 ⇐⇒ RV T < 1, b2 < 0 and b22 − 4b3 b1 > 0, (iv) no endemic otherwise.
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81
(2) Endemic Equilibrium when ρ > 0 and γ = 0 To find the existence of the endemic equilibria of model (2.1) with the C-class omitted, we substitute (4.85) into the third equation of (2.1). The resulting polynomial equation is given by P (I ∗ ) + ρ(c2 I ∗2 + c1 I ∗ + c0 ) = 0, (4.88) where P (I ∗ ) is given in (infe1) and c2 = a4 a5 a9 Λ(a12 − a5 a11 ) + a4 a5 ηΛ(a5 ω − a6 a10 ), c1 = 2a5 a15 Λ − a6 a10 βΛµ, c0 = −a4 a27 a15 Λ.
We note that when ρ > 0, the disease free equilibrium does not exist. This is because the model (2.1) assumes a constant flow of new members into the population of which a specific fraction is infective. Therefore at any time the infected populated is not zero except when ρ = 0. From analysis of the coefficients in (4.88) we note that b3 is positive and c0 is negative. By Descartes’ Rule of signs, we note that there is at least one sign change one sign change in the sequence of coefficients. Hence the model (2.1) with ρ > 0 and γ = 0 has at least a positive root.
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[38] Nu˜ no M., Chowell G. and Gumel A.B. (2006) Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, J. R. Soc. Interface 4(14), 505-521. [39] Parker A.S. Jr. and Bradley S.B. (2001) Influenza: A to B, http : //cme.uf l.edu/media/f lu/index.html, retrieved Sept. 19, 2007. [40] Perko, L. Differential Equations and Dynamical Systems. Springer, New York (1991). [41] Sanofi Pasteur’s Investigational H5N1 Influenza vaccine achieves high immune response at low dosage, http : //online.wsj.com/public/article/P R − CO − 20070918 − 900189.html?mod = crnews, retrieved Sept. 14, 2007. [42] Smith? R.J. and Blower S.M. (2004) Could disease-modifying HIV vaccines cause population-level perversity? Lancet Infect. Dis. 4, 636-639. [43] Spear R. (2002) Mathematical modeling in environmental health, Environ. Health Perspect., 110 A382. [44] United Republic of Tanzania, Ministry of Livestock Development: Avian influenza and human influenza pandemic, Bird Flu Communication Strategy, www.mifugo.go.tz/documents\_storage/ TanzaniaBirdFluCommunicationStrategy.pdf, retrieved Sept. 11, 2007. [45] van den Driessche P. and Watmough J. (2002) Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180, 2948. [46] Wikipedia Influenza pandemic, http://www.en.wikipedia.org/wiki/Influenza_pandemic, trieved Sept. 14, 2007.
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In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 85-131
Chapter 3
A T HEORETICAL A SSESSMENT OF THE E FFECTS OF C HEMOPROPHYLAXIS , T REATMENT AND D RUG R ESISTANCE IN TB I NDIVIDUALS C O - INFECTED WITH HIV/AIDS C.P. Bhunu1,∗ and W. Garira1,2 Modelling Biomedical Systems Research Group, Department of Applied Mathematics, 1 National University of Science and Technology, P. O. Box 939 Ascot, Bulawayo, Zimbabwe 2 Department of Mathematics and Applied Mathematics, University of Venda, South Africa
Abstract Tuberculosis (TB) patients who do not complete TB treatment risk developing antibiotic resistant TB, one of the most serious health problems facing the society today. Effects of TB drug resistance are severe in individuals co-infected with HIV/AIDS. In this paper we develop a two strain TB model, two strain HIV/AIDS model and an HIV/AIDS-TB co-infection model for assessing the impact of treatment and drug resistance in controlling TB in settings with high HIV/AIDS prevalence. In the absence of HIV/AIDS, the TB-only model is shown to exhibit a phenomenon known as backward bifurcation where a stable disease-free equilibrium co-exists with the stable endemic equilibrium when the associated reproduction number is less than unity. On the contrary, the HIV/AIDS-only model shows a globally asymptotically stable disease-free equilibrium when the associated reproduction number is less than unity. The centre manifold theory is used to determine the local asymptotic stability of the endemic equilibria. From the study we conclude that chemoprophylaxis and TB treatment are more effective in controlling TB in the absence of drug resistant TB and HIV/AIDS as shown for some derived critical threshold values. The results of the study show that chemoprophylaxis and treatment of TB infectives are equally effective in controlling TB in co-infected individuals, but not so in cases involving drug ∗
E-mail address:
[email protected],
[email protected]. Corresponding author.
86
C.P. Bhunu and W. Garira resistant TB. Further, we conclude from the study that chemoprophylaxis and treatment of TB are more effective in TB infected individuals co-infected with HIV/AIDS who are not yet on antiretroviral therapy.
Key words: TB, HIV/AIDS, Co-infection, Chemoprophylaxis, Treatment, Antiretroviral therapy, Stability.
1.
Introduction
Tuberculosis (TB) is present in 1.8 billion people world wide (Mukherjee [34]). Its incidence and mortality in Sub-Saharan Africa have reached alarming levels and continue to rise (Cohen et al. [16]). In Sub-Saharan Africa, TB is the leading cause of mortality and in developing countries it accounts for an estimated 2 million deaths which is a quarter of avoidable adult deaths (Raviglione [42]). TB was assumed to be on its way “out” in developed countries until the number of TB cases began to increase in the 1980s. Its control in Sub-Saharan Africa has been hindered by emergence of the Human Immunodeficiency Virus (HIV) and the associated Acquired Immune Deficiency Syndrome (AIDS). HIV/AIDS significantly affects the progression of Mycobacterium tuberculosis(Mtb) infection. In immune-competent individuals Mtb infection rarely leads to disease and usually results in latent non-transmissible infection, but in individuals co-infected with HIV/AIDS the story is different. Mtb infected individuals dually infected with HIV/AIDS tend to develop active TB quickly due to immune suppression and malabsorption of rifampicin and isoniazid/ethambutol in advanced stages of HIV infection, such that it is perhaps counterproductive to conceive that TB can be managed in exactly the same way as immunocompetent people (Perlman [38]). TB treatment has been successful in low HIV/AIDS prevalence areas. Additional measures are necessary to reduce TB incidence in high burden regions (Nunn et al. [37]). In Sub-Saharan Africa the face of HIV/AIDS is TB, HIV/AIDS and TB fuel one another. Preventive therapy of TB in HIV/AIDS infected individuals is highly recommended (WH0 [47]) and could dramatically reduce the impact of HIV on TB epidemiology, but its implementation is limited in developing countries because of complex logistical and practical difficulties (Frieden [23]). HIV/AIDS infected patients given 6 months of anti-TB treatment had a higher relapse rate than those treated longer [39]. From a programme perspective, the ability to use the 6 months short-course anti-TB regimen in both HIV-infected and unifected patients is attractive especially in poor-limited settings. However, given the global estimate of 741 000 HIV-TB cases annually even a low rate of acquired rifampicin resistance can have significant consequences [38]. Apart from the effects of HIV on TB control, multi-drug resistant forms of TB have worsened the situation. Due to high number of TB individuals per health professional, Directly Observed Treatment Strategy (DOTS) has not been successfully implemented in most countries in Sub-Saharan Africa. This has led to incomplete treatment resulting in multidrug resistant cases of TB. Multi-drug resistance is defined as resistance to isoniazid and rifampicin whether there is resistance to other first line drugs (pyrazinamide, ethambutol) or not (Davies [17]). Resistance to isoniazid and streptomycin is the most common form of TB resistance by two drugs. Treatment of multi-drug resistant forms of TB is done us-
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ing second-line drugs (fluoroquinolone, capreomycin, kanamycin, and amikacin) which are more expensive and have more side effects. Unfortunately these second-line drugs are not available in most poor resource settings like those in Sub-Saharan Africa excluding South Africa. People with multi-drug resistant TB can be infectious and pass on the drug resistant bacteria to other people. For those individuals with access to second-line drugs misusing and/or mismanaging them results in creation of extensively drug resistant TB. Erratic supply of second line anti-TB drugs in areas of high HIV/AIDS prevalence has proven to be a fertile ground of extensively drug resistant TB. Extensively drug resistant TB is defined as resistance to any fluoroquine, and at least one injectable second-line drugs (capreomycin, kanamycin, and amikacin), in addition to isoniazid and rifampicin. Demands for the introduction of antiretroviral therapy in Africa have been growing over the past few years. On the face of it seems to be good news, but only a few individuals in need of them in Sub-Saharan Africa have access to these life saving drugs. Thus, like TB, HIV/AIDS has become primarily the disease of the poor (Farmer et al. [21]). If used optimally, HIV treatment could delay the onset of AIDS thus, reducing the burden of HIV/AIDS on TB epidemics. Virus strains with reduced sensitivity to Zidovudine, the first drug used against AIDS were observed in 1989 three years after it was introduced (Lader et al. [28]). Subsequently resistance to every currently licensed antiretroviral drug has been observed (Schinazi et al. [43]). Cross resistance between drugs of the same class is the rule rather than exception implying that drug resistance within an individual is not limited to a single compound (Harrington and Lader [24]). Drug resistance arises by natural selection, mutant HIV strains being selected when virus replicates in sublimiting drug concentrations (Steven et al. [45]). The only way to prevent drug resistance is to use a drug regimen that reduces virus replication to zero [45]. Most resistant strains of HIV are poor at replicating and do not persist in the absence of drugs (Quinones-Mateu and Arts [40]). People infected with drug resistant virus are more likely to have their treatment regimen fail allowing the virus to develop resistance to other drugs. Thus, in the absence of second line drugs for multi-drug resistant TB, HIV/TB co-infection with drug resistance of both infections is a double sword for the poor lives in Sub-Saharan Africa. To avert a looming disaster, there is an urgent need to strengthen both HIV/AIDS and TB control programmes in areas with high rates of HIV-related TB. A number of theoretical studies have been done on the mathematical modelling of coexistence of different pathogens (strains) in the same host [1–3, 5–10, 12, 13, 19, 30, 32, 35, 36]. Castillo-Chavez and Feng [12, 13] studied two strain TB model in the context of treatment, but we differ from [12,13] in that, in our TB submodel we considered exogenous re-infection as well. Naresh and Tripathi [35] did analyse the HIV-TB co-infection model, but we differ from [35] in that we have considered treatment of both infections as well as resistance of anti-retroviral therapy and anti-TB drugs. Bacaer et al. [5], Sharomi et al. [44] did analyse the HIV-TB co-infection model, but we differ from [5, 44] in that in addition to AIDS treatment and TB treatment we have considered resistance to anti-TB drugs and antiretroviral therapy. This paper is organised as follows. In Section 2 we have a two strain TB model description and its analysis. Section 3 presents a description and an analysis of two strain HIV/AIDS model. In Section 4 we present a description and an analysis of full model. Finally, we present the summary and concluding remarks.
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2.
C.P. Bhunu and W. Garira
A Two Strain Tuberculosis Model
In this section we begin by presenting a two strain TB model with treatment for the drug sensitive strain. The model subdivides the human population into the following sub-population of susceptible individuals (S), those exposed to TB, ET1 (drug sensitive) and ET2 (drug resistant), individuals with symptoms of TB, IT1 (drug sensitive) and IT2 (drug resistant), and those who have recovered from drug sensitive TB (RT ). It is assumed susceptible humans are recruited into the population at per capita rate Λ through birth and migration. Susceptible individuals acquire TB following contact with an infectious case at rate λTj , j = 1, 2 where j = 1 and j = 2 correponds to the drug sensitive TB and drug resistant TB, respectively with, βj cT ITj (t) (1) λTj = . NT (t)
Table 1. Model parameters and their interpretations. Definition
Symbol
Estimate(Range)
Recruitment rate
Λ
−1
0.029yr
Mukandavire and Garira [33]
Natural mortality rate
µ
0.02yr−1
Mukandavire and Garira [33]
Contact rate
cT
3yr−1
dT 1 , d T 2
TB induced death rate
Source
Estimate −1
0.3,0.5yr
Dye and Williams [19] −1
Transmission rate
β1 , β 2
0.35 (0.1-0.6)yr
Dye and Williams [19]
Endogeneous reactivation rates
k1 , k2
0.00013
Dye and Williams [19]
(0.0001-0.0003)yr−1 Probability of successful treatment
q
0.2 (0.15-0.25)
Dye and Williams [19]
Treatment rate for the latently infected
r1
0.7yr−1
Dye and Williams [19]
r2
Treatment rate for the infectives
0.88yr
−1
Qing-Song Bao et al. [41]
−1
Estimate
Protective factor
δ1
0.7yr
Protective factor
δ2
0.9yr−1
Estimate
1 − p1
0.01
Estimate
1 − p2
0.1
Estimate
Probability of susceptibles developing drug sensitive fast TB Probability of susceptibles developing drug sensitive fast TB
In equation (1), cT is the per capita contact rate, βj is the average number of susceptible individuals infected by one infectious individual per unit contact of time and NT (t) is the total population size given by, NT (t) = S(t) +
2 X j=1
ETj (t) + ITj (t) + RT (t).
(2)
Susceptibles infected with Mtb enter exposed class, (ETj ) at rate pj λTj and develop fast TB at rate (1 − pj )λTj to enter the infectious class, (ITj ). Individuals in the latently
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infected class, (ETj ) develops active TB as a result endogenous reactivation and exogenous reinfection at rates kj and δj λTj respectively into the infectious class, (ITj ). Individuals infected with the drug sensitive strain in the latently infected class, (ET1 ) and the infectious class, (IT1 ) are treated at rates r1 and r2 respectively. Individuals in ET1 (t) treated will move into RT (t) at rate r1 and of the individuals in IT1 receiving treatment a proportion q responds well to treatment and move into RT (t) and the remainder 1 − q develops drug resistance due to poor administration of treatment and move into ET2 at rate (1 − q)r2 . Individuals in RT (t) are not immune to TB and are reinfected at a rate λTj and individuals in each subgroup have a per capita death rate µ. The per capita disease induced death rate in each infectious class, (ITj ) is given by dTj . Parameters described will assume values in Table 1. The model flow diagram is given in Figure 1.
Figure 1. Structure of model. Putting the formulations together gives the following system of differential equations de-
90
C.P. Bhunu and W. Garira
scribing the model. ′
S =Λ−
2 X j=1
λTj S − µS,
ET′ 1 = p1 λT1 (S + RT ) − (k1 + r1 + µ)ET1 − δ1 λT1 ET1 − λT2 ET1 , IT′ 1 = (1 − p1 )λT1 (S + RT ) + k1 ET1 + δ1 λT1 ET1 − (µ + dT1 + r2 )IT1 , RT′ = r1 ET1 + qr2 IT1 − µRT −
2 X
(3)
λTj RT ,
j=1
ET′ 2 = (1 − q)r2 IT1 + p2 λT2 (S + RT ) + λT2 ET1 − δ2 λT2 ET2 − (µ + k2 )ET2 , IT′ 2 = (1 − p2 )λT2 (S + RT ) + k2 ET2 + δ2 λT2 ET2 − (µ + dT2 )IT2 , Model system (3) has initial conditions given by S(0) = S0 ≥ 0, ETj (0) = ETj0 ≥ 0, ITj (0) = ITj0 ≥ 0, RT (0) = RT0 .
(4)
Based on biological considerations, the model system (3) will be studied in the following region Λ 6 ΩT = (S, ET1 , IT1 , RT , ET2 , IT2 ) ∈ R+ : NT ≤ . (5) µ Theorem 1 assures that model system (3) is well posed in the sense that all solutions with non-negative initial condition remains non-negative for all t ≥ 0, and therefore makes biological sense. Theorem 1. The region ΩT ⊂ R6+ is positively invariant with respect to model system (3). Proof. We prove that the vector field of model system (3) points to the boundary of ΩT and is given by positive S, ETj , ITj and RT -axes. At the start of the process, the vector field, (F1 , F2 , F3 , F4 , F5 , F6 ) of model system (3) restricted to the positive S-axes has the form F1 (S, 0, 0, 0, 0, 0) = Λ − µS, F2 (S, 0, 0, 0, 0, 0) = F3 (S, 0, 0, 0, 0, 0) = · · · = F6 (S, 0, 0, 0, 0, 0) = 0.
(6)
Since F1 (S, 0, 0, 0, 0, 0) > 0 for S > 0 and Λ > µS the vector field point towards the interior of ΩT . On the positive ETj , ITj , (j = 1, 2) and RT -axis we have, F1 (0, ET1 , 0, 0, 0, 0) = F1 (0, 0, IT1 , 0, 0, 0) = F1 (0, 0, 0, RT , 0, 0) (7) = F1 (0, 0, 0, 0, ET2 , 0) = F1 (0, 0, 0, 0, 0, IT2 ) = Λ, implying that the vector field point towards the interior of ΩT on each axis. Thus region ΩT is positively invariant with respect to system (3). Adding the equations in model system (3) together we obtain, 2 X dTj ITj NT′ (t) = Λ − µNT (t) − (8) j=1
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P Λ − µ + 2j=1 dTj NT (t) ≤ NT′ (t) ≤ Λ − µNT (t).
(9)
so that, Thus NT (t) is bounded.
2.1.
Disease-Free Equilibrium and Stability Analysis
Model system (3) has the disease-free equilibrium given by Λ , 0, 0, 0, 0, 0 . ET0 = (S0 , ET10 , IT10 , RT0 , ET20 , IT20 ) = µ
(10)
It can be shown that ET0 attracts the region ΩT0 = {(S, ET1 , IT1 , RT , ET2 , IT2 ) ∈ ΩT : ET1 = IT1 = RT = ET2 = IT2 = 0} . (11) The basic reproduction number R0 , defined as the expected number of secondary infections caused by an infective individual upon entering a totally susceptible population (Anderson and May [4], Diekman et al. [18], van den Driessche and Watmough [46]). The linear stability of ET0 is governed by the reproduction number RTT . Closely following [46], we have
F =
p1 λT1 (S + RT ) (1 − p1 )λT1 (S + RT ) p2 λT2 (S + RT ) (1 − p2 )λT2 (S + RT ) 0 0
and V =
(k1 + r1 + µ)ET1 + (δ1 λT1 + λT2 )ET1 (µ + dT1 + r2 )IT1 − (k1 + δ1 λT1 )ET1 (k2 + µ + δ2 λT2 )ET2 − λT2 ET2 − (1 − q)r2 IT1 (µ + dT2 )IT2 − (k2 + δ2 λT2 )ET2 2 X λTj RT − qr2 IT1 − r1 ET1 µRT + j=1
µS +
2 X j=1
λTj S − Λ
.
(12)
The infected compartments are ET1 , IT1 , ET2 and IT2 . Thus matrices F and V for the new infection terms and the remaining terms are respectively given by 0 p1 β1 cT 0 0 0 (1 − p1 )β1 cT 0 0 and F = 0 0 0 p2 β2 cT 0 0 0 (1 − p2 )β2 CT (13) k1 + r1 + µ 0 0 0 −k1 µ + dT1 + r2 0 0 V = 0 −(1 − q)r2 k2 + µ 0 0 0 −k2 µ + dT2 The dominant eigenvalues of F V −1 are RT2 =
(1 − p2 )β2 cT µ + β2 cT k2 , (µ + k2 )(µ + dT2 ) (14)
RT1
(µ + r1 )(1 − p1 )β1 cT + β1 cT k1 . = (µ + dT1 + r2 )(µ + k1 + r1 )
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Thus the reproduction number for model system (3) is given by ρ(F V −1 ) = RTT = max {RT1 , RT2 } ,
(15)
where RT1 and RT2 are respectively the reproduction numbers for the drug sensitive TB only and drug resistant TB only. Theorem 2 follows from Theorem 2 in [46]. Theorem 2. The disease-free equilibrium ET0 is locally asymptotically stable whenever RTT < 1 and unstable otherwise. In the following section we analyse the endemic equilibria states.
2.2.
Endemic Equilibria
For the model system (3), we have the following steady states. 2.2.1.
Drug Resistant TB-only Equilibrium
The drug resistant TB only equilibrium occurs when ET1 = IT1 = RT = 0 and is denoted by ET∗1 = (S ∗ , 0, 0, 0, ET∗2 , IT∗2 ) (16) where in terms of the equilibrium value of the force of infection λ∗T2 we have S∗ =
IT∗2 =
NT∗
p2 Λλ∗T2 Λ ∗ , , ET2 = ∗ µ + λ∗T2 δ2 λ∗2 T2 + (δ2 µ + k2 + µ)λT2 + µ(µ + k2 ) (k2 + µ + δ2 λ∗T2 )λ∗T2 Λ − µp2 λ∗T2 Λ , ∗ + µ(µ + k ) (µ + dT2 ) δ2 λ∗2 (δ µ + k + µ)λ + 2 2 2 T2 T2
(17)
∗ Λ δ2 λ∗2 T2 + (k2 + µ + p2 dT2 + δ2 µ + δ2 dT2 )λT2 + (k2 + µ)(µ + dT2 ) = . ∗ + µ(µ + k ) (µ + dT2 ) δ2 λ∗2 (δ µ + k + µ)λ + 2 2 2 T2 T2
Substituting equation (17) into expression for λ∗T2 in (1), we have
∗ λ∗T2 f (λ∗T2 ) = λ∗T2 (A1 λ∗2 T2 + B1 λT2 + C1 ) = 0,
(18)
where λ∗T2 = 0 corresponds to the disease free equilibrium and f (λ∗T2 ) = 0 corresponds to the existence of endemic equilibria which implies, p −B1 ± B12 − 4A1 C1 ∗ (19) λT2 = 2A1 where, δ2 k2 + µ + p2 dT2 + δ2 (µ + dT2 ) − β2 cT δ2 , B1 = , (µ + dT2 )(µ + k2 ) (µ + dT2 )(µ + k2 ) and C1 = 1 − RT2 .
A1 =
(20)
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By examining the quadratic equation we see that there is a unique endemic equilibrium if B1 < 0 and C1 = 0 or B12 − 4A1 C1 = 0, there are two if C1 > 0, B1 < 0 and B12 − 4A1 C1 > 0, and there is none otherwise. The coefficient A1 is always positive and C1 is positive or negative if RT2 is less than or greater than one respectively. We therefore rewrite these conditions in Lemma 1. Lemma 1. Model system (3) has (i) precisely one unique endemic equilibrium if C1 < 0 ⇔ RT2 > 1, (ii) precisely one unique endemic equilibrium if B1 < 0 and C1 = 0 or B12 − 4A1 C1 = 0, (iii) precisely two endemic equilibria if C1 > 0, B1 < 0 and B12 − 4A1 C1 > 0, (iv) otherwise there is none. To find the backward bifurcation point, we set the discriminant B12 − 4A1 C1 = 0 and make RT2 the subject of the formulae to obtain RcT2 = 1 −
B12 , 4A1
(21)
from which it can be shown that backward bifurcation occurs for values of RT2 in the range RcT2 < RT2 < 1.
Lemma 2. The endemic equilibrium point ET∗1 exists for RT2 > 1. p −B1 + B12 − 4A1 C1 ∗ Proof. Analysing expression, λT2 = from which is clear that the 2Ap 1 −B1 + B12 − 4A1 C1 disease is endemic only when λ∗T2 > 0 ⇒ > 0 ⇒ B12 − 4A1 C1 > 2A1 B12 ⇒ 4A1 (1 − RT2 ) < 0 ⇒ RT2 > 1 since A1 is positive. RT2 > 1.
Thus ET∗1 exists for
Figure 2 is a graphical representation showing the proportion of drug resistant TB infectives against the reproduction number RT2 . It shows that making the reproduction number less than unity does not necessarily eradicate the epidemic. Using the standard linearisation of the two strain TB model to determine the local asymptotic stability of an endemic equilibrium point is boring and laborious to track mathematically. We now employ the Centre Manifold theory [11] as described in [15] (Theorem 4.1), to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in oder to apply this method. P Let S = x1 , ET1 = x2 , IT1 = x3 , RT = x4 , ET2 = x5 and IT2 = x6 , so that NT = 6n=1 xn . We now use the vector notation X = (x1 , x2 , x3 , x4 , x5 , x6 )T and then, the model system (3) can be written in the dX form = F = (f1 , f2 , f3 , f4 , f5 , f6 )T as shown in Appendix A. The Jacobian matrix of dt system (97) at ET0 in Appendix A is given by −µ 0 0 J(ET0 ) = 0 0 0
0 −(k1 + r1 + µ) k1 r1 0 0
−β1 cT p1 β1 cT (1 − p1 )β1 cT − (µ + dT1 + r2 ) qr2 (1 − q)r2 0
0 0 0 −µ 0 0
0 0 0 0 −(µ + k2 ) k2
−β2 cT 0 0 0 p2 β2 cT (1 − p2 )β2 cT − (µ + dT2 )
.
(22)
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Figure 2. Simulation results showing the backward bifurcation for drug resistant TB cases against RT2 . Bold and dashed lines show stable and unstable states of the equilibrium points respectively. Numerical values used are obtained from Table 1. From (22) it can be shown that RT2 =
(1 − p2 )β2 cT µ + β2 cT k2 , (µ + k2 )(µ + dT2 ) (23)
RT1
(µ + r1 )(1 − p1 )β1 cT + β1 cT k1 = . (µ + dT1 + r2 )(µ + k1 + r1 )
If β2 is taken as a bifurcation point and if we solve RT2 = 1 for β2 we obtain, β2 = β∗ =
(µ + k2 )(µ + dT2 ) . µ(1 − p2 )cT + cT k2
(24)
The linearised system of the transformed equation (97) with β2 = β∗ , has a simple zero eigenvalue. Thus the Centre Manifold theory [11] can be applied in the analysis of the
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dynamics of (97) near β2 = β∗ . The Jacobian of (97) at β2 = β∗ has a right eigenvector associated with the zero eigenvalue given by u = [u1 , u2 , u3 , u4 , u5 , u6 ]T where u1 = −
β1 cT u3 + β∗ cT u6 , µ
u2 =
β1 cT p1 u3 ((µ + dT1 + r2 ) − (1 − p1 )β1 cT ) u3 ⇒ RT1 = 1, u3 = u3 > 0 = k1 k1 + r1 + µ
u4 =
(µ + dT2 − (1 − p2 )β∗ cT ) u6 ((qr2 (k1 + r1 + µ) + p1 β1 cT k1 ) u3 , u5 = , µ(k1 + r1 + µ) k2
u6 =
(1 − q)r2 k2 u3 >0 (k2 + µ)(µ + dT2 ) − β∗ cT k2 − β∗ cT µ(1 − p2 )
for (k2 + µ)(µ + dT2 ) > β∗ cT k2 + β∗ cT µ(1 − p2 ).
(25) The left eigenvector associated with the zero eigenvalue at β2 = β∗ is given by w = [w1 , w2 , w3 , w4 , w5 , w6 ]T where w1 = 0, w2 = w5 =
k1 w3 w3 , w3 = w3 > 0, w4 = 0, k1 + r1 + µ
((µ + k1 + r1 )(µ + dT1 + r2 ) − β1 cT (k1 + r1 + µ − p1 (r1 + µ))) w3 >0 (k1 + µ + r1 )(1 − q)r2
for 1 > RT1 , w6 =
µ + k2 w5 . k2
(26)
Further we use Theorem 3 proven by Castillo-Chavez and Song [15]. Theorem 3. Consider the following general system of ordinary differential equations with a parameter φ dx = f (x, φ), f : Rn × R → R and f ∈ C2 (Rn × R), dt
(27)
where 0 is an equilibrium of the system that is f (0, φ) = 0 for all φ and assume ∂fi A1: A = Dx f (0, 0) = (0, 0) is the linearisation of system (27) around the ∂xj equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts; A2: Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue.
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Let fk be the kth component of f and
a=
n X
vk u i u j
k,i,j=1
b=
n X
k,i=1
∂ 2 fk (0, 0), ∂xi ∂xj (28)
∂ 2 fk (0, 0). vk u i ∂xi ∂φ
The local dynamics of (27) around 0 are totally governed by a and b. i. a > 0, b > 0. When φ < 0 with |φ| 0, and there is none otherwise. The coefficient A2 is always positive and C2 is positive or negative if RT1 is greater than or less than one respectively. We therefore rewrite these conditions in Lemma 3. Lemma 3. Model system (3) has (i) precisely one unique endemic equilibrium if C2 < 0 ⇔ RT1 < 1, (ii) precisely one unique endemic equilibrium if B2 < 0 and C2 = 0 or B22 − 4A2 C2 = 0, (iii) precisely two endemic equilibria if C2 > 0, B2 < 0 and B22 − 4A2 C2 > 0, (iv) none therwise. For this endemic equilibrium we can show that it also exists for RT1 < 1. Lemma 4. The endemic equilibrium point ET∗2 exists for RT1 < 1. p B22 − 4A2 C2 Proof. Analysing the equation = 0 we get = from 2A2 which it is clear that the disease is endemic when λ∗T1 > 0 ⇒ B22 − 4A2 C2 > B22 ⇒ 4A2 (µ + k1 + r1 )(µ + dT1 + r2 )(1 − RT2 ) > 0 ⇒ RT1 < 1. Thus the endemic equilibrium point ET∗2 also exists for RT1 < 1. g(λ∗T1 )
λ∗T1
−B2 +
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Lemma 4 suggests that making the reproduction number less than unity will not result in a decrease of the disease as the disease can even spread when reproduction number is less than unity. The stability of the equilibrium point ET∗2 it can be shown using the Centre Manifold theory manifold theory similar to the analysis of ET∗1 in the previous section but, is not shown here to avoid repetition. 2.2.3.
Co-existence of the Two TB Strains Endemic Equilibrium
This in terms of the force(s) of infection is given by ET∗3 = (S ∗∗∗ , ET∗∗∗ , IT∗∗∗ , RT∗∗∗ , ET∗∗∗ , IT∗∗∗ ) 1 1 2 2
(32)
where, S ∗∗∗ =
ET∗∗∗ = 1
IT∗∗∗ = 1
RT∗∗∗ =
IT∗∗∗ = 2 where,
Λ µ+
λ∗∗ T1
+ λ∗∗ T2
θ1 + θ2 λ∗∗ T1
θ1 + θ2 λ∗∗ T1
,
Λp1 λ∗∗ T1 (µ + dT1 + r2 ) ∗∗2 ∗∗ ∗∗ ∗∗2 , + θ3 λT1 + θ4 λ∗∗ T2 + θ5 λT1 λT2 + θ6 λT2 ∗∗2 ∗∗ ∗∗ y1 λ∗∗ T1 + y2 λT1 + y3 λT1 λT2 ∗∗ ∗∗ ∗∗ ∗∗2 + θ3 λ∗∗2 T1 + θ4 λT2 + θ5 λT1 λT2 + θ6 λT2
(33)
∗∗∗ + R∗∗∗ ) (1 − q)r2 IT∗∗1 + p2 λ∗∗ r1 ET∗∗∗ + qr2 IT∗∗∗ T2 (S T ∗∗∗ 1 1 , , E = P2 T2 δ2 λT∗∗2 + µ + k2 µ + j=1 λT∗∗j
1 ∗∗ ∗∗∗ , (1 − p2 )(S ∗∗∗ + RT∗∗∗ )λ∗∗ T2 + (k2 + δ2 λT2 )ET2 µ + dT2
θ1 = (µ + k1 + r1 )(µ + dT1 + r2 )µ, θ2 = (µ + k1 + r1 )(µ + dT1 + r2 − qr2 + p1 qr2 ) + (µ + dT1 + r2 )(δ1 µ − p1 r1 ) +k1 p1 qr2 , θ3 = δ1 (µ + dT1 + r2 − qr2 ) , θ4 = (µ + dT1 + r2 )(2µ + k1 + r1 ), θ5 = (µ + dT1 + r2 )(1 + δ1 − qr2 + p1 qr2 ), θ6 = µ + dT1 + r2 , y1 = Λ ((µ + dT1 + r2 )(1 − p1 ) + k1 p1 ) , y2 = Λδ1 , y3 = Λ(1 − p1 ),
(34) and is illustrated in Figure 3. Figure 3 shows that chemoprophylaxis and treatment of drug sensitive TB cases result in a significant decrease in the number of drug sensitive latent and active TB cases. A decrease in the number of drug sensitive TB on treatment also results in the decrease in the number of drug resistant TB cases as most cases of multi-drug resistant TB are a result of treatment failure and incomplete treatment. The recovered population starts by increasing following successful chemoprophylaxis and treatment, reaches its peak within the first twenty years and there after it shows a decrease due to a reduction in the number of drug sensitive TB cases. Stability analysis of ET∗3 can be done using the centre manifold theory similar to ET∗1 in Section 2.2.1, but is not shown here to avoid repetition.
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Figure 3. Simulation results showing the co-existence of drug sensitive TB and drug resistant TB in the presence of chemoprophylaxis and treatment of drug sensitive infectives. Numerical values used are obtained from Table 1.
We now illustrate some numerical simulations of the two strain TB model for different values of the basic reproduction number and varying initial conditions in Figure 4. The fourth order Runge-Kutta numerical scheme coded in C++ programming language and parameter values in Table 1 are used to carry out the numerical simulations. In Figure 4(a) we have RT1 > 1 > RT2 , then the drug sensitive TB strain will attain an endemic equilibrium with time while the drug resistant TB will disappear in the population. This is typical in cases where there is little or no treatment coverage of TB. This is in support of the argument that drug resistant TB is a result of incomplete treatment, implying that in the absence of treatment, there is little or no drug resistance. This is in agreement with Castillo-Chavez and Feng [12]. In Figure 4(b), both reproduction numbers are greater than unity (RT2 > RT1 > 1), and the two TB strains attain endemic states and co-exist in the population with time. In Figure 4(c), both reproduction numbers are less than
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(a)
(b)
(c)
(d)
Figure 4. Simulations of model system (3) showing plots of individuals with drug sensitive TB infectives-only and drug resistant TB infectives-only as a function of time with various initial conditions. (a) RT1 > 1 > RT2 , (β2 = 0.1, β1 = 0.6, cT = 40), so that RT1 = 1.23, RT2 = 0.813 and RTT = 1.23. (b) RT2 > RT1 > 1, (β2 = 0.6, β1 = 0.6, cT = 40), so that RT1 = 1.23, RT2 = 4.884 and RTT = 4.884. (c) RT1 < RT2 < 1, (β2 = 0.35, β1 = 0.35, cT = 3), so that RT1 = 0.00891, RT2 = 0.213 and RTT = 0.213. (d) RT2 > 1 > RT1 , (β2 = 0.6, β1 = 0.35, cT = 10), so that RT1 = 0.0297, RT2 = 1.2209 and RTT = 1.2209. Parameter used are as in Table 1. unity (RT1 > RT2 < 1), and both TB strains will with time disappear in the population. This is typical in settings where there is effective monitoring and treatment of drug sensitive TB cases. Strong monitoring and effective treatment will imply no default and incomplete treatment of drug sensitive TB which is a major cause of drug resistance. When RT1 < 1 < RT2 , then drug sensitive TB will disappear in the population and the drug resistant strain TB will exist in the population as shown in Figure 4(d). This is likely to
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happen when the major causes of drug resistance is infection with the drug resistant strain other than inappropriate treatment regimen. Next we illustrate the effects of increasing treatment rates on the population with sensitive and resistant TB strains in Figure 5. Figure
Figure 5. Simulations of model showing plots of total number of individuals with TB. The direction of the arrow shows an increase in treatment rates from r2 = 0.4 with step size 0.1. Parameter values used are in Table 1. 5 shows that increasing treatment rates for drug sensitive TB results in decrease of total TB cases.
3.
A Two Strain HIV/AIDS Only Model
In this section we present and analyse a two strain HIV/AIDS model. The two strain HIV/AIDS model divides the population into the susceptibles S, HIV positive individuals with antiretroviral sensitive strain with no AIDS symptoms IH1 , AIDS individuals who have their health improved as result of using antiretroviral therapy AHt and this group has no AIDS-defining symptoms and do not die from AIDS, AIDS individuals with drug sensitive HIV showing AIDS symptoms AH1 , HIV positive individuals with antiretroviral resistant strain with no AIDS symptoms IH2 and AIDS individuals with antiretroviral resistant HIV showing AIDS symptoms AH2 . The population considered for this model is the sexually active population since HIV/AIDS is predominantly sexually transmitted in Sub-Saharan Africa. Individuals are recruited into the susceptible population at rate Λ and
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are infected with HIV/AIDS following sexual contact with an HIV infected individual at λAi , i = 1, 2 where i = 1 and i = 2, denotes drug sensitive and drug resistant strain respectively with
λ A1 =
βH2 cH (IH2 + ηA AH2 ) (t) βH1 cH (IH1 + ηA (AHt + AH1 )) (t) , λA2 = . NH (t) NH (t)
(35)
where βHi is the probability of getting an HIV following sexual contact with one infectious individual, cH is the number of sexual partners per unit time, ηA > 1 models the fact that HIV positive individuals in the AIDS stage have a higher viral load and therefore are more infectious than HIV positive individuals only and NH (t) is the total population size and is given by
NH (t) = S(t) + AHt (t) +
2 X
(IHi + AHi ) (t).
(36)
i=1
Susceptibles are infected with HIV at rate λAi , enters the HIV positive stage IHi . Individuals in IHi progress to the AIDS stage AHi at rate ρi . AIDS patients with antiretroviral sensitive HIV who are ill and displaying AIDS symptoms are treated at rate α, a proportion ν enters the the AIDS treated class AHt , and the complementary (1 − ν) develop drug resistance and move into AH2 . Treated AIDS move back to AH1 due to treatment failure at a rate θ. Individuals in each stage have a constant natural death rate µ. Individuals in the AIDS stage AHi , displaying AIDS symptoms have an additional disease-induced rate dAi . Parameters described in this section will assume values in Table 2 The model flow diagram is given in Figure 6.
Table 2. Model parameters and their interpretations. Parameter Recruitment rate Natural mortality rate Sexual partners per unit time Proportion of effectively treated Rate of progression to AIDS Modification parameter AIDS related death rate Probability of being infected with HIV Treatment rate for AIDS cases Drug failure rate
Symbol Λ µ cH ν (ρ1 , ρ2 ) η (dA1 , dA2 )
Value 0.029yr−1 0.02yr−1 3 yr−1 0.85 (0.028-0.19)−1 1.02yr−1 (0.333,0.4)yr−1
Source Mukandavire and Garira [33] Mukandavire and Garira [33] Mukandavire and Garira [33] Estimated Hyman et al. [27] Estimated Mukandavire and Garira [33]
(βH1 , βH2 ) α θ
0.011-0.95 0.33yr−1 0.15yr−1
Hyman et al. [27] Mukandavire and Garira [33] Mukandavire and Garira [33]
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Figure 6. Structure of model. Based on these assumptions we have the following system of equations to describe the the model. 2 X λAi S − µS, S ′ (t) = Λ − i=1
′ (t) = λ S − (ρ + µ)I , IH 1 A1 H1 1
A′Ht (t) = ναAH1 − (µ + θ)AHt ,
(37)
A′H1 (t) = ρ1 IH1 − (µ + dA1 + α)AH1 + θAHt , ′ (t) = λ S − (ρ + µ)I , IH 2 A2 H2 2
A′H2 (t) = ρ2 IH2 − (µ + dA2 )AH2 + (1 − ν)αAH1 . All parameters and state variables for model system (37) are assumed to be non-negative
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for t ≥ 0. Consider the region Λ ΩE = (S, IH1 , AHt , AH1 , IH2 , AH2 ) : NH ≤ . µ It can be shown that all solutions of system (37) starting in ΩE remain in ΩE for all t ≥ 0. Thus, ΩE is positively invariant and it is sufficient to consider solutions in ΩE . Existence, uniqueness and continuation results for system (37) hold in this region.
3.1.
Disease-Free Equilibrium and Stability Analysis
Model system (37) has the diseases free equilibrium given by Λ EH0 = (S, IH1 , AHt0 , AH10 , IH20 , AH20 ) = , 0, 0, 0, 0, 0 . µ
(38)
It can be shown that EH0 attracts the region ΩH0 = {(S, IH1 , AH1 , IH2 , AH2 ) ∈ ΩE : IH1 = AH1 = IH2 = AH2 = 0} . The basic reproduction number, RA is determined following [46] to obtain, (ρ1 + µ)IH1 λ A1 S 0 (µ + θ)AHt − ναAH1 (µ + dA1 + α)AH1 − ρ1 IH1 − θAHt 0 F = λA S and V = (ρ2 + µ)IH2 2 (µ + dA2 )AH2 − (1 − ν)αAH1 − ρ2 IH2 0 P2 0 i=1 λAi S + µS − Λ
(39)
.
(40)
The infected compartments are IH1 , AHt , AH1 , IH2 and AH2 . Thus the matrices F and V for the new infection terms and the remaining transfer terms are respectively given by, βH1 cH βH1 cH ηA βH1 cH ηA 0 0 0 0 0 0 0 and, 0 0 0 0 0 F = 0 0 0 βH2 cH βH2 cH ηA 0 0 0 0 0 (41) ρ1 + µ 0 0 0 0 0 µ+θ −να 0 0 . −θ µ + dA1 + α 0 0 V = −ρ1 0 0 0 ρ2 + µ 0 0 0 −(1 − ν)α −ρ2 µ + dA2 The dominant eigenvalues of F V −1 are RA1 =
αβH1 cH (θ + µ) + βH1 cH ηA ρ1 (θ + µ + να) + βH1 cH ((µ + θ)(µ + dA1 ) − θνα) , ((dA1 + α + µ)(θ + µ) − θνα)(µ + ρ1 )
RA2 =
βH2 cH (dA2 + µ) + βH2 cH ηA ρ2 . (dA2 + µ)(ρ2 + µ) (42)
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Thus the reproduction number for model system (37) is the spectral radius ρ(F V −1 ) = RA = max {RA1 , RA2 } ,
(43)
where RA1 and RA2 are reproduction numbers for antiretroviral sensitive HIV only and antiretroviral resistant HIV only respectively. From Theorem 2 in [46] the following holds. Theorem 5. The disease-free equilibrium point EH0 is locally asymptotically stable whenever RA < 1, and unstable otherwise. Following Castillo-Chavez [14] we write model system (37) as X ′ (t) = F (X, Y ) (44) Y
′ (t)
= G(X, Y ), G(X, 0) = 0
where X = (S) and Y = (IH1 , AHt , AH1 , IH2 , AH2 ) with X ∈ R+ (its components) denoting the number of unifected individuals and Y ∈ R5+ (its components) denoting the number of infected individuals. The disease free equilibrium is now denoted by EH0 = Λ (X0 , 0) where X0 = . Conditions (H1) and (H2) in equation (45) below guarantee µ global asymptotic stability of EH0 if met. H1 : For X ′ (t) = F (X ∗ , 0), X ∗ is globally stable (45) b b H2 : G(X, Y ) = AY − G(X, Y ), G(X, Y ) ≥ 0 for (X, Y ) ∈ ΩE ,
where A = DY G(X ∗ , 0) is an M-matrix (the off diagonal elements of A are non-negative). If system (37) satisfies the conditions in (45)then Theorem 6 holds. Theorem 6. The fixed point EH0 is a globally asymptotically stable point of model system (37) provided RA < 1 and assumptions in (45) are satisfied. Proof. Consider F (X, 0) = [Λ − µS],
A=
−(ρ1 + µ) + βH1 cH 0 ρ1 0 0
b and G(X, Y)=
c1 (X, Y ) G c2 (X, Y ) G c3 (X, Y ) G c4 (X, Y )) G c5 (X, Y ) G
βH1 ηA cH −(µ + θ) θ 0 0
βH1 ηA cH 0 να 0 −(µ + dA1 + α) 0 0 −(ρ2 + µ) + βH2 cH (1 − ν)α ρ2
S β c (I + η (A + A )) 1 − A Ht H1 H1 H H1 NH 0 0 = S β c (I + η A ) 1 − H2 H H2 A H2 NH 0
0 0 0 βH2 ηA cH −(µ + dA2 )
.
(46)
b Therefore G(X, Y ) ≥ 0 for all (X, Y ) ∈ ΩA implying that EH0 is globally asymptotically stable for RA < 1.
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C.P. Bhunu and W. Garira
Endemic Equilibria
Model system (37) has three endemic equilibria, the antiretroviral resistant HIV-strain only, antiretroviral sensitive HIV-strain only and the coexistence of both strains equilibria. We start by presenting an analysis of the drug resistant only equilibrium. 3.2.1.
Anti-retroviral Resistant HIV-strain Only Equilibrium
This occurs when IH1 = AHt = AH1 = 0 and is given by ∗ ∗ EH = (S2∗ , 0, 0, 0, IH , A∗H2 ) where 1 2
S2∗ =
∗ NH
Λλ∗A2 ρ2 λ∗A2 Λ Λ ∗ ∗ I = A = , , , H2 µ + λ∗A2 H2 (µ + λ∗A2 )(ρ2 + µ) (µ + λ∗A2 )(ρ2 + µ)(µ + dA2 )
Λ (µ + ρ2 )(µ + dA2 ) + λ∗A2 (µ + dA2 + ρ2 ) , = (µ + λ∗A2 )(µ + ρ2 )(µ + dA2 )
(47) in terms of the force of infection λ∗A2 . Substituting equation (47) into the equation for the force of infection λ∗A2 we have, λ∗A2 h2 (λ∗A2 ) = λ∗A2 (C1 λ∗A2 + C2 ) = 0,
(48)
where λ∗A2 = 0 corresponds to the disease free equilibrium and h2 (λ∗A2 ) = 0 corresponds to the existence of endemic equilibrium point where C1 =
µ + d A2 + ρ 1 , C2 = 1 − RA2 . (µ + dA2 )(µ + ρ2 )
(49)
C1 is always positive and C2 is negative or positive if RA2 is greater than or less than one. ∗ exists whenever R Lemma 5. The endemic equilibrium, EH A2 > 1. 1
Proof. By examining the linear equation C1 λ∗A2 + C2 = 0 we have that λ∗A2 = −
RA2 − 1 C2 = . C1 C1
(50)
But the disease is endemic when the force of infection λ∗A2 > 0 which implies RA2 > 1. ∗ exists whenever R Therefore the endemic equilibrium EH A2 > 1. 1 We now employ the centre manifold theory [11] as described in [15] (Theorem 4.1), to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in order to apply the Center Manifold theory S = x1 , IH1 = P6 x2 , AHt = x3 , AH1 = x4 , IH2 = x5 and AH2 = x6 , so that NH = n=1 xn . We now use the vector notation X = (x1 , x2 , x3 , x4 , x5 , x6 )T . Then model system (37) can be dX written in the form = F = (f1 , f2 , f3 , f4 , f5 , f6 )T , (see Appendix C). The Jacobian dt
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matrix of system (102) at EH0 in Appendix C is given by
J(EH0 ) =
−µ 0 0 0 0 0
−βH1 cH βH1 cH − (ρ1 + µ) 0 ρ1 0 0
−βH1 ηA cH βH1 ηA cH −(µ + θ) θ 0 0
−βH1 ηA cH βH1 ηA cH να −(µ + dA1 + α) 0 (1 − ν)α
−βH2 cH 0 0 0 βH2 cH − (ρ2 + µ) ρ2
−βH2 ηA cH 0 0 0 βH2 ηA cH −(µ + dA2 ) (51)
From which it can be shown that the reproduction numbers are
RA1 =
αβH1 cH (θ + µ) + βH1 cH ηA ρ1 (θ + µ + να) + βH1 cH ((µ + θ)(µ + dA1 ) − θνα) , ((dA1 + α + µ)(ρ1 + µ) − θνα)(µ + ρ1 )
RA2 =
βH2 cH (dA2 + µ) + βH2 cH ηA ρ2 . (dA2 + µ)(ρ2 + µ)
(52) If βH2 is taken as a bifurcation point and if we consider RA2 = 1 and solve for βH2 we get
βH2 = β ∗2 =
(dA2 + µ)(ρ2 + µ) . cH (dA2 + µ + ηA ρ2 )
(53)
Note that the linearised system of the transformed equation (102) with βH2 = β ∗2 has a simple zero eigenvalue. Using the Centre Manifold theory [11] to analyze the dynamics of (102) near βH2 = β ∗2 , the Jacobian of (102) at βH2 = β ∗2 has a right eigenvector associated with the zero eigenvalue given by z = [z1 , z2 , z3 , z4 , z5 , z6 ]T where
z1 = − z2 =
βH1 cH z2 + βH1 ηA cH (z3 + z4 ) + β ∗2 cH z5 + β ∗2 ηA cH z6 , µ
θ (µ + dA1 + α)(µ + θ) βH1 cH ηA (µ + θ + να) z3 = − z3 + z3 ⇒ RA1 = 1, (ρ1 + µ − βH1 cH )να ρ1 ναρ1
z3 = z3 > 0, z4 =
µ+θ z3 , να
z5 =
β ∗2 ηA cH (1 − ν)(ν + θ) z3 , ν ((µ + dA2 )(ρ2 + µ − β ∗2 cH ) − ρ2 β ∗2 cH ηA )
z6 =
β ∗2 ηA cH (1 − ν)(ν + θ)(ρ2 + µ − β ∗2 cH ) z3 . ν ((µ + dA2 )(ρ2 + µ − β ∗2 cH ) − ρ2 β ∗2 cH ηA )
The left eigenvector of J(EH0 ) associated with the eigenvalue at βH2 =
β ∗2
(54) is given by
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y = [y1 , y2 , y3 , y4 , y5 , y6 ]T , where y1 = 0, y2 =
ρ1 (µ + θ) y3 , y3 = y3 > 0, ρ1 βH1 cH ηA + θ(ρ1 + µ − βH1 cH )
y4 =
(µ + θ)(µ + ρ1 − βH1 cH ) y3 , ρ1 βH1 cH ηA + θ(ρ1 + µ − βH1 cH )
y5 =
ρ2 (µ + dA2 ) y6 = y6 , β ∗2 ηA cH ρ2 + µ − β ∗2 cH
(ρ1 + µ − βH1 cH )[(µ + dA1 + α)(µ + θ) − ναθ] − βH1 ηA cH ρ1 (µ + θ + να) y3 . (1 − ν)(ρ1 βH1 ηA cH + θ(ρ1 + µ − βH1 cH )) (55) For the computations of a and b see Appendix D. Using Theorem 3 item (iv), we establish the following result. y6 =
∗ is locally asymptotically stable Theorem 7. If RA1 < 1, the endemic equilibrium point EH 1 for RA2 > 1 but close to 1.
3.2.2.
Antiretroviral Sensitive HIV-strain Only Equilibrium
This occurs when IH2 = AH2 = 0, ν = 1 and is given by ∗ ∗ EH = (S1∗ , IH , A∗Ht , A∗H1 , 0, 0) where 2 1
S1∗
Λλ∗A1 Λ ∗ , , I = = µ + λ∗A1 H1 (µ + λ∗A1 )(ρ1 + µ)
A∗Ht =
A∗H1
αρ1 Λλ∗A1 , (µ + λ∗A1 )(ρ1 + µ) [(µ + θ)(µ + dA1 ) + µα]
(56)
(µ + θ)ρ1 Λλ∗A1 = , (µ + λ∗A1 )(ρ1 + µ) [(µ + θ)(µ + dA1 ) + µα]
in terms of of the force of infection λ∗A1 . Substituting (56) into the equation for the force of infection λ∗A1 we have λ∗A1 h1 (λA1 ) = λ∗A1 (B1 λ∗A1 + B2 ) = 0,
(57)
where λ∗A1 = 0 corresponds to the disease free equilibrium and h1 (λ∗A1 ) = 0 corresponds to the existence of endemic equilibrium point where, B1 = (µ + θ)(µ + dA1 + ρ1 ) + α(µ + ρ1 ), B2 = (ρ1 + µ)[(µ + θ)(µ + dA1 ) + αµ](1 − RA1 ). (58) B1 is always positive and B2 is negative or positive if RA1 is greater than or less than one. ∗ exists whenever R Lemma 6. The endemic equilibrium EH A1 > 1. 2
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Proof. By examining the linear equation B1 λ∗A1 + B2 = 0 we have that λ∗A1 = −
B2 (ρ1 + µ)[(µ + θ)(µ + dA1 ) + αµ](RA1 − 1) = . B1 B1
(59)
But the disease is endemic when the force of infection λ∗A1 > 0 ⇒ RA1 > 1. Therefore the ∗ exists whenever R endemic equilibrium EH A1 > 1. 2 ∗ we use the Centre Manifold theory similar to the of E ∗ in the For stability analysis of EH H1 2 previous section.
3.2.3.
Co-existence of Both HIV Strains Endemic Equilibrium
When both HIV strains co-exist the endemic equilibrium in terms of the force(s) of infection ∗ = (S ∗∗ , I ∗∗ , A∗∗ , A∗∗ , I ∗∗ , A∗∗ ) where is given by EH 2 H1 Ht H1 H2 H2 3 S2∗∗ =
∗∗ IH = 1
A∗∗ Ht = A∗∗ H1 = ∗∗ IH = 2
A∗∗ H2
Λ µ+
∗∗ λA 1
(µ +
+ λ∗∗ A2
,
Λλ∗∗ A1 , + λ∗∗ A2 )(ρ1 + µ)
λ∗∗ A1
(µ + λ∗∗ A1
Λνρ1 αλ∗∗ A1 , + λ∗∗ A2 )(ρ1 + µ) ((µ + dA1 + α)(µ + θ) − νθα)
(µ + λ∗∗ A1
Λρ1 λ∗∗ A1 , + λ∗∗ )(ρ + µ) ((µ + dA1 + α)(µ + θ) − νθα) 1 A2
(60)
λ∗∗ A2 Λ ∗∗ , (µ + ρ2 )(µ + λ∗∗ A1 + λ A2 ) ∗∗ ρ Λ λA 2 2 ∗∗ + λ∗∗ ) (µ + ρ2 )(µ + λA A2 1
1 = µ + dA2
+
(1 − ν)αΛρ1 λ∗∗ A1 + λ∗∗ )(ρ + µ) ((µ + d + α)(µ + θ) − νθα) 1 A 1 A2
(µ + λ∗∗ A1
,
in terms of the equilibrium value of the forces of infection λ∗∗ Ai . ∗ exists whenever R > 1. Lemma 7. The endemic equilibrium EH A 3 ∗ . Since both strains Proof. We have to prove that RA1 > 1 and RA2 > 1 at the point EH 3 exist (S, IH1 , AHt , AH1 , IH2 , AH2 ) > 0. In terms of IH2 , AH1 we have
AH2 =
ρ2 IH2 (1 − ν)αAH2 ρ2 IH2 + , then AH2 ≥ . µ + d A2 µ + d A2 µ + d A2
(61)
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′ (t) in (37) at E ∗ , we have Substituting equation (61) into the equation for IH H3 2 ∗∗ + η A∗∗ )S ∗∗ βH2 cH (IH A H2 3 ∗∗ 2 − (ρ2 + µ)IH = 0, ∗∗ 2 NH ∗∗ + η A∗∗ ) − (ρ + µ)I ∗∗ > 0, but βH2 cH (IH 2 A H2 H2 2
βH2 cH (µ + dA2 + ηA ρ2 implying that IH2 − (ρ2 + µ)IH2 > 0, µ + d A2 giving
(62)
βH2 cH (µ + dA2 + ηA ρ2 ) > 1, hence RA2 > 1. (ρ2 + µ)(µ + dA2 )
In terms of IH1 , AHt we have AH1 =
ρ1 IH1 + θAHt µ+θ AHt and AH1 = να µ + d A1 + α (63)
implying that IH1
(µ + dA1 + α)(µ + θ) − νθα = AHt . ρ1 να
′ in (37)at E ∗ we have, Substituting (63) into the equation for IH H3 1 ∗∗ ∗∗ βH1 cH (IH + ηA (A∗∗ Ht + AH1 )) 1
S3∗∗ ∗∗ ∗∗ − (ρ1 + µ)IH1 = 0, NH
∗∗ + η (A∗∗ + A∗∗ )) − (ρ + µ)I ∗∗ > 0, giving βH1 cH (IH 1 A H1 Ht H1 1
then, βH1 cH A∗∗ Ht >
(µ + dA1 + α)(µ + θ) − ναθ + ρ1 ηA (µ + θ + να) ρ1 να
(ρ1 + µ) ((µ + dA1 + α)(µ + θ) − ναθ) ∗∗ AHt , which gives ρ1 αν
βH1 cH ((µ + dA1 + α)(µ + θ) − να) + βH1 ηA cH ρ1 (µ + θ + να) > 1 ⇒ RA1 > 1. (ρ1 + µ) ((µ + dA1 + α)(µ + θ) − ναθ) (64) ∗ exists whenever R > 1, since R Thus EH > 1 and R > 1 ⇔ R > 1. A A1 A2 A 3 ∗ can also be shown using the centre manifold Stability analysis of the equilibrium point EH 3 ∗ theory similar to the analysis of EH1 in the previous section but is not shown here to avoid repetition.
3.3.
Numerical Simulations
We now illustrate some numerical simulations of the two strain HIV model for different values of the basic reproduction number and varying initial conditions in Figure 7. The fourth
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order Runge-Kutta numerical scheme coded in C++ programming language and parameter values in Table 2 are used in carrying out the numerical simulation.
(a)
(b)
(c)
(d)
Figure 7. Simulations of model (37) showing plots of individuals with sensitive HIV strain only and resistant HIV strain only as a function of time with various initial conditions. (a) RA2 > 1 > RA1 (βH1 = 0.012, βH2 = 0.6), so that RA1 = 0.125, RA2 = 6.448 and RA = 6.448. (b) RA1 > 1 > RA2 (βH1 = 0.6, βH2 = 0.012), so that RA1 = 0.625, R2 = 0.128 and RA = 0.625. (c) RA1 < RA2 < 1 (βH1 = 0.012, βH2 = 0.012), so that RA1 = 0.125, RA2 = 0.128 and RA = 0.128. (d) RA1 > RA2 > 1 (βH1 = 0.2, βH2 = 0.2), so that RA1 = 2.111, RA2 = 1.912 and RA = 2.111. Parameters values used are as in Table 2. When RA2 > 1 > RA1 , resistant HIV strain will exist in the population as shown in Figure 7 (a) and when RA1 > 1 > RA2 , sensitive HIV strain will exist in the population as shown in Figure 7 (b). From the simulations in Figure 7 (a) we notice that when the reproduction number of a strain is less than unity and the other greater than unity, then the later strain
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exists as shown in Figure 7 (a) and (b). If the reproduction numbers of both strains are less than unity ( RA1 < RA2 < 1) then both strains will disappear in the population with time as shown in Figure 7 (c). Simulations for the case where the both reproduction numbers are greater than unity (RA1 > RA2 > 1) show that the two strains can co-exist as shown in Figure 7 (d). Next, we illustrate the effects of increasing antiretroviral treatment rates on the population with sensitive and resistant HIV strains in Figure 8.
(a)
(b)
(c) Figure 8. Simulations of model (37) showing plots of individuals with sensitive HIV strain only, resistant HIV strain only and AIDS patients as a function of time with various treatment rates. (a) population with sensitive HIV strain. (b) population with resistant HIV strain. (c) AIDS patients. The direction of the arrow shows an increase in treatment rates starting from θ = 0.3 with step size 0.1. Parameters values used are as in Table 2. Simulations in Figure 8 illustrates the effects of increasing treatment with amelioration as a single-strategy approach in a community. Figures 8 (a) and (b) show that treatment alone as an intervention strategy results in an increase in the number of infective individuals
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(asymptomatic that is with sensitive and resistant HIV strains) and the arrow show the direction of increase of the infected classes with increase in treatment rates. This suggests that treatment with amelioration intended to lengthen the lives of AIDS patients results in more HIV infections and thus may not benefit the community. This is in agreement with Mukandavire and Garira [33] and Hsu Schmitz [25, 26] that treatment significantly prolongs the incubation period but does not reduce infectiousness. The only way in which such treatment can help the community is when the efficacy of treatment drugs intended for this practice is 100 % [33]. Figure 8 (c) shows treatment alone results in a decrease in number of AIDS patients and the arrow shows the direction of decrease of the AIDS classes with increase in treatment rates. Thus, of course treatment with amelioration as a single-strategy approach enlarges the epidemic but will help the community by reducing HIV/AIDS morbidity/mortality and hence reduces the number of orphans in affected communities. In the next section we give an analysis of the effects of treatment and drug resistance in TB individuals co-infected with HIV/AIDS.
4.
Effects of Treatment and Drug Resistance in TB Individuals Co-infected with HIV/AIDS
In this section, we combine our earlier models (model systems (3) and (37)) to take into account possible HIV/AIDS and TB co-infections. We add the following additional classes EHi Tj (HIV positive exposed to TB), EAi Tj (AIDS individuals exposed to TB), IHi Tj (HIV positive individuals with TB), EAt Tj (treated AIDS cases exposed to TB), AAt Tj (treated AIDS cases with TB), and AAi Tj (AIDS cases with TB) with i = 1 (denoting drug sensitive HIV strain), i = 2 (drug resistant HIV strain), j = 1 (denoting drug sensitive TB) and j = 2 (denoting drug resistant TB). It is assumed that susceptible humans are recruited into the population at per capita rate Λ. Susceptible individuals acquire HIV infection following sexual contact with HIV infected individuls at a rate λHi , λH1
2 2 X X βH1 cH IH1 + IH1 Tj + EH1 Tj + = N j=1 j=1
2 2 X X βH1 cH ηA AH1 + AHt + ηHT (θHT (EA1 Tj + EAt Tj )) , (AA1 Tj + AAt Tj ) + N j=1 j=1 λH2
2 2 2 2 X X X X βH2 cH = (IH2 + IH2 Tj ) + ηA AH2 + ηHT (θHT EA2 Tj ) . EH2 Tj + AA2 Tj + N j=1 j=1 j=1 j=1 (65)
In (65), βHi , cH , are as defined for equation (35), the modification parameter ηHT ≥ 1 accounts for the relative infectiousness of individuals with HIV and in the AIDS stage dually infected with TB (EAi Tj , EAt Tj , AAt Tj and AAi Tj ), in comparison to those solely infected with HIV (IHi ). Further, θHT ≥ 1 models the fact that dually infected people in the AIDS stage of the disease displaying symptoms of TB (AAt Tj , AAi Tj ) are more infectious than the corresponding individuals in the AIDS stage who are only exposed to TB,
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(EAt Tj , EAi Tj ). Finally the parameter ηA > 1 captures the fact that individuals who are in the AIDS stage of infection are more infectious than HIV-infected individuals with no AIDS symptoms. This is so because people in the AIDS stage have a higher viral load compared to other HIV infected individuals with no symptoms and there is a positive correlation between viral load and infectiousness. Susceptibles acquire TB infection following contact with an infectious individual at a rate λTj with, λTj =
β1 cT (IT1 + AAt Tj +
P2
i=1 (AAi Tj
+ IHi Tj ))
N
(66)
,
where βj and cT are as defined for equation (1). The total population size is given by, N (t) = S(t) +
2 X
ETj (t) +
j=1
+ +
2 X
i=1 2 X
AHi +
2 2 X X
2 X
2 X
IHi + AHt
i=1
j=1
EHi Tj
(67)
i=1 j=1
(EAt Tj + AAt Tj ) +
j=1
ITj (t) + RT (t) +
2 X 2 X i=1 j=1
EAi Tj +
2 X 2 X i=1 j=1
IHi Tj +
2 X 2 X
AAi Tj .
i=1 j=1
It is assumed that individuals suffering from drug sensitive TB may recover and enter the recovered class (RT ) at rates r1 and qr2 following chemoprophylaxis for the exposed with drug sensitive Mtb and treatment for the drug sensitive TB infectives. All individuals in different human subgroups suffer from natural death (at a rate µ). Susceptibles infected with Mtb enter the exposed class at rate pj λTj and then progress to active TB at rate kj and the complementary proportion (1 − pj ) develop fast TB. Individuals infected with TB can acquire HIV at rates λHi and δj λHi for the exposed and infectives respectively. Individuals with TB suffer disease-induced death at rates dT1 and dT2 for drug sensitive TB and drug resistant TB, respectively. Individuals infected with HIV only (with no symptoms) is generated following infection at rate λHi . This further reduces following progression to AIDS at rate ρi and through being infected with Mtb to enter the EHi Tj class. Individuals infected with HIV exposed to TB (EHi Tj ) develop (i) active TB at rates ψ2 , ψ4 , ζ2 and ζ4 and (ii) AIDS at rates ψ1 , ψ3 , ζ1 and ζ3 . Individuals in IHi Tj class die due to TB at rate dTj and progress to AIDS at rates nl , l = 1, 2, 3, 4. The population of individuals with AIDS alone is generated following progression to AIDS of people infected with HIV only at rate ρi . Individuals in the AIDS stage with the drug sensitive HIV are treated at a rate α and a proportion ν move into the treated AIDS and the complementary proportion (1 − ν) develop drug resistance and move to the AIDS class with drug resistance. Treated AIDS cases move back to the AIDS class due to waning of the vaccine at rate θ. The population of individuals with AIDS and displaying TB symptoms is generated following progression to AIDS of people dually infected with no AIDS symptoms displaying TB symptoms and/or progression to TB of people with AIDS exposed to TB at rates nl and ml , respectively. Individuals with AIDS die at a constant rate dAj . AIDS individuals displaying symptoms of TB die at an increased death rate υj dAj with υj ≥ 1. The assumptions result in the following differential equations that describe the interaction of HIV and TB.
A Theoretical Assessment of the Effects of Chemoprophylaxis...
2 2 X X dS λHi S − λTj S − µS, =Λ− dt j=1 i=1 2
X dET1 λHi ET1 , = p1 λT1 (S + RT ) − (k1 + r1 + µ)ET1 − (δ1 λT1 + λT2 )ET1 − dt i=1 2
X dIT1 = (1 − p1 )λT1 (S + RT ) + k1 ET1 + δ1 λT1 ET1 − (µ + dT1 + r2 )IT1 − λHi IT1 , dt i=1 2
2
X X dRT λHi RT − λTj RT , = r1 ET1 + qr2 IT1 − µRT − dt j=1 i=1 2
X dET2 λHi ET2 − (µ + k2 )ET2 , = (1 − q)r2 IT1 + p2 λT2 (S + RT ) + λT2 ET1 − δ2 λT2 ET2 − dt i=1 2
X dIT2 = k2 ET2 + δ2 λT2 ET2 + (1 − p2 )λT2 (S + RT ) − (µ + dT2 )IT2 − λHi IT2 , dt i=1 2
X dIH1 λTi IH1 + qr2 IH1 T1 + r1 EH1 T1 , = λH1 (S + RT ) − (ρ1 + µ)IH1 − dt i=1 2
X dAHt = ναAH1 − (µ + θ)AHt − λTj AHt + r1 EAt T1 dt j=1 2
X dAH1 σj λTj AH1 − (µ + dA1 + α)AH1 + θAHt + r1 EA1 T1 + qr2 AA1 T1 , = ρ1 IH1 − dt j=1 2
X dIH2 = λH2 (S + RT ) − (ρ2 + µ)IH2 − λTj IH2 + qr2 IH2 T1 + r1 EH2 T1 , dt j=1 2
X dAH2 δj λTj AH2 − (µ + dA2 )AH2 + (1 − ν)αAH1 + r1 EA2 T1 + qr2 AA2 T1 , = ρ2 IH2 − dt j=1 dEH1 T1 = p1 λT1 IH1 + λH1 ET1 − (µ + r1 + ψ1 + ψ2 )EH1 T1 , dt dEH1 T2 = p2 λT2 IH1 + λH1 ET2 − (µ + ψ3 + ψ4 )EH1 T2 + (1 − q)r2 IH1 T1 , dt dEH2 T1 = p1 λT1 IH2 + λH2 ET1 − (µ + ζ1 + ζ2 + r1 )EH2 T1 , dt dEH2 T2 = p2 λT2 IH2 + λH2 ET2 − (µ + ζ3 + ζ4 )EH2 T2 + (1 − q)r2 IH2 T1 , dt dEA1 T1 = ψ1 EH1 T1 + p1 σ1 λT1 AH1 − (µ + dA1 + r1 + α + m1 )EA1 T1 + θEAt T1 , dt dEA1 T2 = ψ3 EH1 T2 + p2 σ2 λT2 AH1 − (µ + dA1 + α + m2 )EA1 T2 + (1 − q)r2 AA1 T1 + θEAt T2 , dt dEA2 T1 = ζ1 EH2 T1 + p1 δ1 λT1 AH2 − (µ + dA2 + r1 + m3 )EA2 T1 + (1 − ν)αEA1 T1 , dt dEA2 T2 = ζ3 EH2 T2 + p2 δ2 λT2 AH2 − (µ + dA2 + m4 )EA2 T2 + (1 − q)r2 AA2 T1 + (1 − ν)αEA1 T2 , dt
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dEAt T1 dt
= p1 λT1 AHt + ναEH1 T1 − (µ + θ + r1 + γ1 )EAt T1 ,
dEAt T2 = p2 λT2 AHt + ναEH1 T2 − (µ + θ + γ2 )EAt T2 , dt AA t T 1 = (1 − p1 )λT1 AHt + γ1 EAt T1 − (µ + dT1 + r2 + θ)AAt T1 + ναAA1 T1 , dt AA t T 2 = (1 − p2 )λT2 AHt + γ2 EAt T2 − (µ + dT2 + θ)AHt T2 + ναAA1 T2 , dt dIH1 T1 = (1 − p1 )λT1 IH1 + ψ2 EH1 T1 + λH1 IT1 − (µ + dT1 + n1 + r2 )IH1 T1 , dt dIH1 T2 = (1 − p2 )λT2 IH1 + ψ4 EH1 T2 + λH1 IT2 − (µ + dT2 + n2 )IH1 T2 , dt dIH2 T1 = (1 − p1 )λT1 IH2 + ζ2 EH2 T1 + λH2 IT1 − (µ + dT1 + r2 + n3 )IH2 T1 , dt dIH2 T2 = (1 − p2 )λT2 IH2 + ζ4 EH2 T2 + λH2 IT2 − (µ + dT2 + n4 )IH2 T2 , dt dAA1 T1 = (1 − p1 )σ1 λT1 AH1 + θAAt T1 + m1 EA1 T1 + n1 IH1 T1 − (µ + υ1 dA1 + dT1 + r2 + α)AA1 T1 , dt dAA1 T2 = (1 − p2 )σ2 λT2 AH1 + m2 EA1 T2 + n2 IH1 T2 − (µ + υ1 dA1 + dT2 + α)AA1 T2 , dt dAA2 T1 = (1 − p1 )δ1 λT1 AH2 + m3 EA2 T1 + n3 IH2 T1 − (µ + υ2 dA2 + dT1 + r2 )AA2 T1 + (1 − ν)αAA1 T1 , dt dAA2 T2 = (1 − p2 )δ2 λT1 AH2 + m4 EA2 T2 + n4 IH2 T2 − (µ + υ2 dA2 + dT2 )AA2 T2 + (1 − ν)αAA1 T2 . dt
(68)
Based on biological considerations model system (68) will be studied in the following region, D=
Λ µ (69)
(S, ETj , ITj , RT , IHi , AHt , AHi , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ) ∈ R31 + : N (t) ≤
It can be shown that all solutions of model system (68) starting D remain D for all t ≥ 0. Thus, D is positively invariant and it is sufficient to consider solutions in D. Existence, uniqueness and continuation results for system (68) hold in this region.
4.1.
Disease-Free Equilibrium and Stability Analysis
Model system (68) has a disease-free equilibrium given by
Λ G0 = , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 . µ (70) It can be shown that G0 attracts the region D0 = (S, ETj , ITo j , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ) ∈D:S=
Λ µ
,
(71)
.
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and the rest are zeros. The linear stability of G0 is governed by the reproduction number RAT where RAT = max {RA , RTT }
(72)
with RA and RTT as defined in equations (43) and (15) respectively. We now state Theorem 8 whose proof follows from [46] (Theorem 2). Theorem 8. The disease free equilibrium G0 , is locally asymptotically stable for RAT < 1 and unstable otherwise.
4.2.
Endemic Equilibria
For model system (68) there are fifteen possible endemic equilibria states that is (i) drug sensitive TB only, (ii) drug resistant TB only, (iii) coexistence of drug sensitive and resistant TB only, (iv) anti-retroviral sensitive HIV only, (v) anti-retroviral resistant HIV only, (vi) coexistence of anti-retroviral sensitive and resistant HIV only, (vii) coexistence of drug sensive TB and anti-retroviral sensitive HIV, (viii) coexistence of drug sensive TB and anti-retroviral drug HIV, (ix) coexistence of drug sensive TB, anti-retroviral sensitive HIV and anti-retroviral resistant HIV, (x) coexistence of drug resistant TB and anti-retroviral sensitive HIV, (xi) coexistence of drug resistant TB and anti-retroviral resistant HIV, (xii) coexistence of drug resistant TB, anti-retroviral sensitive HIV and anti-retroviral resistant HIV, (xiii) coexistence of drug sensitive TB, drug resistant TB, and anti-retroviral sensitive HIV, (xiv) coexistence of drug sensitive TB, drug resistant TB, and anti-retroviral resistant HIV, (xv) coexistence of drug sensitive TB, drug resistant TB, anti-retroviral sensitive HIV, and anti-troviral resistant HIV endemic equilibria. (i) TB drug sensitive only equilibrium is given by G1 with G1 = (ST∗1 , ET∗1 , IT∗1 , RT∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). (73) The equilibrium point G1 is simply the equilibrium ET∗2 with 0 added to all compartments of drug resistant TB forms, HIV-infected and doubly-infected individuals. (ii) TB drug resistant only is given by G2 with G2 = (ST∗2 , 0, 0, 0, ET∗2 , IT∗2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). (74) The equilibrium point G2 is simply the equilibrium ET∗1 with 0 added to all compartments of drug sensitive TB forms, HIV-infected and doubly-infected individuals. (iii) Coexistence of drug sensitive and resistant TB only is given by G3 with,
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G3 = (S ∗∗∗ , ET∗∗∗ , IT∗∗∗ , RT∗∗∗ , ET∗∗∗ , IT∗∗∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 1 2 2 0, 0, 0, 0, 0, 0, 0). (75) The equilibrium point G3 is simply the equilibrium ET∗3 with 0 added to all compartments of HIV-infected, and those dually infected with HIV and TB. (iii) Anti-retroviral resistant HIV-strain only equilibrium is given by G4 which is ∗ with 0 added to all compartments of TB-infected, those simply the equilibrium EH 1 infected with anti-retroviral sensitive HIV-strain and those doubly-infected and is given by ∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, I ∗ , A∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). G4 = (SH H2 H2 2 (76) (iv) Anti-retroviral sensitive HIV-strain only equilibrium is given by G5 which is simply the ∗ with 0 added to all compartments of TB-infected and those doubly-infected equilibrium EH 2 and is given by ∗ ∗ , 0, 0, 0, 0, 0, 0, IH , A∗Ht , A∗H1 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). G5 = (SH 1 1 (77)
(iv) Coexistence of anti-retroviral sensitive and resistant HIV only is given by G6 which is ∗ with 0 added to the all compartments of TB-infected and those simply the equilibrium EH 3 dually-infected with HIV and TB and is given as ∗∗ ∗∗ ∗∗ ∗∗ G6 = (S ∗∗ , 0, 0, 0, 0, 0, 0, IH , A∗∗ Ht , AH1 , IH2 , AH1 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). 1 (78)
Instead of writing all the other remaining endemic states let us just consider the coexistence of antiretroviral sensitive HIV, antiretroviral resistant HIV, drug sensitive and resistant TB endemic equilibrium which is quite involving to express explicitly in terms of the force(s) of infection, so we just generalise this equilibrium point to be G15 = (S, ETj , ITj , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ), j = 1, 2 and i = 1, 2 such that S, ETj , ITj , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj > 0. (79)
We now state Theorem 9 for the local asymptotic stability G15 whose proof can be shown using the Centre Manifold theory done in the Sections 2 and 3, but is not shown here to avoid repetition. Theorem 9. The endemic equilibrium point G15 is locally asymptotically stable for RAT > 1.
4.3.
Analysis of the Reproduction Number RAT RAT = max{RA , RTT }.
From (80) we have the following scenarios,
(80)
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Case 1: No intervention for all TB and HIV/AIDS infections In the absence of any intervention strategy we have lim
(rj ,θ,α)→(0,0,0)
RAT = max
(1 − pj )βj cT µ + βj cT kj βHi cH (µ + dAi + ηA ρi ) , (µ + kj )(µ + dTj ) (µ + ρi )(µ + dAi )
= R0 ,
which is the pre-treatment reproduction number for model system (68) i = 1, 2; j = 1, 2. (81) Case 2: Only drug sensitive latent and active forms of TB cases are treated In this case α = θ = 0 thus n o lim RAT = lim max {RA1 , RA2 , RTT } = max RAN1 , RA2 , RT (α,θ)→(0,0)
(α,θ)→(0,0)
(82)
where RAN1
βH1 cH (µ + dA1 + ηA ρ1 ) = , (µ + dA1 )(µ + ρ1 )
is the reproduction number for drug sensitive HIV/AIDS in the absence of antiretroviral treatment. In this case the HIV/AIDS epidemic is allowed to grow. Rewrite RA1 as RA1 = H1 RAN1 with H1 ∈ (0, 1) where H1 =
(µ + dA1 ) ((µ + dA1 + α)(µ + θ) − θνα + ηA ρ1 (µ + θ + να)) 1, we want to determine conditions necessary for slowing down the AIDS epidemic. Following Hsu Schmitz [25, 26] we have ∆1 = RAN1 (1 − H1 ) for which ∆1 > 0 is expected if antiretroviral therapy is to reduce the epidemic. Differentiating RA1 partially with respect to α we have, (µ + dA1 )(µ + θ) ((µ + θ)(ν − 1) + νdA1 ) ∂RA1 = . ∂α (µ + dA1 + ηA ρ1 ) ((µ + dA1 + α)(µ + θ) − θνα)2
(84)
∂RA1 < 0 and these ∂α = 1 and solving for the critical
The conditions for slowing the HIV/AIDS epidemic are ∆1 > 0 and occur when 0 < (µ + θ)(1 − ν) − νdA1 . Setting RA1 antiretroviral rate we have, αc =
(µ + dA1 + ηA ρ1 )(µ + θ)(µ + dA1 )(RAN1 − 1)
(µ + θ(1 − ν))(µ + dA1 )(1 − RAN1 ) + ηA ρ1 (µ + θ(1 − ν) − (µ + dA1 )νRAN1 ). (85) Thus antiretroviral therapy will have a positive impact on HIV/AIDS individuals with TB for α > αc . When antiretroviral therapy is administered in a population with both strains of HIV, the long term outcome depends on the reproduction numbers of both strains. Solving
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RA1 = 1, we have, RA2 (µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RA2 )
∗
α c1 =
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RA2 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1
which exists if
RAN1 RA2
>
,
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1. (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)
(86) ∗ If the antiretroviral rate α < αc1 , then all HIV/AIDS cases remain sensitive to antiretrovirals provided there is no primary transmission of anti-retroviral resistant HIV strain to ∗ the susceptibles. If α > αc1 some AIDS cases will become antiretroviral resistant in the long run. If antiretroviral therapy is implemented in a population with (a) drug sensitive TB without any intervention, (b) drug sensitive TB with chemoprophylaxis and treatment, (c) drug resistant TB, we obtain the following critical antiretroviral therapy rates (µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RTN1 )
∗
(a) αc2 =
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RTN1 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1
which exists if
RAN1 RTN1
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RT1 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1
which exists if
RAN1 RT1
>
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RT2 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1
which exists if
RAN1 RT2
>
,
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1, (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)
(µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RT2 )
∗
(c) αc4 =
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1, (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)
(µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RT1 )
∗
(b) αc3 =
>
(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1. (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)
(87)
Case 3: Only drug sensitive AIDS cases are treated In this case rj = 0, (j = 1, 2) thus n o lim RAT = lim max {RA , RT1 , RT2 } = max RA , RTN1 , RT2 ,
rj →0
rj →0
where RTN1
(1 − p1 )β1 cT µ + β1 cT k1 , = (µ + dT1 )(µ + k1 )
(88)
is the reproduction number for the drug sensitive only TB model in the absence of any intervention. In this case the TB epidemic is allowed to grow unchecked. Rewrite RT1 as
,
,
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RT1 = H2 RTN1 with H2 ∈ (0, 1) where H2 =
((1 − p1 )(µ + r1 ) + k1 ) (µ + dT1 )(µ + k1 ) < 1, (µ + dT1 + r2 )(µ + k1 + r1 ) ((1 − p1 )µ + k1 )
(89)
is a factor by which chemoprophylaxis of latent and treatment of active forms of drug sensitive TB reduce the number of secondary TB cases in a community. If RTN1 < 1, then TB may not develop into an epidemic and intervention strategies may not be necessary and for RTN1 > 1 we need to determine conditions for slowing down the TB epidemic. We have that ∆2 = RTN1 (1 − H2 ) > 0 for chemoprophylaxis and treatment to reduce the epidemic and is satisfied for rj ∈ (0, 1), j = 1, 2. Differentiating RT1 partially with respect to r1 and r2 we have, k1 p1 (µ + dA1 )(µ + k1 )RTN1 ∂RT1 =− < 0, ∂r1 (µ + dA1 + r2 )((1 − p1 )µ + k1 )(µ + k1 + r1 )2
(90)
((1 − p1 )(µ + r1 ) + k1 )(µ + dA1 )(µ + k1 )RTN1 ∂RT1 =− 0 are necessary for slowing down the TB epidemic and these are satisfied for rj ∈ (0, 1). Setting RT1 = 1 when r2 = 0 that is when chemoprophylaxis is the only intervention strategy, we have the critical chemoprophylaxis rate as, ((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − 1) , r1c = (1 − p1 )(1 − RTN1 )µ + k1 (1 − (1 − p1 )RTN1 ) (91) (1 − p1 )µ + k1 . which exists if 1 < RTN1 < (1 − p1 )(µ + k1 )
When r1 > r1c , chemoprophylaxis will be able to eradicate the TB epidemic in the community but for r1 < r1c chemoprophylaxis alone will reduce the TB epidemic but not eradicate it and some intervention strategies like treatment, of infectives may be necessary. If chemoprophylaxis is implemented in the presence of the drug resistant TB strain, the long term RT1 = 1 when outcome depends on the reproduction numbers of both strains. Solving for RT2 r2 = 0 for the critical treatment rate we obtain c∗
r11 =
((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RT2 )
(1 − p1 )(RT2 − RTN1 )µ + k1 (RT2 − (1 − p1 )RTN1 )
which exists for 1 < c∗
RTN1 RT2
r11 , then chemoprophylaxis will be able to eradicate TB in communities where drug sensitive and drug resistant TB strains coexist. If chemoprophylaxis is implemented in a population with (a) drug sensitive HIV strains without AIDS
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treatment, (b) drug sensitive HIV strains with AIDS treatment and (c) drug resistant HIV strains without AIDS treatment, the following critical chemoprophylaxis rates are obtained, ((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RAN1 )
c∗
(a) r12 =
(1 − p1 )(RAN1 − RTN1 )µ + k1 (RAN1 − (1 − p1 )RTN1 )
which exists if 1
RAN1 , (a) r2 = (µ + dT1 ) RAN1 (b)
c∗ r23
(c)
c∗ r24
= (µ + dT1 )
RTN1
− 1 which exists if RTN1 > RA1 ,
= (µ + dT1 )
RTN1
− 1 which exists if RTN1 > RA1 .
RA1
RA2
(96)
Figure 9 is graphical representation showing critical treatment rates in 5 different settings
Figure 9. Graphs of the critical treatment rates against RTN1 . Parameter values are obtained from Table 2. Series 1 to 5 denote critical treatment when (a) treatment of TB infectives is implemented in a population with TB resistance, (b) treatment of TB infectives is implemented in a population with antiretroviral sensitive HIV but without treatment, (c) TB treatment is implemented in a population with antiretroviral sensitive HIV/AIDS with treatment, (d) TB treatment is implemented in a population with antiretroviral resistant HIV/AIDS and (e) TB treatment is implemented in a population without HIV/AIDS and TB resistance, respectively. against the no intervention reproduction number RTN1 . This shows that c∗
c∗
c∗
c∗
r2c < r22 < r24 < r23 < r21 for all values of RTN1 > 1 suggesting that treatment of TB infectives is more effective in controlling TB in the absence of anti-TB drug resistance and c∗ c∗ c∗ HIV/AIDS. We note that graphs for r2c , r22 , r24 , r23 are close to one another suggesting that treatment is able to control TB almost equally in individuals co-infected with HIV and c∗ c∗ c∗ those with drug sensitive TB only. The fact that r22 < r24 < r23 suggests that antiretroviral
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therapy have a negative impact in individuals on TB treatment implying that treatment of TB infectives will do better in AIDS individuals who are not on antiretroviral therapy. This suggests that a different treatment regimen for those coinfected with HIV/AIDS if available c∗ will be more appropriate. The fact that r21 is very large shows that in the presence of drug resistant TB, it is difficult to manage TB using first line anti-TB drugs.
5.
Summary and Concluding Remarks
Mathematical models have been presented and studied to assess the the impact of chemoprophylaxis, treatment of drug sensitive TB infectives, drug resistance and antiretroviral therapy on TB cases in areas with high HIV/AIDS prevalence. A two strain TB model incorporating chemoprophylaxis and treatment of infectives for drug sensitive TB was first presented. This was followed by presentation and analysis of a two strain HIV/AIDS model and a full HIV/AIDS-TB co-infection model incorporating drug resistance of both infections. We computed and compared the reproduction numbers of the submodels to assess the effectiveness of chemoprophylaxis and treatment of infectives in the control of TB and antiretroviral therapy in controlling HIV/AIDS. Chemoprophylaxis to prevent progression from latent TB to active TB tuberculosis among those with HIV infection (EHi Tj ) will reduce the local burden of TB disease for several years. This is more pronounced when coverage levels of chemoprophylaxis are high. Effective HIV/AIDS control lessens the impact of HIV on TB epidemics. It is noted from the study that a decrease in number of drug sensitive TB infectives results in a decrease of drug resistant TB cases as most cases of TB drug resistance are due to improper anti-TB drug use. So, a decrease in the number of individuals using TB drugs results in creation of very low rates of drug resistance. Using the same argument, an increase in the number of TB infectives on TB treatment results in an increase of drug resistant TB. The obtained reproduction numbers RTN1 > RT1 and RAN1 > RA1 suggests that chemoprophylaxis and treatment of infectives is effective in controlling TB and that antiretroviral therapy is effective in improving the lives of HIV/AIDS individuc∗ c∗ c∗ c∗ als. The obtained critical treatment rates r2c < r22 < r24 < r23 < r21 for all values of RTN1 > 1 suggests that TB treatment using first line drugs is more effective in controlling TB in settings without or with low levels of drug resistant TB and HIV/AIDS. From c∗ c∗ c∗ the analysis of the critical treatment rates r2c , r22 , r24 , r23 , we conclude that treatment of TB infectives are equally effective in controlling TB in individuals with drug sensitive TB c∗ c∗ c∗ co-infected with HIV/AIDS. We also note from this study that r22 < r24 < r23 implying that treatment of drug sensitive forms of TB in individuals co-infected with HIV/AIDS on antiretrovirals is not as effective as those co-infected with HIV/AIDS and not on antiretrovirals, suggesting that antiretroviral therapy have a negative impact in individuals on TB treatment. However, the scenario is different when there is TB drug resistance as chemoprophylaxis and treatment of infectives fail to contain the epidemic. We note from the study that chemoprophylaxis and treatment of TB with first line drugs alone is not enough in the fight against HIV/AIDS-TB co-infection with resistance, but in the absence of other measures it is important to use the current regimen to control drug sensitive TB in individuals co-infected with HIV/AIDS. To achieve maximum benefit from HIV/AIDS-TB co-infection treatment, it is vital to have a constant reliable supply of drugs as erratic supply results in
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drug resistance of both infections which is quite difficult to manage. Combined HIV/AIDSTB treatment in individuals co-infected results in significant reduction of the TB pandemic and not the same for HIV/AIDS as AIDS treatment results in an increase in the number of AIDS individuals due to a reduction of AIDS related deaths. Thus the succesful control of TB in high-burden HIV settings may require policies that simultenously target the risk of progression when infected by Mtb and the risk of transmission when one has TB.
Acknowledgements This work was made possible through a fellowship received by C. P. Bhunu from the Forgarty International Centre through the International Clinical, Operational and Health Services and Training Award (ICOHRTA).
Appendix A Model system (3) can be written in the form
x′1 (t)
= f1 = Λ −
P2
j=1 βj cT x3j x1 P6 n=1 xn
− µx1 ,
p1 β1 cx3 δ1 β1 cT x3 β2 cT x6 x′2 (t) = f2 = P6 (x1 + x4 ) − (k1 + r1 + µ)x2 − P6 x2 − P6 x2 , n=1 xn n=1 xn n=1 xn x′3 (t) = f3 =
x′4 (t)
(1 − p1 )β1 cT x3 δ1 β1 cT x3 (x1 + x4 ) + k1 x2 + P6 x2 − (µ + dT1 + r2 )x3 , P6 n=1 xn n=1 xn
= f4 = r1 x2 + qr2 x3 − µx4 −
P2
j=1 βj cT x3j x4 , P6 n=1 xn
δ2 βj cT x6 β2 cT x6 p2 β2 cT x6 (x1 + x4 ) + P6 x2 − P6 x5 − (µ + k2 )x5 , x′5 (t) = f5 = (1 − q)r2 x3 + P6 x x x n n n n=1 n=1 n=1
(1 − p2 )β2 cT x6 δ2 β2 cT x6 x′6 (t) = f6 = k2 x5 + P6 x5 + (x1 + x4 ) − (µ + dT2 )x6 . P6 n=1 xn n=1 xn
(97)
Appendix B Computations of a and b: For system (97), the associated non-zero partial derivatives of F at the disease free equilib-
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rium are given by ∂ 2 f2 p1 β1 cT µ + δ1 β1 cT µ ∂ 2 f2 ∂ 2 f2 β∗ cT µ ∂ 2 f2 , , = =− = =− ∂x2 ∂x3 ∂x3 ∂x2 Λ ∂x6 ∂x2 ∂x2 ∂x6 Λ ∂ 2 f2 ∂ 2 f2 ∂ 2 f2 ∂ 2 f2 p1 β1 cT µ = = = =− , ∂x3 ∂x5 ∂x5 ∂x3 ∂x3 ∂x6 ∂x6 ∂x3 Λ 2(1 − p1 )β1 cT µ ∂ 2 f3 δ1 β1 cT µ − (1 − p1 )β1 cT µ ∂ 2 f3 ∂ 2 f3 =− = = , , 2 ∂x2 ∂x3 ∂x3 ∂x2 Λ Λ ∂x3 ∂ 2 f3 ∂ 2 f3 ∂ 2 f3 ∂ 2 f3 (1 − p1 )β1 cT µ = = = =− , ∂x3 ∂x5 ∂x5 ∂x3 ∂x3 ∂x6 ∂x6 ∂x3 Λ ∂ 2 f4 β1 cT µ ∂ 2 f4 ∂ 2 f4 β∗ cT µ ∂ 2 f4 = =− , = =− , ∂x3 ∂x4 ∂x4 ∂x3 Λ ∂x4 ∂x6 ∂x6 ∂x4 Λ ∂ 2 f5 ∂ 2 f5 β∗ cT µ − p2 β∗ cT µ ∂ 2 f5 ∂ 2 f5 p2 β∗ cT µ = = , = =− , ∂x2 ∂x6 ∂x6 ∂x2 Λ ∂x3 ∂x6 ∂x6 ∂x3 Λ 2p2 β∗ cT µ ∂ 2 f6 ∂ 2 f6 ∂ 2 f6 ∂ 2 f6 (1 − p2 )β∗ cT µ ∂ 2 f5 =− , = = = =− , 2 Λ ∂x2 ∂x6 ∂x6 ∂x2 ∂x3 ∂x6 ∂x6 ∂x3 Λ ∂x6 ∂ 2 f6 δ2 β∗ cT µ − (1 − p2 )β∗ cT µ ∂ 2 f6 ∂ 2 f6 2(1 − p2 )β∗ cT µ = = , =− 2 ∂x5 ∂x6 ∂x6 ∂x5 Λ Λ ∂x6
(98)
It follows from (98) that a = ϕ1 + ϕ2 + ϕ3 + ϕ4 ,
k2 + µ + dT2 − (1 − p2 )β∗ cT k2
ϕ1 = −
2w2 Λ
ϕ2 = −
2w3 u2 u3 β1 cT µ(1 − p1 − δ1 ) + u23 (1 − p1 )β1 cT µ + u3 u6 p2 β∗ cT µ , Λ
ϕ3 = − ϕ4 = −
u2 u3 β1 cT µ(p1 + δ1 ) + u2 u6 β∗ cT µ + u3 u6 p1 β1 cT µ
2w5 β∗ cT µ(u3 u6 p2 + u26 − u2 u6 (1 − p2 )) , Λ
2w6 β∗ cT µ (u2 u6 + u3 u6 )(1 − p2 ) + u26 (1 − p2 ) − u5 u6 (δ2 + (1 − p2 )) . Λ
(99)
Thus a < 0 for some ϕi < 0, i = 1, 2, 3, 4 otherwise a > 0. For the sign of b, it is associated with the following non-vanishing partial derivatives of F , ∂ 2 f1 ∂ 2 f5 ∂ 2 f6 = −cT , = p 2 cT , = (1 − p2 )cT . ∂x6 ∂β∗ ∂x6 ∂β∗ ∂x6 ∂β∗
(100)
,
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It follows from (100) that, b=
u 6 w5 c T (k2 + µ(1 − p2 )) > 0 for (k2 + µ)(µ + dT2 ) > β∗ cT k2 + k2
(101)
β∗ cT µ(1 − p2 ), 1 > RT1 .
Appendix C Model system (37) can be written in the form
x′1 (t) = f1 = Λ −
βH1 cH (x2 + ηA (x3 + x4 )) + βH2 cH (x5 + ηA x6 ) x1 − µx1 P6 n=1 xn
βH cH x′2 (t) = f2 = P6 1 (x2 + ηA (x3 + x4 ))x1 − (ρ1 + µ)x2 , n=1 xn x′3 (t) = f3 = ναx4 − (µ + θ)x3 ,
(102)
x′4 (t) = f4 = ρ1 x2 − (µ + dA1 + α)x4 + θx3 , βH cH x′5 (t) = f5 = P6 2 (x5 + ηA x6 )x1 − (ρ2 + µ)x5 , n=1 xn
x′6 (t) = f6 = ρ2 x5 − (µ + dA2 )x6 + (1 − ν)αx4 .
Appendix D Computations of a and b: For the sign of b, it is associated with the following non-vanishing partial derivatives of F , ∂ 2 f1 ∂ 2 f1 ∂ 2 f5 ∂ 2 f5 = −c , = −η c , = c , = ηA cH . H A H H ∂x5 ∂β ∗2 ∂x6 ∂β ∗2 ∂x4 ∂β ∗2 ∂x6 ∂β ∗2
(103)
From (103) it follows that b = y5 cH (z5 + z6 ηA ) > 0 for RA1 < 1.
(104)
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For the sign of a it is associated with the following non-vanishing partial derivatives of F 2βH1 cH µ ∂ 2 f2 ∂ 2 f2 βH cH (1 + ηA )µ ∂ 2 f2 = − , , = =− 1 2 Λ ∂x2 ∂x3 ∂x3 ∂x2 Λ ∂x2 ∂ 2 f2 ∂ 2 f2 βH cH (1 + ηA )µ ∂ 2 f2 ∂ 2 f2 βH cH µ = =− 1 , = =− 1 , ∂x2 ∂x4 ∂x4 ∂x2 Λ ∂x2 ∂x5 ∂x2 ∂x6 Λ ∂ 2 f2 2βH1 cH ηA µ ∂ 2 f2 ∂ 2 f2 βH cH ηA µ ∂ 2 f2 = = − , = =− 1 , 2 ∂x3 ∂x4 Λ ∂x3 ∂x5 ∂x3 ∂x6 Λ ∂x3 ∂ 2 f2 2βH1 cH ηA µ ∂ 2 f2 ∂ 2 f2 βH cH ηA µ = − , = =− 1 , 2 Λ ∂x4 ∂x5 ∂x4 ∂x6 Λ ∂x4 ∂ 2 f5 ∂ 2 f5 β ∗2 cH µ ∂ 2 f5 = = =− , ∂x5 ∂x2 ∂x5 ∂x3 ∂x5 ∂x4 Λ ∂ 2 f5 ∂ 2 f5 β ∗2 cH ηA µ ∂ 2 f5 = = =− ∂x6 ∂x2 ∂x6 ∂x3 ∂x6 ∂x4 Λ ∂ 2 f5 2β ∗2 cH µ ∂ 2 f5 2β ∗2 cH ηA µ β ∗2 cH (1 + ηA )µ ∂ 2 f5 = − = − =− , , . ∂x5 ∂x6 Λ Λ Λ ∂x25 ∂x26
(105)
From (105) it follows that 2µ (z2 + z3 + z4 + z5 + z6 ) y2 βH1 cH (z2 + ηA (z3 + z4 )) + y5 β ∗2 cH (z5 + ηA z6 ) . Λ (106) Thus a < 0 and b > 0. a=−
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[22] Feng Z, Castillo-Chavez C, and Capurro AF, A model for tuberculosis with exogenous reinfection, Theor Pop Bio 57:235-247, 2000. [23] Frieden T, Driver RC, Tuberculosis control: past 10 years and future progress, Tuberculosis 83: 82-85, 2003. [24] Harrington PR, Lader BA, Extent of cross resistance between agents used to treat HIV1 infection in clinically derived isolates, Antimicrob. Agents Chemother. 46: 909-912, 2002. [25] Hsu Schmizt SF, Effects of treatment or/and vaccination on HIV transmission in homosexuals with genetic heterogeneity, Math Biosci 167: 1-18, 2000. [26] Hsu Schmizt SF, Effects of genetic heterogeneity on HIV transmission in homosexuals populations, in Castillo-Chavez C, Blower S, van den Driessche P, Kirshner D, Yakubu A-A (eds), Mathematical Approaches for Emerging and Re-emerging Infectious Diseases: Models, Methods and Theory, Springer-Verlag, 245-260, 2002. [27] Hyman JM, Li J, Stanley EA, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci. 155: 77-109, 1999. [28] Lader BA, Darby G, Richman DD, HIV with reduced sensitivity to zidovudine isolated during prolonged therapy, Science 243: 1731-1734, 1989. [29] Li MY, Muldowney JS, Global stability for the SEIR model in epidemiology, Math. Biosci. 125:155-164, 1995. [30] Martcheva M, Ianelli M, Xue-Zhi-Li, Subthreshold coexistence of strains: The impact of vaccination and mutation, Mathematical Bioscience and Engineering 4(2): 287317, 2007. [31] The management of multi-drug resistant tuberculosis in South Africa, National Tuberculosis Research Programme, South Africa, 1999 [32] May R, Nowak M, Coinfection and the evolution of parasite virulence, Proc. Royal Soc. London 261: 209-215, 1995. [33] Mukandavire Z, Garira W, HIV/AIDS model for assessing the effects of prophylactic sterilizing vaccines, condoms and treatment with amelioration, J. Biol. Syst. 14(3): 323-355, 2006. [34] Mukherjee J, Mukherjee testimony on XDR-TB to the U.S. House, PIH News, April 2007. [35] Naresh R, Tripathi A, Modelling and analysis of HIV-TB co-infection in a variable population size, Mathematical Modelling and Analysis 10(3): 275-286, 2005. [36] Nowak MA, Sigmund K, Super-and coinfection: The two extremes. In: Adaptive dynamics of infectious diseases: In pursuit of virulence management, Cambridge University Press, 2002.
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In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 133-150
Chapter 4
W HEN Z OMBIES ATTACK !: M ATHEMATICAL M ODELLING OF AN O UTBREAK OF Z OMBIE I NFECTION Philip Munz1,∗, Ioan Hudea1,†, Joe Imad2,‡ and Robert J. Smith?3,§ 1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada 2 Department of Mathematics, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada 3 Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada
Abstract Zombies are a popular figure in pop culture/entertainment and they are usually portrayed as being brought about through an outbreak or epidemic. Consequently, we model a zombie attack, using biological assumptions based on popular zombie movies. We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the effects of possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.
1.
Introduction
A zombie is a reanimated human corpse that feeds on living human flesh [1]. Stories about zombies originated in the Afro-Caribbean spiritual belief system of Vodou (anglicised ∗
E-mail address: E-mail address: ‡ E-mail address: § E-mail address: †
[email protected] [email protected] [email protected] [email protected]. Corresponding author.
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voodoo). These stories described people as being controlled by a powerful sorcerer. The walking dead became popular in the modern horror fiction mainly because of the success of George A. Romero’s 1968 film, Night of the Living Dead [2]. There are several possible etymologies of the word zombie. One of the possible origins is jumbie, which comes from the Carribean term for ghost. Another possible origin is the word nzambi which in Kongo means ‘spirit of a dead person’. According to the Merriam-Webster dictionary, the word zombie originates from the word zonbi, used in the Louisiana Creole or the Haitian Creole. According to the Creole culture, a zonbi represents a person who died and was then brought to life without speech or free will. The followers of Vodou believe that a dead person can be revived by a sorcerer [3]. After being revived, the zombies remain under the control of the sorcerer because they have no will of their own. Zombi is also another name for a Voodoo snake god. It is said that the sorcerer uses a ‘zombie powder’ for the zombification. This powder contains an extremely powerful neurotoxin that temporarily paralyzes the human nervous system and it creates a state of hibernation. The main organs, such as the heart and lungs, and all of the bodily functions, operate at minimal levels during this state of hibernation. What turns these human beings into zombies is the lack of oxygen to the brain. As a result of this, they suffer from brain damage. A popular belief in the Middle Ages was that the souls of the dead could return to earth one day and haunt the living [4]. In France, during the Middle Ages, they believed that the dead would usually awaken to avenge some sort of crime committed against them during their life. These awakened dead took the form of an emaciated corpse and they wandered around graveyards at night. The idea of the zombie also appears in several other cultures, such as China, Japan, the Pacific, India, Persia, the Arabs and the Americas. Modern zombies (the ones illustrated in books, films and games [1, 5]) are very different from the voodoo and the folklore zombies. Modern zombies follow a standard, as set in the movie Night of the Living Dead [2]. The ghouls are portrayed as being mindless monsters who do not feel pain and who have an immense appetite for human flesh. Their aim is to kill, eat or infect people. The ‘undead’ move in small, irregular steps, and show signs of physical decomposition such as rotting flesh, discoloured eyes and open wounds. Modern zombies are often related to an apocalypse, where civilization could collapse due to a plague of the undead. The background stories behind zombie movies, video games etc, are purposefully vague and inconsistent in explaining how the zombies came about in the first place. Some ideas include radiation (Night of the Living Dead [2]), exposure to airborne viruses (Resident Evil [6]), mutated diseases carried by various vectors (Dead Rising [7] claimed it was from bee stings of genetically altered bees). Shaun of the Dead [8] made fun of this by not allowing the viewer to determine what actually happened. When a susceptible individual is bitten by a zombie, it leaves an open wound. The wound created by the zombie has the zombie’s saliva in and around it. This bodily fluid mixes with the blood, thus infecting the (previously susceptible) individual. The zombie that we chose to model was characterised best by the popular-culture zombie. The basic assumptions help to form some guidelines as to the specific type of zombie we seek to model (which will be presented in the next section). The model zombie is of the classical pop-culture zombie: slow moving, cannibalistic and undead. There are other ‘types’ of zombies, characterised by some movies like 28 Days Later [9] and the 2004 re-
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make of Dawn of the Dead [10]. These ‘zombies’ can move faster, are more independent and much smarter than their classical counterparts. While we are trying to be as broad as possible in modelling zombies – especially since there are many varieties – we have decided not to consider these individuals.
2.
The Basic Model
For the basic model, we consider three basic classes: • Susceptible (S). • Zombie (Z). • Removed (R). Susceptibles can become deceased through ‘natural’ causes, i.e., non-zombie-related death (parameter δ). The removed class consists of individuals who have died, either through attack or natural causes. Humans in the removed class can resurrect and become a zombie (parameter ζ). Susceptibles can become zombies through transmission via an encounter with a zombie (transmission parameter β). Only humans can become infected through contact with zombies, and zombies only have a craving for human flesh so we do not consider any other life forms in the model. New zombies can only come from two sources: • The resurrected from the newly deceased (removed group). • Susceptibles who have ‘lost’ an encounter with a zombie. In addition, we assume the birth rate is a constant, Π. Zombies move to the removed class upon being ‘defeated’. This can be done by removing the head or destroying the brain of the zombie (parameter α). We also assume that zombies do not attack/defeat other zombies. Thus, the basic model is given by S ′ = Π − βSZ − δS
Z ′ = βSZ + ζR − αSZ
R′ = δS + αSZ − ζR .
This model is illustrated in Figure 1. This model is slightly more complicated than the basic SIR models that usually characterise infectious diseases [11], because this model has two mass-action transmissions, which leads to having more than one nonlinear term in the model. Mass-action incidence specifies that an average member of the population makes contact sufficient to transmit infection with βN others per unit time, where N is the total population without infection. In this case, the infection is zombification. The probability that a random contact by a zombie is made with a susceptible is S/N ; thus, the number of new zombies through this transmission process in unit time per zombie is: (βN )(S/N )Z = βSZ .
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Figure 1. The basic model. We assume that a susceptible can avoid zombification through an altercation with a zombie by defeating the zombie during their contact, and each susceptible is capable of resisting infection (becoming a zombie) at a rate α. So, using the same idea as above with the probability Z/N of random contact of a susceptible with a zombie (not the probability of a zombie attacking a susceptible), the number of zombies destroyed through this process per unit time per susceptible is: (αN )(Z/N )S = αSZ . The ODEs satisfy S ′ + Z ′ + R′ = Π and hence S+Z +R → ∞ as t → ∞, if Π 6= 0. Clearly S 6→ ∞, so this results in a ‘doomsday’ scenario: an outbreak of zombies will lead to the collapse of civilisation, as large numbers of people are either zombified or dead. If we assume that the outbreak happens over a short timescale, then we can ignore birth and background death rates. Thus, we set Π = δ = 0. Setting the differential equations equal to 0 gives −βSZ = 0
βSZ + ζR − αSZ = 0
αSZ − ζR = 0 .
From the first equation, we have either S = 0 or Z = 0. Thus, it follows from S = 0 that we get the ‘doomsday’ equilibrium ¯ Z, ¯ R) ¯ = (0, Z, ¯ 0) . (S, When Z = 0, we have the disease-free equilibrium ¯ Z, ¯ R) ¯ = (N, 0, 0) . (S,
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These equilibrium points show that, regardless of their stability, human-zombie coexistence is impossible. The Jacobian is then −βZ −βS 0 J = βZ − αZ βS − αS ζ . αZ αS −ζ The Jacobian at the disease-free equilibrium is 0 −βN J(N, 0, 0) = 0 βN − αN 0 αN
We have
0 ζ . −ζ
det(J − λI) = −λ{λ2 + [ζ − (β − α)N ]λ − βζN } . It follows that the characteristic equation always has a root with positive real part. Hence, the disease-free equilibrium is always unstable. Next, we have −β Z¯ 0 0 ¯ 0) = β Z¯ − αZ¯ 0 ζ . J(0, Z, αZ¯ 0 −ζ
Thus,
det(J − λI) = −λ(−β Z¯ − λ)(−ζ − λ) . Since all eigenvalues of the doomsday equilibrium are negative, it is asymptotically stable. It follows that, in a short outbreak, zombies will likely infect everyone. In the following figures, the curves show the interaction between susceptibles and zombies over a period of time. We used Euler’s method to solve the ODE’s. While Euler’s method is not the most stable numerical solution for ODE’s, it is the easiest and least timeconsuming. See Figures 2 and 3 for these results. The MATLAB code is given at the end of this chapter. Values used in Figure 3 were α = 0.005, β = 0.0095, ζ = 0.0001 and δ = 0.0001.
3.
The Model with Latent Infection
We now revise the model to include a latent class of infected individuals. As discussed in Brooks [1], there is a period of time (approximately 24 hours) after the human susceptible gets bitten before they succumb to their wound and become a zombie. We thus extend the basic model to include the (more ‘realistic’) possibility that a susceptible individual becomes infected before succumbing to zombification. This is what is seen quite often in pop-culture representations of zombies ([2, 6, 8]). Changes to the basic model include:
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