Advances in Disease Epidemiology [1 ed.] 9781617287664, 9781607414520

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ADVANCES IN DISEASE EPIDEMIOLOGY

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Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

ADVANCES IN DISEASE EPIDEMIOLOGY

JEAN MICHEL TCHUENCHE AND

ZINDOGA MUKANDAVIRE Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITORS

Nova Science Publishers, Inc. New York

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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.

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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 Advances in disease epidemiology / [edited by] Jean Michel Tchuenche. p. ; cm. Includes bibliographical references and index. ISBN 978-1-61728-766-4 (E-Book) 1. Epidemiology--Mathematical models. 2. AIDS (Disease)--Epidemiology--Mathematical models. I. Tchuenche, Jean Michel. [DNLM: 1. HIV Infections--epidemiology. 2. Acquired Immunodeficiency Syndrome-epidemiology. 3. Epidemiologic Research Design. 4. Malaria--epidemiology. 5. Models, Theoretical. WC 503.41 A244 2009] RA652.2.M3A33 2009 614.4--dc22 2009015092

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CONTENTS

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Preface

vii

Chapter 1

An Avian Influenza Model and Its Fit to Human Avian Influenza Cases Joseph Lucchetti, Manojit Roy and Maia Martcheva

Chapter 2

Gender Differences in Heterosexual Transmission of HIV in Urban and Rural Populations Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

31

Chapter 3

A Partnership Network Simulation of the Spread of Sexually Transmitted Infections in Russia Fatemeh Jafargholi and Chris T. Bauch

59

Chapter 4

Malaria Control: The Role of Local Communities as Seen through a Mathematical Model in a Changing Population – Cameroon Miranda I. Teboh-Ewungkem

103

Chapter 5

Application of Optimal Control to the Epidemiology of HIV-Malaria Co-infection F.B. Agusto

141

Chapter 6

Two Strain HIV/AIDS Model and the Effects of Superinfection N.J. Malunguza, S. Dube, J.M. Tchuenche, S.D. Hove-Musekwa and Z. Mukandavire

171

Chapter 7

Modelling the Transmission of Multidrug-resistant and Extensively Drug-resistant Tuberculosis C.P. Bhunu and W. Garira

195

Chapter 8

HIV/AIDS and the Use of Mathematical Models in the Theoretical Assessment of Intervention Strategies: A Review Zindoga Mukandavire, Jean M. Tchuenche, Christinah Chiyaka and Godfrey Musuka

221

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vi

Contents

Chapter 9

A Model for the Spread of HIV/AIDS in a Two Sex Population Ram Naresh, Agraj Tripathi and Dileep Sharma

243

Chapter 10

Fitting Procedure for Host-Parasite Systems Henri E. Z. Tonnang and Jean M. Tchuenche

271

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Index

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PREFACE The field of mathematical biology has produced a sensible advancement in the investigation of disease dynamics. This book is an essential tool for biomathematics courses, especially because there is great emphasis on the dynamics of common diseases in the developing world. We do hope it will inspire research students and provide them with a fresh perspective of recent mathematical approaches to disease epidemiology. The chapters are written by experts as well as young scientists and they offer interested readers a solid understanding of describing, robustly analyzing and interpreting disease models via mathematical applications. Thus, the materials presented herein can be used as a source for projects and supplementary lectures in mathematical biology as well as applications in differential equations courses. The book specifically underscores the importance of finding ways to introduce undergraduate students to research in disease modeling, borrowing techniques and Theorems from mathematical analysis to finding how to curtail epidemic outbreaks by describing and analyzing via elementary methods some models for disease transmission. It is devoted to some of the challenges associated with the mathematical modeling and analysis of the dynamics and control of some diseases of public health importance. Great emphasis is laid on model development, model validation, and model refinement. The reader is expected to have some minimal background in calculus and disease epidemiology as these materials are intended for - use within courses in mathematical biology - local and global stability of system of differential equations and biology courses related to disease dynamics problems that cover the applications of mathematics - research, basically at both advanced undergraduate and graduate levels. The chapters of this book 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 HIV/AIDS and STDs, avian influenza and Tuberculosis. The book is organized as follows: The first Chapter describes the dynamics of avian flu and its fit to human avian influenza cases. In Chapter 2, the impact of counter measures against HIV heterosexual transmission in urban and rural communities is investigated, while the next chapter studies a partnership network of the spread of STIs in Russia. The role of local communities in a changing population as seen through a mathematical model in the transmission dynamics of malaria and control is considered in Chapter 4. By applying optimal control method to a recent co-dynamics model of HIV and malaria, Chapter 5 investigates the effects of treatment and use of insecticide in seeking to

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Jean Michel Tchuenche and Zindoga Mukandavire

reduce the infected as well as the vector populations. Chapter 6 analyses the effect of superinfection in a two strain HIV/AIDS model while Chapter 7 investigates the effects of multidrug resistant and extensively resistant drug in the transmission dynamics of tuberculosis. Chapter 8 gives an overview of some previous works on theoretical models of HIV/AIDS and the assessment of intervention strategies, while Chapter 9 focuses on a two-sex population model for the spread of HIV/AIDS in a community. The last Chapter describes a fitting procedure for host-parasite systems. Thanks to the reviewers and the members of the editorial board of Nova Science Publishers, Inc., for their patience. Finally, despite all the support in the production, at the end, the responsibility for the final product is entirely ours.

Jean M. Tchuenche (Dar es Salaam, TZ) Zindoga Mukandavire (Bulawayo, Zim)

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December, 2008

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Chapter 1

A N AVIAN I NFLUENZA M ODEL AND I TS F IT TO H UMAN AVIAN I NFLUENZA C ASES Joseph Lucchettia,∗, Manojit Royb,† and Maia Martchevaa,‡ a Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611–8105 b Department of Biology, University of Florida, 220 Bartram Hall, PO Box 118525, Gainesville, FL 32611–8525

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Abstract Low Pathogenic Avian Influenza (LPAI) virus, which circulates in wild bird populations in mostly benign form, is suspected to have mutated into a highly pathogenic (HPAI) strain after transmission to the domestic birds. HPAI has recently garnered worldwide attention because of the “spillover” infection of this strain from domestic birds to humans - primarily those in poultry industry - causing significant human fatality and thus creating potentially favorable conditions for another flu pandemic. We use an ordinary differential equation model to describe this complex dynamics of the HPAI virus, which epidemiologically links a number of species in a multi-species community. We include the wild bird population as a periodic source feeding infection to the coupled domestic bird-human system. We also account for mutation between the low and high pathogenic strains. Finally, we fit our model to the actual number of human avian influenza cases obtained from WHO, and estimate the relevant reproduction numbers. We discuss open questions and problems in modeling the complex epidemiology of avian influenza.

Keywords: avian influenza, low-pathogenic avian influenza, high-pathogenic avian influenza, strains, competitive coexistence exclusion, human cases, data fitting. AMS Subject Classification: 92D30, 92D40. ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected]. (Corresponding author.) †

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1.

Joseph Lucchetti, Manojit Roy and Maia Martcheva

The Ecology and Epidemiology of Avian Influenza

Influenza viruses belong to the family Orthomyxoviridae (Greek orthos means straight, and myxo means mucus, implying a disease with respiratory symptoms), and are made up of segmented, negative sense, single stranded RNA genomes [1]. Orthomyxoviridae family currently consists of five distinct genera: Influenza virus A, Influenza virus B, Influenza virus C (also known as Influenza virus types A, B and C – those that cause influenza in vertebrate animals, including birds), Thogotovirus (tick–borne viruses that can also infect mammals) and Isavirus (a viral disease of Atlantic salmon). Among them, only the viruses of Influenza A genus are known to cause Avian Influenza (AI) infection in birds. The RNA genome of these viruses is made up of eight sections corresponding to the proteins they encode, and is encapsulated by a lipid bilayer obtained from host cells. Studding this outer shell are hemagglutinin (HA) and neuraminidase (NA) proteins, which play key roles in fusion of the viral envelope to host cells. Sixteen sufficiently distinct HA molecules (H1– H16) and nine NA molecules (N1–N9) have been observed in influenza A viruses that allow their further classification into subtypes. For example, the most recent AI outbreaks are due to the H5N1 subtype, whereas the previous three major outbreaks that led to pandemics were attributed to H1N1 (“Spanish Influenza”, during 1918–20), H2N2 (“Asian Influenza” of 1957–58) and H3N2 (“Hong Kong Influenza” of 1968–69) subtypes [7]. As of October 22, 2008, there are 387 total reported cases of H5N1 infection in humans worldwide, which resulted in 245 deaths – a case fatality rate of over 60% [8]. When compared with the death toll in the three pandemics mentioned above (an estimated 40– 100 million dead in Spanish Flu, over 4 million in Asian Flu and over 1 million in Honk Kong Flu [16]), the number of H5N1 related cases to date seems miniscule. This is in part due to the timeliness and scale of control measures that are being implemented (for instance, prompt quarantine and large–scale culling of domestic birds after initial infection is detected), but also largely because the virus appears to be quite inefficient in human–to– human transmission, an essential requirement to trigger a pandemic. With a few exceptions of probable limited person–to–person transmission in very close quarters [9, 10, 11, 12, 13, 14], humans seem to get infected only after handling live poultry or consuming severely undercooked poultry products such as raw duck blood [15]. Though it is likely that domestic poultry are often infected due to trafficking of live birds and their by-products, trade alone does not seem sufficient to justify the global spread of the disease. It is expected that migratory birds play a role in the transmission of the disease to domestic birds on the global scale.

1.1.

The Role of Wild Birds

In the past, avian influenza was found primarily in wild birds. All sixteen H subtypes and nine N subtypes are routinely detected in wild bird populations, and are particularly common in Charadriiformes (gulls and shorebirds) and Anseriformes (waterfowl such as ducks, geese and swans) [2, 3]. Until recently, all strains of AI have been either asymptomatic or caused mild respiratory problems in wild birds [3]. These strains are therefore known as low pathogenic AI (LPAI) strains, and wild birds are their natural reservoirs [3]. Prior to 2002, there was only a single reported case of highly pathogenic AI (HPAI) outbreak

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in wild birds; this occurred in 1961 in South Africa where some 1300 common terns died from a strain of influenza A virus of subtype H5N3 [4]. Post 2002, there have been several outbreaks of AI in wild birds accompanied by high mortality rates. Still, relative to the prevalence of AI in wild birds, or the number of AI cases in domestic birds, the instances of wild bird deaths from HPAI infection is quite small [5]. This relatively low frequency of outbreaks in wild birds is one of the characteristics that makes HPAI potentially dangerous. For example, if a strain of H5N1 that is highly pathogenic in domestic birds or humans are asymptomatic, or only mildly symptomatic, in migratory wild birds even for a short duration, the virus could be carried across the globe along migratory pathways [17, 18, 19, 20]. Moreover, direct long–distance movements may not be necessary, and viruses may also spread by sequential contacts among wild birds along their migration routes, and via environmental reservoirs [2, 5, 42, 43]. There are two recent HPAI outbreaks in wild birds that tend to indicate such may be the case: Qinghai Lake outbreak in Western China [21] and Lake Towada outbreak in Japan [43]. In 2005, approximately 6000 wild birds, mostly bar-headed geese, were found dead around Qinghai Lake [21]. While such HPAI outbreaks typically occur in the vicinity of domestic poultry farms, thereby indicating that these may be due to “spillover” infection from domestic to wild birds, this particular outbreak occurred in the Qinghai Lake Natural Protection Zone, far away from any such farm. Phylogenetic analysis of isolates of viruses from outbreaks in Europe [39] and Africa [40] in 2006 revealed that the Qinghai Lake virus was their likely ancestor. However, among the 390 strains of H5N1 isolated from poultry farms and bird markets in Southern China in 2005 and 2006, only one was found to be genetically similar to the Qinghai Lake virus [41]. These observations provide compelling evidence that migratory birds might have been the cause of transmission. Similarly, on April 21, 2008, four whooper swans were found dead in Lake Towada in Japan, all infected with H5N1 strains [43]. Though there have been poultry outbreaks of H5N1 across Japan, there was none since the beginning of 2008, and phylogenetic analysis indicated that the viruses that killed these swans are genetically distinct from the earlier domestic strains [43]. Limited outbreaks of H5N1 in Australia further suggest that migration may play a role in the transmission of AI, since many of the birds that migrate to Australia do so over regions with high H5N1 activity [44]. The hypothesis that wild birds, and in particular migratory birds, are a significant contributor to the global spread of HPAI is, however, not without scrutiny. Feare [5] analyzes several outbreaks between 2002 and 2007, and notes that many of these neither follow migratory paths nor occur during the season when birds typically migrate. The case at Towada Lake appears to lend support to Feare’s argument, since the migratory pattern of whooper swans indicates they might have already been in Japan several months prior to their death. The most likely explanation is that other wild birds infected these swans while they were residing in Japan [43], an apparent case of sequential infection (mentioned earlier). Further, even though H5N1 strains that are highly pathogenic to poultry and humans are found to be non-pathogenic in some wild birds in laboratory experiments [45, 46, 47, 48], there is insufficient evidence that this will remain so in the field. The few times such HPAI infection in apparently healthy migratory birds have been reported, for instance at Poyang Lake in China [22] and Lake Chany in Russia [23], the methodology and sampling employed in these works are questioned [24]. Moreover, the physiologically demanding task of long-

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distance flight appears to induce immunosuppression in wild birds, whose migratory performance is thereby compromised by HPAI infection [25]. Thus, it seems unlikely that wild birds can remain asymptomatic to HPAI infection long enough to spread the viruses over long distances. From these observations, a scenario seems to be emerging where the migratory birds perform at least a moderately important role in the geographic distribution of HPAI virus. Because they are natural reservoirs of, and therefore mostly asymptomatic to, the LPAI strains, there is little doubt that they have been carrying low pathogenic strains during migration. Recent phylogenetic analysis of LPAI isolates collected in Alaska from the northern pintail ducks, a species that migrate between North America and Alaska, shows significantly high frequency (45%) of intercontinental genetic exchange between Asian and North American virus lineages, nearly 7× larger than previously reported [26] (the authors suspect this may still be an underestimate). This is an important observation given that northern pintails are one of the rare east–west transcontinental migrants, and from this finding it is possible that eventual point mutations of these North American HPAI strains may retain substantial genetic similarity to their Asian counterparts.

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1.2.

The Role of Domestic Birds

Domestic poultry play several crucial roles in the evolutionary dynamics and transmission of HPAI. Studies seem to indicate that HPAI strains evolved by point mutations from the LPAI strains after their spillover infection of domestic birds from the wild birds [1, 3]. Prior to the first outbreak of human cases of H5N1 in 1997, it was determined that amino acid changes in AI viruses did not cause selective benefits in wild birds. This indicated that AI in the wild may have been at an evolutionary equilibrium [2, 49]. This is most likely not the case for domestic poultry, which are novel hosts to many subtypes of AI. This allows selective pressures to act, possibly creating highly pathogenic strains. Because of the likely infection of these HPAI strains from domestic birds back to the wild bird populations (which are also novel hosts to these evolved strains), studies have shown that many strains of AI are currently under positive selective pressure [40, 50]. The processes that trigger such point mutations of LPAI strains are not yet well known. In some cases the mutations appeared to have happened soon after introduction of the LPAI strains into domestic birds, whereas in others the LPAI viruses circulated in poultry for months before mutating into HPAI [1]. Even though it is impossible to predict if and when this mutation will occur, it seems clear that the longer and wider LPAI circulates in poultry, the higher the chance that mutation to HPAI will occur [1]. Once mutant strains of AI arise, domestic birds play a further role in the maintenance and transmission of the virus. One of the most interesting developments in the analysis of AI in domestic birds occurred in 2005 when Hulse-Post et al., [45] investigated the pathogenicity of H5N1 in ducks. It is widely known that some strains of H5N1 that are fatal in chickens can be asymptomatic in ducks. However, there are many strains of H5N1 that cause high mortality in ducks. Hulse-Post et al., found that if a duck is infected with a strain that eventually causes mortality, the virus that is shed by the infected duck during the late stages of infection can exhibit low pathogenicity to other ducks, while still being highly pathogenic to chickens and mice. This suggests that ducks may play a crucial role in

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maintaining and transmitting mutant strains of AI that are highly pathogenic to poultry and humans [45]. The role of ducks in transmission of AI is further explored by a study of the spatial spread of outbreaks in Thailand during 2004 and 2005. It was found that these outbreaks were strongly correlated with the abundance of free-grazing ducks. Such ducks rotate through recently harvested rice paddy fields every few days, carrying any pathogens they may be infected with to new fields. These fields are also a source of food for many migratory birds [51]. These observations, taken in conjunction with Hulse-Post et al.,’s findings and the hypothesis that wild birds may transmit HPAI, suggest a complex picture for the spread of AI via inter–specific interactions. One possible scenario is that wild birds carry AI either short or long distances and infect ducks, which convert the highly virulent strain of AI into a form that is asymptomatic in other ducks. These ducks could then freely infect chickens and humans without being overburdened by the disease themselves. Anthropogenic factors, such as movement of poultry, poultry manure (as agriculture and aquaculture fertilizers), poultry by–products, accidental transfer of infected material from poultry farms (e.g., contaminated water, straw, or soil in vehicles during transfer), legal and illegal trades of live animals etc. are also linked to many AI outbreaks all over the world, particularly in South–East Asia [22]. For example, a multivariate analysis of risk factors associated with H5N1 outbreaks in individual farms in China during 2002 identified one of the strong factors to be whether a farm had been visited by someone from a live bird market [52]. Even though poultry are not transferred from live bird markets to farms, bird buyers, catchers, trucks and cages all could carry the virus in that direction [52].

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1.3.

Transmission to Humans and Control Measures

Because all HA and NA influenza subtypes are found in wild birds and have the potential to add new virulent mutations and combinations to the existing human flu arsenal, a high degree of human infection risk from avian influenza remains. However, the rarity of direct interaction between wild birds and humans (their ecological habitats do not often overlap) creates an epidemiological bottleneck that the infectious strains must first pass through domestic bird populations before they can reach human populations; that is, humans are at the end of the interaction chain wild birds → domestic birds → humans. While the mechanisms of bird–to–bird transmission of LPAI and HPAI strains follow complex interaction pathways between wild and domestic birds, and are still poorly understood [1, 3], most human infections with HPAI viruses have occurred from direct and prolonged contact with poultry. The likely pathways are proximity to contaminated air or water, inhalation of infectious aerosol droplets, exposure to infected poultry and butchering of birds, and consumption of duck’s blood or undercooked poultry, even though the relative efficiency of these different routes has not been determined [15]. The first known example of the direct transmission of HPAI from domestic birds to humans was recorded in Hong Kong in 1997, when 18 people were infected with H5N1 strains, six of whom died [27]. Since then, human cases have been reported mainly in China, Thailand, Indonesia, and Vietnam; almost all of them are from poultry–related infection. There have also been occasional instances where such bird–to–human transmission was observed outside mainland Asia. For example, in 2003 poultry workers in the Netherlands contracted viral conjunctivitis, after being infected with

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the HP H7N7 strain from poultry flocks [28]. Timely intervention of international authorities, including the OIE (World Organization for Animal Health), resulted in its successful eradication. However, in Southeast Asia, livestock husbandry practices that include housing domestic birds in close proximity to humans and the use of live–bird markets make this region a high–risk place for human infection with HPAI strains. Judging from the available data, no case of transmission via aerosols has been confirmed, risks to health care workers appear to be minimal, and blood tests of persons in contact with human AI–infected patients have been negative [29]. These facts seem to suggest that the virus has not yet evolved into becoming broadly transmissible from human to human [30]. There are, however, instances where probable person–to–person transmission has been reported among humans in very close quarters, for example within family clusters [9, 10, 11, 12, 13, 14]. For instance, the 2003 H7N7 outbreak in the Netherlands appeared to have caused secondary human–to–human transmission between poultry workers and their immediate families in at least 30 households [12, 31]. From December 2005 to January 2006, a cluster of 8 confirmed cases, belonging to 3 households within 1.5 km of each other, was detected in eastern Turkey [13]. In 2006, a cluster of 8 human cases, all members of the same extended family of a 37–year old woman who originally became infected with HPAI H5N1, was detected in northern Sumatra [32]. However, none of these reported human–to–human transmission cases seem to be definitive, and moreover, their frequency is still much smaller than the confirmed bird–to–human transmission cases. Control of AI outbreaks in domestic birds typically involves mass culling and quarantine of infected poultry. Globally several hundred million birds have been destroyed so far, causing economic damage estimated at over US$10 billion in Asian poultry sector alone [33]. In 2004, more than 62 million birds were either killed from H5N1 infection or culled in Thailand, whereas Vietnam saw culling of over 50 million birds since 2004 [33]. In the 2003 Dutch H7N7 outbreak, a total of 255 flocks became infected during a period of nine weeks, and more than 30 million birds were culled as a control measure [34]. Besides mass culling, vaccination of flocks against HPAI infection is now also adopted as an effective, and popular, alternative strategy [35]. Several vaccination initiatives have been implemented in Mexico, Italy, The Netherlands, and different places in Asia [33, 36, 37]. However, vaccinating poultry can make monitoring and surveillance difficult, and using a single vaccine strain may drive evolution of AI strains into new genetic variants [38]. These issues apart, AI vaccination appears to be here to stay and promises to be effective in reducing HPAI outbreaks [3].

1.4.

Epidemiological Models of Avian Influenza

The models developed so far to describe the dynamics of avian influenza fall into two basic categories: stochastic and deterministic. With the exception of one paper, all of the stochastic models surveyed are intended to describe the spatial spread of H5N1 after its hypothetical mutation into a strain that can efficiently transmit from human to human. Germann et al., [53] used a stochastic model to show that if a human–to–human transmittable form of AI starts at a single location and has a low basic reproduction number (R0 < 1.9), early detection and preventive measures (most importantly vaccination and restricting mobility of people) are likely to work in all but a small portion of the population. Further, for

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such values of R0 , the study concludes that vaccination alone will be able to eradicate any pandemic if sufficient stockpiles of vaccine (55 million courses for R0 = 1.8) exist. This is in agreement with an earlier study by Longini et al., [54] that concluded that, for small values of R0 (< 1.4), containment with 100,000 courses of vaccine could be sufficient to control the pandemic, whereas larger values of R0 (> 1.7) will likely require restricting the mobility of the population in order to control the spread of the disease if only limited stockpiles of vaccine are available. Mills et al., [55] disagrees with this mode of analysis. Indeed, given the way that AI spreads through wild and domestic birds, it is unlikely that human–to–human transmission will occur only once or even several times. Such control measures will only prolong the time until a control measure fails. Rao et al., [56] instead ran stochastic simulations assuming that human–to–human transmission is not yet efficient, and focused their model on wild and domestic birds. They concluded that without efficient human–to–human transmission, it is improbable that a large pandemic would occur. The second class of models are deterministic, typically based on differential or difference equations. Relatively little seems to have been done for this class of models. The model that seems most relevant to our work here is due to Iwami et al., [57]. Their model is a basic SIR model with the following additional features: 1) compartments are provided for susceptible and infected domestic birds, 2) mutation is allowed to occur within the human population that causes the virus to become human–to–human transmittable, and 3) the model assumes that infection with either strain of the virus results in permanent immunity from both strains. However, once such mutant viruses are created, the model assumes that they stay within the human population. The authors are able to obtain reasonable definitions of the basic reproduction numbers for the bird virus strain as well as the human virus strain and obtain global stability of the appropriate equilibria. In a later paper [58], the authors analyze the model’s predicted effect of human quarantine and culling of domestic birds. While quarantine is found to be always beneficial, culling is detrimental for certain values of the parameters. However, since this model neglects to take into account the complicated wild bird-domestic bird dynamics and does not allow for the possible transmission of mutant viruses in bird populations, it is unclear how realistic the conclusions are.

1.5.

Structure and Organization of This Article

The chapter is organized as follows. In section 2, we introduce our deterministic (ODE– based) model of AI transmission dynamics, incorporating the epidemiological interactions between wild and domestic birds and humans, which include LPAI→HPAI mutation within the domestic bird population. Section 3 presents a preliminary analysis of the model and describes some of its mathematical properties. This section derives the reproduction numbers of LPAI and HPAI in domestic birds, as well as the invasion reproduction number of LPAI. Invasion of HPAI may occur only in the case when there is no LPAI→HPAI mutation, and the respective invasion reproduction number is also derived in this case. In section 4, we describe the curve fitting algorithm using MATLAB software package, and use it to fit the human HPAI infection data obtained from WHO database. We discuss the possibility for projections of the cumulative number of cases based on the model introduced in section 2. We conclude the chapter in section 5 with a discussion of some open questions and problems in AI modeling.

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2.

The Model

2.1.

Description

We introduce an ordinary differential equation model for the wild bird→domestic bird→ human pathway of avian influenza transmission. The main objective of our model is to capture the epidemiology of domestic bird–human interaction dynamics, where wild birds serve only as a background source of LPAI injection into domestic bird populations. Therefore, we model the number of LPAI–infected wild birds as a parameter, denoted by Iw . Transmission of LPAI virus to domestic birds (from infected wild birds) occurs at a rate βLw Iw , where βLw is the transmission coefficient; likewise, βLd denotes the transmission coefficient for transmission from LPAI–infected domestic birds. Because LPAI is typically mild in domestic birds and rarely causes mortality, in the model we assume that for LPAI–infected birds there is no disease–induced death. Domestic birds recover from LPAI infection at a rate αd . From the available evidence discussed earlier, we include the possibility that LPAI strains can mutate into HPAI within domestic bird populations at a per capita rate m. By definition, domestic birds suffer a high mortality rate from infection with HPAI strain; we denote this rate by µIHd . Both susceptible and LPAI-infected domestic birds are infected by HPAI strains with the transmission coefficient βHd . The most favored control strategy adopted today consists of destroying all birds in the farm whenever HPAI infection is detected. The very high mortality from HPAI together with culling cause very few domestic birds to survive HPAI. Accordingly, our model does not incorporate recovery from HPAI infection. Furthermore, we will assume that the rate at which culling of the domestic bird population takes place is proportional to the number of birds that are infected with either LPAI or HPAI, and that birds from all disease classes are culled at the same per capita rate µcull (ILd + IHd ). Humans appear to contract only HPAI, and primarily through contact with infected domestic poultry. We denote the corresponding transmission coefficient by βHu . Roughly 40% of infected humans do recover (presumably because of health care), and we denote the per capita recovery rate by αHu . Since there are very few confirmed cases of human– to–human transmission of H5N1 so far, we do not consider human-to-human transmission dynamics in our model. We model the 60% human case fatality (from cumulative WHO data) by incorporating a disease-induced death rate µIHu . Thus, we use a standard SIR model for the human epidemiology with the modification that susceptible humans are infected at a rate proportional to the number of HPAI–infected domestic birds, rather than the number of infected humans. Figure 1 shows the observed cumulative number of human cases of H5N1 infection provided by WHO. One objective of our model is to find values of the model parameters (described above) that best fit this data in the least squared sense. A glance at Figure 1 shows that the rate of new infections appears to oscillate with a period close to 365 days (one year). The model described below may not be capable of intrinsically generating such cycles. Therefore, we introduce an external periodic forcing to the dynamics, in terms of a periodic LPAI injection into the domestic bird population from infected wild birds. There is a strong biological justification for such a periodic forcing, as explained below.

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Cummulative Data for H5N1 Infections

Cummulative Number of Infected Humans

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Figure 1. Graph of the cumulative number of human infections. Data taken from WHO’s web-site.

We model LPAI transmission from wild to susceptible domestic birds as the per capita 2π rate βLw Iw (sin((t + ω) ∗ 365 )c1 + c2 ), where c1 ≤ 1 and c2 ≥ 1 influence how strongly this periodic function affects the transmission rate (the condition c2 ≥ c1 prevents negative values for a density function). This periodic transmission may be interpreted as due to 2π )c1 + c2 ), either a periodicity in the infected wild bird population size, Iw sin((t + ω) ∗ 365 2π or a periodicity in the transmission term, βLw sin((t + ω) ∗ 365 )c1 + c2 ). The first type of periodicity may be a result of the seasonal breeding patterns of migratory wild birds, which seasonally boosts the number of susceptible wild birds that in turn leads to a surge in the number if LPAI-infected wild birds. The second type of periodicity may arise in the periodically enhanced contact rates between wild and domestic birds during seasonal migration of the former.

2.2.

Summary of Simplifying Assumptions

Modeling the complex interactions involving wild bird species, domestic bird species and humans requires introducing a large number of parameters, few of which are known or even easily determined from available information. Therefore, we must make several assumptions in order to reduce the number of parameters. This in turn will reduce the number of parameters to be estimated through fitting. 1. Because the initial period of time over which we fit the data is fairly small ( 0 such that lim inf ILd (t) ≥ η

for all

t

ILd (0) > 0.

Similarly, we will say that the HPAI is uniformly strongly persistent if there exists η > 0 such that lim inf IHd (t) ≥ η for all IHd (0) > 0. t

Uniform strong persistence says that if the initial number of domestic birds infected with LPAI is positive, the number of domestic birds infected with LPAI will remain positive and long term will not approach zero. To see the uniform strong persistence, notice that the number of susceptible domestic birds satisfy the following inequality Sd′ ≥ Λd − pd Sd Λd d where pd denotes the following constant: pd = µcull Λ µd + µd + (βHd + βLd ) µd + (c1 + c2 )Iw βLw . The above inequality implies the persistence of the susceptible population, a result that would hold independently of the presence of the external source:

lim inf Sd (t) ≥ t

Λd > 0. pd

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The second equation in (2.1) leads to the following inequality for the number of domestic birds infected with LPAI: IL′ d (t) ≥ ΛLd − pLd ILd (t) Λd Λd d where pLd = µcull Λ µd +µd +m+αd +βHd µd and ΛLd = c2 βLw Iw 2pd . We note that ΛLd is positive only in the case when Iw is positive. The above inequality implies the persistence of the LPAI: ΛLd lim inf ILd (t) ≥ > 0. t pLd

LPAI would persist in the domestic bird population without additional conditions only in the case that the wild bird population is a constant external source of the pathogen. Similar reasoning applied to the fourth equation in (2.1) gives ′ IH (t) ≥ m d

ΛLd − pHd IHd (t) 2pLd

which leads to persistence lim inf IHd (t) ≥ t

mΛLd > 0. 2pLd pHd

HPAI persists in the domestic bird population without additional conditions only when mutation of LPAI into HPAI takes place on a continuous basis and the wild bird population is an external source of LPAI. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Since system (2.1)–(2.2) has a periodic forcing term, it may be possible to establish that it has one (or more) periodic solution(s) of the same period as the source term. Most simulations result in solutions that converge to a periodic solutions. None of the human epidemiological classes in system (2.2) is persistent in any sense. A continuous infection process coming from a persistent infection in the domestic bird population depletes the pool of susceptible humans. Because the model assumes no birth, the number of susceptible humans, and hence the number of infected and recovered humans, decline in time to zero. This means that the human component of our model is only valid short term.

3.2.

Avian Influenza in the Domestic Bird-Human System

Strains of LPAI were first introduced to the domestic bird populations from the wild bird populations. However, sampling for LPAI H5N1 strains in wild bird populations, particularly in North America, shows very low prevalence [61]. We would like to investigate whether the LPAI and HPAI strains can persist in the domestic bird-human system if there is no continuous inflow from the wild bird population. Because we estimate the transmission rate of LPAI strains from wild to domestic birds as very small (see Table 2), and this transmission rate can be made even smaller through control measures, we set βLw = 0. In this case the system becomes closed (there is no influx) and autonomous. We can define the reproduction numbers of the LPAI as RL =

βLd Λd µd (µd + m + αd )

and the HPAI as

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RH =

βHd Λd . µd (µd + µIHd )

The reproduction numbers measure the number of secondary cases of LPAI (HPAI respectively) that one infected domestic bird with LPAI (HPAI respectively) will produce in an entirely susceptible bird population. With our least-squared estimated parameters in Table 2 we have RL = 1.42857, RH = 2.71781. Recall that these are reproduction numbers in the absence of external infection source. Typically, LPAI would persist in the absence of HPAI if RL > 1. Similarly, HPAI would persist in the absence of LPAI if RH > 1, as in each case, each infected bird replaces itself with more than one new infected bird. We need both reproduction numbers to be larger than one so that strains of both kinds may persist, if we assume no mutation of LPAI into HPAI. Since we assume continuous mutation (although very small) of the LPAI into the HPAI, we only need RL > 1 to guarantee persistence of both LPAI and HPAI in the domestic bird population. In other words, if LPAI persists in the domestic bird population and continuously mutates into the HPAI, the HPAI will also persist. All those observations can be rigorously established mathematically as we did before about the non-autonomous system but will be omitted. System (2.1) with βLw = 0 controls the behavior of system (2.2). Avian flu will remain in the human population if it persists in the domestic bird population. We focus on system

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(2.1) to understand the behavior of the solutions. Ordering the variables in the order of the equations in system (2.1), system (2.1) has a disease-free equilibrium where there are only susceptible birds but no pathogen of any kind. The disease-free equilibrium is given d by E0 = ( Λ µd , 0, 0, 0). Concerning the endemic equilibria, since our estimated value of the mutation rate m is very small, it is worth considering two cases which lead to somewhat different equilibria of the LPAI and HPAI. In this section, we continue with the case m 6= 0. The case m = 0 will be taken up in the next section. In the case m 6= 0, there is no equilibrium of the LPAI alone. That is, we cannot have non-zero numbers of infected individuals with LPAI but zero infected individuals with HPAI. This is in agreement with our previous observation that persistence of LPAI in the presence of continuous mutation leads to persistence of HPAI as well. Thus, single-strain equilibria are only of the HPAI. This means that HPAI can be present in the domestic bird population whether LPAI is present or not. Any HPAI–only equilibrium has the form EH = ∗ ) where (we omit the stars) (Sd∗ , 0, 0, IH d Sd =

µcull IHd + µd + µIHd βHd

and IHd is the solution of the following equation µcull IHd + µd + µIHd

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βHd

=

βHd IHd

Λd . + µcull IHd + µd

(3.3)

The above equation has a positive solution if and only if RH > 1. The solution, if it exists, is necessarily unique, since the left hand side above is an increasing function of IHd while the right hand side is a decreasing function of IHd . This gives a unique equilibrium EH of HPAI alone. It can be shown that this endemic equilibrium is locally asymptotically stable. It can also be shown that if RL < 1 and RH < 1, then, both LPAI and HPAI will disappear from the domestic bird population. The invasion reproduction number of the LPAI at the equilibrium of the HPAI is given by ˆL = R

∗ µcull IH d

βLd Sd∗ ∗ +µ +m+α . + βHd IH d d d

ˆ L > 1 (and that implies If RH > 1 and LPAI can invade the equilibrium of HPAI, that is R that RL > 1), then we expect that the two AI strains coexist. In fact, it should be possible to establish uniform strong persistence of both LPAI and HPAI. In this case, the endemic equilibrium EH for HPAI is unstable. If RH > 1 and LPAI cannot invade the equilibrium ˆ L < 1, then, the expectation is that HPAI persists but LPAI dies out, even of HPAI, that is R if RL > 1. This last statement is not easy to show, and more complex scenarios might be possible. The biological significance of this last scenario is that HPAI can displace ˆ L that corresponds (competitively exclude) LPAI strain in domestic birds. The value of R ˆ to the parameter estimates that are listed in Table 2 is RL = 0.501173. This value suggests that in the absence of external inflow of LPAI from the wild bird population, HPAI will displace and eliminate LPAI in the domestic bird population. We note that culling as a control strategy in model (2.1) does not affect the reproduction number RH because we assume that the per capita culling rate is proportional to the number

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of infected birds. In other words, in a completely susceptible bird population, there is no culling (this is equivalent to the fact that culling is not needed if infection is not detected). So culling has no effect on the reproduction number. There are other types of reproduction numbers, such as the effective reproduction number. The effective reproduction number depends on the number of susceptible birds at time t, and will depend on culling: RH (t) =

βHd Sd (t) . µd + µIHd

Looking at equation (3.3) one can see that the equilibrium number of HPAI infected do∗ declines as µ mestic birds IH cull increases. The maximum number of infected birds which d occurs with µcull = 0 is given by max IH = d

µd (RH − 1) . βHd

ˆ L on the culling rate µcull The dependence of Sd∗ and the invasion reproduction number R seems non-monotone, and possibly complex.

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3.3.

Invasion of HPAI. The Case m = 0

Wild bird populations are reservoir of LPAI strains. It is currently believed that strains of the H5N1 AI underwent antigenic drift in the mid-1990’s and became adapted to domestic birds. The HPAI occurred as a result of reassortment in the domestic birds, and now is spreading among the domestic bird population [6]. It appears that continuous mutation from LPAI to HPAI does not occur – a scenario which seems to agree with our estimates on m which are nearly zero. In this subsection, we consider the mathematically distinct case m = 0 and address the question under what conditions a newly emerged HPAI strain can invade the domestic bird population. In this case, the reproduction numbers of LPAI and HPAI are given by the same expressions as before with m = 0. In the case m = 0, besides the disease-free equilibrium E0 that we found with m 6= 0, there is a unique LPAI-only equilibrium and a unique HPAIonly equilibrium. The HPAI exclusive equilibrium exists if an only if RH > 1 and is ∗ ) with values of the non-zero quantities given by the expressions in EH = (SdH , 0, 0, IH d the case m 6= 0. In addition to the HPAI exclusive equilibrium, in the case m = 0 there is also an LPAI exclusive equilibrium which exists if and only if RL > 1 and is given by EL = (SdL , IL∗ d , R∗ , 0). The non-zero values in that equilibrium are SdL and

IL∗ d

=

µcull IL∗ d + µd + αd βLd

is the unique positive solution of the equation: βLd IL∗ d

µcull IL∗ d + µd + αd Λd = βLd + µcull IL∗ d + µd

which exists if and only if RL > 1. The value of R∗ is given by R∗ =

αd IL∗ d µcull IL∗ d + µd

.

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The invasion capabilities of an emergent HPAI strain are measured by the HPAI invasion reproduction number at the equilibrium of LPAI. The HPAI invasion reproduction number is given by βHd (SdL + IL∗ d ) ˆH = R . (3.4) µcull IL∗ d + µd + µIHd It can be rigorously established that if RL > 1 so that the equilibrium EL exists and if ˆ H > 1, that is HPAI can invade the equilibrium of LPAI, then HPAI persists in the R domestic bird population. Depending on the invasion reproduction number of LPAI, the following options are possible: ˆ H > 1, and if, in addition, RH > 1 so that the equilibrium EH 1. If RL > 1 and R ˆ L < 1, so that HPAI cannot invade the equilibrium of HPAI, then the exists and if R ∗ as t → ∞. This expectation is that ILd (t) → 0 as t → ∞, while IHd (t) → IH d global result is not easy to establish. ˆ H > 1, and if, in addition, RH > 1 so that the equilibrium EH 2. If RL > 1 and R ˆ L > 1, so that LPAI can also invade the equilibrium of HPAI, then exists and if R ¯ I¯H ). there is a coexistence equilibrium E ∗ = (S¯d , I¯Ld , R, d

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To see this last statement, we set the derivatives equal to zero. From the first equation in (2.1) we have Λd . Sd = µcull (ILd + IHd ) + βHd IHd + βLd ILd + µd The right-hand side of the expression above is a function of IHd and ILd which we denote by S(ILd , IHd ). The function S is a decreasing function of each of its arguments when the other argument is held fixed. From the second and forth equation in (2.1) we get the system:  F (ILd , IHd ) = 1, (3.5) G(ILd , IHd ) = 1, where the two functions F and G are given by  βLd S   F (ILd , IHd ) = ,   µcull (ILd + IHd ) + βHd IHd + αd + µd       G(ILd , IHd ) =

(3.6)

βHd (S + ILd ) . µcull (ILd + IHd ) + µIHd + µd

The function F is a decreasing function of both ILd and IHd . The function G is a decreasing function of IHd when ILd is held fixed. For each fixed ILd , the equation F (ILd , IHd ) = 1 has a unique solution. Since the derivative of F with respect to its second argument is strictly negative for non-negative values of the arguments, the implicit function theorem implies that there is a continuous, differentiable function f such that IHd = f (ILd ). Define H(IHd ) = G(ILd , f (ILd )). If IˆHd is the unique solution of F (0, IˆHd ) = 1, then ∗ , where I ∗ is the value in the HPAI-only equiIˆHd = f (0). We note that IˆHd > IH Hd d librium EH . The function G(ILd , IHd ) is a decreasing function of IHd and therefore,

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∗ ) > G(0, Iˆ ) = H(0). On the other hand, since F (I ∗ , 0) = 1, then 1 = G(0, IH Hd Ld d f (IL∗ d ) = 0. Consequently,

H(IL∗ d ) =

βHd (S(IL∗ d , 0) + IL∗ d ) µcull IL∗ d + µd + µIHd

ˆ H > 1. =R

This implies that the equation H(ILd ) = 1 has a solution I¯Ld satisfying 0 < I¯Ld < IL∗ d and I¯Hd = f (I¯Ld ) with 0 < I¯Hd < IˆHd . That establishes the existence of a coexistence equilibrium. The coexistence equilibrium that occurs when both invasion reproduction numbers are larger than one, is usually locally asymptotically stable. We now move on to fitting our differential equation model to the cumulative number of human case data of AI as given by WHO [8]. Fitting a dynamic model to data has a threefold purpose: (1) Validation of the model, (2) Estimating the parameters, and (3) Projection of future number of cases. We address all three purposes, but first we describe our fitting procedure.

4.

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4.1.

Fitting Model to Data General Method

The model fitting is implemented using MATLAB’s lsqcurvefit function. Cumulative HPAI infection data for humans are obtained from the WHO database [8]. We discard data points between the days from January 28 through October 25, 2004, because of lack of precision in reporting (and also their seeming mismatch with the rest of the data – see Figure 1). All points near the beginning and end of the apparent period (time unit=500 and time unit=850 in Figure 1) are included since the outbreak pattern seems to change rapidly near it. For the remainder of the days between January 7, 2005 (time unit=341), and April 11, 2007 (time unit=1165), roughly one point per month is selected from the available data whenever possible. We carried out our initial curve fitting shortly after April 11, 2007, based on the data points available up to that time. Data from April 11, 2007, to September 10, 2008 (most recent data available at the time of writing this article), was later collected from the updated WHO database, and used to determine the predictive capabilities of the model. We fit our model to the data using initial guesses for all model parameter values, and obtained better estimates of the same parameters from the fit. The function that MATLAB fit to the data was the numerical solution of the model (2.1)–(2.2). That is, MATLAB computed the numerical solution of the model with a different set of parameters every time lsqcurvefit evaluated the function with which it was attempting to fit data. Unfortunately, MATLAB’s curve fitting procedures are not particularly good at determining global minima, so one must manually determine reasonable values of the parameters. If the initial guesses for the parameters are poor, MATLAB will find a local minimum for the least squared error that is far from the global minimum. Once sufficiently good estimates of the parameters were determined, a script was run which iteratively executed lsqcurvefit. After each execution, one or several parameters were perturbed in order to attempt to find alternative smaller local minima near the previous best-fit parameters.

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4.2.

19

Specific MATLAB Code

Before the curve fitting can begin, MATLAB must be supplied with the model. To code a basic system of differential equations in MATLAB, one creates a function that takes as parameters a real number (the time variable) and an array (which specifies the value of each state variable). The function should then generate an array that specifies the value of the derivative of each of the state variables.1 Further, since the model’s parameters will be changing within the program, this function should also take in an array which specifies the value of each parameter. Suppose that AI model(t,x,c) is such a function, where t is the time variable, x is the array which specifies a value for each state variable (say x=[Sd , ILd , Rd , IHd , SHu , IHu , RHu , Itotal ]) and c is the array of parameters. Now, this function is neither what the differential equation solver expects as input (since it requires an array of parameters to be supplied), nor is it what lsqcurvefit expects (since it is the model itself, not the model’s solution). So, we need another function that fills this gap. This function should take as input an array of values of the independent variable and an array of parameters. It should return as output an array whose ith entry is the numerical solution of the model using the specified array of parameters evaluated at the ith entry of the array of the independent variable. In particular, the model will be fit to data that was obtained for Itotal , so the 8th coordinate of the numerical solution should be returned. The following function performs this task:

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function output=call AI model(c,tdata) %Input - c : An array of parameters for the model % tdata : An array of data points of the independent variable %Output % The numerical solution to AI model evaluated at times specified by tdata x0=[c(15),c(16),c(17),c(18),c(19),0,13,47]; AI model constants=@(z0,z1)(AI model(z0,z1,c)); [t,s]=ode15s(AI model constants, tdata, x0); output=s(:,8); lsqcurvefit can call this function since it takes in the proper variables and generates the proper array as output. It calls ode15s (one of MATLAB’s numerical ODE solvers) to solve AI model with parameters coming from c. So, we can now use lsqcurvefit to call this function using the following command: [new params,error] = lsqcurvefit(@call AI model,initial params, tdata, Infected data, param lower bound, param upper bound) We then place this command into a couple of embedded loops which slightly perturb the new parameters after each execution of lsqcurvefit. 1

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4.3.

Joseph Lucchetti, Manojit Roy and Maia Martcheva

Results from the Fitting. Extending the Fit

As mentioned before, we fit the model (2.1)–(2.2) in April, 2007, to the then available data on the cumulative number of human cases. We will call this set of data our calibration data set. This calibration data set consists of a total of 41 data points. The main criterion used for the goodness of fit to the calibration set of data was minimizing the least squared error Ec2 . The smallest least squared error that we obtained with our fit at that time had a value 283.08 across the 41 data points. The results of our original fit are presented in Figure 3.

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Figure 3. The curve that resulted from the curve fitting plotted alongside the WHO data

The initial fit was performed with all parameters left free for MATLAB to determine them from the best fit. Our main thrust was that a better fit will be obtained if MATLAB’s optimization routines have more degrees of freedom. MATLAB obtained the following values for the parameters from that fit (Table 2). One problem with this method is that for those parameters whose values can be independently obtained from elsewhere (e.g., available literature), the fitted estimates computed by MATLAB may not agree well with these values. For example, one parameter whose value can be obtained from the literature is the lifespan of poultry, which for chickens is 5 to 10 years when kept under favorable conditions. We estimate µd = 0.001678 day−1 which corresponds to a lifespan of 1.6 years. Similarly, WHO [8] data on the mortality of humans infected with avian influenza give a mean case-fatality proportion (CFP) [62] of approximately 0.6. That is, about 60% of the infected humans die from avian influenza. The probability of dying when infected with avian influenza, as given by the model, is Pd =

µIHu , µIHu + αHu

Pr =

αHu . µIHu + αHu

while the probability of recovery is

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Based on CFP, Pd ≈ 0.6 and Pr ≈ 0.4, whereas Table 2 gives Pd = 0.52 and Pr = 0.48. Thus, a value of Pd = 0.52 underestimates the observed probability of human death. (A better estimate of the disease-induced death rate µIHu may be obtained if we fit the cumulative number of dead individuals as reported by WHO [8].)

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Table 2. Parameters determined via the original curve fit of the model Variable Λd µIHd βHd βLw αd µIHu ω αHu Sd0 IHd0 µd µcull βLd m βHu c1 c2 Iw ILd0 SHu0

Value 1.7711 0.08912 0.0002338 3.1599e-007 0.03495 0.2845 7.1453 0.2669 746.0352 0.3384 0.001678 0.001207 4.9575e-005 6.6097e-009 0.001098 0.8517 1.0490 80.7456 14.1744 357.9811

Units individuals · day−1 day−1 individual−1 · day−1 individual−1 · day−1 day−1 day−1 days day−1 individuals individuals day−1 individual−1 · day−1 individual−1 · day−1 day−1 individual−1 · day−1 unitless unitless individuals individuals individuals

Despite the fact that Table 2 underestimates the poultry lifespan and the probability of human death from infection, these estimates are still reasonably close to the observed values. MATLAB optimization routines have the ability to keep a parameter fixed at a predefined value, or search for an optimal value of the parameter in a predefined range. Using these options would guarantee that known parameters have values within expected ranges. The least squared errors obtained from a fit when some parameters are kept fixed, or within range, are expected to be larger. In November 2008, we revisited the problem. WHO has continued to collect and update its database on the cumulative number of human HPAI infection cases. We will call this new data set between April 2007 and November 2008 the test data set. The main question that we asked was: If we use the model developed in April 2007 to predict this recent test data, how good would our predictions be? We extrapolated the best fit curve obtained from the model (2.1)–(2.2), with the estimated parameters from Table 2, up to November 2008 and

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compared it to our test data. In this early attempt (not shown here) the model appears to seriously underestimate the observed number of human cases. For our initial fit to the calibration data set, we chose the model with the smallest least squared error Ec2 , which appears to have done poorly with the new test data set. We refitted the model to the calibration data starting with larger numbers of susceptible humans and allowing for larger error Ec2 . In Figure 4 we show the results of our fit to the calibration data (red dots), with least squares error Ec2 of 370. The projection of the fit to the present day (blue line) as well as the new test data set from WHO (blue stars) are also shown for comparison. As can be seen, the model continues to underestimate the recently observed number of human cases between April 2007 and November 2008.

400

Cumulative Number of Infected Humans

350 300 250 200 150 100 Curve Fit Data Used to Fit Curve Data Available After Original Fit

50

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0 200

400

600

800 1000 1200 Time (Days Since 2/1/2004)

1400

1600

1800

Figure 4. The curve that resulted from the curve fitting the calibration set of data with least squares error of 370, extended to November 2008 and plotted alongside the WHO data

Figure 5 presents another attempt at predicting the test data set using an initial fit with even larger least squared error: Ec2 = 574. This time the model does a far better job of capturing the recent patterns of human cases. We show a sample of the model’s fit to the calibration data set, and its prediction of the test data set, in Table 3. Of course, in the future, if we use the model with least squared error of 574 (Figure 5) to predict the cumulative number of human HPAI cases after November 2008, our model would likely underestimate this future data as well. From the requirement of starting with higher and higher initial susceptible humans for better fit with future data sets, it appears that one likely reason for the underestimation is the absence of a recruitment term for susceptible humans in our model. However, continuing to increase the number of susceptible humans would lead to an even larger least squared error over the calibration set. Moreover, there are now nearly four years worth of WHO data on the human cases of avian influenza. Any model that fits those and predicts into the future would have to run over a period of 4+ years.

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An Avian Influenza Model and Its Fit to Human Avian Influenza Cases

23

450

Cumulative Number of Infected Humans

400 350 300 250 200 150 100 Curve Fit Data Used to Fit Curve Data Available After Original Fit

50 0 200

400

600

800 1000 1200 Time (Days Since 2/1/2004)

1400

1600

1800

Figure 5. The curve that resulted from the curve fitting the calibration set of data with least squares error of 574, extended to November 2008 and plotted alongside the WHO .

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This is no longer short enough period so that human demography can be totally ignored. Our next step will be to incorporate human births and natural deaths and refit the model. We close this section with some reflections on our curve fitting procedure. One of the problems with this method is that MATLAB’s routines assume that the function that is being fit to the data is continuous. That is, MATLAB assumes that the solution of the model is continuous in the parameters of the model. One explanation why some trial and error was required to find good values of the parameters is that this continuity requirement is not satisfied. However, lsqcurvefit did in general perform reasonably well at some points. The fit we obtained is, however, far from perfect. Consider the behavior of the points in Figure 3 near time unit=500 and time unit=850. The data points immediately before and after these times lie below the curve, whereas those at these times lie above the curve. Further, consider Figure 4, especially in the interval from 500 to 850 days. It looks as though this curve is typically increasing except at the beginning/end of this period. We believe a slight modification on the model (2.1)–(2.2) would result in a better fit. For instance, instead of our assumption that the death rate (of domestic birds) due to culling is simply proportional to the number of infected domestic birds, it seems as though they should have a more complex non-linear relationship. This is because an HPAI infection is not typically noticed immediately after it has occurred, and instead only once the infection becomes noticeable is culling implemented. Also, our model ignores vaccination as an alternative control measure. However, as noted in the beginning, vaccination is increasingly being employed in many places. Future efforts should focus on determining appropriate functional forms for these various effects.

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Joseph Lucchetti, Manojit Roy and Maia Martcheva Table 3. Results of curve fit.

Extended Prediction

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As Originally Fit

Date January 7, 2005 February 2, 2005 March 11, 2005 April 4, 2005 May 4, 2005 June 8, 2005 July 27, 2005 August 5, 2005 September 16, 2005 October 10, 2005 November 9, 2005 December 9, 2005 January 10, 2006 February 9, 2006 March 10, 2006 April 3, 2006 May 4, 2006 June 6, 2006 July 14, 2006 August 14, 2006 September 14, 2006 October 16, 2006 November 13, 2006 December 27, 2006 January 22, 2007 February 19, 2007 March 19, 2007 April 11, 2007 May 16, 2007 June 29, 2007 July 25, 2007 August 31, 2007 October 2, 2007 December 4, 2007 January 2, 2008 February 1, 2008 March 4, 2008 April 2, 2008 May 28, 2008 June 19, 2008 September 10, 2008

Days (Since 2/1/2004) 341 367 404 428 458 493 542 551 593 617 647 677 709 739 768 792 823 856 894 925 956 988 1016 1060 1086 1114 1142 1165 1200 1244 1270 1307 1339 1402 1431 1461 1493 1522 1578 1600 1683

Predicted CN (Dates Used for Fitting) 47.00 56.85 66.76 75.672 87.658 99.919 111.84 113.51 120.45 124.28 129.62 136.37 146.33 159.32 175.30 189.88 207.66 222.41 233.38 239.08 243.38 247.42 251.29 259.45 266.36 276.09 287.79 297.85

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Projected CN

Observed CN

311.43 322.90 327.10 331.26 334.17 340.63 344.96 351.12 359.77 368.74 383.74 387.69 395.63

47 55 69 79 89 100 109 112 113 117 125 137 147 166 176 190 206 225 230 238 246 256 258 261 269 274 280 291 306 317 319 327 329 336 348 357 370 376 383 385 387

An Avian Influenza Model and Its Fit to Human Avian Influenza Cases

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5.

25

Discussion

We developed an ordinary differential equation model to describe the complex epidemiology of the LPAI and HPAI strains, involving multi–species interaction of wild bird, domestic bird and human populations. Our model particularly focused on how the HPAI strain, after having mutated from the LPAI strain within domestic birds, is passed on to humans. We assumed wild birds to be a periodic source feeding seasonally pulsed LPAI infection to the domestic birds, in order to incorporate the approximately 1-year period oscillation observed in the cumulative human HPAI infection cases (the assumption of an external periodic forcing is biologically justifiable because of the seasonal migratory behavior of wild birds, which may change their population size during breeding season, and/or increase their contact rates with domestic birds during migration). Our model further incorporates a continuous mutation of the LPAI virus into HPAI strain within the domestic bird population, a feature again rooted on strong biological reasoning (to our knowledge, previous AI models have not taken such mutation into account). We mathematically analyzed the persistence and co-existence of LPAI and HPAI strains within the domestic bird population under different conditions by deriving appropriate reproduction numbers. Because there is no recruitment of humans in our model, all human disease classes approach zero asymptotically. Finally, we carried out a least-squared fit of our model to the cumulative human cases of HPAI infection obtained from the WHO database and estimated the model parameters, including the reproduction numbers, from the best fitted model. The interacting bird–human system of transmission of avian influenza seems ideal for developing and testing mathematical models of complex epidemiological systems that have predictive properties. The fact that WHO provides regularly updated cumulative number of human cases can help modelers fine–tune our models, and thereby increase their predictive powers. The examples shown here illustrate some of the challenges involved in this endeavor of model fitting. Based on the real–time updating of the WHO data, one can calibrate the model using the data available until this moment, and use this calibrated model to predict future cases. Later on, one can revisit the model and compare its predictions with the actual new set of data as reported by WHO. Prediction using these complex epidemic systems is uncertain and inherently risky. One can even expect models with good predictive powers to work over only small periods of time. The best strategy appears to be to fit the model to the known set of data, and predict future occurrences over short time windows. As new data become available, the model has to be refit and fine-tuned, and possibly used again for short-term prediction. One of the most pressing concerns today is the possibility that the HPAI strain can genetically recombine with the human influenza strain (that causes year–round mild flu symptoms in humans) within a co–infected host, such as pigs or humans that have similar cellular receptors for both flu strains and also often live in close proximity to domestic poultry [3, 60]. Such a recombination process can create a new subtype that has both the high pathogenicity of HPAI and high human–to–human transmissibility of human influenza. Because human immunity against such a novel strain will be minimal, a global pandemic with a high level of mortality could then occur. Indeed, a recent conservative estimate suggests a 3.9% probability of a flu pandemic occurring in any given year, with a 95% support interval of 0.7% – 7.6% (see Figure 1, [60]). Given this persistent threat, an urgent focus of the

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Joseph Lucchetti, Manojit Roy and Maia Martcheva

modeling efforts should be in understanding the dynamics of such a co–infection of, for example, the human hosts. This would require extending the model (2.1)–(2.2) to include the human influenza strain co–infecting humans that are simultaneously infected with HPAI strains (from birds), and a recombination process that creates the evolved strain within a human host. The model by Iwami et al., [57] considers human infection by a “mutant” AI strain that originates from point mutation of the original HPAI strain, and therefore ignores the biological realism of genetic recombination within a co–infected host as the more likely source of this novel strain. Developing suitable models for the next possible flu pandemic is a task for the future.

Acknowledgments The authors acknowledge support from the NSF under grants DGE-0801544 and DMS0817789.

References [1] D. J. Alexander, An overview of the epidemiology of avian influenza, Vaccine, 25 (2007), 5637–5644. [2] R. Webster, W. Bean, O. Gorman, T. Chambers, Y. Kawaoka, Evolution and ecology of infuenza A viruses. Microbiol. Rev. 56, (1992), p. 152–179. [3] L. Clark, J. Hall, Avian influenza in wild birds: status as reservoirs, and risks to humans and agriculture. Ornithol Monogr. 60, (2006), p. 3–29.

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[4] W.B. Becker, The isolation and classification of Tern virus: Influenza A-Tern South Africa-1961, J. Hyg. 64 (1966), p. 309–320. [5] C. Feare, The role of wild birds in the spread of HPAI H5N1. Avian Dis. 51 (2007), p. 440–447. [6] An early detection system for HPAI in wild migratory birds, http://www.doi.gov/issues/birdflu strategicplan.pdf [7] T. Horimoto, Y. Kawaoka, Pandemic threat posed by avian influenza A viruses. Clin. Microbiol. Rev. 14 (2001), p. 129–149. [8] World Health Organization, http://www.who.int/csr/disease/avian influenza/country/cases table 2008 09 10/en/index.html [9] S. Olsen, K. Ungchusak, L. Sovann, et al., Family clustering of avian influenza A (H5N1). Emerg. Infect. Dis. 11 (2005), p. 1799–1801. [10] K. Ungchusak, P. Auewarakul, S. Dowell, et al., Probable person–to–person transmission of avian influenza A (H5N1). N. Engl. J. Med. 352 (2005), p. 333–340. [11] WHO Avian Influenza – situation in Indonesia – update 16, http://www.who.int/csr/don/2006 05 31/en/. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

An Avian Influenza Model and Its Fit to Human Avian Influenza Cases

27

[12] M. Boven, M. Koopmans, M. Holle, et al., Detecting emerging transmissibility of avian influenza virus in human households. PLoS Comp. Biol. 3 (2007), p. 1394– 1402. [13] Y. Yang, E. Halloran, J. Sugimoto, I. Longini, Detecting human–to–human transmission of avian influenza A (H5N1). Emerg Infect. Dis. 13 (2007), p. 1348–1353. [14] H. Wang, Z. Feng, Y. Shu, et al., Probable limited person–to–person transmission of highly pathogenic avian influenza A (H5N1) virus in China. Lancet 371 (2008), p. 1427–1434. [15] J. Beigel, J. Farrar, A. Han, et al., Avian influenza A (H5N1) infection in humans. N. Eng. J. Med. 353 (2005), p. 1374–1385. [16] J. Peiris, et al., Avian influenza virus (H5N1): a threat to human health. Clin. Microbiol. Rev. 20 (2007), p. 243–267. [17] J. Liu, H. Xiao, F. Lei, et al., Highly pathogenic H5N1 influenza virus infection in migratory birds. Science 309 (2005), p. 1206. [18] M. Gilbert, X. Xiao, J. Domenech, J. Lubroth, V. Martin, J. Slingenbergh, Anatidae migration in the Western Palearctic and spread of highly pathogenic avian influenza H5N1 virus. Emerg. Infect. Dis. 12 (2006), p. 1650–1656.

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[19] A. Kilpatrick, A. Chmura, D. Gibbons, R. Fleischer, P. Marra, P. Daszak, Predicting the global spread of H5N1 avian influenza. Proc. Natl. Acad. Sci. U.S.A. 103 (2006), p. 19368–19373. [20] A. Peterson, B. Benz, M. Papes, Highly pathogenic H5N1 avian influenza: entry pathways into North America via bird migration. PLoS One 2 (2007), p. 1–6. [21] H. Chen, G. Smith, S. Zhang, et al., Avian flu: H5N1 virus outbreak in migratory waterfowl. Nature 436 (2005), p. 191–192. [22] H. Chen, K. Li, J. Wang, et al., Estblishment of multiple sublineages of H5N1 influenza virus in Asia: implications for pandemic control. Proc. Nat. Acad. Sci. U.S.A. 103 (2006), p. 2845–2850. [23] D. Lvov, M. Schelkanov, P. Deriabin, et al., Isolation of influenza A/H5N1 virus strains from poultry and wild birds in west Sibaria during epizooty (July 2005) and their depositing to the state collection of viruses (August 2005). Vopr. Virusol. 51 (2006), p. 11–14. [24] C. Feare, M. Yasu´e, Asymptomatic infection with highly pathogenic avian influenza H5N1 in wild birds: how sound is the evidence? Virology J. 3 (2006), p. 1–4. [25] T. Weber, N. Stilianakis, Ecologic immunology of avian influenza (H5N1) in migratory birds. Emerg. Infect. Dis. 13 (2007), p. 1139–1145. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[26] A. Koehler, J. Pearce, P. Flint, C. Franson, H. Ip, Genetic evidence of intercontinental movement of avian influenza in a migratory bird: the northern pintain (Anas acuta). Mol. Ecol. 17 (2008), p. 4754–4762. [27] E. Class, et al., Human influenza A H5N1 virus related to a highly pathogenic avian influenza virus. Lancet 351 (1998), p. 472–477. [28] R. Fouchier, P. Schneeberger, F. Rozendaal, et al., Avian influenza A virus (H7N7) associated with human conjunctivitis and a fatal case of acute respiratory distress syndrome. Proc. Natl. Acad. Sci. U.S.A. 101 (2004), p. 1356–1361. [29] L. MacKellar, Pandemic influenza: a review. Pop. Devel. Rev. 33 (2007), p. 429–451. [30] N. Liem, W. Lim, Lack of H5N1 avian influenza transmission to hospital employees, Hanoi, 2005. Emerg. Infect. Dis. 11 (2005), p. 210–215. [31] M. Du Ry van Beest Holle, A. Meijer, M. Koopmans, C. de Jager, Human–to–human transmission of avian influenza A/H7N7, The Netherlands, 2003. Eur. Surveill. 10 (2005), p. 264–268. [32] D. Butler, Family tragedy spotlights flu mutations. Nature 442 (2006), p. 114–115. [33] WHO Successful strategies in controlling avian influenza. http://www.who.int /foodsafety/fs management/No 04 AvianInfluenza Aug06 en.pdf.

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[34] A. Stegeman, A. Bouma, A. Elbers, M. de Jong, et al., The avian influenza (H7N7) epidemic in The Netherlands in 2003. Course of the epidemic and effectiveness of control measures. J. Infect. Dis. 190 (2004), p. 2088–2095. [35] T. Ellis, et al., Vaccination of chickens against H5N1 avian influenza in the face of an outbreak interrupts virus transmission. Avian Pathol. 33 (2004), p. 405–412. [36] I. Capua, S. Marangon, Vaccination policy applied for the control of avian influenza in Italy. Dev. Biol. 114 (2003), p. 213–219. [37] C. Villarreal–Chavez, E. Rivera–Cruz, An update on avian influenza in Mexico. Avian Dis. 47 (2003), p. 1002–1005. [38] D. Suarez, et al., Recombination resulting in virulence shift in avian influenza outbreak, Chile. Emerg. Infect. Dis. 10 (2004), p. 693–699. [39] S. Weber, et al., Molecular analysis of highly pathogenic avian influenza virus of subtype H5N1 isolated from wild birds and mammals in northern Germany. J. Gen. Virol. 88 (2007), p. 554–558. [40] M. Ducatez, et al., Molecular and antigenic evolution and geographical spread of H5N1 highly pathogenic avian influenza viruses in western Africa. J. Gen. Virol. 88 (2007), p. 2297–2306. [41] G. Smith, et al., Emergence and predominance of an H5N1 influenza variant in China. Proc. Natl. Acad. Sci. U.S.A. 103 (2006), p. 16936–16941. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[42] A. Lang, A. Kelly, J. Runstadler, Prevalence and diversity of avian influenza viruses in environmental reservoirs. J. Gen. Virol. 89 (2008), p. 509–519. [43] Y. Uchida, M. Mase, K. Yoneda, et al., Highly pathogenic avian influenza virus (H5N1) isolated from whooper swans, Japan. Emerg. Infect. Dis. 14 (2008), p. 1427– 1429. [44] J. Tracey, et al., The role of wild birds in the transmission of avian influenza for Australia: an ecological perspective. Emu 104 (2004), p. 109–124. [45] D. Hulse-Post, et al., Role of domestic ducks in the propagation and biological evolution of highly pathogenic H5N1 influenza viruses in Asia. Proc. Natl. Acad. Sci. U.S.A. 102 (2005), p. 10682–10687. [46] J. Brown, D. Stallknecht, J. Beck, D. Suarez, D. Swayne, Susceptibility of North American ducks and gulls to H5N1 highly pathogenic avian influenza viruses. Emerg. Infec. Dis. 12 (2006), p. 1663–1670. [47] D. Kalthoff, A. Breithaupt, J. Teifke, et al., Highly pathogenic avian influenza virus (H5N1) in experimentally infected adult mute swans. Emerg. Infec. Dis. 14 (2008), p. 1267–1270. [48] J. Keawcharoen, D. van Riel, G. van Amerongen, et al., Wild ducks as long–distance vectors of highly pathogenic avian influenza virus (H5N1). Emerg. Infec. Dis. 14 (2008), p. 600–607.

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[49] O. Gorman, et al., Evolution of influenza A virus nucleoprotein genes: implications for the origins of H1N1 human and classical swine viruses. J. Virol. 65 (1991), p. 3704–3714. [50] R. Chen, E. Holmes, Avian influenza virus exhibits rapid evolutionary dynamics. Mol. Biol. Evol. 23 (2006), p. 2336–2341. [51] M. Gilbert, et al., Free-grazing ducks and highly pathogenic avian influenza, Thailand. Emerg. Infect. Dis. 12 (2006), p. 227–234. [52] N. Kung, et al., Risk for infection with highly pathogenic influenza A virus (H5N1) in chickens, Hong Kong, 2002. Emerg. Infect. Dis. 13 (2007), p. 412–418. [53] T. Germann, et al., Mitigation strategies for pandemic influenza in the United States. Proc. Natl. Acad. Sci. U.S.A. 103 (2006), p. 5935–5940. [54] I. Longini, et al., Containing pandemic influenza at the source. Sci. Express 309 (2005), p. 1083–1087. [55] C. Mills, et al., Pandemic influenza: risk of multiple introductions and the need to prepare for them. Public Libr. Sci. Med. 3 (2006), p. e135. [56] D. Rao, et al., Modeling and analysis of global epidemiology of avian influenza. Environ. Model. Software 24 (2009), p. 124–134. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[57] S. Iwami, Y. Takeuchi, X. Liu, Avian–human influenza epidemic model. Math. Biosci. 207 (2007), p. 1–25. [58] S. Iwami, et al., Prevention of avian influenza epidemic: what policy should we choose? J. Theor. Biol. 252 (2008), p. 732–741. [59] An early detection system for highly pathogenic H5N1 avian influenza in wild migratory birds U.S. interagency strategic plan (2006). http://www.usda.gov/documents/wildbirdstrategicplanpdf.pdf [60] T. Day, J. Andr´e, A. Park, The evolutionary emergence of pandemic influenza. Proc. Roy. Soc. B 273 (2006), p. 2945–2953. [61] B. Olsen, V. J. Munster, A. Wallensten, J. Waldenstram, A. D.M.E. Osterhaus, R. A.M. Fouchier, Global patterns of influenza A virus in wild birds, Science 312 (2006), p. 384–388.

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[62] G. Chowell, H. Nishiura, Quantifying the transmission potential of pandemic influenza, Physics of Life Reviews 5 (2008), p. 50–77.

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 31-57

ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 2

G ENDER D IFFERENCES IN H ETEROSEXUAL T RANSMISSION OF HIV IN U RBAN AND RURAL P OPULATIONS

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Bernhard P. Konrada,∗, Robert J. Smithb,† and Frithjof Lutscherc,‡ a Faculty of Mathematics, The University of Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany b Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada c Department of Mathematics, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada

Abstract We explore effect of disease spread in both urban and rural populations for heterosexual transmission. We develop a two-sex model for the spread of HIV using ordinary differential equations. We then use two methods to calculate the basic reproductive ratio (R0 ) and demonstrate that one is more biologically reasonable than the other. Furthermore, including gender differences can have a large quantitative effect on our choice of intervention strategies. We use numerical simulations to explore the impact of several possible intervention strategies against the disease. These results suggest that focusing on the “weaker” sex, i.e., the sex with the higher risk of being infected, will have a greater impact on slowing the spread of the disease than focusing on the more infectious sex. We also demonstrate that infection in the rural population can be sustained by sexual mixing in urban centers.

Keywords: HIV, mathematical model, differential equations, gender, urban, rural.

1.

Introduction

Currently, HIV infects approximately 33 million individuals worldwide, 68% of whom are in sub-Saharan Africa [1]. The epidemic in southern Africa, which is spreading largely ∗

E-mail address: [email protected] E-mail address: [email protected]. (Corresponding author.) ‡ E-mail address: [email protected] †

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32

Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

through heterosexual exposure, is driven by high rates of labour migration, concurrent sexual partnerships and gender inequalities [2]. Women are infected in greater numbers than men in sub-Saharan Africa [3]. Intravaginal practices, such as washing, douching, wiping and inserting substances into the vagina have also been associated with a higher prevalence of HIV [4, 5]. The highest risk group in rural Uganda are young married women [6]. Factors such as economic dependence, gender discrimination and neglect of women’s sexuality increase a woman’s risk for HIV [3]. Marriage has been implicated as a risk factor for HIV infection in young African women [7], while HIV acquisition is significantly higher for pregnant women [8]. In many developing countries, shifting population demographics have driven rural men to seek work in urban centres [9]. Prevalence and risk factors of HIV-1 and HIV-2 infections vary in urban and rural areas [10]. Migrant men in South Africa are 2.4 times more likely to be HIV infected than non-migrant men [11]. While HIV prevalence in rural areas is relatively low, rural men engaging in risky sex in urban areas subsequently infect women in rural areas, who can then infect further rural men [12]. A number of mathematical models have been developed to account for these effects. Blower et al. [13] demonstrated that the allocation of HIV medication between urban and rural settings has a significant effect on the outcome. Optimal public-health outcomes can be achieved by allocating all resources in the urban centres. However, this is not ethical and unlikely to be implemented. Consequently, the authors demonstrate that there is an optimal division of resources between urban and rural settings that facilitates equitable access to medication for all infected individuals. Renton et al. [14] use a two-sex model to demonstrate the importance of promoting sexually transmitted disease control as a major element of HIV prevention. Robinson et al. [15] used a simulation model to asses the impact of a variety of intervention strategies in rural Uganda. Gregson et al. [16] used mathematical models to analyse sexual mixing patterns in rural Zimbabwe. Mekonnen et al. [9] modelled the demographic impact of HIV in urban Ethiopia. Coffee et al. [17] modelled the impact of migration in South Africa and showed that frequent return of migrants is an important risk factor for HIV. Here, we develop a mathematical model to examine the effects of gender differences in urban-rural populations. We model heterosexual sex, since that is the primary route of adult HIV transmission in Africa [12]. We pose the following research questions: 1. Can urban transmission sustain infection in rural areas? 2. Which intervention strategies will have the greatest effect on the outcome? 3. How do gender differences affect our choice of intervention strategy? This chapter is organised as follows. In section 2., we introduce the mathematical model. In section 3., we analyse the model. In section 4., we illustrate the results with numerical simulations. In section 5., we explore the effect of possible intervention strategies. Finally, in section 6., we discuss the implications of the results. Matlab codes used to generate the figures are displayed in the Appendix.

2.

The Model

The model consists of an urban community, in equations indicated by an index u, and a rural community, indicated by r. In each division, the population N is split into female (X) and Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Gender Differences in HIV Models

33

male (Y ). Susceptible individuals and infected ones are denoted by S and I, respectively. The removal rate from the sexually active population is denoted by µ and the (constant) rate in which individuals join the sexually active population is η. The parameter β is the product of the contact rate between the two genders and the transmission probability. Nicolosi et al. [18] point out that the probability of transmission depends on which partner is infected. For convenience, this will be reflected by the parameter α, the degree of differential infection, that occurs in all the female terms. Critical to the model is the assumption that individuals from the rural community travel (for example to work) to the urban community and have sexual contact there. We denote the mixing probability by c, such that cy , for example, is the probability that a male from the rural community has contact with an urban female within the urban region. The much more unlikely case of an urban individual having contacts in the rural community is not considered. We will now introduce the model for the urban region. The change rate of the susceptibles consists of three parts. First, there is the constant inflow η u . Second, we have the rate by which (healthy) individuals conclude their sexually active phase, µs . Third, there is the rate that describes the loss towards the infected population. This is a product of the contact rate β and the probability of actually having an infected partner. An urban woman may meet either an urban man, or a rural man, who has the probability cy of having a contact in the city. Thus, the chance that an urban woman encounters an infected man in the city is Iyu + cy Iyr . Y u + cy Y r Hence, the urban model is

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N u = Xu + Y u X

u

=

Y

u

=

dSxu dt dIxu dt dSyu dt dIyu dt

Ixu Iyu

+ +

(2.1)

Sxu Syu

= η u − µsx Sxu −

(2.2) (2.3) Iyu Yu

+ +

cy Iyr cy Y r

· α · β · Sxu

Iyu + cy Iyr · α · β · Sxu − µix Ixu Y u + cy Y r I u + cx Ixr = η u − µsy Syu − xu · β · Syu X + cx X r Ixu + cx Ixr = · β · Syu − µiy Iyu . X u + cx X r =

(2.4) (2.5) (2.6) (2.7)

The equations are similar in the rural community. The difference is that rural individuals may have both rural contacts as well as urban contacts. Additionally, they could also meet rural individuals in the urban region. Thus, the chance that a rural woman encounters an infected urban man is Iyu + cy Iyr Iyr + c . x u Yr Y + cy Y r

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34

Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher Hence, the rural model is N r = Xr + Y r X

r

=

Y

r

=

dSxr dt dIxr dt dSyr dt dIyk dt

Ixr Iyr

+ +

(2.8)

Sxr Syr

(2.9) (2.10) Iyr Yr

Iyu cx u Y

 cy Iyr cy Y r

+ = η r − µsx Sxr − + · α · β · Sxr +  r  Iy Iyu + cy Iyr = + cx u · α · β · Sxr − µix Ixr Yr Y + cy Y r   r Ix Ixu + cx Ixr r r + cy u · β · Syr = η − µsy Sy − Xr X + cx X r  r  Ix Ixu + cx Ixr = + cy u · β · Syr − µiy Iyr . Xr X + cx X r 

(2.11) (2.12) (2.13) (2.14)

The model is illustrated in Figure 1.

3.

Analysis

We will now derive the basic reproductive ratio for our model. Suppose the system is in a state where almost no individuals are infected. Let the proportion of females in the infected population be denoted by p ∈ [0, 1], i.e. Ixu = p · I u . Then, for the urban population, we find

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dI u dt

dIxu dIyu + dt u dt r  Iy + cy Iy Ixu + cx Ixr u u = · α · Sx + u S β − (µix Ixu + µiy Iyu ) Y u + cy Y r X + cx X r y   Iyu Iyr = + cy u · α · β · Sxu Y u + cy Y r Y + cy Y r   Ixu Ixr + + cx u · β · Syu X u + cx X r X + cx X r −(µix · p + µiy · (1 − p))I u   Sxu Sxu u r · I + cy u ·I ·α·β = Y u + cy Y r y Y + cy Y r y   Syu Syu u r · I + cx u ·I ·β + X u + cx X r x X + cx X r x −(µix · p + µiy · (1 − p))I u =

Approximation 3.1. Since the number of rural individuals is much smaller than the number of urban individuals, we set Sxu Sxu ≈ Y u + cy Y r Yu

and

Iyr Ixr = ≈ 0. Yu Xu

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We thus have   Syu dI u Sxu ≈ α · (1 − p) · u + p · u · β · I u − (µix · p + µiy · (1 − p))I u . dt Y X

Figure 1. The diagram visualises the relationships between the eight compartments. A dashed arrow indicates the inflow rate which affects only the susceptible population. A solid thin arrow shows where the susceptible people may convert to; the probability is proportional to β and αβ, respectively, depending on the sex. The thick arrows point out which infected population influences which susceptible population. In order to not overload the diagram, we left out the removal rates, which would affect every compartment. The removal rates differ between the sexes, and between the infected population and the susceptible population. Approximation 3.2. We make the further approximation that the male and female population are roughly equal in size. Thus Sxu ≈1 Yu

and

p=

1 . 2

The same approximations are also valid if x and y, or X and Y , are interchanged. Hence,

1 dI u ≈ ((α + 1)β − µix − µiy ) I u ≡ r0,1 · I u . dt 2

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Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

The value of the intrinsic growth rate r0,1 is critical for the behaviour of the disease. If r0,1 is positive, then, the number of infected people increases. Rephrasing, this means that the disease will persist if

R0,1 =

(α + 1)β µix + µiy

is greater than 1, whereas the disease will be eradicated if R0,1 < 1. The same calculation can be done for the rural population and, using Approximations 3.1 and 3.2 accordingly, leads to the same result. In general, the important parameter R0 is defined as the average number of susceptible people that an infected person infects during their sexually active phase. Then, it is clear that the total number of infected people can only decrease if R0 < 1. However, as Heffernan et al. [19] make clear, one should be very careful when it comes to such a concrete meaning for R0 ; it is often not clear what the appropriate choice would be. Surrogate thresholds like R0,1 , although not necessarily the average number of secondary infections, retain the same threshold property: persistence results if the value is greater than 1 and eradication results if the value is less than 1.

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However, as we shall see, there are potential issues with this value. Thus, we shall try another approach to find a better R0 . A widely used method is invasion analysis with Jacobian matrices. The interested reader can find a useful introduction in [20]. To do invasion analysis, let J be the 8 × 8 Jacobian matrix for our system. Then, J = (J1 |J2 |J3 |J4 ) with



J1

J2

Iyu +cy Iyr Y u +cy Y r · α · Iyu +cy Iyr Y u +cy Y r · α · β Ixu +cx Ixr u (X u +cx X r )2 · β · Sy u r Ix +cx Ix u (X u +cx X r )2 · β · Sy

−µsx −

0

β

       =    0   0  r u  cy Iux +cx Ixr 2 · β · S r y  (X +cx X ) Ixu +cx Ixr −cy (X u +cx X r )2 · β · Syr I u +cy I r (Y uy+cy Y yr )2 · α · β · Sxu u r − Iy +cy Iy · α · β · S u x (Y u +cy Y r )2 Ixu +cx Ixr · β −µ − u r sy X +cx X Ixu +cx Ixr u +c X r · β X = u rx cx Iuy +cy Iyr 2 · α · β · Sxr (Y +c Y ) y Iyu +cy Iyr −cx (Y u +c Y r )2 · α · β · Sxr y 0 0

−µix (X u +cx X r )−(Ixu +cx Ixr ) − · β · Syu (X u +cx X r )2 u r u r (X +cx X )−(Ix +cx Ix ) · β · Syu (X u +cx X r )2 0 0 (X u +cx X r )−(Ixu +cx Ixr ) · β · Syr (X u +cx X r )2 (X u +cx X r )−(Ixu +cx Ixr ) cy · β · Syr (X u +cx X r )2

−cy



(Y u +cy Y r )−(Iyu +cy Iyr ) · α · β · Sxu (Y u +cy Y r )2 u r u r (Y +cy Y )−(Iy +cy Iy ) · α · β · Sxu (Y u +cy Y r )2

−

0 −µiy Y u +cy Y r −(Iyu +cy Iyr ) · α · β · Sxr (Y u +cy Y r )2 Y u +cy Y r −(Iyu +cy Iyr ) cx · α · β · Sxr (Y u +cy Y r )2

−cx

0 0

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0 0 cx (Ixu +cx Ixr ) u (X u +cx X r )2 · β · Sy u r cx (Ix +cx Ix ) · β · Syu − (X i h ru +cx X r )2u I I +c I r = −µsx − Yyr + cx Y yu +cyy Yyr · α · β h r i Iy Iyu +cy Iyr ·α·β + c r u r x Y +cy Y h rY i Ix Ixu +cx Ixr r + c y (X u +cx X r )2 · β · Sy h(X r )r2 i I u +c I r I − (X xr )2 + cy (X ux+cxxXxr )2 · β · Syr

37

0 0 cx (X u +cx X r )−(Ixu +cx Ixr )cx · β · Syu (X u +cx X r )2 u r u r cx (X +cx X )−(Ix +cx Ix )cx · β · Syu (X u +cx X r )2

−

0 −µix h

− h

(X u +cx X r )cx −cx (Ixu +cx Ixr ) + cy (X u +cx X r )2 i (X u +cx X r )cx −cx (Ixu +cx Ixr ) + cy u r 2 (X +cx X )

1 Xr

1 Xr

i

· β · Syr

· β · Syr

and J4 = cy (Iyu +cy Iyr ) u (Y u +cy Y r )2 · α · β · Sx u r cy (Iy +cy Iy ) u − (Y u +c r 2 · α · β · Sx yY ) 0 0 h r i Iy cy (Iyu +cy Iyr ) r + c r 2 u r 2 x (Y +cy Y ) i· α · β · Sx h(Y )r r u cy (Iy +cy Iy ) − (YIry)2 + cx (Y · α · β · Sxr u +c Y r )2 y i h Ixu +cx Ixr Ixr −µsy − X · β · Syr r + cy X u +c X r h r ix u r Ix Ix +cx Ix r X r + cy X u +c X r · β · Sy

(Y u +cy Y r )cy −cy (Iyu +cy Iyr ) · α · β · Sxu (Y u +cy Y r )2 u r u r (Y +cy Y )cy −cy (Iy +cy Iy ) · α · β · Sxu (Y u +cy Y r )2

−

0 0 − h

h

Y r −Iyr cy (Y u +cy Y r )−cy (Iyu +cy Iyr ) + c r 2 x (Y ) (Y u +cy Y r )2 i cy (Y u +cy Y r )−cy (Iyu +cy Iyr ) Y r −Iyr + c x (Y r )2 (Y u +cy Y r )2

i

· α · β · Sxr

· α · β · Sxr

0

−µiy

x



        .       

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The disease-free state satisfies

(Sxu , Ixu , Syu , Iyu , Sxr , Ixr , Syr , Iyr )

=



 ηu ηr ηr ηu , 0, , 0, , 0, ,0 . µsx µsy µsx µsy

Then, the Jacobian evaluated at this point becomes J(Sxu , 0, Syu , 0, Sxr , 0, Syr , 0) = (J5 |J6 ), where



J5

      =       

−µsx 0 0 − Y u +c1 y Y r · β · Sxu 1 u 0 −µix 0 Y u +cy Y r · β · Sx 1 u 0 0 − X u +cx X r · β · Sy −µsy 1 u 0 0 −µiy X u +cx X r · β · Sy 0 0 0 −cx Y u +c1 y Y r · β · Sxr 0 0 0 cx Y u +c1 y Y r · β · Sxr 0 −cy X u +c1 x X r · β · Syr 0 0 1 r 0 0 0 cy X u +cx X r · β · Sy

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Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

and

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c 0 0 0 − Y u +cyy Y r · β · Sxu c 0 y u 0 0 Y u +cy Y r · β · Sx cx u 0 − X u +c 0 0 r · β · Sy xX c u x 0 · β · S 0 0 y X u +cx X r h i c J6 = −µsx 0 0 − Y1r + cx Y u +cyy Y r · β · Sxr h i cy 1 r 0 −µ 0 + c ix x Y u +cy Y r · β · Sx Yr i h cx · β · Syr −µsy 0 − X1r + cy X u +c 0 r xX i h cx 1 r 0 −µiy 0 X r + cy X u +cx X r · β · Sy

                

Four of the eigenvalues are −µsx , −µsy , −µsx and −µsy , and are thus negative. For the u u r r behaviour of the steady state ( µηsx , 0, µηsy , 0, µηsx , 0, µηsy , 0), we need to find the eigenvalues of H = (H1 |H2 ), where  µsy ηu · α · β · −µix u r µsx η +cy η u  µsx η  µsy · ηu +cx ηr · β −µiy  H1 =  µsy ηr 0 c · α · β · u r x  µsx η +cy η r η cy µµsx · · β 0 u r η +cy η sy  µsy ηu · ηu +c 0 cy µsx r · α · β η y  ηu  cx µµsx · 0 u +c η r · β η sy x  h i r H2 =  µsy η −µix 1 + c c · α · β  x y η u +cy η r µsx h i  r µsx η −µiy µsy 1 + cy cx ηu +cx ηr · β In order to simplify matters, we will use the following approximation:

Approximation 3.3. As there are many more urban individuals than rural, the constant urban inflow rate η u is much bigger than the rural inflow rate, η r . Furthermore, cx and cy are both much smaller than 1, so we shall use the approximations ηu ≈1 η u + cb η r

and

ηr ≈0 η u + cb η r

for b ∈ {x, y}. For convenience, abbreviate γ = µµsx . Then, H simplifies to sy  −µix αβγ −1 0 cy αβγ −1  βγ −µiy cx βγ 0 ˜ =  H  0 0 −µix αβγ −1 0 0 βγ −µiy

   

with eigenvalues λ1,2 =

1 2

  q −µix − µiy ± (µix − µiy )2 + 4αβ 2

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(each one with algebraic multiplicity 2). As the expression under the square root is nonnegative, all eigenvalues will be real. Thus, we have six negative eigenvalues. In order for the seventh and eighth (they are the same) to be negative, we would need q (µix − µiy )2 + 4αβ 2 > µix + µiy ⇐⇒

µix µiy > αβ 2 .

Thus, we define R0,2 ≡

αβ 2 αβ β = · µix µiy µiy µix

Again, the value R0,2 = 1 is the critical point. If R0,2 > 1, then we expect the disease to spread as we have an unstable disease-free steady state. The value of R0,1 has a threshold when

1 µiy



β − µix + αβ − µiy = 0    1 β β −1 + −1 = 0. µix µix µiy

However, when R0,1 = 1, it is not true in general that R0,2 = 1. For example, if β = 9, µix = 10, α = 111/9 and µiy = 100, then, 12 > 1 11 = 0.999 < 1 .

R0,1 =

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R0,2

We shall argue that R0,2 is more biologically meaningful. The first term of R0,2 is the number of females that an infected male infects in his active phase, the second term the number of males a female infects. Suppose, for example, a male infects four females and a female infects two males, on average. Then, the proper R0 in this process will be 8, as shown in Figure 2.

Figure 2. An example of R0,2 .

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Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher 6

4

x 10

3.5

3

Population

2.5

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

Time (Months)

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Figure 3. Ten year urban timecourse without intervention. The top curve represents the total urban population, the middle one shows the number of susceptible urban individuals and the bottom curve shows the number of infected urban individuals. The curves for the rural population are similar, but on a much smaller scale (not shown). Of course, the same result is gained if, in Figure 2, we had started from an infected female. Interestingly enough, this example shows that, for the reproductive ratio, it does not matter whether the factor α is associated with either males or females. Thus, henceforth, when we refer to R0 , we are referring to R0,2 .

4. 4.1.

Numerical Simulations The Choice of Parameter Values

A typical African city, like Durban in KwaZulu-Natal, has around 3.5 million inhabitants. Blower et al. [13] suggest 400-4,000 villages, with populations ranging from 1,300 to 13,000 individuals. For convenience, we will choose 3,500 inhabitants. The average number of new sex partners per person per year is given as 0.5-1.5 in [13], but ranges from 0-18.1 in [14]. The latter also suggests a per-partnership transmission probability ranging from 0.01 to 0.1, so we model the worst-case scenario by using the latter. The time period that susceptible and infected individuals are sexually active is 30 and 10

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years, respectively, as suggested in [14]. For the urban-rural contact rate, [13] uses several fc random variables, namely c = 1+d , where f ∈ (0, 1), c ∈ (0.5, 1.5) and d ∈ (10, 100). As 0.5·1 a mean, we will use 50 = 0.01 (per year), or 0.00085 per month. As it is rare that females travel to the city, we will set cx = 0. See Table 1. Table 1. Abbreviations and parameter values Variable

Meaning

Value

Source

Na

Total number of individuals in community a ∈ {u, r}

N u (0) = 3, 500, 000 N r (0) = 3, 500

[13]

a

Total number of females in community a ∈ {u, r}

50%

[21]

Ya

Total number of males in community a ∈ {u, r}

50%

[21]

Sab

Total number of susceptibles in community a ∈ {u, r}, gender b ∈ {x, y}

Sbu = 0.87 × B u Sbr = 0.91 × B r

[21]

Total number of infected in community a ∈ {u, r}, gender b ∈ {x, y}

Ibu = 0.13 × B u Ibr = 0.09 × B r

[21]

2

[18]

= 1/6 partners per month × 0.1 transmission probability

[13]

ηu = 7, 500; ηr = 7.5

[21]

X

Iab

α β

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ηa µsb µib

cb

The degree of differential infection (contact rate between the two genders) × (transmission probability)

1 60

Constant rate of individuals in community a ∈ {u, r} joining the sexually active phase Constant removal rate of susceptibles of gender b ∈ {x, y}

30 active years ⇒ µsb =

1 360

[14]

Constant removal rate of infected of gender b ∈ {x, y}

10 active years ⇒ µib =

1 120

[14]

Degree of sexual cross-mixing of individuals of gender b ∈ {x, y} between urban and rural areas

cy = 0.00085; cx = 0

[13]

Using the values in Table 1, we have 1 (2 + 1) 60 (α + 1)β = 1 1 =3 µix + µiy 120 + 120 = 8.

R0,1 = R0,2

4.2.

Numerical Results

We use the parameters given in Table 1 and let the system run for ten years. The method used for all numerical analysis is Runge-Kutta (3,4) with adaptive stepsize. The results are Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

42

Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

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Table 2. Changes in infection after ten years Parameter

Starting value

Ending value

Sxu Ixu Syu Iyu Sxr Ixr Syr Iyr Nu Nr S I pux pui prx pri

1.52 · 106 2.28 · 105 1.52 · 106 2.28 · 105 1.59 · 103 1.58 · 102 1.59 · 103 1.58 · 102 3.50 · 106 3.50 · 103 3.05 · 106 4.55 · 105 50.00% 13.00% 50.00% 9.00%

9.34 · 105 7.88 · 105 1.17 · 106 6.15 · 105 1.13 · 103 6.62 · 102 1.34 · 103 5.05 · 102 3.51 · 106 3.63 · 103 2.11 · 106 1.40 · 106 49.06% 39.97% 49.26% 32.15%

shown in Figure 3. The numerical changes are given in Table 2, where pab , a ∈ (u, r), b ∈ (x, y) is the percentage of individuals of a given gender in a given region. It is interesting to note that pux and prx , the percentage of the female population in the city and the village, respectively, stays close to 50%, although females are infected twice as easily as men.

4.3.

Long-Term Behaviour and a Second Steady State

In order to understand the dynamics of the system better, we simulated fifty years of the epidemic. The results are shown in Figure 4. We see that the trend indicated in the first ten years, will be continued for quite a while and that infected individuals will eventually outnumber susceptible individuals. However, at some point the number of susceptible individuals stops decreasing so drastically and seems to asymptotically approach a steady state from above. Also, the infected population seems to approach a steady state from above. Note that the graph for the infected population overshoots and reaches its peak slightly before the decrease of the susceptible population slows down. This phenomenon is typical for SIR-models. Any long timescale simulation we performed showed the effect of all three functions (the total, the susceptibles and the infected population) seemingly approaching a steady state. However, we haven’t shown that this state is actually a stable steady state, so we will refer to it as a “quasi steady state”. From a biological point of view, however, such a long timescale is not realistic. The

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parameters may very well change over time and some new dynamics may enter the system to change it completely. That is the reason why the focus in this chapter is on the shorter timescale of ten years. 6

4

x 10

3.5

3

Population

2.5

2

1.5

1

0.5

0

0

100

200

300

400

500

600

700

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Time (Months)

Figure 4. Fifty year urban timecourse without intervention.

5. 5.1.

Intervention Strategies Modelling Intervention Strategies

We wish to include a variety of intervention strategies, in order to compare their effectiveness. When modelling intervention strategies, there are two main approaches. The first is to alter our differential equations, such as including a recovery term or subtracting a certain term from the transmission probability. The advantage of this method is that it provides a general approach. The disadvantage is that it complicates the model. The second method is to change some parameter values. While this approach cannot give us a general picture, it is straightforward and allows for a variety of possible intervention strategies to be explored numerically. From our analysis of R0 , the parameters β, µix and µiy are clearly the most important. For comparison purposes, we will always make changes in these parameters in such a way that R0,2 reduces to 2. Although this value is not below one (and will thus not lead to erad-

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44

Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

ication), it is nevertheless a significant reduction from 8. Consequently, we are comparing intervention strategies that reduce the prevalence of the disease, but which do not lead to eradication. AIDS-awareness education and/or an increased condom use will decrease the transmission probabilty and/or the contact rate, resulting in a decreased β. If AIDS tests were available that could inform infected people about their status, we could hope that infected people stop having unsafe contact with others and thus drop out of our model. The result would be an increase of µix and/or µiy . Antiretroviral drugs would decrease β, while simultaneously increasing µix and µiy . A critical question is whether or not any of these changes would influence the system in the long run. We can obviously hope that the disease will spread slower with a smaller R0 , but will it in the long run still reach this second steady state we saw in the previous section? In order to address this question, let us look at Figure 5. 6

4

x 10

3.5

3

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Population

2.5

2

1.5

1

0.5

0

0

200

400

600

800

1000

1200

1400

Time (Months)

Figure 5. The system with doubled removal rates. The solid curves show the behaviour of the system without intervention; the thinner red lines give the same functions for a doubled µix and µiy . Thus, if we double both µix and µiy and compare it to the behaviour of the system without intervention, we see a significant difference. While we still have the overshooting of the infected population, the actual numbers are drastically different. The infected population never outnumbers the susceptible population. Note that the overall population is lower,

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since we are assuming many more people are removed from the system. Changes resulting from an increased β are shown in Figure 6. As expected, the disease spreads much quicker with the higher β. Besides that, the functions behave similarly. Again, we see an overshoot of the infected population and both systems approaching a steady state in the long run. However, we see that these two steady states differ and that with the smaller β the approached steady state is much better, due to a smaller amount of infected people and a larger amount of susceptibles. 6

4

x 10

3.5

3

2.5

2

1.5

1

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0.5

0

0

100

200

300

400

500

600

700

Figure 6. The system with β halved. The thin red curves show the behaviour of the system without intervention, the solid black curves give the same functions for β halved.

5.2.

Strategy Comparison

We want to compare the following intervention strategies: - Strategy I: Halving β through increasing education. - Strategy II: Doubling µix and µiy , due to more widespread testing. The result are shown in Figure 7. Clearly, strategy I is superior to strategy II. The total number of susceptible individuals increases in strategy I, while it decreases in strategy II. Even in the total number of infected individuals, strategy II shows a worse performance

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Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher

although infected individuals leave the system twice as often! These two effects, taken together, result in a huge difference in the AIDS prevalence after ten years. The actual numbers are found in the Table 3. 6

4

x 10

3.5

3

Population

2.5

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

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Time (Months)

Figure 7. Comparison of the two intervention strategies. The solid black curves show the system without intervention. The dotted red lines show the effects of halving β (strategy I). The dashed blue curves show the effects of doubling µix and µiy (strategy II). While the number of infected individuals increases in both strategies, it should nevertheless be noted that R0 is still greater than 1. If strategy I is implemented instead of strategy II, there is a difference of approximately 80,000 fewer infected after 10 years; the percentage reduction in urban infectionis 20.5% under strategy II, but only 15.4% under strategy I.

5.3.

The Effect of Gender Differences

We now propose intervention strategies that affect only one gender, in order to determine some of the effects of gender on the outcome. Instead of doubling the removal rates uniformly, as in strategy II, we instead propose quadrupling only one gender-specific removal rate. This reflects the situation where testing of one gender results in them ceasing to find new sexual partners and thus leaving the sexually active pool.

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Table 3. Changes in infection after ten years, using strategies I and II.

Sxu Ixu Syu Iyu Sxr Ixr Syr Iyr Nu Nr S I pux pui prx pri

Starting value

No intervention

Strategy I

Strategy II

1.52 · 106 2.28 · 105 1.52 · 106 2.28 · 105 1.59 · 103 1.58 · 102 1.59 · 103 1.58 · 102 3.50 · 106 3.50 · 103 3.05 · 106 4.55 · 105 50.00% 13.00% 50.00% 9.00%

9.34 · 105 7.88 · 105 1.17 · 106 6.15 · 105 1.13 · 103 6.62 · 102 1.34 · 103 5.05 · 102 3.51 · 106 3.63 · 103 2.11 · 106 1.40 · 106 49.06% 39.97% 49.26% 32.15%

1.53 · 106 3.29 · 105 1.64 · 106 2.47 · 105 1.66 · 103 2.43 · 102 1.74 · 103 1.81 · 102 3.74 · 106 3.83 · 103 3.17 · 106 5.77 · 105 49.57% 15.40% 49.69% 11.08%

1.19 · 106 3.73 · 105 1.38 · 106 2.88 · 105 1.38 · 103 3.00 · 102 1.53 · 103 2.26 · 102 3.23 · 106 3.43 · 103 2.57 · 106 6.61 · 105 48.45% 20.48% 48.84% 15.30%

- Strategy III: Available AIDS tests for males only to quadruple µiy .

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- Strategy IV: Available AIDS tests for females only to quadruple µix . We compare all four intervention strategies in Figure 8. In order to get a better view of the development of the number of infected individuals, we zoom in. See Figure 9. The values are given in Table 4, where pui and pri are the percentages of infected individuals in the urban and rural areas, respectively. When comparing the percentage of infected people in the population, we see that strategy III and strategy IV perform about equally (approximately 18%), which is between strategy I (15.4%) and strategy II (20.5%). However, if we concentrate on the total amount of infected people, we see that strategy II (6.61 × 105 ) performs worse than strategy IV (5.77 × 105 ), which actually gives us the same result as strategy I. So the difference between strategy I and strategy IV really is only in the number of susceptible individuals. The number of infected people in strategy IV is always less than the number of infected people of strategy I for the first ten years. The number of infected people in strategy III is less than that in strategy I for five years and thereafter is greater. Strategy II always produces more infected individuals than any other strategy.

5.4.

Redefining “Success”

The previous discussion raises the question of which measurement of “success” is appropriate. On the one hand, we want as few infected people to be sexually active as possible. On the other hand, the lower the percentage of infected individuals in the sexually active phase, Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Bernhard P. Konrad, Robert J. Smith and Frithjof Lutscher Table 4. Changes in infection after ten years, using all four strategies Starting value

Sxu Ixu Syu Iyu Sxr Ixr Syr Iyr u

N Nr S I pux pui prx pri

6

1.52 · 10 2.28 · 105 1.52 · 106 2.28 · 105 1.59 · 103 1.58 · 102 1.59 · 103 1.58 · 102 3.50 · 106 3.50 · 103 3.05 · 106 4.55 · 105 50.00% 13.00% 50.00% 9.00%

No Intervention 5

9.34 · 10 7.88 · 105 1.17 · 106 6.15 · 105 1.13 · 103 6.62 · 102 1.34 · 103 5.05 · 102 3.51 · 106 3.63 · 103 2.11 · 106 1.40 · 106 49.06% 39.97% 49.26% 32.15%

Strategy I

Strategy II

6

6

1.53 · 10 3.29 · 105 1.64 · 106 2.47 · 105 1.66 · 103 2.43 · 102 1.74 · 103 1.81 · 102 3.74 · 106 3.83 · 103 3.17 · 106 5.77 · 105 49.57% 15.40% 49.69% 11.08%

1.19 · 10 3.73 · 105 1.38 · 106 2.88 · 105 1.38 · 103 3.00 · 102 1.53 · 103 2.26 · 102 3.23 · 106 3.43 · 103 2.57 · 106 6.61 · 105 48.45% 20.48% 48.84% 15.30%

Strategy III 6

1.37 · 10 4.42 · 105 1.36 · 106 1.60 · 105 1.53 · 103 3.38 · 102 1.52 · 103 1.28 · 102 3.33 · 106 3.51 · 103 2.73 · 106 6.02 · 105 54.43% 18.07% 53.18% 13.27%

Strategy IV 1.13 · 106 2.18 · 105 1.49 · 106 3.59 · 105 1.33 · 103 1.80 · 102 1.63 · 103 2.71 · 102 3.20 · 106 3.40 · 103 2.62 · 106 5.77 · 105 42.13% 18.05 % 44.27% 13.26%

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the lower the chance for an individual to actually have sexual contacts with an infected person. Besides the percentage or overall number of infected people in the population, another way to measure success is to look at the total number of newly infected people in a given time period. Rephrasing this, we want to count the cumulative number of susceptible people who become infected. Consequently, we introduce a measure function M , that satisfies dM dt

=

Iyu + cy Iyr Ixu + cx Ixr u · α · β · S + · β · Syu x Y u + cy Y r X u + cx X r    r  r Iyu + cy Iyr Iy Ixu + cx Ixr Ix r + cx u · α · β · Sx + + cy u · β · Syr + Yr Y + cy Y r Xr X + cx X r

with M (0) = 0. The four summation terms in the derivative of M are the number of new infections in, respectively, urban men, urban women, rural men and rural women. In Figure 10, we see the change in M when there are no intervention strategies, while Figure 11 compares the different measure functions for the four different strategies. While strategies II, III and IV are more or less equally “successful”, strategy I is significantly better. Here, the number of new infections is less than half as big as the amount in any other strategy and is one third of the amount in the system without intervention. It follows that, in terms of preventing new infections in all groups, strategy I (halving β) is clearly superior. However, the next best intervention is strategy IV (quadrupling the female removal rate), which is significantly better than quadrupling the male removal rate. This shows that gender differences have an important impact on the outcome.

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6

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Time (Months)

Figure 8. Comparison of all intervention strategies. The solid black curves show how the system behaves without intervention. The dotted red curves represent the system under the influence of strategy I. The dashed blue curves represent strategy II. The dashed pink curves represent strategy III and the dot-dashed green curves represent strategy IV.

6.

Conclusion

The best strategy for reducing the impact of the epidemic is reducing the infection probability. Failing that, the next best strategy is to increase the removal rate of females from the sexually active pool. This may be achieved through education, increased testing, improved condom use or prevention awareness. Interestingly, increasing the removal rate of females is significantly more likely to improve the outcome than increasing the removal rate of males (Figure 11). Our model thus demonstrates that gender differences can have a significant effect on the outcome. We also demonstrated that transmission occurring predominantly in urban areas results in an increase from 13% to 40% in urban areas and an increase from 9% to 32% in rural areas after ten years (Table 2). It follows that urban transmission can sustain infection in rural areas. We also analysed the effects of calculating the basic reproductive ratio under two scenarios. In the first, we assumed that the number of rural individuals is significantly smaller than the number of urban individuals, and also that the male and female populations are

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Time (Months)

Figure 9. Comparison of intervention strategies for infected individuals. The dotted red curve shows the number of infected people under strategy I, the dashed blue curve shows the number of infected people under strategy II. The solid pink curve represents the number of infected people under strategy III and the dot-dashed green curve represents the number of infected people under strategy IV. similar in size. In the second, we assumed that the inflow of urban individuals is large, while the inflow of rural individuals is small. In each case, the value depends only on the transmission probability, the removal rates of each gender and the degree of differential infection. While each value satisfies the threshold condition that the disease persists if R0 > 1 and is eradicated if R0 < 1, we showed that the second approximation led to a path-independent product of the individual reproductive numbers for each gender. This demonstrates the care that needs to be taken when calculating surrogate R0 -like thresholds from mathematical models (see [19] for more discussion). Furthermore, the second calculation of R0 demonstrated that the choice of whether the degree of differential infection is applied to males or females is arbitrary. Thus, the results can be generalised to note that removing the “weaker” sex – i.e., the one with the highest risk of being infected – has a greater impact on the outcome. This “removal” could be achieved via targeted education strategies, increased testing for one gender or through community organisations. Of course, education campaigns or AIDS testing should attempt to encompass both genders if possible, but the realities of existing cultural structures may

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60 80 Time (Months)

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Figure 10. The model without intervention (black curves) and the measure function M (red curve). make one gender more receptive to some stategies than others. While the first two approximations lead to a “first guess” for R0 , this value is not biologically meaningful. By refining our approximations, we derived a biologically useful threshold condition. Approximation 3.3 implies that HIV can be sustained in urban areas alone. By using this key approximation, we were able to simplify an 8 × 8 system to two 2 × 2 systems. This makes the system mathematically tractable and allows the derivation of the second, more useful, R0 . We are primarily interested in transient behaviour of the system. While analytical methods may determine long-term phenomena such as equilibria, the timescale of such behaviour may be much longer than the lifespan of an infected individual. Consequently, we use numerical simulations to examine the short-term dynamics of the system. In Tables 2 and 4, our starting values are a long way from the disease-free equilibrium and do not evolve to stable behaviour in a ten-year period. While stable behaviour is eventually seen, it takes approximately 50 years to reach (Figure 4), much longer than the timecourse of the disease in individuals. It should be noted that this model is only a partial snapshot of the epidemic, as it ignores many other important routes of transmission. Specifically, homosexual transmission, needle sharing and vertical transmission are not modelled. In summary, the effects of gender differences can have a significant impact on which intervention strategy should be applied. Not all intervention strategies will have the same effect, even if they remove the same number of total individuals. Furthermore, the effects of urban and rural mixing have a significant effect on the outcome for rural individuals,

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2

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Figure 11. The measure function for all intervention strategies. The solid black and red curves show the system without intervention and with strategy I, respectively. The dotted curve demonstrates strategy II, the dotted-dashed curve strategy III and the dashed curve strategy IV.

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who suffer a proportionally greater increase in prevalence than do urban areas. It follows that, before intervention strategies are implemented, the population-level impact should be carefully assessed.

Appendix Below are the codes used in generating the figures. Associated function file (aidsf.m) function z = aidsf(t,x0); global beta alpha etau usx usy cx cy uix uiy etar z(1,:) = etau - usx*x0(1) - alpha*beta*((x0(4) + cy*x0(8))/(x0(4) + x0(3) + cy*(x0(8) x0(7))))*x0(1); z(2,:) = alpha*beta*((x0(4) + cy*x0(8))/(x0(4) + x0(3) + cy*(x0(8) + x0(7))))*x0(1) - uix*x0(2); z(3,:) = etau - usy*x0(3) - beta*((x0(2) + cx*x0(6))/(x0(2) + x0(1) + cx*(x0(6) + x0(5))))*x0(3); z(4,:) = beta*((x0(2) + cx*x0(6))/(x0(2) + x0(1) + cx*(x0(6) + x0(5))))*x0(3) - uiy*x0(4); z(5,:) = etar - usx*x0(5) - alpha*beta*(x0(8)/(x0(8) + x0(7)) + cx*(x0(4) + cy*x0(8))/(x0(4) x0(3) + cy*(x0(8) + x0(7))))*x0(5); z(6,:) = alpha*beta*(x0(8)/(x0(8) + x0(7)) + cx*(x0(4) + cy*x0(8))/(x0(4) + x0(3) + cy*(x0(8) x0(7))))*x0(5) - uix*x0(6); z(7,:) = etar - usy*x0(7) - beta*(x0(6)/(x0(6) + x0(5)) + cy*(x0(2) + cx*x0(6))/(x0(2) + x0(1) cx*(x0(6) + x0(5))))*x0(7);

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+ + +

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z(8,:)= beta*(x0(6)/(x0(6) + x0(5)) + cy*(x0(2) + cx*x0(6))/(x0(2) + x0(1) + cx*(x0(6) + x0(5))))*x0(7) - uiy*x0(8); end The main file (aids.m) function aids(varargin); % There are two special features included: The function ’aids’ takes one or two optional arguments. % They allow you to implement the counter strategies I,II,III or IV and allow you to choose the % running time. If no argument is give the system runs on strategy 0 (= no counter measures) % for Tmax = 120 months. If only one optional argument is given, that stands for the counter % strategy: % % 0 = no counter measures % 1 = Strategy I (half beta) % 2 = Strategy II (double µix , µiy ) % 3 = Strategy III (quadruple µix ) % 4 = Strategy IV (quadruple µiy ) % % If two optional arguments are given, the first one stands for the counter strategy and the second % one for the time (in months) the simulation shall run. Examples: % % >> aids; % no counter strategy, ten years % >> aids(2); % counter strategy II, ten years % >> aids(3,240); % counter strategy III, twenty years % >> aids(0,1200); % no counter strategy, one hundred years % % The second special feature is that you can plot several runs into one diagram. MatLab % automatically plots the old graphs red and the new one blue, so you always know where you’re % at. % global parameter values global beta alpha cx cy usx usy uix uiy etau etar; beta = 1/60; alpha = 2; cx = 0; cy = 0.00085; usx = 1/360; usy = 1/360; uix = 1/120; uiy = 1/120; etau = 7500; etar = 7.5; % further parameter values strategy = 0; Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Tmax = 120; pui = 0.13; pri = 0.09; Nu = 3500000; pux = 0.5; Nr = 3500; prx = 0.5;

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% starting values Sux = (1-pui)*pux*Nu; Iux = pui*pux*Nu; Suy = (1-pui)*(1-pux)*Nu; Iuy = pui*(1-pux)*Nu; Srx = (1-pri)*prx*Nr; Irx = pri*prx*Nr; Sry = (1-pri)*(1-prx)*Nr; Iry = pri*(1-prx)*Nr; M = 0; % handling the optional argument if nargin == 1 strategy = varargin1; end if nargin == 2 strategy = varargin1; Tmax = varargin2; end if nargin > 2 disp(’Too many arguments’); end % resetting the parameter values according to the chosen strategy if strategy == 1 beta = .5*beta; end if strategy == 2 uix = 2*uix; uiy = 2*uiy; end if strategy == 3 uiy = 4*uiy; end if strategy == 4 uix = 4*uix; end

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% solving the ode j=1; tau=0.1; t0=0; x0=[Sux,Iux,Suy,Iuy,Srx,Irx,Sry,Iry];

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for i=1:Tmax/tau tspan=[t0 t0+tau]; [t,x] = ode45(@aidsf,tspan,x0); n=length(t); xnew=x(n,:); for k=1:8 u(k,j)=xnew(k); end time(j)=t(n); t0=t0+tau; x0=x(n,:); j=j+1; end % plotting the results hold on grid on set(findobj(’Type’,’line’),’color’,’r’); plot(time,u(1,:) + u(2,:) + u(3,:) + u(4,:)); % urban population plot(time,u(2,:) + u(4,:)); % infected urban population plot(time,u(1,:) + u(3,:)); % suspective urban population plot(time,u(5,:) + u(6,:) + u(7,:) + u(8,:)); % rural population plot(time,u(6,:) + u(8,:)); % infected rural population plot(time,u(5,:) + u(7,:)); % suspective rural population end

Acknowledgments We thank Isabella Graf, Shoshana Magnet, Sarah Berry and Jeremy Kloet for technical discussions. BPK acknowledges the Ontario/Baden-W¨urtemberg exchange program for enabling him to spend a year abroad at the Department of Mathematics in Ottawa and the Baden-W¨urtemberg Stipendium for financial support. RJS? and FL are supported by NSERC Discovery Grants and Ontario Early Research Awards. RJS? also receives funding from MITACS.

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References [1] UNAIDS press release (2007). http://data.unaids.org/pub/EPISlides/2007/ 071119_epi_pressrelease_en.pdf [2] Beyrer, C (2007). HIV epidemiology update and transmission factors: Risks and risk contexts - 16th International AIDS Conference Epidemiology Plenary. Clinical Infectious Diseases 44, 981–987. [3] Susser I, Stein Z (2000). Culture, sexuality, and women’s agency in the prevention of HIV/AIDS in Southern Africa. American Journal of Public Health 90:7, 1042–1048. [4] Myer L, Denny L, De Souza M, Barone MA, Wright TC, Kuhn L (2004). Intravaginal practices, HIV and other sexually transmitted diseases among South African women. SexTransm Dis 31, 174–179. [5] McClelland RS, Lavreys L, Hassan WM, Mandaliya K, Ndinya-Achola JO, Baeten JM (2006). Vaginal washing and increased risk of HIV-1 acquisition among African women: a 10-year prospective study. AIDS 20, 269–273. [6] Nunn AJ, Kengeya-Kayondo JF, Malambda SS, Seeley JA, Mulder DW (1994). Risk factors for HIV-1 infection in adults in a rural Ugandan community: a population study. AIDS 8, 81–86.

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[7] Clark S (2004). Early marriage and HIV risks in sub-Saharan Africa. Stud Fam Plann 35, 149–60. [8] Gray RH, Li X, Kigozi G, Serwadda D, Brahmbhatt H, Wabwire-Mangen F, Nalugoda F, Kiddugavu M, Sewankambo N, Quinn TC, Reynolds SJ, Wawer MJ (2005). Increased risk of incident HIV during pregnancy in Rakai, Uganda: a prospective study. The Lancet 366:9492, 1182–1188. [9] Mekonnen Y, Jegou R, Coutinho RA, Nokes J, Fontanet A (2002). Demographic impact of AIDS in a low-fertility urban African setting: Projection for Addis Ababa, Ethiopia. J Health Popul Nutr 20:2, 120–129. [10] Mnyika KS, Klepp KI, Kvale G, Nilssen S, Kissila PE, OleKing’ori N (1994). Prevalence of HIV-1 infection in urban, semi-urban and rural areas in Arusha region, Tanzania. AIDS 8:10, 1477–81. [11] Lurie MN, Williams BG, Zuma K, Mkaya-Mwamburi D, Garnett GP, Sturm AW, Sweat MD, Gittelsohn J, Abdool Karim SS (2003). The impact of migration on HIV1 transmission in South Africa: a study of migrant and nonmigrant men and their partners. Sex Transm Dis 30, 149–156. [12] Hayes R, Mosha F, Nicoll A, Grosskurth H, Newell J, Todd J, Killewo J, Rugemalila J, Mabey D (1995). A community trial of the impact of improved sexually transmitted disease treatment on the HIV epidemic in rural Tanzania: 1. Design. AIDS 9, 919– 926.

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[13] Blower SM, Kahn JO, Wilson DP (2006). Predicting the epidemiological impact of antiretroviral allocation strategies in KwaZulu-Natal: The effect of the urban-rural divide. PNAS, 103, 14228–14233. [14] Renton AM, Whitaker L, Riddlesdell M (1998). Heterosexual HIV transmission and STD prevalence: predictions of a theoretical model. Sexually Transmitted Infections 74, 339–344. [15] Robinson NJ, Mulder DW, Auvert B, Hayes RJ (1995). Modelling the impact of alternative HIV intervention strategies in rural Uganda. AIDS 9, 1263–1270. [16] Gregson S, Nyamukapa CA, Garnett GP, Mason PR, Zhuwau T, Cara¨el M, Chandiwana SK, Anderson RM (2002). Sexual mixing patterns and sex-differentials in teenage exposure to HIV infection in rural Zimbabwe. The Lancet 359:9321, 1896– 1903. [17] Coffee M, Lurie MN, Garnett GP (2007). Modelling the impact of migration on the HIV epidemic in South Africa. AIDS 21:3, 343–350. [18] Nicolosi A, Corrˆea Leite ML, Musicco M, Arici C, Gavazzeni G, Lazzarin A (1994). The efficiency of male-to-female and female-to-male sexual transmission of the Human Immunodeficiency Virus: A study of 730 stable couples. Epidemiology 5:6, 570–575. [19] Heffernan JM, Smith RJ, Wahl LM (2005). Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface 2, 281–293.

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[20] Murray JD (2001). Mathematical Biology. Springer-Verlag, New York. [21] South Africa HIV & AIDS Statistics http://www.avert.org/safricastats.htm

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 59-101

ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 3

A PARTNERSHIP N ETWORK S IMULATION THE S PREAD OF S EXUALLY T RANSMITTED I NFECTIONS IN RUSSIA Fatemeh Jafargholi and Chris T. Bauch∗ Department of Mathematics and Statistics, University of Guelph Guelph, Ontario N1G 2W1, Canada

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Abstract We develop a heterosexual partnership network model based on Russian survey data. Our purpose is to study the spread of sexually transmitted infections (STIs) such as HIV and gonorrhea and to assess prevention strategies. The key ingredients of the model are age-structured sexual behavior, casual versus steady partnerships, a core group, and condom usage. In the first part, we attempt to reproduce observed STI prevalence to validate the model through correct parametrization. In the second part, we study possible prevention strategies. The results show that the rate of spread varies according to how many individuals use condoms and how many core group members there are. An STI can disappear from the population if it disappears from the core group, and increasing condom usage in the core group has disproportionate impacts on STI prevalence. The model predicts highest HIV incidence in the 25-34 age class, which is consistent with Russian data.

1.

Introduction

Mathematical models and computational simulations are important tools in furthering our understanding of the transmission of sexually transmitted infections (STI). Sexually transmitted infections are also known as sexually transmissible diseases and sexually transmitted diseases (STDs), and are defined as diseases or infections that have a significant probability of transmission between humans by means of sexual contact. Sexual behaviors vary between countries and cultures (1) and therefore different models may be appropriate for different countries. For studying the spread of STIs, a partnership network for Russian ∗

E-mail address: [email protected]. Corresponding author.

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society is developed and simulated in this chapter. The transmission of human immunodeficiency virus (HIV) and Gonorrhea through the partnership network are studied. The model is based on a discrete time stochastic process of pair formation and a stochastic process of disease transmission. Behavioral changes (using condoms) are also explored in this study. We describe HIV and Gonorrhea epidemiology in the following subsections.

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1.1.

HIV/AIDS

In 1981, clinical investigators in New York and California observed among young, previously healthy, homosexual men an unusual clustering of cases of rare diseases, notably Kaposi’s sarcoma (KS), a very rare skin neoplasm, and opportunistic infections such as Pneumocystis Carinii Pneumonia (PCP), a lung infection caused by a pathogen to which most individuals are exposed with no undue consequences. They also observed cases of unexplained, persistent lymphadenopathy (2). These events signalled the arrival of a new disease in the population, which later became known as human immunodeficienty virus (HIV). HIV is a retrovirus that primarily infects vital components of the human immune system. It directly and indirectly destroys cells that are required for the proper functioning of the immune system. When enough of these cells have been destroyed by HIV, the immune system functions poorly, leading to the syndrome known as as Acquired Immune Deficiency Syndrome (AIDS). There are three stages for HIV infection. The transmission rate is highest in the first stage of infection (lasting about 2.5 months) and lowest in the second stage (lasting about 7-9 years). The last stage, called AIDS, lasts about 2 years if untreated and infected people die at the end of this stage (4). AIDS is the final and most serious stage of the disease caused by the human immunodeficiency virus. AIDS happens concurrently with numerous opportunistic infections and tumors that are normally associated with HIV infection. Although treatments for HIV/AIDS exist that slow the virus’ progression, there is no known cure. Most patients die from opportunistic infections or malignancies associated with the progressive failure of the immune system (5). Since the beginning of the pandemic, three main transmission routes of HIV have been identified: the sexual route, the blood or blood product routes, and the mother-to-child route. Here we study the sexual route. Although researchers have found HIV in the saliva of infected people, there is no evidence that the virus is spread by contact with saliva. Laboratory studies reveal that saliva has natural properties that limit the power of HIV to infect, and the amount of virus in saliva appears to be very low. Research studies of people infected with HIV have found no evidence that the virus is spread to others through saliva by kissing. The lining of the mouth, however, can be infected by HIV, and instances of HIV transmission through oral intercourse have been reported. There has also been found no evidence that HIV is spread through sweat, tears, urine, or feces. The majority of HIV infections are acquired through unprotected sexual relations between partners. Unprotected receptive sexual acts are riskier than unprotected insertive sexual acts, and the risk for transmitting HIV from an infected partner to an uninfected partner through unprotected insertive anal intercourse is greater than the risk for transmission through vaginal intercourse or oral sex (6). The United Nations Programme on HIV/AIDS (UNAIDS) and the World Health Or-

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ganization (WHO) estimate that AIDS has killed more than 25 million people since it was first recognized in 1981, making it one of the most destructive epidemics in recorded history. Despite recent, improved access to antiretroviral treatment and care in many regions of the world, the HIV epidemic claimed an estimated 3.1 million (between 2.8 and 3.6 million) lives in 2005 of which more than half a million were children (3). Sub-Saharan Africa remains by far the worst-affected region, with an estimated 23.8 to 28.9 million people currently living with HIV. More than 60% of all people living with HIV are in subSaharan Africa, as are more than three quarters (76%) of all women living with HIV. South and South East Asia are the second worst affected with 15%. AIDS has accounted for the deaths of 500,000 children since the start of the epidemic (8). There is also growing concern about a rapidly growing epidemic in Eastern Europe and Central Asia, where an estimated 0.99-2.3 million people were infected as of December 2005. In this region, the rate of HIV infections began to grow rapidly from the mid-1990s, due to social and economic collapse, increased levels of intravenous drug use and increased numbers of prostitutes. The AIDS epidemic in Eastern Europe and Central Asia shows no signs of abating. Some 230,000 people were infected with HIV in 2003, bringing the total number of people living with the virus to 1.5 million. By 2004, the number of reported cases in Russia was over 257,000, according to the World Health Organization, up from 15,000 in 1995 and 190,000 in 2002; some estimates claim the real number is up to five times higher, over 1 million. Ukraine and Estonia also had growing numbers of infected people, with estimates of 500,000 and 3,700 respectively in 2004. AIDS claimed an estimated 30,000 lives in 2005; worst-affected are the Russian Federation, Ukraine, Estonia and Latvia (8). There are predictions that the infection rate in Russia will continue to rise quickly, because education there about AIDS is almost non-existent. The epidemic is still in its early stages in this region, which means that prevention strategies may be able to halt and reverse this epidemic. During 1999–2001, the estimated number of cases of HIV reported officially in the Russian Federation increased approximately 16-fold, from 11,000 to 177,000 (10). The first official case of HIV in the Union of Soviet Socialist Republics (USSR) was recorded in the end of 1986, in a Russian who contracted the virus in Africa and then infected 15 Soviet soldiers with whom he had homosexual relationships. This was immediately publicized in a mass media campaign which proclaimed that HIV/AIDS was a disease of a corrupt lifestyle. The USSR was not ready socially, ideologically, or economically for a serious prevention campaign. At that time, homosexuality was illegal, issues related to reproductive health were not considered appropriate themes for public discussion, and the country was reeling from the instability of perestroika (the restructuring of the Soviet economy and bureaucracy that began in the mid 1980s). The public gave little consideration to the threat of HIV during the period of the late 1980s and early 1990s, which is often associated with Russia’s sexual revolution, an increase in intravenous injection (IV) drug use, and a surge in prostitution. The challenges involved in containing the growing epidemic in Russia are greatly compounded by the stigma and discrimination experienced by drug users, street children, and those with HIV, who in many places are refused treatment despite laws to the contrary, and who remain reluctant to come forward or accept what support is available (11). By some estimates, there could be as many as 3 million injecting drug users in the Russian Federation alone. Most of these

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Figure 1. HIV/AIDS surveillance in Europe. End-of-year report. Data compiled by the European Center of the Epidemiological Monitoring of AIDS, from national AIDS programmes (2002). drug users are male and many are very young. In St Petersburg, studies found that 30% of them were under 19 years of age, while in the Ukraine, 2% were still teenagers. A survey of Moscow youth aged 15-18 found that 12% of the males had injected drugs. In the Russian Federation, 8% of HIV cases due to injecting drug use are in young persons under 30. Because most injecting drug users are young and sexually active, a significant share of new infections is occurring through sexual transmission (often when injecting drug users or their HIV-infected partners engage in unsafe sex) (8).

1.2.

Gonorrhea

Gonorrhea is one of the most common curable sexually transmitted diseases in the world and is caused by the Neisseria gonorrhoeae bacterium (9). These bacteria grow well only on mucous membranes and die in seconds outside the human body. Untreated gonorrhea can cause serious and permanent health problems in both women and men. In men, epididymitis, prostatitis and urethral stricture can result from untreated gonorrhoea. In women, Bartholinitis and abscess formation (causing trouble walking), pelvic inflammatory disease (PID) and Fitz-Hugh-Curtis syndrome can occur. The most common result of untreated gonorrhea is PID, a serious infection of the female reproductive tract. PID causes scarring of the fallopian tubes which leads to increased risks of causing an ectopic pregnancy as a fertilized egg may not be able to pass through the narrowed, scarred fallopian tube. Ectopic pregnancies are serious conditions which are potentially life-threatening to the mother (12). When initially infected, the majority of men have some signs or symptoms. Symptoms and signs include a burning sensation when urinating and a yellowish white discharge from the penis. Sometimes, men with gonorrhea get painful or swollen testicles (12). In women, the early symptoms of gonorrhea are often mild, and many women who are

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infected have no symptoms of infection. Even when a woman has symptoms, they can be so non-specific as to be mistaken for a bladder or vaginal infection. The initial symptoms and signs in women include a painful or burning sensation when urinating and a vaginal discharge that is yellow or occasionally bloody. Women with no or mild gonorrhea symptoms are still at risk of developing serious complications from the infection (12). Symptoms usually appear within two to 10 days after sexual contact with an infected partner. A small number of people may be infected for several months without showing symptoms. Past infection of gonorrhea does not make a person immune to gonorrhea and previous infections with gonorrhea may allow complications to occur more rapidly and increase the risk of HIV (14). As gonorrhea is an STI, proper use of barrier contraceptives such as latex condoms will significantly reduce the risk of getting gonorrhea and its complications. Educational programs either in clinics or in the media might make the sexually active population more aware of the symptoms and seriousness of gonorrhea so that people who suspect that they might be infected would seek examination and treatment sooner. Since there is currently no vaccine which can prevent gonorrhea, other control procedures must be applied. Contact investigation or contact tracing, an example of a commonlyused control procedure, attempts to identify contacts of known infectives and to encourage contacts to be checked as soon as possible. Contact tracing for gonorrhea sometimes consists of educating known infectives about the seriousness of gonorrhea and asking them to encourage their contacts to be examined. In the early to mid-1970s, most European countries saw a peak in cases of gonorrhoea. It is thought that the advent of AIDS/HIV infection in the 1980s led to safer sex and accelerated the reduction in gonorrhoea.

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1.3.

Previous STI Models

In modeling transmission dynamics of a communicable disease, it is common to divide the population into disjoint classes (compartments) whose sizes change with time. The infection status of any individual in a population can be Susceptible, when the person is healthy and susceptible to the disease (denoted by S), Exposed, when the person is in a latent period but not yet infectious (denoted by E), Infected, when the individual carries the disease and is infectious (denoted by I), or Removed, when the person has recovered and is at least temporarily immune or has died because of the disease (denoted by R). In some diseases such as HIV, there is no recovery. In other diseases, if an infected person recovered he/she may be susceptible again. A sequence of letters, such as SEIR, describes the movement of individuals between the classes: susceptibles become latent, then, infectious and finally recover with immunity. To model diseases which confer permanent immunity and which are endemic because of births of new susceptibles, SIR or SEIR models with vital dynamics are suitable. Vital dynamics is needed to avoid explosion of the population size. Models of SEIRS or SIRS types are used to model diseases with temporary immunity and in cases where there is no immunity, models are named SIS or SEIS. The last S points the individual becoming susceptible again, after recovery. Such models may be appropriate for gonorrhea, for instance. Not all epidemic models are suitable for STIs since the sexual network plays an important role in spread of disease, and thus the populations are not mixing homogeneously. Different mathematical modeling approaches to STI modeling have been applied, including

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ordinary differential equation models, pair-formation models, and network models. Next, we briefly review some of these models.

1.4.

Ordinary Differential Equation Modeling

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STIs have been modeled by systems of ordinary differential equations (ODEs). If the characteristics of the disease, sexual behavior of the population and prevention strategies are known, one can design a compartmental system by dividing the population into different groups and making assumptions about the interactions between these groups. Process dynamics are modeled in terms of mass fluxes between the compartments, inflow and outflow. If the mass flow from compartment i into j is assumed to be linearly proportional to the mass in i, then, one obtains a system of n linear ODEs, where n, the number of compartments, can be written in the form of X˙ = A(t)X(t) + U (t), where X ∈ Rn , A ∈ Rn×n , U ∈ Rn . X is the vector of compartments’ mass, A is the matrix of flows between compartments and U is the outflow from the compartments and inflow to the compartments. Compartmental epidemic models, by comparison, are usually nonlinear. In the simplest model of SIS type, we assume there is no difference between disease characteristics for female and male, and there is no immigration or vital process. Let d be the average infectious period and hence an average infective has a d1 chance of recovering on any day, and after recovery is susceptible again. Assume the population size is N and the fractions of the population that are infected or susceptible at time t are S(t) and I(t). Let λ denote the average constant rate of disease transmission from an infective to a susceptible (the rate at which a susceptible gets the infection). Since S(t) = 1 − I(t), the changes in number of infected individuals in the population is: N I(t) d (N I(t)) = λN I(t)(1 − I(t)) − , I(0) = I0 . dt d The model could be made much more complex by defining more compartments and epidemiologic factors. These models can predict the existence of equilibrium points or stable cycles in disease prevalence. Many such models have been used for epidemic problems, and examples can be found in (14). Compartmental models were traditionally applied to diseases transmitted through respiratory infection such as measles. In the case of STIs, the assumption of homogeneous (mass-action) mixing behind such compartmental models is less accurate. Rather, since an STI can only be received from a small subset of the population, it may be more appropriate to model the transmission as occurring on a network of contacts, or between distinct pairs of individuals. In such cases where the individual-level structure of the population is important, pair-formation models or network models are more useful.

1.5.

Pair Formation Models

Pair formation models are a type of ordinary differential equation models that have sometimes been used to study STI transmission in populations. They incorporate the repeated contacts within partnerships which happen frequently in real sexual networks. They were first developed in 1988 by Dietz et al., (15) to study STIs in monogamous partnerships. In

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this model, if two susceptible individuals form a pair then they can be considered temporarily immune as long as they do not separate and have no contacts with other partners. This aspect influences transmission dynamics considerably, especially when the disease is first introduced, since the vast majority of existing pairs are susceptible. A pair formation model can be described by the following equations. Let x and y denote the density of single females and single males, and let p denote the density of pairs. Let κx , κy be the birth rates of females and males, and µx , µy the corresponding death rates and σ the pair separation rate. Pair formation is a nonlinear process, denoted by φ(x, y). All parameters are nonnegative. Then we have x˙ = κx − µx x + (µy + σ)p − φ(x, y) y˙ = κy − µy y + (µx + σ)p − φ(x, y)

(1)

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p˙ = −(µx + µy + σ)p + φ(x, y) where x0 is the number of susceptible single females, y0 is the number of susceptible single males, x1 is the number of infected single females, y1 is the number of infected single males, p00 is the number of pairs, both partners susceptible, p01 is the number of pairs, only male infected, p10 is the number of pairs, only female infected, and p11 is the number of pairs, both partners infected, and φij (i, j=0, 1) is the pair formation rate between individuals with different infection statuses, and is a function of x0 , y0 , x1 , y1 . Let hx be the probability that an infected female infects a susceptible male and hy the corresponding probability for an infected male. These transmissions could be defined either by a probability per time step or probability per relationship. For the case of curable diseases, let γx and γy be the rates of recovery for an infected female or male respectively. One can write the equations for susceptible and infected densities with different φ functions and solve them for the stationary point. Heesterbeek, using the model of Dietz (15), extends a calculation for compartmental STI models to pair STI models (16) (cited in (17)). Although pair formation models have been utilized for different problems, they are not able to simulate concurrent relationships, which occur frequently in real populations.

1.6.

Network Modeling

In network models, the population is studied on the individual level. The demographic and epidemiologic status of every individual is explicitly represented in the model simulation. Connections between individuals (figure 2), create a network in which each node represents an individual and connections represent interactions with other individuals in the population. An interaction is represented by a connection in the network if there is a possibility of disease transmission from an infected individual to a healthy one. Types of interactions (sexual, social, school, business, etc.) that can result in transmission, and hence connection architecture, vary depending on the disease being modeled. Any two individuals who are connected together are usually called neighbors. The number of neighbors of an individual is called the degree of the node. Designing a network model for STI transmission requires data from real populations and real diseases. Some researchers carry out questionnaire surveys to estimate real world social interactions (18, 19) and some researchers collaborate with clinics and hospitals to determine the biological characteristics of the disease (4, 20). While studying a network

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Figure 2. A network model. model for epidemics, a wide range of different networks can be created and this also gives an opportunity to compare the effect of network structure on spread of disease. Diseases that require intimate contact for transmission are more strongly affected by network biases, such as mixing rules, than diseases passed by casual contact, which can often be modeled by models that assume homogeneous mixing. Network structure is now recognized to have an important role in the sexual spread of HIV and other STIs. Three important aspects of network architecture that influence disease spread are summarized below, from Ref. (21): 1. Core groups: groups where members have high levels of risky behavior and associate preferentially with one another. They contribute a disproportionate share of incidence of HIV/STIs, and can fuel sustained transmission in the general population (22). Although this is a widely-applied concept, there is no unique definition for the core group, in that some individuals may also exist in a ’gray area’ between core and non-core behavior (23).

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2. Loops: Nodes in the graph that are directly or indirectly connected, making a loop. The number of these closed loops in the network influences how the disease spreads and may vary between different societies. 3. Degree distribution: The distribution of number of neighbors of individuals can also vary in networks and is an important parameter of networks (24). Some common concepts in the field of STI modeling include: 1. Infection status: Individuals in a population can be infected (I), susceptible (S), exposed (E), removed (R). More explanation is mentioned at the beginning of section 1.3.. 2. Exclusively monogamous: A pair of individuals who do not have interactions with any other nodes of the network, so if they are susceptible, they remain so always. 3. Serial monogamists: Persons who have a succession of monogamous relationships. If they have unprotected sex, they have a higher risk of HIV/STDs than exclusively monogamous persons, since the infection can pass through pairwise series of numerous serial monogamous relationships over time (22). 4. Risky behavior: Having many neighbors or being a member of a loop or a cluster that increases the risk of infection is called risky behavior. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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5. Basic reproductive number: The number of secondary infections produced by an infected individual in an otherwise susceptible population, denoted by R0 and also refered to as the Basic Reproductive Ratio (25). If R0 > 1, then, a disease may spark an epidemic, but if R0 < 1, it may only produce a small localized outbreak that will immediately fade on its own. A basic principle for all diseases with an endemic equilibrium, is that the effective reproductive number is one when the disease is at endemic equilibrium. This principle is an average result since some infectives might infect several susceptibles and some might infect no susceptibles. Understanding the network structure of a population can lead researchers to better model the spread of epidemic diseases and suggest prevention strategies. Some databases exist which document sexual behavior, such as the Russian Longitudinal Monitoring Survey, and can be used for such a purpose.

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1.7.

Other Considerations

For all of these types of models, birth, aging and death processes in the population can be introduced. Birth and death determine population demographics and aging is important for behavioral changes of individuals. One can simulate death by replacing every dead individual with a new young one, hence births exactly equal deaths and the population size is constant over time (28). Otherwise, the average rate of births and deaths can be equal, so the population size is constant except for some stochastic variation around a mean. In other cases, the simulated period of time is short in comparison to demographic time scales, or the time scale over which disease mortality acts, so birth, death and aging can be ignored (29). Another issue concerns the formation of new partnerships and separation of current partnerships. One very simple approach is to assume that the probability of forming a new partnership is some constant value per time step between any two individuals of opposite sex in the population, regardless of age. Alternatively, a rule can dictate preferential mixing with certain ages, as usually occurs in reality (28,29). For partnership breakup, the simplest assumption is that there is a constant probability of breakup per time step. This approach is simple but probably unrealistic. Finally, the population can also be stratified according to level of behavioral risk. In most models, a fraction of population is assigned to have high risk behavior, such as through having a high number of concurrent (simultaneous) partners. This group plays an important role in spread of STIs. Most models treat the actual disease transmission process in the same way, as a probability of disease transmission per time step, from an infected individual to his/her susceptible partner. The probability of transmission depends on the disease and often the gender of the infected individual as well.

1.8.

Data Sources Used

To parameterize our model, we used a Russian survey on sexual behavior, intended to be representative of the Russian population (2001). We review this survey below. In the Russian Longitudinal Monitoring Survey there are 157 questions about personal information, sexual behavior and general knowledge about STIs. A set of questions about details of the three most recent partners in the last 12 months are asked (Questions 18 to 71), but for those who have more than 3 partners in the last 12 months there are no data about these additional

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relationships. In the questions about three most recent partners, information on the type of relation was requested. After asking several questions about the three most recent partners, the respondent was asked for the total number of people whom he/she had sexual relations with, in the last 12 months (question 76). The following data can also be obtained from the survey: marital status, age, whether the respondent had sex in the last 12 months, the number of sexual partners in the last 12 months, the type of relation with the last 3 partners, the date of last and first sexual act with each of these three partners, and how/whether condoms were used. Another important source of data is the demographic yearbook of the United Nations. The death rate per year for each age/gender group was obtained from the Demographic Yearbook 2001 for Russia. Parameters related to the probability of infection transmission were obtained from other literature sources (4). The procedure by which parameters were derived from these data sources is described in section 2.. Model parameters were chosen either to fit simulation results to available data or to satisfy some other assumption.

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1.9.

Chapter Objectives and Outline

The objective of this chapter is to use data from the Russian Longitudinal Monitoring Survey and other sources to develop and study a network model for HIV and gonorrhea transmission. This partnership network can be used to study other STIs as well. Model simulations are analyzed to shed light on the impact of condom usage in various groups, the impact of age structure, the impact of size of core group in a population, and the positive and negative interactions between HIV and gonorrhea. Section 2. describes the network model, the disease transmission model, and how these were parameterized with available data. In section 3., the results are presented and discussed. Results include simulations for the actual (real-world) parameters as well as results for hypothetical scenarios. Section 4. presents the discussion and section 5. the conclusion.

2.

Model Description

The model is a stochastic, individual-based network model. There are 1500 individuals in the model population. Individuals are born into the 15 year old age class and exit the sexually at-risk population at 55. The population is assumed to be heterosexual with two kinds of relationships, steady partnerships (long term) and casual partnerships (short term). Individuals choose partners randomly with a probability influenced by their age and their current number of partners. It is assumed that no one can have more than three concurrent steady relations at the same time, and except for the case of individuals with the maximum number of steady relations, there is no upper limit to the number of casual partners in the model. The number of new casual partners gained per time step is chosen from a binomial distribution. The partnership network is designed to have a Poisson distribution of number of partners per person. Since the two kinds of relationships have different properties, the Poisson distributions used for each relationship type are different. The duration of each relationship is picked on the first day of relationship and is related to the type of relationship and age of both parties. The duration of the relationship is sampled from a Weibull

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Table 1. Death rates per capita per year, taken from the Demographic Yearbook of Russia, 2001 Age class Age Male Female

1 15 to 24 0.00266 0.00080

2 25 to 34 0.00529 0.00128

3 35 to 44 0.0096 0.0026

4 45 to 54 0.01891 0.00595

Table 2. Birth rates per capita per year, chosen to yield a stable population size. Gender Male Female

Birth Rate 0.01606 0.01582

distribution. The population is also stratified into a small high risk (core) group and a low risk (general population) group. More details are given in the following subsections.

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2.1.

Birth/Death Rates

We assume that the population size is constant in the absence of deaths caused by disease. This simulation assigns a constant per capita birth rate to each person. A constant agespecific per capita death rate is also assumed. Since these are probabilistic parameters, the model population size is not exactly constant, but rather is fluctuating around the average population size, in the absence of infection deaths. Since birth and death rates are constant, the age distribution also remains constant in the absence of deaths caused by infections. Death rates for each age class, in the absence of disease, are the same as the real death rates in Russia in 2001, from UN demographic data (table 1). In order to have an approximately constant population size the birth rate, β, is chosen to balance the death rate, table 2. We have different birth rates for males (denoted by βm ) and females (denoted by βf ). Every year, βf × N , new females, and βm × N , new males are born at age 15, which means they are added to the model population. Immigration of individuals into the population was not taken into account. We note that these assumptions are relevant to the situation before disease introduction. Diseases which cause death, such as HIV, decrease the population size, and in such cases the ratio of females to males also changes since infection rates are different for males and females.

2.2.

Age Groups

Since the survey covers people from 15 to 54 years old, and in other studies the sexually active population has about the same age range, the simulated population was assumed to be between 15 and 54 years old. The sexual behavior of population members is specific to age classes. There are 4 groups: 15-24, 25-34, 35-44, 45-54. At the beginning of each year in the simulation, all the individuals are moved up (aged) by one year. Approximately 40 years after some arbitrary initial conditions on the population distribution, the population

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has reached a demographic equilibrium. In the simulation, the sizes of the age classes are set by the birth and death rates defined in the previous section. At equilibrium, each age class constitutes about 25 percent of the total population.

2.3.

Partnership Dynamics

In the model, relationships are divided into two types: Steady, which are long-lived and usually monogamous, and casual, which are short-lived and often concurrent. Official marriages, unofficial marriages, friends with sexual relations who live together and old friends with sexual relations who do not live together are counted as steady relations for the purposes of our model (questions 18, 37 and 56 of the survey, section 1.8.). Newer friends who do not live together and casual acquaintances with sexual relations, relations in exchange for gifts, and relations in which one would be paid are counted as casual relations. The partnership formation process is described in sections 2.5. to 2.7.. Before introducing the partnership formation process used in the model, we describe here a much simpler version that, while being unrealistic, permits closed form solutions.

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2.3.1. Derivation of a Simple Model In this simple model, each box labeled ‘i’ in figure 3 represents the number of individuals with i partners. Every day each individual in the population has a chance of forming a new relationship with any other given individual in the model population. So the rate that an individual makes a new relationship is ρ times the size of the population, N . On the other hand, each current partnership is broken up on a given day, with probability σ.

Figure 3. The schematic of a simple model for sexual partnership dynamics. In each time step with probability ρ × (N − i), an individual with i partners forms a new relationship and enters the compartment of individuals with i + 1 partners. From now, we suppose i is negligible compared to N (i ≪ N ), so we estimate N − i with N . In each time step each current partnership also breaks up with probability σ, hence a person with i + 1 partners leaves that compartment and enters the compartment of individuals with i partners with probability (i + 1)σ. When the partnership network is in its steady state, the size of each box should not change. Hence, the number of people leaving the box equals the number of people joining it. Let Ni denote the number of people with i partners, that is the size of box i. Assuming the system is in its steady state, the following balance equations

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are obtained from the model: → N1 =



Nρ σ



N2 2σ + N0 ρN = N1 ρN + N1 σ → N2 =



Nρ σ

2

N0 2

N3 3σ + N1 ρN = N2 ρN + N2 σ → N3 =



Nρ σ

3

N0 2×3

N0 ρN =

N1 σ

...

N0

( Nσρ )i Ni!0

Ni =

The last equation is a Poisson distribution with mean λ = ( Nσρ ) and. The probability of Pi=N Ni existence of NP i individuals with i partners is N , denoted by Pi . We know i=0 Pi = 1, which means i Ni = N . On the other hand, X

Ni =

X N ρ N0 ( )i σ i! i

i

and i=N X i=0

λn ≈ eλ n!

for large N. Hence, we have

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Pi = (

λi N ρ i N0 e−λ = e−λ . ) σ i! N0 i!

The problem with this simple model is that it assumes the probability of partnership breakup is independent of the time since the relationship has been formed, whereas in general a partnership lasts some characteristic length of time, and it does not distinguish according to age. Hence, in the model used to generate results for this chapter, there is an age mixing rule and people are distinguished as having either a high level or a low level of sexual activity.

2.4.

Distribution of Number of Partners per Person in Survey

For a given partnership type, the survey data also exhibit a distribution similar to the Poisson distribution (figures 4 and 5). The Poisson distribution is a one-parameter (λ) distribution that, according to our simple model, is related to the rate of forming (ρ) and breaking (σ) partnerships.

2.5.

Calculating ρ

Although the distribution of number of partnerships per person can be Poisson for a given type of partnership, we have seen that a simple closed form solution can only be derived for a homogeneous population, whereas heterogeneity is important in the spread of HIV. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 4. Distribution of number of casual partnerships per single person, from survey data.

Figure 5. The distribution of number of steady partnerships per person, from survey data.

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Table 3. The estimated rate ρS of forming new steady partnerships, per year Age class Male Female

1 0.0130 0.0227

2 0.0338 0.0302

3 0.0205 0.0166

4 0.0162 0.0158

Table 4. The estimated rate ρM 0C,jk of forming casual partnerships for single men of different ages, per year

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Age class men, 1st 2nd 3rd 4th

women, 1st 2.31691 1.65878 1.14577 0.33793

2nd 0.18770 0.92696 0.65473 0.42241

3rd 0.01173 0.04879 0.25721 0.59138

4th 0.00000 0.01626 0.04677 0.33793

Instead of using the foregoing simple model, we develop a more sophisticated model based on the survey data as follows. The rate that an individual forms a partnership with some other given person in the population on each day is ρ. We estimate ρ from survey responses as the average number of newly formed relationships of an individual, reported in a given period of time, divided by the number of people whom the individual can form a partnership with (in the case of a heterosexual society, this is N/2 where N is the total population size). This period of time is taken to be 10 years for the steady relationships and 1 year for the casual relationships M that may even last for a few days. Let ρM C and ρS denote the rate that a male individual forms a new casual and steady partnership in each time step, respectively, and likewise ρFC and ρFS for females (estimates are shown in tables 4 and 5). The pattern of age-preferential mixing is also observed from the survey for casual relations and used to parameterize the model. Let ρM iC,j denote the probability per time step at which a man in age class j with i steady partners forms a casual partnership with a given female. Moreover let ρM iC,jk denote the probability per time step at which a man in age class j with i steady partnerships forms a casual partnership with a given female in age class k. Table 4 shows the values of ρM 0C,jk determined from the survey. It is clear that there is a strongly assortative pattern, with individuals preferring to mix with other individuals

Table 5. The estimated rate ρF0C,jk of forming casual partnerships for single women of different ages, per year Age class women, 1st 2nd 3rd 4th

men, 1st 1.18035 0.35783 0.10390 0.00000

2nd 0.53435 0.92283 0.41564 0.14242

3rd 0.04785 0.52733 0.72736 0.56970

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4th 0.00000 0.07533 0.38100 0.85454

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Table 6. The estimated rate ρM (i>0)C,jk of forming casual partnerships for married men of different ages, per year Age class men, 1st 2nd 3rd 4th

women, 1st 0.46208 0.42329 0.20201 0.09558

2nd 0.12938 0.20010 0.06734 0.11947

3rd 0.00000 0.01539 0.03741 0.00000

4th 0.00000 0.00000 0.00748 0.00000

Table 7. The estimated rate ρF(i>0)C,jk of forming casual partnerships for married women of different ages, per year Age class women, 1st 2nd 3rd 4th

men, 1st 0.19392 0.10359 0.05053 0.08732

2nd 0.15756 0.15937 0.01685 0.04366

3rd 0.01212 0.03187 0.06738 0.00000

4th 0.01212 0.01594 0.03369 0.00000

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of their own age. However, in table 4 the lower triangle is larger than the upper triangle, but in table 5 the upper triangle is larger than the lower one. This means that men tend to be older than their partners. Age-structured mixing is not included for steady partnership formation, since there were no respondents data in the survey to divide the data into 32 groups of all ages and genders. Hence, individuals do not consider age when forming steady partnerships.

2.6.

Core Group

A core group is a group whose members have high risk sexual activities, more concurrent relations and higher mean number of casual partners. They often form highly connected clusters in the sexual partnership network, tending to mix with themselves more than with the general population. For instance, a core group could simulate prostitutes and their clients, or highly active homosexual populations. For some diseases, the presence of a core group sustains a disease that would otherwise not be able to sustain itself in the general population alone (30). At the beginning of a simulation, a number Nc of individuals are chosen at random to be members of the (minority) core group, and the remaining Nn = N − Nc individuals are members of the general population. The core group has different partnership formation rates than the non-core group, and hence different behavior. Since the Russian survey (section 1.8.) contains no information about mixing between any inferred core and non-core groups, both the size of the core group, and mixing patterns between them, are arbitrary parameters in the simulation. However, we wish the average partnership formation rates across the entire population to remain the same once the core group is introduced. Let ρL denote the probability of partnership formation (for low risk activities) of the

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non-core group and let ρH denote the probability of partnership formation (for high risk activities) of the core group. To maintain a constant value of ρ across the whole population, the following relation is considered (equation 2). N H = pC × N NL = (1 − pC ) × N N ρ = N H ρH + N L ρL

(2)

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where N in the total population size, NH is the core group size, NL is the non-core group size and pC is the proportion of the population that is in the core group. In the simulation, the core group makes up 2.5% of population (pC = 0.025). Equation 2 still has a free variable and so it can not be solved until we introduce a further assumption for the relationship between ρL and ρH . Since core groups have higher levels of sexual activity, we assume that ρH = αρL and assume that αc = 5 for casual relationships and αs = 0.1 for steady relationships. Hence, core group members form steady relationships 10 times more slowly than non-core individuals, but they form casual partnerships five times more quickly than non-core individuals. Moreover, each individual tends to form most of their relationships with their own group and there are relatively few connections between groups. To build such a network, ρL and ρH are further divided into two other values. ρL is divided to ρLL and ρLH , and ρH to ρHH and ρHL . Here ρLL denotes the probability per time step that a non-core individual forms a partnership with another non-core individual, and ρHH , ρHL and ρLH respectively denote probability per time step of a core individual pairing with a core individual, a core individual pairing with a non-core individual, and a non-core individual pairing with a core individual. Equations 3 describe how ρL and ρH are broken down into these new parameters. N ρL = NH ρLH + NL ρLL N ρH = NH ρHH + NL ρHL ρLL = χL ρLH ρHH = χH ρHL

(3)

LL for casual relations, χL,C = 250, In the simulation, the following values are used: ρρLH ρLL ρHH for steady relations, χ = 1000, for casual relations, χH,C = 3, ρρHH for steady L,S ρLH ρHL HL relations, χH,S = 500. These values are chosen to ensure that relationships between core and non-core individuals are relatively rare.

2.7.

Simulating Matchmaker

In the simulation, we use the law of rare events (31), which states that the Binomial distribution converges to the Poisson distribution when N , the population size, is large enough. We know that   λ N lim 1 − = e−λ . N →∞ N

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In the Binomial distribution limN →∞ Nk pk (1 − p)N −k , let N be large enough, in comparison to k in the Binomial distribution. Then we have:    k   N k N! λ λ N −k N −k lim p (1 − p) = lim 1− N →∞ k N →∞ (N − k)!k! N n

= lim

N →∞

 |

N N



N −1 N



N −2 N {z



···



N −k+1 N

    λk λ N λ −k 1− 1− . k! N N } | {z } | {z }| {z }



As N approaches ∞, the expression over the first under brace approaches 1; the second remains constant since N does not appear in it at all; the third approaches e−λ , and the fourth expression approaches 1. Consequently the limit is

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λk e−λ k!, which is the probability function of a Poisson distribution, with λ a random variable. More generally, whenever a sequence of binomial random variables with parameters N and pN is such that limN →∞ N PN = λ, the sequence converges in distribution to a Poisson random variable with mean λ. The number of new partners of each person in a given time step is determined by sampling from a Binomial distribution on each day in the simulation (section 2.5.). The Binomial distribution is parameterized with the set of rates of forming various types of new partnerships with another given person in the population on each day, as obtained from the survey, the rates for which are given in tables 3 to 7. For each person, there is a chance to form new steady relationships with individuals of any age, and new casual relationships with other individuals from all four age groups, according the social rules presented in the previous subsection. Once the number of new partners of a person is determined from sampling the Binomial distribution, another procedure is run to search for the newly acquired partners of the person. These new partners are chosen randomly from among all individuals of the opposite sex and in case of new casual partners, according to the age-structured mixing matrix (tables 4 and 5).

2.8.

Relationship Duration

For each newly formed relationship, the duration of the relationship is specified by sampling from a Weibull distribution, with different scale parameters and location parameters for both types of relationships, steady and casual. The Weibull distribution has the following formula with two controlling parameters γ and α: γ

f (t) = γα × tγ−1 e−αt , t > 0, γ > 0, α > 0.

(4)

The response from the survey are censored because they report time of last sexual contact, not necessarily the time that the partnership was ended. If it is known which data points in a survey are censored and which are not, then there exist statistical methods to estimate uncensored distributions. But in our case, there is no way to know which relations are finished at the time of the survey. So to estimate the duration distribution, we assumed

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Table 8. Characteristic parameters for the relation duration distributions Type of relationship γ α

Steady 0.0463 0.839

Casual 0.149 0.649

Table 9. The average relation duration Type of relationship Survey Simulation

Steady 10.5 years 8.3 years

Casual 400 days 410 days

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that if the time since the last reported sexual contact was less than 22% of the time since the casual partnership started, that the partnership was ongoing. Otherwise it was over. The percentage for steady partnerships was 13 %. These percentages were chosen to allow for a sufficient number of uncensored data in the survey. Using these assumptions, we used SAS to fit the survey data to a Weibull distribution and hence obtain estimates of the Weibull distribution parameters (table 8). Duration distribution differs by type of relationships (steady or casual), (table 8). The estimated duration of casual relations for those who are married is only slightly different than the duration of casual relations for single people. Hence, in the simulation, the marital status of an individual who has acquired a casual partnership does not influence the duration of the casual partnership and the parameters values of γ and α are always used for the duration. Table 9 compares the mean durations of censored relations in the simulation at equilibrium to the (censored) durations of relations reported from the survey. The censored simulation duration is the time between the start of the relationship and some arbitrary sampling time t, when the model individuals are asked for the duration of their relationships, in common for all relations of current individuals. It means relationships of those who have died before time t, are not taken into account; this is in order to have comparable values from the simulation and the survey. The reported and simulated values are similar. In the simulation the relation duration for the steady type is less than the duration of same type in the survey. This seems natural since steady durations are long enough that one of the partners might die before the expected duration and it makes the duration less than what was expected. This can also happen for the casual relationships, but it is infrequent since the casual relationships are much shorter than the average life time of a model individual. On the other hand, since casual durations for married individuals (those who are in one or more steady relationships) obey the same distribution for casual relationships of single individuals (in reality it is usually less than casual duration for single individuals), the mean casual duration in the simulation is slightly larger than what is found in the survey.

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Fatemeh Jafargholi and Chris T. Bauch Table 10. Average number of new relationships formed in last 12 months Type of relationship In the survey From the simulation

New casual relationships 2 1.4

New steady relationships 0.15 0.07

Table 11. Percentage of condom users observed in the survey, by the type of relation Steady 14%

2.9.

Casual 44%

Mean Number of Oartners

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The mean number of partners at a given time is not a model parameter itself, since it is the outcome of formation and breakup rates. However, we can compute the mean number of partners as a test of agreement between the model and the survey data. Table 10 shows how the agreement between the mean numbers of newly formed casual/steady partners in the last 12 months in the simulation is roughly comparable with the same parameters in the survey. As was mentioned before, in subsection 2.2., the model individuals in the population are between 15 and 54 years old. The population is heterosexual and there are two kinds of relationships, Steady and Casual, as discussed previously. After allowing 40 years for network equilibration, the disease is introduced to the completely susceptible population: a number of randomly selected individuals are infected. In the simulation, every infected individual transmits disease to his/her partners with probability dependent upon disease, behavioral, and gender specific characteristics, as described in the following sections.

2.10.

Condom Users

Condom use is generally low among young people, including those at highest risk of HIV transmission in Eastern Europe and Central Asia. According to one survey in the Russian Federation, fewer than half of teenagers aged 16-20 used condoms when having sex with casual partners. The percentage of sex workers reporting consistent condom use has seldom topped 50%, while, among injecting drug users, fewer than 20% on average report consistent condom use (8). In the simulation, condom users are assigned randomly according to their gender. Simulated individuals do not change their behavior over time, and a condom user uses condoms consistently and correctly. According to the survey, the ratio of female condom users to male condom users is 0.69 and the number of people who use condoms in casual relations are more than those who use condoms in their steady relations (table 11). The model is parameterized to have the same proportion of condom users, and ratio of male to female users, as in the survey. We also make the reasonable assumption that people who use condoms in their steady relations use it in their casual relations as well, but the converse does not hold. Each individual is recruited a condom user or a non-condom user with some probability.

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A newborn male individual will be assigned as a condom user for both kinds of relation with probability (0.14 × 1.18 = 0.16) and otherwise he is a condom user only in his casual relations with probability ((0.44 − 0.14) × 1.18 = 0.35). For a newborn female individual these probabilities are (0.14 × 0.82 = 0.11) for both types of relationship and ((0.44 − 0.14) × 0.82 = 0.25) for only casual relations. The probabilities 0.14 and 0.44 are, respectively, the percentage of steady condom users and casual condom users in the survey. Assuming all steady condom users are also casual condom users, (0.44 − 0.14) is the percentage of condom users for only casual relations but not for steady relations. The 0.82 = 0.69, determines the ratio of female condom users and male condom users. quantity 1.18 Before starting the process of disease spread, condom users are chosen as described, and after that, any newly recruited individual will be a condom user, with a probability the same as the percentage of condom users in population. When someone is recognized as a condom user for a given type of relation, the probability of infection transmission, through that type of relationship, is zero for him/her.

2.11.

Epidemiological Parameters

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Each individual is in one of several possible disease states: susceptible, infected and also recovered for the case of curable diseases with immunity (section 1.6.). The first epidemiological parameter is the duration of the infection D, i.e., the time between infection and recovery or death. In the simulation, it is assumed that HIV infected individuals have the same death rate as susceptible individuals, although they die from AIDS ten years after being infected. And it is also assumed that the duration of infection for Gonorrhea is 10 days for infected men and 20 days for infected women, after which they become susceptible once again (28). The second epidemiological parameter is the rate of transmission from an infected person to a susceptible partner (denoted by λ ). This is estimated as the probability of transmission per coital act times average number of coital acts per day.

2.12.

Probability of HIV Transmission per Coital Act

In the case of HIV, there are three stages of infection (section 1.1.). The probability of transmission per coital act for each stage can be estimated from the literature (4) and (20). In table 12), which is reported by Wawer et al., (4), many stages for HIV infection have been listed that we summarize in three stages. The probability of HIV transmission per coital act in table 12 is much higher in the first 2.5 months than at other times. We use this value for the first stage of HIV infection in the simulation, which lasts 5 months. The rates for the period of time between the sixth month since infection and 16 months before death are likewise close together, so we take the average of the probabilities during these periods, which is 0.0007, as the second stage of HIV infection in the simulation which lasts 24 months. And the probability of transmission for the last 24 months in the simulation is the average of the reported values of the last 25 months in Figure 2.12.. In the Wawer et al., study, no significant association was seen between the probability of transmission and the sex of the HIV positive partner (4). However, for other STIs, it is believed that the probability of transmission is gender specific, and this will be applied in this simulation.

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Figure 6. Table from (4) : HIV Transmission rates per coital act, for details see (4)

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Table 12. The probability of transmission of HIV per coital act, used in the simulation Stage Male to Female Female to Male

2.13.

First 0.0082 0.0041

Second 0.0007 0.0003

Third (AIDS) 0.0036 0.0018

Probability of Gonorrhea Transmission per Coital Act

In the case of gonorrhea (section 1.2.) which is a curable STI, the probability of transmission per coital act is also available from the literature (32). In reality, a few gonorrhea infected individuals may be immune after treatment, but this is not taken into account in this simulation. In the simulation, gonorrhea infected individuals become susceptible after Dg days (duration of infection of gonorrhea), which means we need a SIS model for simulating spread of gonorrhea in the population. The integration of HIV/AIDS and other STIs into a single model is important because HIV transmission is facilitated if a partner carries another STI (20). It was assumed that the probability of HIV transmission (table 12) is increased fivefold per coital act in individuals with a gonorrhea infection (20, 37).

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Table 13. The probability of transmission of Gonorrhea per coital act, used in simulation Gender Male to Female Female to Male

Rate 0.60 0.25

Table 14. Frequency of coital act per day Age group Steady relations Casual rel. for single people Casual rel. for married people

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2.14.

1 0.61 0.19 0.07

2 0.57 0.2047 0.1045

3 0.48 0.17 0.13

4 0.48 0.23 0.12

Probability of Coital Act per Day

To get the frequency of coital acts per day for each age class and each type of relationship, data from the Russian survey were used: questions 20, 39 and 58, which time since the last coital act in his/her relationship, were used. The average inverse of these answers is taken to be the frequency of coital acts per day for the different age groups. The frequency of coital act per day times the probability of disease transmission per coital act gives us the probability of disease transmission per day. The frequencies seem reasonable (table 14). In the simulation, these frequencies have been changed according to the individuals’ membership in the core or non-core groups. Two partners from the core group should have more coital acts per unit time than two partners from the non-core group. When both partners are core group members, the rate of coital acts (reported in the table 14) was multiplied by 2. When one partner is from the core group and one is from the non-core group, the coital rates are multiplied by 1.1. When both partners are from the non-core group, the coital rates are multiplied by 0.8. These multipliers do not have a statistical basis and have been chosen to make a reasonable difference in sexual behavior levels between different groups.

2.15.

Spread of Disease

The partnership dynamics and network structure, transmission rate, duration of infectiousness, and known proportions of condom users completely specify the transmission dynamics of the disease and hence the transmission process of the disease. When the number of infected individuals is small, the disease may disappear through stochastic fade-out. There is also a possibility that the first infected persons will be individuals in a core group, in which case the percentage of infected people grows much faster than usual. In the simulation, once the partnership network has reached equilibrium, HIV is introduced in two non-condom users of core group and Gonorrhea in five non-condom users of core group. Considering these assumptions, the initially infected people pass their diseases to their partners by some rate that is related to the stage of the infection, their type of relations and their

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age (the last two parameters affect the frequency of coital acts). As for other infectious diseases, the course of STI epidemics are determined by a balance between an increase in the size of the group of infected persons and a decrease in the size of the group of susceptible persons, respectively increasing and decreasing the risk of new infections.

3.

Results

In the following subsections, we first present simulated epidemic curves for parameter values taken from the survey or other sources of data (”real-world” parameters). We then present simulated epidemic curves for hypothetical parameters values (”experimental” parameters) in order to understand the effect of changes in condom usage, or changes in core group members, for instance.

3.1.

HIV Prevalence in Population

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Figure 7 shows the epidemic curve of the number of HIV+ individuals for the real-world parameters. This plot is the average result of 5 runs. The incidence grows for about 30 years to a maximum and then it decreases to a local minimum (perhaps even zero). This prevalence decrease can be the result of a decrease in number of susceptible non-condom users, i.e., a decrease in the effective reproductive number (section 1.6.). When a large number of infected individuals die because of AIDS, the percentage of condom users increases and in this simulation, condom users are supposed to be immune. The simulated epidemic curve is a ‘classic’ epidemic curve for a disease introduced into a fully susceptible population.

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Figure 8 shows the HIV epidemic curve over 30 years from the beginning of the spread of HIV. This plot, figure 9 and figure 10 are the average result of 40 runs. The prevalence grows for 11 years to the maximum, then it decreases for a while, and then it grows again. The decrease after 11 years is due to the assumption that individuals die after 11 years of being infected by HIV (section 1.1., (4)). In reality, we note that this infectious period has some variance around the mean. From figure 8, we see that HIV prevalence increases more sharply in the first years of HIV introduction to the completely susceptible population, before entering a phase of slower, apparently linear growth. This may partly be due to the fact that the first infected individuals are members of the core group, thus the disease tends to spread at first only among core group members, who are sexually more active. It may also reflect the fact that the probability of HIV transmission is highest in the first stage of infection.

Figure 8. HIV prevalence, over 30 years, for real-world parameters. In figure 8, HIV prevalence increases starts to rise more rapidly at about the ninth year after the introduction of infection, which is just before the start of AIDS-related deaths. At this time, those who were infected in the first years of HIV introduction have entered the last stage of infection, AIDS. In this stage, the infection transmission probability is higher than in the middle stage, so they infect more susceptibles and prevalence climbs more rapidly up to the eleventh year. As can be seen in the graph, after 14 years the HIV prevalence among adults is 1.4% of the model population. The actual HIV prevalence among Russian adults was 1.1% of population in 2001, 14 years after the first case of HIV-positive in USSR (34). Hence, the model prevalence is close to the actual prevalence. It should be noted that the model does not take into account the contribution to prevalence from the drug-addicted population, which constitutes a large proportion of HIV prevalence in Russia. The difference between predicted and observed prevalence could be due to the non-inclusion of IV drug users transmission in the simulation model, or the assumptions behind the core group in the model (either in the

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size of the core group or its relations to the non-core group), or the result of assumptions made about the relation duration for core group members (section 2.8.).

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Figure 9. HIV prevalence among the core group. Figure 9 shows the epidemic curve of HIV-infection among the core group for the realworld parameters, in which the core group consists of 2.5% of the population. The HIV prevalence is much higher in the core group than the whole population. In the core group, the prevalence can be as high as 45% due to their higher sexual activity. Comparing figure 8 and figure 9 we can see the strong effect of HIV prevalence among the core group on the whole population; HIV prevalence in the whole population increases and decreases as prevalence in the core group increases and decreases. The effect of the core group on the HIV prevalence of the whole population is discussed more in section 3.7.. Figure 10 shows the epidemic curve of HIV-infection among the non-core group. The HIV epidemic grows more slowly in the non-core group compared to the whole population, but it also seems to have monotonous growth, which is alarming.

3.2.

Gonorrhea Prevalence

Gonorrhea is a curable disease which does not cause death. So in the simulation, no one dies because of gonorrhea and each infected individual recovers after a while, becoming susceptible to the disease once again (a SIS model). Figure 11 shows a gonorrhea epidemic over 80 years, after a 10-year initialization. The smaller graph inside figure 11 shows the gonorrhea prevalence in the period of initialization and 5 years after that. This plot, figure 12 and figure 13 are the average result of 20 runs. The gonorrhea prevalence fluctuates between 2.5% to 4% and seems to reach an equilibrium. These stochastic fluctuations in the prevalence of gonorrhea are a common feature of infectious disease systems and have been studied thoroughly. Gonorrhea prevalence in the core group, figure 12, reaches 50%,

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Figure 10. HIV prevalence among the non-core group. while among the non-core group, figure 13, prevalence remains between 1.5% and 3%. Gonorrhea rates among adolescents are reported to be about 600 per 100,000 in the Russian Federation (35) or about 0.6%. For some STIs, such as gonorrhea, patients are more likely to avoid the stigma attached to the state service and to seek treatment in the growing private sector or, alternatively, to treat themselves (36). Based on studies by Waugh et al., it is believed that only 1 in 20 cases of gonorrhea is reported in eastern Europe (36), hence the actual prevalence of gonorrhea is likely to be much higher than the reported 0.6%, and probably exceeds the model predicted value of 1.5% to 3%. If the estimate of 20:1 under-reporting for Eastern Europe is close to the actual under-reporting rate for Russia, the simulated prevalence in the whole population is approximately a third of the actual gonorrhea prevalence of the Russian population. Distinguishing asymptomatic and symptomatic gonorrhea infected individuals in the model could improve the result.

3.3.

Interaction between Gonorrhea and HIV

It is believed that the presence of STI infection increases the likelihood of HIV transmission by two to five times (37) (38) (39) (40). In the simulation, it is assumed that infectivity of HIV is increased fivefold per coital act by gonorrhea. Since Wawer et al., have not mentioned the presence of gonorrhea cases in their study (4), it is assumed that the HIV transmission rate to a gonorrhea infected individual is five times the values reported by Wawer et al., for each stage of HIV infection (table 12). Of course this assumption increases the HIV epidemic in the simulation and AIDS death also affects the gonorrhea epidemic. In the simulation, gonorrhea is first introduced to the population and then after 10 years, when it reaches the equilibrium, HIV begins to spread. Figure 14, the HIV prevalence, and figure 15, the gonorrhea prevalence, are the average result of 30 runs of this simulation,

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Figure 11. Gonorrhea prevalence in the population (%).

Figure 12. Gonorrhea prevalence among the core group.

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Figure 13. Gonorrhea prevalence among the non-core group.

Figure 14. HIV prevalence, over 50 years, in a population where gonorrhea is also present

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while both diseases exist in the population. As expected, the HIV prevalence is increased to 6.5% at the maximum point. It reaches a higher maximum than when gonorrhea is absent (figure 7); it also has a sharper decrease. Since a larger number of non-condom users get HIV, when AIDS deaths start, suddenly the number of non-condom users decreases and the percentage of condom users in the population increases. Hence, the effective reproductive number decreases and so the HIV prevalence decreases.

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Figure 15. Gonorrhea prevalence, over 50 years, in a population where HIV is also present. On the other hand, since gonorrhea infected individuals are more at the risk of HIV infection, when the HIV positive population decreases due to AIDS deaths, gonorrhea prevalence decreases to zero. During the first ten years, the period before AIDS deaths, gonorrhea prevalence shows the same behavior as when there is no HIV infectives in the population (figure 11). When AIDS deaths occur, a sharp drop in the gonorrhea prevalence appears.

3.4.

Age Structure of Infected Population

In the simulation, sexual behavior varies according to age. In particular, the number of coital acts per day and the rate at which new partnerships are formed are age-dependent. Age structure is an important feature of real populations, impacting how diseases spread and how they are controlled. The speed with which an epidemic invades a population can vary depending on the age of the first infected individuals, since mixing and sexual behavior are age-dependent. Therefore, it is important to study the effects of age structure on epidemic spread. Studying age structure will also allow us to see how the burden of disease varies among age groups. In order to do so, the simulation is run with different initial conditions for the age class of the first infected individuals. In the first (respectively second, third, fourth) scenario, the first HIV-infected individuals are from the first (respectively second,

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third, fourth) age class. Gonorrhea is also introduced into the population, but not with agedependence. As in section 3.3., the simulation is run with only gonorrhea until it reaches the equilibrium (to simulate the endemic state of gonorrhea in Russia), and then HIV is introduced. The results are presented in figure 16, which shows the average relative HIV prevalence during the tenth year after introduction (average of 4 runs, error bars are one standard deviation). This is the last year before the start of AIDS deaths. In all scenarios, individuals of the second age class have the highest relative prevalence of HIV and the oldest groups have the lowest relative prevalence of HIV infection. However, this pattern does not seem to vary much across the four scenarios (although there is no significance testing in these plots). We should keep in mind that during these ten years, individuals grow up and they leave their age class to the next age class. Also, the oldest individuals at the time of HIV introduction will have left the sexually active population within ten years. Individuals who were in the third age class are either members of the fourth age class or dead. The older a member of the population is, the higher the death rate (table 1). It is not clear if the differences between scenarios are significant. Also, the largest difference between age classes appears to be between the second age class and all other age classes. The very high prevalence in the second age class requires explanation, which we attempt in the following paragraphs.

Figure 16. Age structure of the HIV-positive population, for different scenarios. Since the rates of forming new casual relationships in the single population (ρM 0,Cjk and are much higher than the rates of forming new casual relationships in the married F M population (ρM i>0,Cjk , ρi>0,Cjk ) and the rate of forming new steady relationships (ρS and ρFS ) (section 2.5.), and since casual relationships have shorter durations (section 2.8.), it is likely that casual relationships have the strongest effects on spread of STIs (24), contributing disproportionately to the rapid spread across all age classes. Disease transmission rates for HIV are higher from males to females in comparison to females to males. This could explain why the HIV prevalence among females is higher in the simulation, and perhaps in the HIV-positive population whom are infected through sexual contacts. Likewise, the gonorrhea transmission rate from an infected man to a susρF0,Cjk )

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Table 15. Ratio of the number of HIV infected females to the number of HIV infected males

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Disease HIV ( in the absence of Gonorrhea) HIV (in the presence of Gonorrhea) Gonorrhea (in the presence of HIV)

ratio 1.4 1.6 1.7

ceptible woman is higher than the transmission rate from an infected woman to a susceptible man (section 2.13.). Moreover, an individual with gonorrhea is five times more likely than a healthy person to get HIV. As a result, women are even more at risk of HIV infection in a sexual network with the presence of gonorrhea. Table 15 compares the HIV prevalence in both genders; the results presented in the table are the average of the first ten years after HIV introduction. There is also a possibility that men, especially young men of the first age class, exaggerate the number of their partners and this may make the male ρ values less reliable. Also, since women have higher HIV rates for the reasons described above, their choice of partners has a bigger effect on the age structure of HIV prevalence than the male choice of partners. In the female table of partnership formation rates (table 5), the second age class values are larger than the other age classes and also they form partnerships with a wider variety of ages. By comparison, the male values of partnership formation rates (table 4) reveal an unusually large value for the relationships between the youngest females and the youngest males (ρM 0C,11 ). The ρ values for men in the second age class are higher than the third and fourth age classes, and similar to the values in the first age class. However, the ρ values in the second age class are spread more evenly across the age groups than in the first age class, so again we observe that this age group should experience a higher infection rate than the first age group, which tends to mix only with itself. For example, for both genders we have j=4 X j=1

ρ0C,1j ≈

j=4 X

ρ0C,2j ,

j=1

but we see that ρ0C,14 = 0. The tendency of both males and females in the second age class to form relationships with a broad spectrum of ages makes a high potential for infection in the second age class, regardless of the age of initial infectives. The age-dependent frequency of coital acts per day (table 14, reproduced below in table 18) may also be important in age-related differences. This value times the disease transmission probability per coital act gives us the disease transmission rate per day, which is important in the disease prevalence. Among the first three age classes, the frequency of coital act per day is higher for individuals in the second age class. Although the frequency of coital acts per day for the oldest group is higher than others, it seems the effect of death rate and the reduced partnership formation rate is stronger than other factors for this age class. In Russia, the most heavily HIV affected groups are young adults aged 20-30 years old (41, 42), which are the second half of our first age class and the first half of the second

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Table 16. Values of ρM 0C,jk for casual partnerships of single men for different ages, per year. Age class 1 2 3 4

1 2.31691 1.65878 1.14577 0.33793

2 0.18770 0.92696 0.65473 0.42241

3 0.01173 0.04879 0.25721 0.59138

4 0.00000 0.01626 0.04677 0.33793

Table 17. Values of ρF0C,jk for casual partnerships of single for different ages, per year

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Age class 1 2 3 4

1 1.18035 0.35783 0.10390 0.00000

2 0.53435 0.92283 0.41564 0.14242

3 0.04785 0.52733 0.72736 0.56970

4 0.00000 0.07533 0.38100 0.85454

age class. Hence, the simulation is consistent with real-world age-specific epidemiological patterns. Another interesting observation is that the total HIV prevalence in the whole population 10 years after introduction declines monotonically with age class, being highest when introduced in age class 1 and lowest when introduced in age class 4 (results not shown). Due to the death rate, older infectives have less time to infect their partners and also have fewer casual partners. Because mixing is primarily assortative (within the same age class), the age class in which the virus is initially introduced has a large impact on the course of the epidemic, even 10 years after introduction.

3.5.

Simulation Results for ”experimental” Parameters

In the previous section, we studied simulation output for the real-world parameters in order to gain some insight into the observed patterns of spread. In this section, we vary key parameters in an experimental approach in order to understand the patterns of spread in alternative situations. First, we vary the number of condom users in the population at large (section 3.6.). Second, we vary the number of condom users in the core group only (section 3.6.). Third, we vary the size of the core group and study the impact on disease prevalence (section 3.7.).

Table 18. Frequency of coital act per day for casual relationships of single population Age group Frequency of coital act per day

1 0.19

2 0.2047

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4 0.23

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3.6.

Fatemeh Jafargholi and Chris T. Bauch

Influence of Percentage of Condom Users in Population

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To study the effects of the number of condom users on spread of STIs, seven different scenarios are designed. In these scenarios, the percentages of condom users in the population at large are 0.25, 0.50, 0.75, 1.25, 1.50, 1.75 and 2 times the real percentage of condom users. Figures 17 and 18 show respectively the mean HIV prevalence in the first 15 years and the maximum HIV prevalence in the population over 300 years. These two plots are the average result of 5 runs. In general, there is an almost exponential relationship between the number of condom users and the reduction in maximum of HIV incidence.

Figure 17. The effect of changes in the proportion of condom users relative to the empirical case in the whole population (horizontal axis) on average HIV prevalence over the first 15 years (vertical axis). In another experiment, it is supposed that the number of condom users in the non-core group is the same as the real-world simulation, and changes in the percentage of condom users in the core group alone are studied. Seven scenarios are studied, where the percentages of condom users are 0.25, 0.50, 0.75, 1.25, 1.50, 1.75 and 2 times of the real percentage of condom users for core group members. Figure 19 shows the mean HIV prevalence in the first 15 years (top panel) and the maximum prevalence of HIV-infected individuals (bottom panel) in the population over 300 years, against the percentage change in condom users in the core group only. These plots are the average result of 5 runs. Again, there is an exponential relationship between the coverage level in the core group and reduction in the maximum of HIV incidence. Naturally, the prevalence decreases more quickly by increasing the number of condom users in the whole population, than by increasing the number of condom users just in the core group. But, it is interesting that this difference is not very large. This means that changes in condom-using behaviour in the core group have a massive impact on HIV prevalence, relative to condom-using changes in the non-core group. Since the core group is much smaller, the total number of individuals who must change their behaviour is also smaller. A large impact can be achieved by changing the behaviour of a very small number of individuals.

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Figure 18. The effect of changes in the proportion of condom users relative to the empirical case in the whole population (horizontal axis) on maximum HIV prevalence in the first 300 years (vertical axis). This insight can be a key for prevention strategies.

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3.6.1. A Simple Model for Prevention Strategies The fact that almost all of the reduction in HIV prevalence due to increased condom usage (figure 17) is due to increased condom usage in the core group (figure 19) suggests that prevention budgets should be focused on changing behavior in members of the core group. Suppose a budget of Y rubles is available to be spent in this matter. Let xp denote the average cost for a member of the population (core or non-core) to be converted from non-condom user to condom user. This cost may include any activities that convince an individual to consistently use condoms in his or her sexual contacts. Similarly, let xc denote the average cost for a core group member to be converted from non-condom user to condom user. Let αp and αc denote the proportion of condom users in the whole population and in the core group respectively, that are created by these programmes. Finally, let N denote the size of the whole population and let C denote the size of the core group only. If the budget is spent on the whole population (respectively, on the core group), the total number of individuals who change their behavior would be αp × N (respectively αc × C). Then, since a budget of Y rubles will produce Y /xp (respectively Y /xc ) changed individuals, we clearly have: αp =

Y Y , αc = xp N xc C

(5)

If xp and xc are of the same order of magnitude, (perhaps xp is smaller than xc ), then x N we have that xpc C >> 1, since C/N > αp , so the same Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 19. Top Panel: The effect of changes in the proportion of condom users relative to the empirical case in the core group only (horizontal axis) on average HIV prevalence over the first 15 years. Bottom Panel: the effect of changes in the proportion of condom users relative to the empirical case in the core group only (horizontal axis) on maximum HIV prevalence in the first 300 years.

expenditure creates a larger percentage change in condom using behavior in the core group than in the general population. As depicted in figures 17 and 19, a given percentage increase in the number of condom users in the core group makes almost as big an impact on HIV prevalence as the same percentage increase in the number of condom users in the whole population! So, focusing on the core group provides as much leverage as focusing on the general population, both from epidemiologic and economic viewpoints. As a result, focusing on the core group seems more reasonable when the goal is increasing the number of condom users.

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Figure 20. Influence of the size of the core group (horizontal axis) on average HIV prevalence over the first 15 years (vertical axis).

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3.7.

Influence of the Size of Core Group in Spread of HIV

The core group has a strong impact on the spread of STIs (section 2.6.). The size of the core group could be a determinant in controlling the spread of STIs. For instance, the core group size could be changed through educational campaigns. In this set of simulations, the size of the core group is varied and the impact on HIV prevalence is studied. The size of the core group is varied from 0.75% to 5% of the total population size. Figures 20 and 21 show respectively the mean HIV prevalence in the first 15 years and the maximum prevalence of HIV over 300 years in the whole population. These plots are the average result of 5 runs. The larger the size of the core group, the higher the prevalence. The maximum HIV prevalence appears to increase exponentially (or perhaps linearly) with the size of core group. Relatively small changes in the size of the core group have a very large impact on maximum HIV prevalence. The maximum HIV prevalence increases sixfold as the size of the core group is varied from 1.5% to 5%. The impact of the core group on the whole population is clear in this and previous simulations. Controlling the size of core group without changing their casual behaviors, affects the HIV prevalence in the whole population. In countries with large core groups or countries where having several simultaneous partners is a widespread cultural practice, and where condom usage is not widespread, HIV prevalence is extremely high. For instance, in Sub-Saharan Africa, individuals have several extramarital sexual relationships at a time and they do not use or they do not have access to condoms, so they have a highly connected sexual network and the HIV prevalence is as high as 40% in certain countries (UNAIDS). These observations suggest that HIV could fade out if the core group transmission rate is low enough. The importance of condom usage and core group size are apparently so significant from these simulations that there should be more studies on them.

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Figure 21. Influence of the size of the core group (horizontal axis) on maximum HIV prevalence in the first 300 years (vertical axis).

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4.

Discussion

In this chapter, an age-structured heterosexual partnership network with a core group was simulated based on data from a sexual behavior survey. The purpose was to study the spread of STIs in Russia, specifically, HIV and gonorrhea. In order to make the model more representative of the Russian population, the Russian Longitudinal Monitoring Survey and Russian demographic data from the United Nations yearbook were used. The simulation was a stochastic individual-based network similar to those of other STI modeling studies, where nodes are individuals and links are partnerships. Mortality and recruitment of new individuals into the sexually active population were assumed to occur at some constant rate. There was no immigration, so all new susceptible individuals arrived in the sexually active population by joining the youngest age class. Partnerships were defined either as steady (long-term and often exclusive) or casual (short-term and often concurrent with other casual partnerships). The distribution of the number of partners per person was Poisson, and the duration of relations were described by a Weibull distribution. Age-structure was an important feature of the model, and certain sexual behaviors were age-dependent, such as the rate of forming new casual relationships, and selection of new partners. Individuals tended to form partnerships with other individuals in the same age class, with interesting exceptions. Individuals were also assigned to either a core group or a non-core group (general population). The members of the core group had higher levels of sexual activity and hence were at higher risk of getting STIs. Individuals were also assigned to be either a condom user or a non-condom user. Condom users were assumed to be immune to infection. The percentage of condom users, for both genders, was designed to be the same as the reported percentage of condom users in the Russian survey. For disease transmission, an age-gender-disease specific transmission rate was defined as the probability of transmission per coital acts (from non-Russian studies) multiplied by the number of coital act per day (from the Russian survey).

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The most novel aspect of this work was that real-world data were applied to make the network simulation as close as possible to the behavior of a real population. Most previous modeling studies have made idealized assumptions. Although lack of data for certain aspects of the model and computational limitations in memory and time required us to make some unrealistic assumptions, the model still reproduces many important characteristics of a real society. Like any model, this model makes a number of assumptions that may limit its validity and accuracy for the Russian population or applicability to other populations. We describe some of these limitations in this paragraph. The size of the simulated population, which fluctuated according to birth and death events, was about 1500 individuals. Increasing the number of individuals might have improved the signal to noise ratio, but also would have increased the required memory and run time for the simulations. Considering these limitations, 1500 individuals in the simulation was felt to be sufficiently large to distinguish interesting deterministic patterns. The model also assumed that individuals older than 54 years have safe sexual behavior and do not play a role in the spread of STIs, which is perhaps somewhat idealistic. For simplification, it was also assumed that the relation duration for the core group members is the same as for the non-core group members, which is probably not true. In fact, core group members probably have shorter casual relationships. Naturally, if the core group has the same relationship duration as the non-core group, the disease prevalence and rate of spread will appear to be higher than is actually the case. Another assumption which could have influenced the results involves the core group definition and its relation to the non-core group. We did not have enough data about the membership of the core group or mixing between these groups; as a result, we made some arbitrary choices about how many individuals are in the core group and how partnerships are formed between core and non-core individuals. Although the assumptions are reasonable, we do not know how close they are to the real behavior of Russian society. We also assumed that when someone starts as a member of the core group, they remain a member of the core group for their entire life, and of course in reality individuals may enter or leave the core group. Another important limitation has been alluded to, namely, the non-inclusion of HIV infection as the result of intravenous drug users (IDU). Unlike the case in many countries, much HIV infection in Russia is a result of IDU transmission, and this is simply not accounted for in the model, which therefore will tend to underestimate HIV prevalence and rate of spread. Finally, the model assumes that individuals do not change their behaviour once infected with HIV, or once HIV becomes prevalent in the population. Most models also make this erroneous assumption, however, modellers are taking the first steps to incorporate human behaviour into their models. Again, this is due to lack of data on human behavior. One of the most important motivations for simulating the spread of STIs is examining the impact of different prevention strategies, and especially determining which ones are most effective from both an epidemiologic and economic point of view. When a sexual network model captures the most important characteristics of a population, then it is ready to be used to assess prevention strategies and to inform policy. Future work could extend this model to allow it be used for such purposes, for instance by including homosexual and injecting drug users populations, and by finding better data on core group members and their interactions.

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5.

Fatemeh Jafargholi and Chris T. Bauch

Conclusions

An age-structured heterosexual network simulation for the Russian population has shown that (i) the presence of gonorrhea in the population significantly increases the prevalence of HIV, and hence gonorrhea prevention is also HIV prevention, (ii) HIV prevalence may be higher in the second age class (25-34 years old) due to their slightly higher rates of partnership formation and especially their stronger tendency to form partnerships with individuals from all age classes, not just their own, (iii) increasing the proportion of condom users in the core group has almost the same effect on STI prevalence as the same proportional increase of condom users in the entire population (a much larger number of individuals), and hence (iv) focusing intervention on core groups provides much greater reductions in disease burden for a much lower cost, and finally, (v) the size of the core group also has a very significant influence on STI prevalence. This chapter shows that realistic sexual network models can be a useful aid in understanding STI transmission patterns and determining the optimal methods for their control.

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[2] National institute of allergy and infectious diseases, A brief history of the emergence of AIDS., http://www.niaid.nih.gov/publications/hivaids/4.htm, 2003. [3] F.J. Palella, K.M. Delaney, A.C. Moorman, M.O. Loveless, J. Fuhrer, G.A. Satten, D.J. Aschman, S.D. Holmberg, Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, HIV Outpatient Study Investigators. New England Journal of Medicine 338, 13, 853–860, MAR 26, 1998. [4] M.J. Wawer, R.H. Gray, N.K. Sewankambo, D. Serwadda, X. Li, O. Laeyendecker, N. Kiwanuka, G. Kigozi, M. Kiddugavu, T. Lutalo, F. Nalugoda, F. WabwireMangen, M.P. Meehan, T.C. Quinn, Rates of HIV-1 Transmission per coital Act, by stage of HIV-1 Infection, in Rakai, Uganda. Journal of Infection Disease, 191, 9, 1403–1409, MAY.1.2005. [5] M.A. Johnson, C.I. Lipman, R.W. Baker, An Atlas of Differential Diagnosis in HIV Disease. Taylor & Francis, page 7, 2004. [6] A. Lazzarin, A. Saracco, M. Musicco, A. Nicolosi, Man-to-woman sexual transmission of the human immunodeficiency virus. Risk factors related to sexual behavior, mans infectiousness, and womans sesceptibility. Archives of Internal Meedicine 151, 12, 2411–2416, 1991. [7] Internet website, Wikipedia. http://www.wikipedia.org/en [8] T. Dougherty, UNAIDS: AIDS epidemic update., December, 2003. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[9] M.S. Cohen, J.G. Cannon, A.E. Jerse, L. Charniga, Human Experimentation with Neisseria gonorrhoeae: Rationale, Methods, and Implications for the Biology of Infection and Vaccine Development. Journal of Infectious Diseases, 169, 3, 532, 1994. [10] V.V. Pokrovski, N.N. Ladnaya, E.V. Buratzova, Moscow, Russian Federation: Russian Federal Center for Prevention of AIDS. HIV Infection Bulletin, 2002. [11] T. Dougherty, IV epidemic in Russia. The Lancet, 366, 9490, 983–984, 2005. [12] Internet Website, Centers for Disease Control and Prevention (CDC). http://www.cdc.gov/ [13] M.W. Adler, A.Z. Meheus, Epidemiology of sexually transmitted infections and human immunodeficiency virus in Europe. European Academy of Dermatology and Venereology, 14, 370-377, 2000. [14] H.W. Hethcote, J.A. Yorke, Gonorrhea transmission dynamics and control. Lecture Notes in Biomathematics, Springer-Verlag, 1984. [15] K. Dietz, K.P. Hadeler, Epidemiological models for sexually transmitted diseases. Journal of Mathematical Biology, 26, 1–25, 1988. [16] H. Heesterbeek, R0 . Ph.D. thesis, Centrum voor Wiskunde en Informatica, Amesterdam, 1992.

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[17] C.T. Bauch, Moment closure approximations in epidemiology. Ph.D. thesis, The University of Warwick, 2000. [18] W.J. Edmunds, C.J. O‘Callaghan, D.J. Nokes, Who mixes with whom? A method to determine the contact patterns of adults that may lead to the spread of airbone infection. Proceedings of the Royal Society B, 264, 949–957, 1997. [19] W. Farr, Progress of epidemics. In second report of the registrar general for England, 1840. [20] J.N. Wasserheit, Epidemiological synergy. Interrelationships between human immunodeficiency virus infection and other sexually transmitted diseases. Sexually Transmitted Diseases, 19, 61–77, 1992. [21] O. Diekmann, J.A.P. Heesterbeek, Mathematical epidemiology of infectious diseases. Wiley Series in Mathematical and computational biology, 2000. [22] M. Morris, How do sexual networks affect HIV/STD prevention? 641–648, April 11, 1997.

AIDS, 11, 5,

[23] F. Liljeros, C.R. Edling, L.A. Nunes, H.E. Stanley, Y. Aberg, The web human sexual contacts. Nature, 411, 6840, 907-908, JUN 21, 2001. [24] C. Bauch, D.A. Rand, A moment closure model for sexually transmitted disease transmission through a concurrent partnership network. Proceedings of the Royal Society B, 267, 1456, 2019–2027, 2000.

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[25] R.C. Brunham, Core Gruop Theory: A Central Concept in STD Epidemiology. Venereology, 10, 1, 34–39, 1997. [26] A.C. Ghani, G.P. Garnett, Risk of acquiring and transmitting sexually transmitted diseases in sexual partner networks. Sexually Transmitted Diseases, 27, 10, 579– 587, 2000. [27] D.J. Watts, S.H. Strogatz, Collective dynamics of small-world networks. Nature, 6684, 440–442, 1998. [28] M. Kretzschmar, Y.T.H.P. Van Duynhoven, A.J. Severijnen, Modeling prevention strategies for gonorrhea and Chlamydia using stochastic network simulations. American Journal of Epidemiology, 144, 3, 306–317, 1996. [29] M. Morris, M. Kretzschmar, Concurrent partnerships and spread of HIV. AIDS, 11, 641–648, 1997. [30] R. Brunham, Core group theory: a central concept in STD epidemiology. Venereology, 10, 1, 34–39, 1997. [31] N.C. Giri, Introduction to Probability and Statistics. Marcel Dekker, second edition, page 61, 1993. [32] R.B. Rothenberg, M. Scarlett, C. Del Rio, D. Reznik, C. O‘Daniels, Oral transmission of HIV. AIDS 12, 16, 2095–2105, 1998.

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[33] M. Komsomolets, Russia First HIV Patient Tells Story of Alienation and Persecution. Mosenews, 01.12.2005. [34] Internet website, CIA world factbook. https://www.cia.gov/cia/publications/factbook/ [35] C. Panchaud, S. Singh, D. Feivelson, J.E. Darroch, Sexually transmitted diseases among adolescents in developed countries. Family Planning Perspectives, 32, 1, January/February, 2000. [36] M.A. Waugh, Task force for the urgent response to the epidemics of sexually transmitted diseases in eastern Europe and central Asia. International Journal of STD and AIDS, London, 10, 1, 60–63, 1999. [37] C. Van Vliet, E.I. Meester, E.L. Korenromp, B. Singer, R. Bakker, J.D.F. Habbema, Focusing strategies of condom use against HIV in different behavioral settings: an evaluation based on a simulation model. Bulletin World Health Organanization, 79, 5, 442–454, 2001. [38] P. Gibson, Risk, HIV, and STD Prevention. Focus, 18, 10, 1–5, 2003. [39] E.J. Erbelding, D. Stanton, T.C. Quinn, A. Rompalo, Behavioral and biologic evidence of persistent high-risk behavior in an HIV primary care population. AIDS, 14, 3, 297–301, 2000. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[40] B. Varghese, J.E. Maher, T.A. Peterman, B.M. Branson, R.W. Steketee, Reducing the Risk of Sexual HIV Transmission:Quantifying the Per-Act Risk for HIV on the Basis of Choice of Partner, Sex Act, and Condom Use. Sexually Transmitted Diseases, 29, 1, 38–43,2002. [41] M. Feshbach, HIV/AIDS in the Russian Military. UNAIDS Meeting, Copenhagen, Denmark, February 22-23, 2005.

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[42] A. Rakhmanova, G.V. Volkova, A.A. Yakouleu, L.N. Kryga, N.A. Chaika, V.R. Shelukhina, E.N. Vinogradova, HIV/AIDS epidemiological situation in St Petersburg, Russia, in 1999. International Conference of AIDS, Durban, Jul 9-14, 2002.

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 103-140

ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 4

M ALARIA C ONTROL : T HE R OLE OF L OCAL C OMMUNITIES AS S EEN THROUGH A M ATHEMATICAL M ODEL IN A C HANGING P OPULATION – C AMEROON Miranda I. Teboh-Ewungkem Department of Mathematics, Lafayette College, Easton, PA, 18042 USA

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Abstract Despite efforts to combat malaria, the disease remains a serious public health risk in the African regions where it is endemic. Amongst several factors, one that needs attention is the attitude by individuals in not completing malaria treatment thereby lowering their recovery rate and staying longer as infectious individuals. A Susceptible Exposed Infectious (SEI) differential equation model is used to explore the behaviour of the disease with variable host (human) and vector (mosquito) populations based on published data from Cameroon. With a base set of parameters, the basic reproductive number, R0 , is computed and the model realistically reproduces endemic stable equilibrium and show that R0 is high when the recovery rate is low. Moreover, when the contact rates between mosquitoes and humans are relatively high, a reduction in the recovery rate from 0.0014 to 0.0056, a value much less than clinically observed recovery rates, resulted in an 80% reduction in the stable endemic proportion of infectious human, a 71.4% reduction in the maximum proportion of infectious humans, and a 52.2% reduction in the stable endemic proportions of infectious mosquitoes. This information suggests that if local communities greatly increase their efforts in educating and informing the public of the risks involved in not completing malaria treatment, and the benefits of completing malaria treatment, a significant impact can be made in controlling the disease. Further results indicate that control measures designed to target the contact rates between mosquitoes and humans will be more efficient at reducingR0 and hence initial disease transmission, and also reduce the endemic infectious steady state human proportion. On the other hand, control measures designed to target the recovery rate will be more efficient at reducing the maximum proportion of infectious individuals, and for areas with very high contact rates and very low recovery rates, it will be more efficient at reducing the endemic infectious steady state human proportion. Hence in an endemic region as Cameroon, the results obtained signify that a good strategy for control, from the perspective of individuals and local communities,

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Miranda I. Teboh-Ewungkem should include a combination of education; enforcement of the completion of malaria treatment; use of prescribed non-resistant efficient drugs; reduction of contacts between mosquitoes and humans via the use of insecticide-treated bed nets and the use of mosquito repellent that are safe to use on humans; and reduction of the mosquito recruitment rate by minimizing breeding sites around living areas.

Keywords: Incomplete malaria treatment, malaria in Cameroon, mathematical model; ordinary differential equation, recovery rate.

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1. Introduction Malaria is a tropical disease caused by protozoan parasites of the genus plasmodium. There are at least 300 million acute cases of malaria each year globally, resulting in more than one million deaths. This is about 2700 deaths a day and about 2 deaths a minute. Around 90% of these deaths occur in Africa, mostly in young children. Malaria is Africa’s leading cause of under-five mortality (20%) and constitutes 10% of the continent’s overall disease burden (see [1] for more detailed statistics). It remains one of the principal public health problems in the African region and is one of the principal causes of poverty in endemic regions [2]. With the goals: to Roll Back Malaria (RBM) by the year 2010; to ensure that malaria-related mortality is reduced by 50% of the 2000 figures, by the year 2010 [2]; and for malaria to cease to have significant public health impact by the year2030 [3], it is important to revisit this disease and look at some of the conditions that may impact transmission and ultimately help drive down the malaria burden. Much information has been gathered from biological and epidemiological points of view, and exercises to eradicate the disease are strong and ongoing. However, where malaria is endemic, prevention and necessary health precautions are not often a high priority. Sometimes, due to negligence, poverty or the fear of drugs, the inhabitants are slow to start treatment or do not complete their treatment and hence stay longer with the disease thereby decreasing the recovery rate. There are four species of plasmodium, transmitted by the female anopheles mosquito, that can produce the disease in its various forms: plasmodium falciparum, plasmodium ovale, plasmodium vivax, and plasmodium malariae. The most commonly found species in Cameroon is plasmodium falciparum and it is the most dangerous because it can lead to fatal cerebral malaria if untreated. It is transmitted by the female Anopheles gambiae complex. Other anopheles species found in Cameroon are the Anopheles funestus and Anopheles bancocki complexes, (see [4]). Female anopheles mosquitoes feed on sugar sources for energy but require a blood meal for egg development. When a female takes a blood meal, they will rest for a few days during which time the blood is digested and eggs are developed. This process depends on temperature and takes a few days under tropical conditions. Once the eggs are fully developed, the female lays them and resumes host seeking, a cycle that is repeated until the female dies. Females can survive up to a month (or longer in captivity) [5]. Their lifespan depends on temperature, humidity, its internal characteristics and the presence of natural enemies especially during a blood meal. For more on the ecology of anopheles mosquitoes, see [5] and [6].

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When a female anopheles mosquito takes a blood meal from an infected malaria individual, it may pick up the sexual forms, mature male and female gametocytes, of the parasite. These gametocytes (male and female) undergo gametogenesis (creation of the male and female gametes that will undergo fertilization), followed by fertilization and then, the development into sporozoites in the mosquitoes gut. These sporozoites then migrate to the salivary glands of the mosquito where they may pass into the blood stream of humans during a blood meal. At this stage the mosquito is said to be infective. When the infective mosquito bites a human, it releases sporozoites into the human blood stream, which circulate briefly in the blood until they find, invade and infect hepatocytes (liver cells) in the human liver. Upon invasion of a hepatocyte, a sporozoite transforms into a trophozoite which undergoes asexual reproduction via the process of schizogony, in which the parasite cell divides and separates into numerous merozoites. These merozoites cause the host hepatocyte to burst, freeing the merozoites into the circulating blood. The merozoites then invade erythrocytes (red blood cells) in the blood stream infecting them. Then, they transform into trophozoites and undergo another round of schizogony producing more merozoites that cause the host erythrocyte to rupture, releasing these merozoites. The erythrocyte generated merozoites may either transform into a trophozoite and repeat the cycle of schizogony and merozoite production, or may transform into trophozoites which undergo gametocytogenesis, generating either a male or female gametocyte within the host erythrocyte. Once these male and female gametocytes mature, the human is said to be infective and a mosquito on contact with the infectious human can pick up these mature male and female gametocytes. The breaking down of the red blood cells results in bouts of fever, anaemia, shivering, pain in the joints and headache in the infected individual. See [7] for more on the life cycle of the malaria parasite. The incubation period in human depends on the malaria strain and the immune response of the host. This period can be as short as 7 days or as long as 30 days depending on the type of plasmodia involved. Commonly, clinical symptoms occur after 7 − 14 days for P. falciparum, 7–30 days for P. malariae and 8–14 days for P. vivax and P. ovale, (see [8]). On average, however, the incubation period is about 11 to 12 days in humans forplasmodium falciparum and about 10 days in mosquitoes [9]. Mathematical models to study malaria, pioneered by Ross in 1909 and 1911 [10, 11], and later updated by MacDonald in 1957 [12], have been analyzed and gave useful insights which led to the eradication of malaria in some parts of the world. The next major study was the work by Kermack and McKendrick [13-15], where they considered endemic diseases and extended greatly the concept of the basic reproductive number. Their work has been extended and refined over five decades by others to either incorporating the exposed class; the concept of temporary and/or variable immunity against reinfection; superinfection; age-structure; and also spatial heterogeneity [16-20]. Despite these advances, malaria continues to be a burden in the African continent. Moreover, most of the models described assume constant population size, which is valid when studying diseases of short duration with limited effects on mortality. Ngwa and Shu in 2000, [9], modeled malaria dynamics in a changing population and showed that under the assumption of zero disease induced deaths, the disease-free equilibrium always exist and is globally stable when the basic reproductive number is below unity and that the endemic equilibrium, when it exists, is unique and globally stable.

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Here, a deterministic Susceptible-Exposed-Infectious (SEI) mathematical model with variable mosquito and human populations that closely modelplasmodium falciparum malaria in an endemic region as Cameroon, is used to study the disease dynamics, and to examine and quantify the impacts of the recovery rate, the contact rates and the recruitment rate on prevalence and initial disease transmission. The model is formulated such that both the total host and vector populations are non-constant. It takes into account disease-induced death rate and density dependent deaths for the human population, which in most models are not accounted for. In general, in the fight to control malaria, an important question to ask is: How much of an impact can local communities and individuals have in enabling the successful eradication of malaria? In an endemic region like Cameroon, the recovery rate, the contact rates and the recruitment rate are possible parameters that can be controlled based on changes in individual attitudes and behaviours within local communities as follows: 1. Early diagnosis and the start and completion of malaria treatment are factors that can help reduce the infectious period and increase the recovery rate of an infectious individual. In [21], the patent period1 for plasmodium falciparum under treatment is given to be about 4-6 weeks and without treatment about 18 months. These numbers could be higher, to about 23 and a half months, based on the recovery rates quoted in [22]. Biological studies and some mathematical studies have shown that low recovery rates do stabilize endemic malaria even in areas of low transmission [22, 23]. However, most of the discussions on increasing recovery rates have always been focused on early diagnoses, which is great. However, in endemic regions like Cameroon, strong emphasis needs to be placed on treatment completion, which has not always been of great focus. Because of some bad cultural practices and lack of information, malaria medications are sometimes taken (without completion) for preventive purposes, even when the individual may be infectious. From experience also, many sick individuals start their malaria treatment but do not complete them, many at times because their symptoms (like fever and joint aches) subside and the individuals believe that they are no longer ill, though may still be infectious. This may lead to prolong developmental stages of the parasite within the individuals, hence the very long patent periods for malaria. This is a serious issue in areas of Cameroon and other endemic regions and may be one of the causes of drug resistant parasite. Therefore, it is important to quantify this impact and see the effect it has on malaria transmission and prevalence. 2. The use of non-toxic environmentally save anti-mosquito repellent during the early hours or early evening periods around houses and farming areas as a means to reduce the average number of bites a mosquito takes and hence reduce the contact rates between mosquitoes and humans. 3. Reduction or elimination of breeding sites around houses and hence a reduced recruitment rate of mosquitoes. 1

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In the simulation, realistic feasible published data mostly derived from the Cameroon region will be used as follows: (1) to examine, measure and quantify the impact reducing the recovery rate can have on initial disease transmission via the basic reproductive number R0 ; and on disease prevalence via the endemic infectious steady state proportions and the maximum infectious human proportions; (2) to examine the individual and combined effects increasing the recovery rate of infectious humans and decreasing the contact rates between humans and mosquitoes have on initial disease transmission and on disease prevalence; (3) to examine and analyze the impact reducing the mosquito recruitment rate has on disease dynamics. To quantify the effects of prolonged malaria, the recovery rate will be controlled and its effects quantified. The assumption is that very low recovery rates (that are within experimentally verified results) are due to prolonged infectious disease periods in humans which mostly occur because of incomplete malaria treatment. It could also be due to numerous exposure and contacts with infectious mosquitoes. The information to be obtained from the results can be used to inform and mobilize local communities and individuals as ”malaria disease control agents”, to be more aggressive in educating the public, and will suggest strategies for efficient malaria control in an endemic region like Cameroon The basic mathematical model is developed in section 2, and in section 3 the model is analyzed, the steady states obtained and the basic reproductive number computed. Using base parameters specific to Cameroon, numerical results are presented in section 4. In that section, the numerical values for R0 , and the coordinates of the steady states are computed for different values of the recovery rate. In addition, the linear stability of these steady states are discussed and the plots of the solution curves presented and analyzed. Finally the paper ends with a conclusion and discussion section, which gives a summary of the findings, suggest possible control strategies and discuss the impact of local communities in the fight against malaria.

2. The Mathematical Model Here, a Susceptible-Exposed-Infectious-Removed (SEIS) criss - cross model of malaria fever transmission is presented that will enable us to study and explore the behaviour and dynamics of malaria in an endemic region like Cameroon. Only populations involved in disease transmission are presented. These include only the adult female mosquitoes (vector) and humans of all ages and sexes (host). Transmission is between the host and vector populations which are non constant. Each of the host and vector populations is divided into classes (which are the compartments to be referred to as state variables) representing disease status. Compartments for the human populations at time t, (in days), are: Susceptible (Sh ) humans who do not have any form of the parasite in their blood stream; Exposed (Eh ) humans who are incubating the parasite but are not yet infectious to others; Infectious (Ih ) - humans with the parasite (in particular male and female gametocytes) in their blood which can be picked up by a mosquito on contact. Those who die due to the disease or naturally can be collected into a removed class denoted by Rh . Natural death, here, is death due to other causes. Compartments for the mosquito population are: Susceptible (Sv ) - mosquitoes that do not have any form of the parasite in their stomach lining, their gut or their salivary glands; Exposed (Ev ) - mosquitoes that are incubating the parasite but can not pass it on to

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Host (Human) H

Vector (Mosquito) V Recruitment, K

Sh lh

rh lh lh

Eh

µh αh

Sv

µv

αv µh

Ev

µv

υv

υh

Ih

µh

Iv

µv

γh

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Figure 1. Schematic representation of the model.

a human on contact; Infectious (Iv ) - mosquitoes with the parasite in their salivary glands that can be transmitted to humans on contact. Once mosquitoes become infectious they are assumed to remain so until death. Hence, the effective total populations, at timet, for both the human and the vector populations are respectively Nh (t) = Sh (t) + Eh (t) + Ih (t) and Nv (t) = Sv (t) + Ev (t) + Iv (t). The conceptual framework of the model is shown in the schematics Figure 1. The model takes into account births for humans, adult eclosion for mosquitoes (i.e., act of an insect emerging from its pupal case or the hatching of an insect larva from an egg) and deaths for each species. The model assumes that for the human population, all new births are recruited into the susceptible compartment, i.e., no vertical transmission, with a per capita birth rate denoted by λh > 0 (day−1 ). For the mosquito population, recruitment into the susceptible compartment is assumed to occur at the rate K > 0 (Number per day). To estimate K, several factors can be taken into consideration among which are: (1) the number of eggs laid by a female mosquito, (2) the combined percentage of eggs larva and pupa that survive and mature into adult mosquito, (3) the sex ratio of survived male to female mosquitoes, (4) climatic and seasonal conditions (5) the survival rate of adult mosquitoes and (6) the availability of breeding sites. Deaths, other than disease induced deaths, occur in all classes at rates µh (Nh ) and µv (Nv ) for both the human and mosquito populations. For the mosquito population, µv (Nv ) is assumed to be constant so that µv (Nv ) = µv (days−1) . For the human population, the death rate, µh (Nh ), is taken to be density dependent and for simplicity, taken to be proportional to the total population Nh at the relevant time, i.e., µh (Nh ) = µh Nh where µh (with units days−1 ∗ N umber −1 ) is a constant. The model does not take into account migration, since the net migration rate for the Cameroon region is0 [24]. Flow from the susceptible to exposed compartments (exposure rate) for each species depends on the level of contact with the other species. Once a member of a susceptible class comes in contact with a member of an infectious class, the susceptible member joins

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the exposed class. In the presence of the malaria disease, the transmission rate in humans depends on the interaction between the human and the mosquito populations. The level of contact is determined by the biting rate of mosquito (average number of bites per day), the transmission probabilities (probability that an infectious vector will transmit the parasite to a susceptible host in the course of contact) as well as the number of individuals in the susceptible and infectious compartments for each species. Let bi represent the average biting rate of an infectious mosquito. A fraction, Sh /Nh , of their meals is taken from susceptible humans, and each leads to an infection with probability pvh (i.e., uniform mixing is assumed). Hence the average transmission probability (contact rate) from vector to host is Sh cvh = bi pvh and the incidence function for humans isωh = cvh N Iv . Similarly, if bs repreh sents the average biting rate of a susceptible mosquito biting a fractionIh /Nh of infectious humans, and each mosquito becomes infected with probability phv . Then, the incidence function for mosquitoes is ωv = chv NIhh Sv , where chv = bs phv is the average transmission probability (contact rate) from host to vector. Exposed members from both the human and mosquito populations, respectively become infectious at rates νh = 1/ (average intrinsic incubation period) and νv = 1/(average extrinsic incubation period), where νh > 0 and νv > 0. Assuming little conference of immunity to malaria, humans can either recover from the disease and join the susceptible class at rate rh (the recovery rate) or they can die from the disease at rateγh (the disease induced death rate). Mosquitoes, when they become infective, remain so until they die.

2.1. The Differential Equations

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Using standard mass action laws (that is, the rate of change of a class is directly proportional to the density of the state variables in the given class), the above flow and transmission process is modeled by the nonlinear system of differential equations given below. For the human population: Sh dSh = λh Nh − (µh Nh ) Sh − cvh Iv + rh Ih dt Nh

(2.1)

Sh dEh = cvh Iv − (µh Nh + νh ) Eh dt Nh

(2.2)

dIh = νh Eh − (µh Nh + rh + γh ) Ih dt

(2.3)

dNh = (λh − µh Nh ) Nh − γh Ih , (2.4) dt where Nh = Sh + Eh + Ih is the total effective human population and the differential equation describing its change, eqn. (2.4), is obtained by adding the corresponding equations for Sh , Eh ,and Ih . Notice that the total population is non constant over time. The removed population (those that die naturally and those that die due to the disease) will be collected into the class denoted by Rh which satisfies dRh /dt = γh Ih + (µh Nh ) Nh . This equation decouples since the value of Rh can be obtained once the values of Sh , Eh and Ih are known. This equation, though not of central importance, does give a direct means of calculating the mortality rate in the human population.

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Miranda I. Teboh-Ewungkem For the mosquito population, the system is Ih dSv = K − µv Sv − chv Sv dt Nh

(2.5)

dEv Ih = chv Sv − (µv + νv ) Ev (2.6) dt Nh dIv (2.7) = νv Ev − µv Iv dt dNv = K − µv Nv . (2.8) dt Similarly, the total effective mosquito population, Nv = Sv + Ev + Iv , is also not constant over time and its rate of change, eqn. (2.8), is obtained by adding eqns. (2.5-2.7). The system, eqns. (2.1-2.8) is a well-posed autonomous system. Since the system monitors the dynamics of humans and mosquito, all the associated variables and parameters are assumed to be non-negative for all t ≥ 0. The functions on the right hand side of this system are smooth and admit no singularity in the space of functions considered and under specified reasonable initial conditions, has a unique solution satisfying the initial conditions for all t ≥ 0. One of the goals of this model is to see how many new cases of infective humans and mosquitoes arise due to the introduction of a single infective mosquito into a wholly susceptible population of mosquitoes and humans. It is assumed that the single mosquito does transmit the disease. Hence the initial conditions to be specified for both the mosquito and human populations are

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Sh (0) = Sh0 , Eh (0) = 0, Ih (0) = 0, Sv (0) = Sv0 , Ev (0) = 0, Iv (0) = Iv0 (0).

(2.9)

3. Analysis 3.1. Non-dimensionalization The susceptible, exposed and infectious human populations are each normalized byNv0 = K/λh (number per day/per day), while the susceptible, exposed and infectious mosquito populations are each normalized by Nh0 = cvh K/λ2h ((per day * number per day)/(per day * per day)). Hence the normalized SEI populations are U=

Eh Ih Sv Ev Iv Sh ,V = ,W = ,X = ,Y = ,Z = . Nh0 Nh0 Nh0 Nv0 Nv0 Nv0

The time scale, t, is normalized by 1/λh (day), so that τ = λh t. Hence the normalized parameters are α=

rh γh νh chv µv νv µh Nh , β = ,γ = ,δ = ,κ = ,σ = ,θ = . λh 0 λh λh λh λh λh λh

In dimensionless form, the system for both the human and mosquito populations become UZ dU = (1 − αN )U + V + (1 + β) W − dτ N

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dV UZ = − (αN + δ)V dτ N

(3.11)

dW = δV − (αN + β + γ) W dτ

(3.12)

dN = (1 − αN ) N − γW dτ

(3.13)

dX κXW =1− − σX dτ N

(3.14)

dY κXW = − (σ + θ)Y dτ N

(3.15)

dZ = θY − σZ dτ

(3.16)

dN = 1 − σN , dτ

(3.17)

with initial conditions U (0) = U 0 , V (0) = 0, W (0) = 0, K = K 0 , X(0) = X 0 , Y (0) = 0, Z(0) = Z 0 . The variables N = U + V + W and N = X + Y + Z are respectively the total host and vector populations in dimensionless forms. Hence, the principal equations of interest are (3.10-3.12) and (3.14-3.16), since the equations for the total host and vector populations (3.13) and (3.17) can be obtained by adding equations (3.10-3.12) and (3.143.16) respectively.

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3.2. Parameter Space Since the model eqns. (3.10-3.12) and (3.14-3.16) also monitor the dynamics of humans and mosquito, all the associated variables and parameters are assumed to be non-negative for all τ ≥ 0. In addition, the rate of progression to infectiousness is greater than the human birth rate so that, realistically, the non-dimensional parametersδ = νh /λh > 1 and θ = νv /λh > 1. Moreover, realistically sustainable human populations occurs if the disease induced death rate, γh , is less than the human birth rate, λh so that γ = γh /λh < 1. Hence the following proposition. Proposition 1. The parameter space is defined as Π = (α, β, γ, δ, κ, σ, θ) ∈

6 +

: α > 0, β ≥ 0, κ ≥ 0, σ > 0, δ > 1, 0 ≤ γ < 1, θ > 1 .

3.3. Boundedness and Positivity of Solutions Theorem 2. 3.16) with ally

All the solution initial conditions

enter the attracting 1o 1 . 0≤N ≤ ,0≤N ≤ α σ

set Ψ

of in =

the system R6+n are

(3.10-3.12) and (3.14bounded and eventu-

(U, V, W, X, Y, Z)

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R6+

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Proof. Let (U, V, W, X, Y, Z) ∈ R6+ be any solution of the model, eqns. (3.10-3.12) and (3.14-3.16) with non-negative initial conditions. Then, since γ > 0 and N = U + V + W satisfying equation (3.13), it follows that dN ≤ (1 − αN ) N dτ so that 0≤N ≤

(3.18)

N (0) , αN (0) + e−t

where N (0) is the initial dimensionless total human population. Hence, asτ → ∞, 0≤N ≤

1 . α

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Therefore, all feasible solutions to the human component of the model, eqns. (3.10-3.12), enter the region,   1 3 . ΨH = (U, V, W ) ∈ R+ : 0 ≤ N ≤ α For the mosquito component of the model, eqns. (3.14-3.16), N = X + Y + Z and 1 satisfies eqn. (3.17). If eqn. (3.17) is solved, then, is an upper bound of N (τ ) provided σ 1 1 that N(0) ≤ . Further, if N (0) > , then, N (τ ) will decrease to this level. Hence, the σ σ feasible region for the mosquito population is   1 3 , ΨV = (X, Y, Z) ∈ R+ : 0 ≤ N ≤ σ n so that all feasible solutions enter the region, Ψ = (U, V, W, X, Y, Z) ∈ R6+ : 0 ≤ N ≤ 1o 1 . ,0≤N ≤ α σ

3.4. Disease-Free Equilibrium (DFE) and the Basic Reproductive Number R0 In the absence of malaria in the population, the entire host and vector populations,N and N respectively, are susceptible and hence the model, eqns. (3.10-3.12) and (3.14-3.16) has a disease-free steady state given as  
1 1 , 0, 0, , 0, 0 . (3.19) S0 = (U0 , V0 , W0 , X0 , Y0 , Z0 ) = N, 0, 0, N , 0, 0 = α σ In dimensional form, this DFE equilibrium is  λh K , 0, 0, , 0, 0 . S0 = (Sh0 , Eh0 , Ih0 , Sv0 , Ev0 , Iv0 ) = (Nh , 0, 0, Nv , 0, 0) = µh µv (3.20) To analyze the linear stability of the DFE (3.19 or 3.20), a derivation of the basic reproductive number, R0 , is first performed. R0 is the average number of secondary infections

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in humans that arise from a single primary infection in the humans due to the introduction of an infection in a wholly susceptible population of mosquitoes and humans. To derive a value for R0 , disease transmission from humans to mosquitoes, and vice versa is considered and a calculation of the number of infections that result from a single newly infected individual of the other type carried out at the onset of the disease (i.e., at the DFE). At the disease-free equilibrium (DFE), the rate at which humans infect mosquitoes Sv Ih is chv . Hence, a single infectious human gives rise to mosquito infections at Nh Sv 1 rate chv . The average length of time a human is infectious is , Nh (µh Nh + rh + γh ) Sv chv Nh infections in the mosquito popwhich gives rise to an average of (µh Nh + rh + γh ) ulation. However, not all exposed individuals become infectious and so multiplying this number by the probability of moving from an exposed class to an inνh (the average length of time a human is fectious class, which is, (νh + µh Nh ) exposed times the rate at which the exposed human becomes infectious), gives 1 Sv νh chv . However, Sv = Nv , at the disease-free equilib(µh Nh + rh + γh ) Nh (νh + µh Nh ) 1 Nv chv × rium, so that one newly infected human gives rise to R0hv = (µh Nh + rh + γh ) Nh νh newly infected mosquitoes. In a similar manner, the rate at which mosquitoes (νh + µh Nh ) Sh infect humans at the DFE is cvh Iv , in which case a single infectious mosquito gives rise Nh Sh to human infections at the rate cvh . The average length of time a mosquito is infecNh 1 Sh 1 so that there is an average of cvh infections in the human population. tious is µv µv Nh When this number is multiplied by the probability of a human surviving their incubation νv period, , it can be seen that at the DFE where Sh = Nh , each of the newly (µv + νv ) 1 νv cvh newly infected humans. infected mosquito, R0hv , gives rise to R0vh = µv (µv + νv ) Hence, the basic reproductive number, R0 , is obtained by multiplying R0hv and R0vh to 1 νh 1 νv Nv chv N cvh . At obtain R0 = R0hv R0vh = h (ν + µ N ) (µ ) (µh Nh + rh + γh ) (µ v v + νv ) h h h the disease-free equilibrium, (eqn. 3.20), Nh = λh /µh and Nv = K/µv so that R0 =

chv cvh Kµh νh νv 2 µv λh (λh +rh + γh ) (νh + λh ) (µv

+ νv )

.

(3.21)

This formula agrees with the work of Newton and Reiter [25], where a model was developed to study the transmission of dengue fever. In non-dimensional terms, this value is given as R0 =

κθδ ακθδ N = 2 . (3.22) N σ (σ + θ) (αN + δ) (αN + β + γ) σ (σ + θ) (1 + δ) (1 + β + γ)

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3.4.1. Stability of the DFE Theorem 3. The DFE is Locally asymptotically Stable (LAS) if R0 ≤ 1 and unstable if R0 > 1. Proof. To proof this, it suffices to look at the Jacobian matrix J at the DFE, given as 

     J0 =      

1 − 2αN 0 0

1 − (δ + αN ) δ

0

0

0

0

0

0

1+β 0 − (αN + β + γ) κU0 X0 κN − =− 2 N N κN N 0

0 0 0

0 0 0

−1 1 0



      −σ 0 0 ,   0 − (θ + σ) 0   0 θ −σ (3.23)

1 1 and N = are the total populations at the disease-free equilibrium. The α σ 6th order characteristic polynomial for this matrix is found by solving

where N =

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|J0 −λI| = 1−2αN −λ 1 1+β 0 0 −1 0 − (δ+αN )−λ 0 0 0 1 0 δ − (αN +β +γ)−λ 0 0 0 κN −σ−λ 0 0 0 0 − N κN 0 − (θ+σ) − λ 0 0 0 N 0 0 0 0 θ −σ − λ (3.24) = (1 − 2αN − λ) (−σ − λ) Je0 −λI = 0, where



− (δ + αN ) 0 0 1  δ − (αN + β + γ) 0 0  Je0 =  κN  − (θ + σ) 0 0  N 0 0 θ −σ Hence, with the substitution N =



  .  

1 , the characteristic polynomial is: α


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(3.25)

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where R0 is as defined in equation (3.22) and the coefficients A1 , A2 , A3 and A4 are A1 = (2σ + θ) + (2αN + β + γ + δ) = (2σ + θ) + (2 + β + γ + δ) > 0, A2 = σ (σ + θ) + (2αN + β + γ + δ) (2σ + θ) + (αN + β + γ) (δ + αN ) > 0 = σ (σ + θ) + (2 + β + γ + δ) (2σ + θ) + (1 + β + γ) (δ + 1) > 0, A3 = σ (σ + θ) + (2σ + θ) (2αN + β + γ + δ) (δ + αN ) = σ (σ + θ) + (2σ + θ) (2 + β + γ + δ) (δ + 1) > 0, A4 = σ (σ + θ) (αN + δ) (αN + β + γ) = σ (σ + θ) (1 + δ) (1 + β + γ) > 0. (3.26) If R0 ≤ 1, then, A1 , A2 , A3 and A4 are all positive and by the Descartes rule of signs [26], the characteristic polynomial for the submatrix Je0 , which is
λ4 + A1 λ3 + A2 λ2 + A3 λ + A4 (1 − R0 ) , has no sign change and hence, no positive real roots. Moreover, if we apply the RouthHurwitz criterion [27, 28] to this polynomial, it can be shown that A1 A2 A3 > A23 + A21 A4 (1 − R0 ) so that all the four eigenvalues have negative real parts when R0 ≤ 1. The straightforward but lengthy calculations are omitted. Hence all the eigenvalues of the matrix, eqn. (3.23), with characteristic polynomial, eqn. (3.25), are all negative or have negative real parts and the DFE is LAS.

3.5. Endemic Equilibrium In the presence of the disease, examination of the system of equations (3.10-3.12) and (3.14-3.16) leads to the endemic steady state defined as S = (U1 , V1 , W1 , X1 , Y1 , Z1 ),

(3.27)

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where (αN1 + δ) (αN1 + β + γ) σ (σ + θ) (κ (1 − αN1 ) + σγ) N12 , γδθκ (αN1 + β + γ) (1 − αN1 ) N1 V1 = , γδ (1 − αN1 ) N1 γ , X1 = , W1 = γ (κ (1 − αN1 ) + σγ) κ (1 − αN1 ) κθ (1 − αN1 ) , Z1 = , Y1 = (σ + θ) (κ (1 − αN1 ) + σγ) σ (σ + θ) (κ (1 − αN1 ) + σγ)

U1 =

(3.28)

and N1 is obtained by adding the coordinates U1 , V1 and W1 . Adding U1 , V1 and W1 leads to (3.29) B4 N14 + B3 N13 + B2 N12 + B1 N1 + B0 = 0, the 4th order polynomial in the steady state total host population,N1 , with B0 B1 B2 B3 B4

= κθ [δ (γ − 1) − γ − β] < 0, for all parameters from the set Π. = ακθ (δ − 1 + γ + β) − σ (σ + θ) (δ) (β + γ) (κ + σγ) = ασ (σ + θ) κδ (β + γ) − ασ (σ + θ) δ (β + γ + δ) (κ + σγ) + α2 κθ = σ (σ + θ) α2 (κβ + κγ + κδ − κ − σγ) = σ (σ + θ) α3 κ > 0, for all parameters from the setΠ.

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(3.30)

116

Miranda I. Teboh-Ewungkem

Both from a physical and a realistic stand point, a positive solution for N1 obtained from eqn. (3.29) should exist in Ψ. If such exists, then, the endemic steady state, eqn. (3.27), with coordinates defined in eqn. (3.28) are all positive and thus exist inΨ, and their values could be computed. In fact, in the parameter space Π, the coefficient B0 < 0 since δ > 1 and γ < 1. Moreover, since the parameters are all positive, B4 > 0. Therefore, from the sequence of coefficients {B0 , B1 , B2 , B3 , B4 } there is at least one sign change (either one or three) irrespective of the signs of B1 , B2 and B3 . Thus by Descartes rule of signs, (see [26]), there is at least one real, positive steady state total population. With this positive value of N1 , the values of U1 , V1 , W1 , X1 , Y1 and Z1 can be obtained. It should be noted that using realistic feasible parameters (see result section), when R0 > 1, only one such positive real value of N1 occurs in Ψ.

The local stability of this steady state, eqn. (3.27), can be analyzed by looking at the Jacobian matrix (at this steady state) that arises from the system (3.10-3.12) and (3.14-3.16). This matrix is defined as



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   J1 =    

 1−α (N1 +U1 )−A 1+B 1+β+B 0 0 −R A − (δ+αN1 )−B −B 0 0 R   αW1 δ − αW1 − (αN1 +β +γ +W1) 0 0 0   P P −Q −T −σ 0 0   −P −P Q T − (θ+σ) 0  0 0 0 0 θ −σ

where U1 , V1 , W1 , X1 , Y1 and Z1 are as given in eqn. (3.28) and A =

Z1 (V1 + W1 ) , N12

κW1 X1 κ (U1 + V1 ) X1 U1 κW1 U1 Z1 ,P = ,Q= ,R= ,T = . The 6th order N1 N1 N12 N12 N12 characteristic polynomial (See the Appendix) for this matrix is

B =


(−σ − λ) −λ5 − C1 λ4 + C2 λ3 + C3 λ2 + C4 λ + C5 = 0, Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(3.31)

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where C1 = (a1 + a2 − a3 + a4 + a5 ) C2 = a2 (a3 − a4 − a5 ) + a5 (a3 − a4 ) + a4 (a3 − a4 − a1 ) + a1 (a3 − a2 − a5 ) + (1 + B) A + (1 + β + B) αW1 C3 = a2 (a1 + a5 ) (a3 − a4 ) + a5 [a1 (a3 − a2 ) + a4 (a3 − a1 )] + a3 a4 (a1 + a2 ) + (1 + B) A (θ + σ + T + σ + αN1 + β + γ + W1 ) + (1 + β + B) αW1 (θ + σ + T + σ + δ + αN1 + B) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] C4 = a1 a2 a5 (a3 − a4 ) + a3 a4 (a1 a2 + a1 a5 + a2 a5 ) + (1+B) A ((θ+σ) (T +σ)+(θ+σ) (αN1 +β +γ +W1)+(T +σ) (αN1 +β +γ +W1 )) + (1 + β + B) αW1 ((θ + σ) (T + σ) + (θ + σ) (δ + αN1 + B) + (T + σ) (δ + αN1 + B)) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] (θ + σ + T + σ) + θR [P (δ − αU1 ) + Q (δ − 2QαW1 )] C5 = a1 a2 a3 a4 a5 + (1 + B) A ((θ + σ) (T + σ) (αN1 + β + γ + W1 )) + (1 + β + B) αW1 ((θ + σ) (T + σ) (δ + αN1 + B)) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] (θ + σ) (T + σ) − θR (1 − αN1 ) (Q + P ) δ + θR (P γδ + 2P αW1 + 2QαW1 + P δW1 − P αβU1 − P αγU1 + QαδU1 + 2P αβW1 ) +
− θR QαδW1 + P αU1 W1 + P α2 N1 U1 + 2Qα2 N1 W1 + Qα2 U1 W1 ,

with a1 = (θ + σ) ;

a2 = (T + σ) ;

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a4 = (δ + αN1 + B) ;

a3 = 1 − α (N1 + U1 ) − A

a5 = αN1 + β + γ + W1 ;

a6 = (1 − (δ + αN1 )) .

From this polynomial, eqn. (3.31), the eigenvalues, λ, that corresponds to the endemic steady state, eqn. (3.27), can be obtained.

4. Simulation and Results Simulations of the model eqns. (3.10-3.12) and (3.14-3.16), performed using a reasonable set of baseline parameter values that are in line with the literature on malaria transmission in an endemic region as Cameroon, [4, 9, 22, 24, 29, 30], and as shown on Table 1 are carried out in this section. For these parameter sets, it is numerically and graphically demonstrated that the model does indeed posses a globally and asymptotically stable endemic steady state when R0 > 1, for the initial conditions prescribed on Table 2. A FORTRAN code is used to solve the system of equations, eqns. (3.10-3.12) and (3.14-3.16) to obtain the numerical and graphical results. Each ordinary differential equation is solved using first order Euler’s method with a very small step size, h, chosen to minimize the error made. Here, a step size of h = 5 × 10−6 is used in the Euler scheme to achieve accurate results. 2

The induced death rate for Malaria in Cameroon for under five is much higher and has a 2006 estimated value of 602/100000 per year (Globalis (2000 estimate) ). But this value does not alter the analysis presented in the Simulation section. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Miranda I. Teboh-Ewungkem Table 1. Baseline parameter pata used in the numerical simulation

Base Parameter Birth rate of host, λh (days)−1 Density dependent death rate of host, µh (days ∗ N umber)−1 Death rate of vector, µv (days)−1 Disease induced death rate, γh (days)−1 Average intrinsic incubation period, vh (days)−1 Average extrinsic incubation period, vv (days)−1 Contact rate from Iv to Sh , cvh (days)−1 Contact rate from Sv to Ih , chv (days)−1 Recovery rate, rh (days)−1 Recruitment Rate, K (days−1 ∗ N umber)

Value 34.59 1000∗365 12.41 1000∗365∗18,467,692 1 24

References [24, 29] [24, 29]

1/12

[9]  2 30 [9]

1/10

[22]

0.45 0.27 0.0056 104

[22] [22] Will be varied guess (see sec. 4.3.2.)

108 100,000∗365

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Table 2. Initial conditions used in the numerical simulation Initial Conditions Susceptible host, Sh (0) Susceptible vector, Sv (0) Exposed Host, Eh (0) Exposed Vector, Ev (0) Infectious Host, Ih (0) Infectious Vector, Iv (0)

Value (in Numbers) 105 2.0 × 105 0 0 0 1

4.1. Quantification of the Impact Increasing the Recovery Rate Will Have on Malaria Dynamics The first aim is to quantify the impact the recovery raterh , has on prevalence and on the basic reproductive number. In his paper, [31], Trape estimated the values of the daily recovery rates, in the Congo region, for plasmodium falciparum to be in the range 0.004 − 0.020 and in the range 0.012 − 0.043 for plasmodium malariae. These are the two most prominent plasmodia parasite in the Cameroon region, a neighboring country to Congo. Although his estimates were carried out in Congo, it gives an indication of the possible ranges for the recovery rate in neighboring regions to Congo that also experience endemic malaria disease dynamics such as Cameroon. In addition, Nakul et al., [22] gave ranges 0.0014 − 0.017 for rh for plasmodium falciparum malaria in endemic regions. Here, different values of the recovery rate that fall within the ranges of 0.0014 − 0.02 will be used in the numerical simulation, together with the baseline parameters given in Table 1. Numerical values for the basic reproductive number, the endemic steady state and the corresponding eigenvalues will be computed and the linear stability analysis of this endemic steady state discussed.

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Starting with a recovery rate value, rh = 0.0014 (which corresponds to an infectious period of about 23.5 months), the non-dimensional parameters become α = 9734.322, β = rh /λh = 14.773, γ = 0.003, δ = 879.349, κ = 2849.089, σ = 439.674, θ = 1055.218. Using these parameters together with the dimensionless form of the initial conditions shown on Table 2, there is only one endemic steady state, eqn. (3.27), with coordinates, eqn. (3.28), given as: U1 = 5.811 × 10−5 , V1 = 7.639 × 10−7 , W1 = 4.254 × 10−5 , X1 = 6.117 × 10−4 , Y1 = 4.890 × 10−4 , Z1 = 1.174 × 10−3 . The values for the total host and vector populations are respectively obtained to be N1 = 1.014 × 10−4 and N 1 = 2.274 × 10−3 . Notice that the coordinates of this steady states are all positive and their original values can be obtained by multiplying each of U1 , V1 and W1 by Nh0 = 5.011 × 1011 and each of X1 , Y1 and Z1 by Nv0 = 1.055 × 108 . The time t (in days) can be obtained by multiplying the nondimensional time, τ , by T0 = 10552. Note that the values Nh0 = 5.011 × 1011 , Nv0 = 1.055 × 108 and T0 = 10552 will not change in this section. The eigenvalues obtained from the characteristic equation, eqn. (3.31), are: −0.117, −23.479, −439.674, −904.290, −1436.200 and −1674.000, all negative. Hence, the steady state

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5.811 × 10−5 , 7.639 × 10−7 , 4.254 × 10−5 , 6.117 × 10−4 , 4.890 × 10−4 , 1.174 × 10−3




is locally asymptotically stable to small perturbations. The basic reproductive number is computed to be R0 = 6.401 > 1. Figures 2 and 3 show the solution curves for the susceptible, exposed and infectious vector populations in the non-dimensional variables for this parameter set. Figure 2 shows the solution curves for the proportions of the susceptible, exposed and infectious host populations over time while Figure 3 shows the solution curves for the mosquito population. At the onset of the disease, the susceptible populations drop initially to a low nonzero value (which leads to an increase in the exposed and infectious populations), but eventually increases to its stable value. For the human population, Figure 2, the infectious population proportion reach its peak within the time interval τ ∈ [9, 11], while for the mosquito population, this peak is reached very early on, Figure 3. In addition, at this low recovery rate value of 0.0014, the stable infectious mosquito population proportion is always higher than the stable susceptible mosquito population proportion. This reflects the degree of difficulty at eradicating the disease, which is also reflected by the high value of the basic reproductive number,R0 = 6.401. Keeping all other rates constant and increasing the recovery rate, rh four folds from 0.0014 to 0.0056, (i.e., β = 59.092, corresponding to an infectious period of a little under six months), there is again only one endemic steady state solution, eqn. (3.27), with coordinates: U1 = 9.339 × 10−5 , V1 = 5.810 × 10−7 , W1 = 8.498 × 10−6 , X1 = 1.479 × 10−3 , Y1 = 2.338 × 10−4 and Z1 = 5.612 × 10−4 . These steady state coordinates are again all positive and their original values can also be obtained by multiplying each of U1 , V1 and W1 by Nh0 , each of X1 , Y1 and Z1 by Nv0 and the time (in days) also obtained by multiplying τ by T0 , as earlier defined. The corresponding eigenvalues also all have negative real parts implying that the steady state solution, (9.339×10−5 , 5.810×10−7 , 8.498×10−6 , 1.479×10−3 , 2.338×10−4 , 5.612×10−4 ), is linearly stable to small perturbations. More importantly, the basic reproductive number is R0 = 1.683 > 1, but much less than the previous value of R0 = 6.401. The plots of the solution curves are shown in Figures 4 and 5. Figure 4 shows the solution curves for the exposed and infectious host population pro-

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Miranda I. Teboh-Ewungkem

Susceptible, Exposed and Infectious Host Populations

0.00006

0.00005

0.00004

Susceptible Host Exposed Host

0.00003

infectious Host

0.00002

0.00001

0 0

5

10

15

20

25

30

35

40

Time

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Figure 2. Solution curves for the Susceptible, Exposed and Infectious human population proportions when the recovery rate, rh , is 0.0014 and all other parameters are as shown on Table 1.

portions at this recovery rate value of 0.0056. There is about an 80% decrease in the steady state proportion of infectious humans in the population and a71.4% decrease in the maximum proportion of the infectious human population, which occurs within the first 10 time units. This is reflected by the reduction of the basic reproductive number, R0 . For the mosquito population, not only does the steady state population proportion of infectious mosquito decrease by about 52.2%, the stable population proportion of susceptible mosquitoes in the community is greater than the stable population proportion of infectious mosquitoes, Figure 5. If the recovery rate is further increased to rh = 0.0098 (i.e., β = 103.411, corresponding to an infectious period of a little less that three and a half months), while keeping all other parameters fixed, the basic reproductive number drops to R0 = 0.969 < 1 and the long term stable steady state is the disease-free
steady state (DFE), eqn.(3.19), with values 1.0273 × 10−4 , 0, 0, 2.2744 × 10−3 , 0, 0 . Figures 6 and 7 respectively show the solution curves for the infectious host and vector population proportions at this recovery rate, and graphically demonstrate that the disease-free equilibrium (DFE) is globally stable. Notice that the maximum number of humans in the infectious class at any time within the population decreases by about 43.3% from the previous value obtained when rh = 0.0056, and in addition, disease eradicated is achieved by 15 time units. From the above results, it is seen that the recovery rate plays a significant role in reducing the malaria burden in an endemic region like Cameroon, which agrees with the work by others [22, 23]. The results, however, further quantifies this impact and shows that the steady state population proportion of infectious humans can be reduced by up to80%; the maximum population proportion of infectious humans reduced by up to 71.4%; and the steady state population proportion of infectious mosquito reduced by up to52.2%, by increasing the recovery rate from 0.0014 to higher values that have been clinically observed. This parameter is one that can be easily targeted and reduced by a change in the habits

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0.0015

Susceptible, Exposed and Infectious Vector Populations

0.0014 0.0013 0.0012 0.0011 0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004

Susceptible Vector Exposed Vector Infectious Vector

0.0003 0.0002 0.0001 0 0

5

10

15

20

25

30

35

40

Time

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Figure 3. Solution curves of the proportions of Susceptible, Exposed and Infectious mosquito populations when the recovery rate, rh , is 0.0014 and all other parameters are as shown on Table 1.

and behaviours of individuals and local residents by completing their malaria treatment, once diagnosed, and taking approved non-resistant prescribed drugs that can potentially completely kill the parasite. This is crucial since many malaria patients or their parents, in most endemic regions like Cameroon, do not complete their treatment and purchase nonprescribed malaria drugs without adequate information about their dosage or resistance to the parasite. Hence, a serious and active community based campaign is needed to educate the public about the risks involved in such bad practices and to encourage them to seek immediate intervention once they realize that they have been infected with the malaria parasite, and to purchase prescribed non-resistant drugs and completely take them following the recommended dose. Notice that in the above analysis, the recovery rate value of 0.0098 is still less than observed values of about 0.02 (as quoted in [31]), which is equivalent to an infectious period of about seven weeks. Achieving a higher recovery rate does mean a lower percentage of individuals that ultimately are infectious, and a shorter time at disease eradication, implying relative ease at controlling the disease. In fact, a recovery rate value of0.0154 (which is equivalent to an infectious period of about 9 weeks) gave R0 = 0.6186 < 1 (i.e., the DFE is at the steady state equilibrium) and led to a 37.7% reduction in the maximum infectious individuals and a shorter time to achieve disease eradication.

4.2. Quantification of the Impact the Contact Rates have on Malaria Prevalence and on Initial Disease Transmission In endemic regions, an individual is subjected to many bites from mosquitoes. Hence, though increasing the recovery rate is crucial, it is important to also include the impact of reducing the contact rates by reducing contacts between humans and mosquitoes, at a realistically and experimentally feasible recovery rate value. In this section, a recovery rate

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Miranda I. Teboh-Ewungkem 0.000013 0.000012 0.000011

Exposed and Infectious Host Populations

0.00001 0.000009 0.000008 0.000007 Exposed Host Infectious Host

0.000006 0.000005 0.000004 0.000003 0.000002 0.000001 0 0

5

10

15

20

25

30

35

40

Time

Figure 4. Long term population proportion profiles for the Exposed and Infectious host populations for the recovery rate, rh = 0.0056, with all other parameters as shown on Table 1.

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value rh = 0.0056 will be used in the analysis with the other parameters as shown on Table 1. For the analysis, the contact rates are measured by the average number of bites a mosquito makes (i.e., bites by susceptible mosquito on an infectious human or by an infectious mosquito on a susceptible human), times the probability that a bite will lead to an infection. The average number of bites can be reduced by preventing or reducing human contacts with mosquitoes. However, the probability that a bite (from a susceptible mosquito to an infectious human or from an infectious mosquito to a susceptible human) will lead to an infection is highly variable and can be a difficult target or control value. Here, a set of possible realistic feasible contact rate values (chv ∈ (0, 0.27] and cvh ∈ (0, 0.45], which includes values quoted by [4, 9 and 22] will be used. These values could be higher as quoted by others, see for example [ 22, 32]. In the simulations performed, the base contact rates are chv = 0.27 and cvh = 0.45, as shown on Table 1. Figures 8 and 9 show the profiles of the proportion of the host and vector populations over time for rh = 0.0056, and when the base contact rates are each reduced by 25%. At this recovery rate value, all the non-dimensional parameters stay the same as earlier defined, except for a reduction in α and κ. Their new reduced values are α = 7300.741 and κ = 2136.817. The basic reproductive number also drops from the previously obtained value of R0 = 1.683 > 1 to 0.946 < 1 (a drop of about 44%) and the stable equilibrium is the DFE. Moreover, in comparing the graphs obtained by using the base contact rates, Figure 4, to that obtained by using the 25% reduced contact rates, Figure 8, it can be seen that there is a slight decrease in the maximum number of infected humans in the population and this drop is about 10.7%. This seems to indicate that the contact rates may have a greater impact at reducing initial disease transmission (since R0 decreased by a higher percentage). If this is the case, then, for control strategy, targeting the contact rates by reducing contacts between mosquitoes and humans and thereby reducing the average number of bites a mosquito makes on a human might be a good strategy to control initial disease

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0.0016 0.0015

Susceptible, Exposed and Infectious Vector Populations

0.0014 0.0013 0.0012 0.0011

Susceptible Vector Exposed Vector Infectious Vector

0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 0

5

10

15

20 Time

25

30

35

40

Figure 5. Solution curves for the Susceptible, Exposed and Infectious mosquito population proportions for the recovery rate, rh = 0.0056. All other parameters are shown on Table 1.

transmission.

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4.3. Sensitivity Analyses To better understand the relative importance of the recovery rate and the contact rates to disease transmission and to disease prevalence in the Cameroon region, a sensitivity analyses is carried out on the model equations, eqns. (3.10-3.12) and (3.14-3.16), in which different feasible values of the recovery rate parameter and different sets of feasible values of the contact rate parameters are used in the simulation of the model equations. The corresponding basic reproductive numbers, endemic steady state proportions and maximum proportions of infectious hosts are computed and the results graphically shown and discussed. 4.3.1. Effects of the Recovery Rate and Contact Rates on R0 ; on the Equilibrium Infectious Host Proportion; and on the Maximum Infectious Host Proportion Effects on R0 : To determine what the best efforts of local communities in endemic regions like Cameroon are in their role to control malaria, it is important to know the relative effects and importance of the recovery rate and the contact rates in the control of malaria transmission and prevalence. The transmission at the onset of the disease is directly related to R0 and the prevalence of the disease to the endemic steady state prevalence and also to the maximum prevalence. The relationships between R0 and the recovery rate and also between R0 and the contact rates are examined and discussed below. In addition, the relationships between these rates to the equilibrium proportions and to maximum proportions of infectious humans at any given time are also examined and discussed below. The goal is to understand and identify the impact each of the parameters mentioned above have on initial disease transmission (via R0 ) and disease prevalence (via the infectious endemic host equilibrium and the maximum infectious host ) in an endemic region like Cameroon, using published data as is related to the region. For the simulation and analysis, all other

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Miranda I. Teboh-Ewungkem 0.000007

Exposed and Infectious Host Populations

0.000006

0.000005

0.000004

0.000003

Exposed Host Infectious Host

0.000002

0.000001

0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Time

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Figure 6. Long term population proportion profiles of the Exposed and Infectious host populations for the recovery rate, rh = 0.0098, with all other parameters as shown on Table 1.

parameters are kept fixed as shown on Table 1, with the base set of contact rates being {cvh = 0.45, chv = 0.27} (as given on Table 1). Figures 10-12 show the plots and profiles of the basic reproductive number, R0 , for different values of the recovery rate, rh , and for different sets of contact rates. Figure 10 shows the values of R0 as a function of the recovery rate and the contact rates, while Figure 11 shows the contour plots. Figure 12 shows the profiles ofR0 for the zoomed out recovery rate interval [0.00252, 0.0042]. From the graphs, it can be seen that decreasing the contact rates between mosquitoes and humans have a greater impact in reducing R0 and hence initial disease transmission than increasing the recovery rate. In fact, for fixed contact rates, doubling the recovery rate from 0.0014 to 0.0028 (same as reducing the infectious disease period from about 24 months to about 12 months) led to a 48.3% reduction in R0 , which for the base contact rates is a reduction of R0 from 6.40 to 3.31. However, at any fixed recovery rate, halving the contact rates from the base set {cvh = 0.45, chv = 0.27} to the set {cvh = 0.225, chv = 0.135} led to a 75% reduction in R0 , which for rh = 0.0014 is a reduction of R0 from 6.40 to 1.60. Moreover, even a 10% reduction in the base contact rates had a larger impact than a 10% increase in the recovery rate from rh = 0.0028, in reducing R0 , as evidenced by the profiles in Figure 12 (the width between any two adjacent curves is larger than the slope from rh = 0.0028 to rh = 0.00308 for any of the curves) The contour plots, Figure 11, show that for relatively low recovery rates (for examplerh = 0.0028) and for reasonably high contact rates (for example the base contact rates), disease eradication is possible, since R0 < 1, though this will be achieved over a longer time frame. Hence, reducing the contact rates have a higher impact at reducing initial disease transmission, and should be a target value in achieving this. Effects on the maximum infectious human population proportion: Figures 13 and 14 show the plots and profiles of the maximum proportion of infectious humans which occurs

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0.0024

Susceptible, Exposed and Infectious Vector Populations

0.0022 0.002 0.0018 0.0016 0.0014 Susceptible Vector Exposed Vector Infectious Vector

0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Time

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Figure 7. Solution curves for the Susceptible, Exposed and Infectious mosquito population proportions for the recovery rate, rh = 0.0098. All other parameters are shown on Table 1.

at some given time, for different values of the recovery rate, rh , and for different sets of contact rates. Figure 13 shows the values (as heights) in the non-dimensional variables of the maximum proportion of infectious humans while Figure 14 shows the profiles as a function of the recovery rate in the zoomed out interval[0.00252, 0.0042], for different sets of contact rates. If rh is kept constant, a reduction in the contact rates have very little impact in reducing the maximum proportion of infectious humans. However, this impact increases with smaller contact rates as seen in the profiles of Figure 14. Moreover, this impact is much less than the impact an increase in the recovery rate has on reducing the maximum proportion of infectious humans, over constant sets of contact rates. In fact, there is more than a 50% reduction in the maximum proportion of infectious individuals if the recovery rate is doubled from 0.0014 to 0.0028 but much less than that if the base contact rates are halved, Figure 13. It is important to note, however, that for any fixed set of contact rates, this impact decreases with increase in the recovery rate. Hence, increasing the recovery rate has a much higher impact in reducing the maximum proportions of infectious individuals in the community at a given time in the course of the infection, prior to reaching the endemic steady states, and should be a target value in achieving this. Effects on the equilibrium infectious human population: For this case, the impacts that can be discussed are those for which the DFE is unstable, i.e., R0 > 1, hence, a restricted range of values for rh is used. Figures 15 and 16 show the plots and profiles of the values of the endemic steady state for different sets of contact rates and for different recovery rate values in the interval [0.00252, 0.0042]. From Figure 15, it can be seen that the profiles for the endemic infectious human steady state proportions for a recovery rate value are concave down. Hence the effects the contact rates have on reducing the endemic infectious steady state human proportions increases for smaller sets of contact rates. Also, for a fixed set of contact rates, the profiles for the endemic infectious human steady state proportions with respect to increasing recovery rate values are concave up, indicating that effects the

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Miranda I. Teboh-Ewungkem 0.000011

Exposed and Infectious Host Population Proportions

0.00001 0.000009 0.000008 0.000007 0.000006 Exposed Host

0.000005

Infectious Host

0.000004 0.000003 0.000002 0.000001 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Time

Figure 8. Long term population proportion profiles of the Exposed and Infectious host populations for rh = 0.0056, chv = 0.2020 and cvh = 0.3375, with all other parameters as shown on Table 1.

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recovery rates have on reducing the endemic infectious steady state human proportions are much larger for smaller recovery rates. In addition, from both Figures 15 and 16, it can be seen that for high contact rates and very low recovery rates (in this case base contact rates and higher with about a 10 months infectious period and higher), a slight10% increase in the recovery rate impacts and affects (by reducing) the value of the equilibrium proportions of infectious individuals more than a corresponding 10% decrease in the contact rates between mosquitoes and humans. However, as the contact rates is reduced, by even only an additional 10% below the base contact rates and the recovery rate increases, the impact of decreasing the contact rates then have a stronger impact at reducing the equilibrium proportions of the infectious population. Hence, in general, after a long time and at a relatively higher recovery rate and smaller sets of contact rate values, the parameters with a larger impact in reducing disease prevalence by reducing the endemic equilibrium infectious value are the contact rates. In conclusion, reducing the contact rates between mosquitoes and humans will lead to a stronger impact at reducing R0 and initial disease transmission and hence, a better target for control measures. However, during the course of infection, the recovery rate has a much stronger impact at reducing disease prevalence by decreasing the maximum proportion of infectious humans in the population and strategically, should be a better target. However, over longer periods, the impact on the endemic infectious human proportion is mixed depending on how small the recovery rate is and how large the contact rates are between humans and mosquitoes. But, for modestly smaller contact rates and larger recovery rates, targeting the contact rates will lead to a more efficient control of the endemic infectious steady proportion of humans and hence prevalence. The results suggest that a combined effort based on using insecticide treated bed nets and mosquito repellent (safe for use by humans especially children) together with individual responsibilities of completing malaria

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127

Susceptible, Exposed and Infectious Vector Population Proportions

0.0024 0.0022 0.002 0.0018 0.0016 0.0014 Susceptible Vector Exposed Vector Infectious Vector

0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Time

Figure 9. Solution curves for the proportions of Susceptible, Exposed and Infectious mosquito populations for rh = 0.0056, chv = 0.2020 and cvh = 0.3375, with all other parameters as shown on Table 1.

treatment once diagnosed and seeking early diagnosis is the best control scheme on the part of local communities to achieve better disease control and possible malaria eradication.

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4.3.2. Effects of Recruitment Rate on R0 , on the Equilibrium Infectious Human Proportion, and on the Maximum Infectious Human Proportion In all of the above analyses, the recruitment rate,K, was taken to be 104 (days−1 ∗Number). This rate can vary depending on a number of conditions such as climatic and seasonal conditions, availability of mates, availability of breeding sites, survivability of eggs, larva, pupa and adult mosquitoes. The rate value of 104 was estimated from the formula K = b nζρ2 , where ρ1 represent the survival rate of adult female mosquitoes; N b the number ρ1 N of viable (those that successfully can be fertilized) female mosquitoes; n the number of eggs laid by a female mosquito; ρ2 the survival probability of the combined developmental stages from egg to larvae to pupae; and ζ the proportion of eggs that eventually develop into female mosquitoes. This formula is just an estimated formula of the mosquito recruitment rate. Here, the assumption is that the above mentioned conditions are the primary major factors affecting mosquito recruitment some of whose parameters are known. An adult female anopheline mosquito lays between 50 − 200 eggs (see [5] and [33]). Hence, a reasonable value for n is 125, the average of the given range. Charlwood et al., recorded survival rates between 0.834 and 0.849 per day for Anopheles gambiae (one of the anopheles species found in Cameroon) in Sao Tome and Principe Islands located in West Africa [34] and between 0.77 and 0.84 for Anopheles gambiae in Tanzania [35]. However, this range can differ greatly depending on the circumstances and the country and hence, the lower guess choice of ρ1 = 0.75. Also, if we assume that at any time, the number of b = 104 (a number taken to be less than the initial viable female mosquitoes present is N susceptible mosquito population as shown on Table 2), and that 50% of the eggs laid will

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Miranda I. Teboh-Ewungkem

6.5 6 5.5 5 4.5 4 R0

3.5 3 2.5 2 1.5 1 0.5

0.014

base contact rates reduced by10 percent base contact rates

0.0154

0.0112

base contact rates reduced by 20 percent

0.0126

recovery rate (rh)

base contact rates reduced by 30 percent

0.0098

0.007

base contact rates reduced by 50 percent base contact rates reduced by 40 percent

0.0084

0.0042

0.0056

0.0028

0.0014

0

Figure 10. R0 as a function of the recovery rate rh and different sets of contact rates with all other parameters as shown on Table 1. The base set of contact rates is {chv = 0.27, cvh = 0.45} .

base contact rates reduced by 50 percent

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base contact rates reduced by 40 percent

R0 1 C2 = a2 (a3 − a4 − a5 ) + a5 (a3 − a4 ) + a4 (a3 − a4 − a1 ) + a1 (a3 − a2 − a5 ) + (1 + B) A + (1 + β + B) αW1 C3 = a2 (a1 + a5 ) (a3 − a4 ) + a5 [a1 (a3 − a2 ) + a4 (a3 − a1 )] + a3 a4 (a1 + a2 ) + (1 + B) A (θ + σ + T + σ + αN1 + β + γ + W1 ) + (1 + β + B) αW1 (θ + σ + T + σ + δ + αN1 + B) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] C4 = a1 a2 a5 (a3 − a4 ) + a3 a4 (a1 a2 + a1 a5 + a2 a5 ) + (1 + B) A ((θ + σ) (T + σ) + (θ + σ) (αN1 + β + γ + W1 ) + (T +σ) (αN1 +β +γ +W1 )) + (1 + β + B) αW1 ((θ + σ) (T + σ) + (θ + σ) (δ + αN1 + B) + (T + σ) (δ + αN1 + B)) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] (θ + σ + T + σ) + θR [P (δ − αU1 ) + Q (δ − 2QαW1 )] C5 = a1 a2 a3 a4 a5 + (1 + B) A ((θ + σ) (T + σ) (αN1 + β + γ + W1 )) + (1 + β + B) αW1 ((θ + σ) (T + σ) (δ + αN1 + B)) + [(1 + β + B) A (δ − αW1 ) − (1 + B) BαW1 ] (θ + σ) (T + σ) − θR (1 − αN1 ) (Q + P ) δ + θR (P γδ + 2P αW1 + 2QαW1 + P δW1 − P αβU1 − P αγU1 + QαδU1 + 2P αβW1 ) +
− θR QαδW1 + P αU1 W1 + P α2 N1 U1 + 2Qα2 N1 W1 + Qα2 U1 W1 ,

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a1 = (θ + σ) ;

a2 = (T + σ) ;

a4 = (δ + αN1 + B) ;

a3 = 1 − α (N1 + U1 ) − A

a5 = αN1 + β + γ + W1 ;

a6 = (1 − (δ + αN1 )) .

References [1] World Health Organization (WHO) Report : Division and Control of Tropical Diseases, 2001-2010, United Nations Decade to Roll Back Malaria. (latest update2008). http://www.rbm.who.int/cmc upload/0/000/015/370/RBMInfosheet 3.htm [2] WHO/AFRO Press release, Roll Back Malaria (RBM): An Initiative in the Right Direction. (August–September, Online 2000). http://www.afro.who.int/textonly/press/english/2000/rc/rc5007.html [3] ATM-WHO/AFRO, Malaria: An online resource. (January, 2008). http://www.afro.who.int/malaria/vision mission.html [4] Wanji S., Tanke T., Atanga S. N., Ajonina C. and Nicholas T. (2003). Anopheles Species of the Mount Cameroon Regions: Biting Habits, Feeding Behaviour and Entomological Inoculation Rates. Tropical Medicine and International Health, 8(7), 643-649.

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[5] Center for Disease Control and Prevention (CDC): Anopheles Mosquitoes. (2008 update). http://www.cdc.gov/malaria/biology/mosquito/#eggs [6] Anderson, R.M. (1993). Epidemiology, In Modern Parasitology. (2nd ed.), (Edited by F.E.G.COX), pp 75-116. Oxford: Blackwell scientific publications. [7] Oakes, S.C. (1991). Malaria: Academies Press.

Obstacles and Opportunities, 25-36, National

[8] The Victorian Government Health Information, Blue book-Guidelines for the Control of Infectious Diseases: Malaria. (2005). http://www.health.vic.gov.au/ideas/bluebook/malaria.htm [9] Ngwa, G.A. and Shu, W.S. (2000). A Mathematical Model for Endemic Malaria with Variable Human Host and Mosquito Populations. Mathl. Comput. Modelling, 32, 747-763. [10] Ross, R. (1909). Report on the Prevention of Malaria in Mauritius, London, Churchhill. [11] Ross, R. (1911). The Prevention of Malaria, 2nd ed., Murray, London. [12] MacDonald, G. (1957). The Epidemiology and Control of Malaria, Oxford University Press, Oxford, UK. [13] Kermack, W.O. and McKendrick, A.G. (1927). Contributions to the Mathematical Theory of Epidemics. Proc. Royal Soc. London, 115, 700-721.

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[14] Kermack, W.O. and McKendrick, A.G. (1932). Contributions to the Mathematical Theory of Epidemics, part II. Proc. Royal Soc. London, 138, 55-83. [15] Kermack, W.O. and McKendrick, A.G. (1933). Contributions to the Mathematical Theory of Epidemics. part III, Proc. Royal Soc. London, 141, 94-112. [16] Bailey, N.T.J. (1957 2nd edition.in 1975). The Mathematical Theory of Infectious Diseases, Hafner, New York. [17] Dietz, K., Molineaux, L. and Thomas, A. (1974). Malaria model tested in the African Savannah. Bulletin of the World Health Organization, 50, 347-357. [18] Nedelman, J. (1984). Inoculation and recovery rates in the model of Dietz, Molineaux and Thomas. Math. Biosci. 69, 209-233. [19] Wikan, A. and Mjolhus, E. (1995). Periodicity of 4 in Age-Structured Population Models with Density Dependence. J. Theo. Biol., 173(2),109-119. [20] Lloyd, A.L. and May, R.M. (1996). Spatial Heterogeneity in Endemic Models. J. Theo. Biol., 176, 1-11. [21] Encyclopedic reference of Parasitology, Disease. Treatment. Therapy. (2nd Ed.), (2001). Edited by Heinz Mehlhorn. Springer, pp 302. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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139

[22] Nakul C., Hyman, J.M. and Cushing, J.M. (2008). Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model. Bulletin of Mathematical Biology. 70: 1272–1296. [23] Gu, W., Mbogo, C.M., Githure, J.I., Regens, J.L., Killeen, G.F., Swalm, C.M., Yanf, G. and Beier, J.C. (2003). Low recovery rates stabilize malaria endemicity in areas of low transmission in coastal Kenya. Acta Tropica, 86(1), 71-81. [24] Index Mundi, Home of the Internet’s most Complete Country Profiles. (2008 estimates). http://www.indexmundi.com [25] Newton, E.A. and Reiter, P. (1992). A Model of the Transmission of Dengue Fever with an Evaluation of the Impact of Ultra-Low volume (ULV) Insecticide Applications on Dengue Epidemics, Am. J. Trop. Med. Hyg., 47(6), 709-720. [26] Johnson, L.W. and Reisz, R.D. (1982). Numerical Analysis, (2nd Ed). Addison Wesley, pp 185-189. [27] May, R. (1973). Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, N.J. [28] Edelstein-Keshet, L. (2004). Mathematical Models in Biology. SIAM . [29] Central Intelligence Agency, CIA—The World Factbook. (July 2008 estimates). https://www.cia.gov/library/publications/the-world-factbook/index.html

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[30] Globalis-Cameroon: Malaria-related mortality rate: All ages, 2000 estimate. (Viewed, 2008. http://globalis.gvu.unu.edu/indicator detail.cfm?IndicatorID=72&Country=CM [31] Trape, T.J. (1993). Estimation of the Parasitological Incidence and Recovery Rates of Plasmodium falciparum, P. Malariae and P. Ovale among Children Living in a Malarial Holoendemic Zone in Congo. Bull. Soc. Pathol. Exot, 86(4), 248-253. [32] Teboh-Ewungkem, M.I., Podder, C.N. and Gumel, A.B. (submitted 2008). Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics. [33] Encyclopedic reference of Parasitology, Biology. Structure. Function. (2nd Ed.), (2001). Edited by Heinz Mehlhorn. Springer, pp 381. [34] Charlwood, J.D., Pinto, J., Sousa, C.A, Ferreira, C., Petrarca, V. and do E Rosario, V. (2003). A mate or a meal’ – Pre-gravid Behaviour of Female Anopheles gambiae from the islands of S˜ao Tom´e and Pr´ıncipe, West Africa. Malaria Journal, 2(9), 1-11. [35] Charlwood, J.D., Smith, T., Billingsley, P.F., Takken, W., Lyimo, E.O.K. and Meuwissen, J.H.E.T. (1997). Survival and Infection Probabilities of Anthropophagic Anophelines from an Area of High Prevalence of Plasmodium falciparum in Humans, Bulletin of Entomological Research, 87, 445-453.

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[36] Foko, D.G. A., Messi, J. and Tamesse, J.L. (2007). Influence of Water Type and Commercial Diets on the Production of Anopheles gambiae Giles, under Laboratory Conditions, Pakistan Journal of Biological Sciences, 10(2), 280-286.

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[37] Anderson, R.M. and May, R.M. (1991). Infectious Diseases of Humans: InDynamics and Control. Oxford, Oxford University Press.

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 141-170

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Chapter 5

A PPLICATION OF O PTIMAL C ONTROL TO THE E PIDEMIOLOGY OF HIV-M ALARIA C O - INFECTION F.B. Agusto Department of Mathematical Sciences, Federal University of Technology Akure, Akure, Nigeria

Abstract

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Optimal control theory is applied to a system of ordinary differential equation describing HIV-malaria co-infection in individuals. Seeking to reduce the groups of individuals infected with malaria and the vector population, control representing treatment and use of insecticide is applied. The existence and uniqueness results of the solutions are discussed. The optimal controls are characterized in terms of the optimality system, which is solved numerically for several scenarios.

Keywords: Optimal control, malaria, HIV/AIDS, optimality system. AMS Subject Classification: 92B05, 93A30, 93C15.

1. Introduction HIV (human immunodeficiency virus) is the virus that causes AIDS (Acquired Immune Deficiency Syndrome). It kills or damages cells in the body’s immune system, gradually destroying the bodys ability to fight infection. HIV has killed more than 25 million people since it was first recognized in 1981, making it one of the most destructive epidemics in recorded history [31]. It remains one of the leading cause of death in the world, with its effect most devastating in Sub-Saharan Africa, where HIV prevalence can range between 12% to 42% [27]. One of the key factors that fuels the high incidence of HIV in SubSaharan Africa is the dual infection with malaria [1]. Malaria and HIV are among the two most important global health problems of the present time [32]. Together, they cause more than four million deaths per year. Malaria accounts for more than a million deaths each year, of which about 90% occur in tropical Africa, where malaria is the leading cause of mortality in children below five years. Aside from young children, pregnant women are

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among the most affected by the disease. Constituting 10% of the overall disease burden, malaria places a substantial strain on health services and costs Africa about USD 12 billion in lost production each year [32]. HIV has been shown to increase the risk of malaria infection and accelerate the development of clinical symptoms of malaria, with the greatest impact in immune-suppressed persons. Conversely, malaria has been shown to induce HIV-1 replication in vitro and in vivo. A biological explanation for these interactions lies in the cellular-based immune responses to HIV and malaria. Studies have shown that when HIV-infected individuals are attacked by malaria, their body immune system weakens significantly, creating a conducive environment for the HIV virus to replicate (virtually unchallenged), resulting in an increase in the viral load (the amount of HIV virus in the body). Hence, since viral load is correlated with infectiousness [34], such a process (co-infection with malaria) leads to an increase in the number of new HIV cases in the population. HIV-positive patients with low CD4 cell counts tend to have an increased risk of malaria treatment failure when compared to both HIV-negative patients and HIV-positive patients with higher CD4 cell counts [28]. Emerging evidence indicates that antimalarial drugs may be less efficacious in people living with HIV. A small study in Ethiopia found that HIV-infected adults had an increased parasite and fever clearance time following artemisinin treatment for uncomplicated malaria [23] consistent with the view that the host immune response to parasites is important in determining response to therapy. Furthermore, a low CD4-cell count may predict a poorer response to antimalarial treatment. In a randomized controlled trial in Zambia, HIV-infected adults with CD4 counts less than 300 cells/ml who received antimalarial drugs for uncomplicated malaria had significantly higher rates of parasitological treatment failure to both sulfadoxine-pyrimethamine and artemetherlumefantrine [33]. Macdonald [22] identified mosquito vector longevity as the single most important variable in the force of transmission, if mosquitoes life-span is reduce by the use of insecticide, the incidence of Malaria can be minimized. There are a number of interventions available for people living with HIV/AIDS that can prevent the devastating effects of malaria such as sleeping under an insecticide-treated bed net (ITN), which repels and kills malariatransmitting mosquitoes [6] and receiving treatment with effective antimalarial medications. In this paper, we consider (time dependent) optimal control strategies associated with improving response to antimalarial drugs and use of insecticide to reduce the vector population for the model of Mukandavire et.al. [24]. However, this model did not consider the dynamical control strategies since their discussions are based on prevalence of disease at equilibrium. The time dependent control strategies have been applied in various studies of HIV models [16], tuberculosis models [14] and others [34]. Two control mechanisms representing insecticide use and improve drug response are introduced into the model. The insecticide use is incorporated by adding a control term that reduces a fraction of the vector population. The improved drug response is incorporated by adding a control term that increases treatment response rate of malaria infected HIV/AIDS individuals. Our objective functional balances the effect of minimizing the singly and dually infected individuals with malaria and minimizing the cost of implementing the controls. To proceed, model [24] is modified by including three classes, that is, class of recovered individuals singly infected with malaria and those with dual infection, as well as the class of removed vectors. This chapter is organized as follows: Section 2. modifies the model in [24] followed by

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analysis on the modification to gain insights into the qualitative features of its associated equilibria. Section 3. is concern with the existence and characterization of the optimal controls. Section 4. includes some numerical simulations of the optimal controls and discusses the results. The numerical codes are included in the Appendix

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2. Model Formulation The model in [24] is modified to include recovery classes, that is, class of recovered individuals singly infected with malaria and those with dual infection. The notations and the analysis in this section are closely related to those in [24]. The class RH of recovered individuals singly infected with malaria is generated following recovery of singly infected individuals with malaria. The populations is diminished by loss of malaria immunity (at the rates αM ) and by natural death (at the rate µH ). The class RAM of recovered individuals dually infected with HIV and malaria is generated following recovery of dually infected individuals with HIV and malaria. The populations is diminished by loss of malaria immunity (at the ratesαH and αA , where αH is the rate of immunity loss by HIV individuals and αA is the rate of immunity loss by individuals displaying AIDS) and by natural death (at the rateµH ) and AIDS induced death (at the rate δH ). Controls u1 (t) and u2 (t) are introduced to the above model, where u1 (t) ( 0 ≤ u1 (t) ≤ 1) is the control functions representing spray of insecticide aimed at reducing(1−p) fraction of the mosquitoes sub-populations. And (1 + u2 (t)) (0 ≤ u2 (t) ≤ 1) is the control on malaria treatment to improve response to malaria drugs. The class RV M of removed vectors is generated following removal of susceptible, exposed and infected vectors on applying control u1 (t). The population is diminished by natural death (at the rate µV ). Putting the above formulations and assumptions together gives the following system of differential equations for the modified model, which is given below as: dSH = ΛH + αM RH − λH SH − λM SH − µH SH , dt dEM = λM SH − λH EM − (γH + µH )EM , dt dIM = γH EM − σλH IM − (φ1 + δM + µH )IM dt dIH = λH SH + αH RAM − ϑλM IH − (κ + µH )IH , dt dEHM = λH EM + ϑλM IH − (γH + κ + µH )EHM , dt dIHM = σλH IH + γH EHM − ((1 + u2 (t))φ2 + ξκ + µH + τ δM )IHM , dt dAH = κIH + αA RAM − ϑλM AH − (µH + δH )AH , dt dEAM = ϑλM AH + κEHM − (γH + µH + δH )EAM , dt

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(1)

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F.B. Agusto

dAHM = γH EAM + ξκIHM − ((1 + u2 (t))φ3 + µH + ψδH + τ δM )AHM , dt dRH = φ1 IM − (αM + µH )RH , dt dRAM = (1 + u2 (t))φ2 IHM + (1 + u2 (t))φ3 AHM − (αA + αH + µH + δH )RAM , dt dSV = ΛV − λV SV − u1 (t)(1 − p)SV − µV SV , dt dEV = λV SV − u1 (t)(1 − p)EV − (γV + µV )EV , dt dIV = γV EV − u1 (t)(1 − p)IV − µV IV , dt dRV M = u1 (t)(1 − p)SV + u1 (t)(1 − p)EV + u1 (t)(1 − p)IV − µV RV M . dt

2.1. Analysis of the HIV-Malaria Model 2.1.1. Invariant Regions Model (1) monitors human and vector populations and so all variables and parameters of the model are non-negative. The HIV-malaria model (1) will be analyzed in a biologicallyfeasible region as follows. The system (1) is split into two parts, namely the human population (NH ; with NH (t) = SH + EM + IM + IH + EHM + IHM + AH + EAM + AHM + RH + RAM ), and vector population (NV ; with NV = SV + EV + IV + RV M ). Consider the feasible region 4 Γ = ΓH ∪ ΓV ⊂ R11 + × R+

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with, ΓH

=

and ΓV

=

(

(

(SH , EM , IM , IH , EHM , IHM , AH , EAM , AHM , RH , RAM ) ∈ R11 +

ΛV (SV , EV , IV , RV M ) ∈ R4+ : NV (t) ≤ µV

ΛH : NH (t) ≤ µH

)

.

The following steps are followed to establish the positive invariance ofΓ (i.e., solutions in Γ remain in Γ for all t > 0). The rate of change of the total vector population is obtain by adding the first eleven equations and the last four equations of the model (1) and this gives dNH (t) = ΛH − µH NH (t) − δM (IH (t) + τ IHM (t) + τ AHM (t)) dt −δH (AH (t) + EAM (t) + ψAHM (t) + RAM ), dNV (t) = ΛV − µV NV (t). dt and it follows that dNH (t) ≤ ΛH − µH NH (t), dt dNV (t) = ΛV − µV NV (t). dt

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(2)

(3)

)

Application of Optimal Control...

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A standard comparison theorem [20] can then be used to show that NH (t) ≤ −µH t ) and N (t) ≤ N (0)e−µV t + ΛV (1 − e−µV t ). In parH NH (0)e−µH t + Λ V V µH (1 − e µV

ΛV ΛH ΛV H ticular, NH (t) ≤ Λ µH and NV (t) ≤ µV , if NH (0) ≤ µH and NV (0) ≤ µV , respectively. Thus, the region Γ is positively-invariant. Hence, it is sufficient to consider the dynamics of the flow generated by (1) in Γ. In this region, the model can be considered as being epidemiologically and mathematically well-posed [12]. Thus, every solution of the basic model (1) with initial conditions in Γ remains in Γ for all t > 0. Therefore, the ω-limit sets of the system (1) are contained in Γ. This result is summarized below. 4 Lemma 1. The region Γ = ΓH ∪ ΓV ⊂ R11 + × R+ is positively-invariant for the basic model (1) with non-negative initial conditions in R15 +.

2.1.2. Analysis of the Sub-models Before analyzing the full model (1), it is instructive to gain insights into the dynamics of the models with HIV only (HIV-only model) and malaria only (Malaria-only model). HIV-only Model The model with HIV only (obtained by setting EH = IM = EHM = IHM = EAM = AHM = RH = RAM = SV = EV = IV = RV M = 0 in (1)) is given by dSH = ΛH − λH SH − µH SH , dt

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dIH = λH SH − (κ + µH )IH , dt

(4)

dAH = κIHH − (µH + δH )AH , dt where, λH =

βH (IH +ηAM AH ) , NH

DH

and NH = SH + IH + AH . Consider the region

  NH 3 = (SH , IH , AH ) ∈ R+ ≤ . µH

It can be shown that all solutions of the system (4) starting in DH remain in DH for all t ≥ 0. Thus, DH is positively-invariant (hence, it is sufficient to consider the dynamics of (4) in DH ). Stability of the Disease-Free Equilibrium (DFE) The HIV-only model (4) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by ! Λ H ∗ ∗ , IH , A∗H ) = , 0, 0 . E0H = (SD µH

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F.B. Agusto

The linear stability of E0H can be established using the next generation operator method on the system (4). Using the notations in [30], the matrices F and V , for the new infection terms and the remaining transfer terms, are, respectively, given by,     0 βH βH ηAM (κ + µH ) F = and V = . −κ (µH + δH ) 0 0 It follows that the basic reproduction number of the HIV-only system (4), denoted by RH , is given by βH (κηAH + µH + δH ) , (5) RH = ρ(F V −1 ) = (κ + µH )(µH + δH ) Further, using Theorem 2 in [30], the following result is established. Lemma 2. The DFE of the HIV-only model (4), given by E0H , is locally asymptotically stable (LAS) if RH < 1, and unstable if RH > 1. The basic reproduction number (RH ) measures the average number of new infections generated by a single infected individual in a completely susceptible population [2, 9, 12, 30]. Thus, Lemma 2 implies that HIV can be eliminated from human population (when RH < 1) if the initial sizes of the sub-populations are in the basin of attraction of the DFE, E0H . Malaria-only Model Consider the Malaria-only model (obtained by settingIH + EHM + IHM + AH + EAM + AHM + RAM = 0 in (1)), given by

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dSH = ΛH + αM RH − λM SH − µH SH , dt dEH = λM SH − (γH + µH )EM , dt dIM = γH EM − (φ1 + µH + δM )IM dt dRH = φ1 IM − (αM + µH )RH , dt (6) dSV = ΛV − λV SV − u1 (1 − p)SV − µV SV , dt dEV = λV SV − u1 (1 − p)EV − (γV + µV )EV , dt dIV = γV EV − u1 (1 − p)IV − µV IV . dt dRV M = u1 (1 − p)SV + u1 (1 − p)EV + u1 (1 − p)IV − µV RV M . dt Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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where, NH = SH + EM + IM + RH + RHM and NV = SV + EV + IV . Consider the region DM = DH ∪ DV ⊂ R5+ × R3+ with, DH

=

and DV

=

(

(

(SH , EM , IM , RH , RHM ) ∈

(SV , EV , IV ) ∈

R3+

R5+

ΛH : NH (t) ≤ µH

ΛV : NV (t) ≤ µV

)

)

,

.

It can be shown that all solutions of the system (6) starting inDM remain in DM for all t ≥ 0. Thus, DM is positively-invariant (hence, it is sufficient to consider the dynamics of (6) in DM ) Stability of the Disease-Free Equilibrium (DFE) The Malaria-only model (6) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by ! ΛH ΛV ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , 0, 0, 0, 0, , 0, 0 . E0M = (SH , EM , IM , RH , RHM , SV , EV , IV ) = µH µV The matrices M and Q, for the new infection terms and the remaining transfer terms, are, respectively, given by,

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

0  0 M =  0 0 and



0 0 βV bM µH ΛV ΛH µV

k1  −γH Q=  0 0

0

 0 βM bM  0 0   0 0 0 0

0 0 k2 0 0 k3 0 −γV

 0 0  . 0  k4

k1 = γV + µH , k2 = φ1 + µH + δM , k3 = u1 (1 − p) + γV + µV , k4 = u1 (1 − p) + µV It follows that the basic reproduction number of the Malaria-only system (6), denoted by RV , is given by s b2M βV βH γH γV µH ΛV . (7) RM = ρ(M Q−1 ) = ΛH µV k1 k2 k3 k4 Further, using Theorem 2 in [30], the following result is established. Lemma 3. The DFE of the Malaria-only model (6), given by E0M , is locally asymptotically stable (LAS) if RM < 1, and unstable if RM > 1.

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2.2. Local Stability of the DFE of the HIV-Malaria Model The HIV-Malaria model (1) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , EM , IM , IH , EHM , IHM , A∗H , EAM , A∗HM , RH , RAM , SV∗ , EV∗ , IV∗ , RM E0 = (SH V) ! ΛH ΛV = , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, , 0, 0, 0 . µH µV

It is easy to show, using the next generation method (as in Sections 2.1.2. and 2.1.2.), that the associated reproduction number for the full HIV-malaria model (1) (denoted byRHM ) is given by RHM = max{RH , RM }

(8)

where, RH and RM are as defined before, so that the following result is established using the Theorem 2 in [30]. Lemma 4. The DFE of the HIV-Malaria model (1), given by E0 , is locally asymptotically stable (LAS) if RHM < 1, and unstable if RHM > 1.

3.

Analysis of Optimal Control

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In order to achieve the aim of the study, we give the objective functional which minimizes the number of individuals showing symptoms of malaria and the cost of applying the controls u1 , u2 as  Z tf  2 2 J = min AIM + BIHM + CAHM + Du1 + Eu2 dt (9) u1 ,u2

0

where A, B, C, D, E are positive weights. With the given objective function J(u1 , u2 ); our goal is to minimize the number of infective IM (t), IHM (t) and AHM (t), while minimizing the cost of control u1 (t), u2 (t). We seek an optimal control u∗1 , u∗2 such that J(u∗1 , u∗2 ) = min{J(u1 , u2 )|u1 , u2 ∈ U}

(10)

where U = {(u1 (t), u2 (t)) | (u1 (t), u2 (t)) measurable, ai ≤ (u1 (t), u2 (t)) ≤ bi , i = 1, 2, t ∈ [0, tf ]} is the control set.

3.1.

Existence of Optimal Control

The boundedness of solution of the system (1) for the finite time interval is used to prove the existence of an optimal control using a result by Fleming and Rishel [11]. Theorem 1. Consider the control problem associated with the model system (1). There exists (u∗1 , u∗2 ) ∈ U such that min

(u∗1 ,u∗2 ) ∈ U

J(u1 , u2 ) = J(u∗1 , u∗2 ).

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Proof. To use an existence result from [11], we check the following properties: 1. The set of controls and corresponding state variables is nonempty. 2. The control set U, stated above, is convex and closed. 3. The right hand side of the state system (1) is bounded by a linear function in the state and control variables. 4. The integrand of the objective functional (9) is convex onU. 5. There exists constants c1 , c2 > 0 and β > 1 such that the integrand L(y, u, t) of the objective functional satisfies, L(y, u, t) ≤ c1 + c2 (|u1 (t)|2 + |u2 (t)|2 )β/2 .

(11)

In order to verify these conditions, we use a result by Lukes [21] to give the existence of solutions of the ODE system of equations (1) with bounded coefficients, which gives condition 1. We note that the solutions are bounded. Our control set satisfies condition 2. Since our system is bilinear in (u1 , u2 ), the right hand side of system(1) satisfies condition 3, using boundedness of the solutions. Note that the integrand of our objective function is convex. Also we have the last condition needed, L(y, u, t) = AIM + BIHM + CAHM + Du21 + Eu22 ≤ c1 + c2 (|u1 (t)|2 + |u2 (t)|2 ) (12) where c1 depends on the upper bounded of y, and c2 = sup(u1 , u2 ) since β = 2. We conclude that there exist an optimal control.

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3.2.

Characterization of Optimal Controls

The necessary conditions that an optimal pair must satisfy come from the Pontryagin’s Maximum Principle[26]. This principle converts (1) and (9) into a problem of minimizing pointwise a Hamiltonian H, with respect to (u1 , u2 ). First we formulate the Hamiltonian from the cost functional (9) and the governing dynamics (1) to obtain the optimality conditions. H = AIM + BIHM + CAHM + Du21 + Eu22   + λSH ΛH + αM RH − λH SH − λM SH − µH SH   + λEM λM SH − λH EM − (γH + µH )EM   + λIM γH EM − σλH IM − (φ1 + µH + δM )IM   + λIH λH SH + αH RAM − ϑλM IH − (κ + µH )IH   + λEHM λH EM + ϑλM IH − (γH + κ + µH )EHM

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150

F.B. Agusto   + λIHM σλH IH + γH EHM − ((1 + u2 )φ2 + ξκ + µH + τ δM )IHM   + λAH κIH + αA RAM − ϑλM AH − (µH + δH )AH   + λEAM ϑλM AH + κEHM − (γH + µH + δH )EAM   + λAHM γH EAM + ξκIHM − ((1 + u2 )φ3 + µH + ψδH + τ δM )AHM   + λRH φ1 IM − (αM + µH )RH ,   + λRAM (1 + u2 )φ2 IHM + (1 + u2 )φ3 AHM − (αA + αH + µH + δH )RAM   + λSV ΛV − λV SV − u1 (1 − p)SV − µV SV   + λEV λV SV − u1 (1 − p)EV − (γV + µV )EV   + λIV γV EV − u1 (1 − p)IV − µV IV   (13) + λRM V u1 (1 − p)SV + u1 (1 − p)EV + u1 (1 − p)IV − µV RV M .

where the λSH , λEM , λIM , λIH , λEHM , λIHM , λAH , λEAM , λAHM , λRH , λRAM , λSV , λEV , λIV , λRM V are the associated adjoints for the states SH , EM , IM , IH , EHM , IHM , AH , EAM , AHM , RH , RAM , SV , EV , IV , and RM V , respectively. The system of equations is found by taking the appropriate partial derivatives of the Hamiltonian (13) with respect to the associated state variable. ∗ , E∗ , I ∗ , I ∗ , E∗ ∗ Theorem 2. Given optimal control u∗1 , u∗2 and solutions SH M M H HM , IHM , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ AH , EAM , AHM , RH , RAM , SV , EV , IV , RM V of the corresponding state system (1) that minimizes J(u1 , u2 ) over U, there exist adjoint variables λSH , λEM , λIM , λIH , λEHM , λIHM , λAH , λEAM , λAHM , λRH , λRAM , λSV , λEV , λIV , λRM V satisfying

−

∂H dλi = dt ∂i

(14)

and with transversality conditions λi (tf ) = 0, where i = SH , EM , IM , IH , EHM , IHM , AH , EAM , AHM , RH , RAM , SV , EV , IV , RM V (15)

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u∗1 = min

b1 , max a1 ,

(1 − pp)(SV λSV + EV λEV + IV λIV ) − (1 − pp)(SV + EV + IV )λRM V 2D

u∗2 = min

b2 , max a2 ,

IHM λIHM + AHM λAHM − (φ2 IHM + φ3 AHM )λRAM 2E

,

. (16)

Proof. Corollary 4.1 of [11] gives the existence of an optimal control pair due to the convexity of the integrand of J with respect to u1 , u2 , a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control pair. Then the adjoint system can be written as, −

dλSH dt

∂H , ∂SH

=

λSH (tf ) = 0,

··· dλRAM − dt

=

∂H , ∂RAM

dλSV dt

=

∂H , ∂SV

−

λRAM (tf ) = 0, λSV (tf ) = 0,

··· dλRV M − dt

=

∂H , ∂RV M

λRV M (tf ) = 0,

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evaluated at the optimal control pair and corresponding states, which result in the stated adjoint system (14) and (15). By considering the optimality conditions,

0=

∂H ∂H and 0 = . ∂u1 ∂u2

Solving for u∗1 , u∗2 , subject to the constraints, the characterization (16) can be derived and we have ∂H = 2Du∗1 − (1 − p)(SV∗ λSV + EV∗ λEV + IV∗ λIV ) + (1 − p)(SV∗ + EV∗ + IV∗ )λRM V ∂u1 ∂H ∗ ∗ 0= = 2Eu∗2 − IHM λIHM − A∗HM λAHM + (φ2 IHM + φ3 A∗HM )λRAM . ∂u2 (17) Hence, we obtain (see [19]) 0=

u∗1 = u∗2

(1 − p)(SV∗ λSV + EV∗ λEV + IV∗ λIV ) − (1 − p)(SV∗ + EV∗ + IV∗ )λRM V , 2D

∗ I ∗ λI + A∗HM λAHM − (φ2 IHM + φ3 A∗HM )λRAM = HM HM . 2E

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(18)

152

F.B. Agusto Let

A = B =

(1 − p)(SV∗ λSV + EV∗ λEV + IV∗ λIV ) − (1 − p)(SV∗ + EV∗ + IV∗ )λRM V , 2D ∗ IHM λIHM

+

A∗HM λAHM

∗ (φ2 IHM

− 2E

+

φ3 A∗HM )λRAM

(19)

.

Then, by Standard control arguments involving the bounds on the controls, we conclude  a A ≤ a1    1 A a1 < A < b1 , (20) u∗1 =    b1 A ≥ b1 In compact form u∗1

= min 1,

(1 − p)(SV∗ λSV + EV∗ λEV + IV∗ λIV ) − (1 − p)(SV∗ + EV∗ + IV∗ )λRM V 2D

,

Similarly, we conclude that  a    2 u∗2 = B    b2

B ≤ a2 a2 < B < b2

(21)

B ≥ b2 .

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In compact form   ∗ ∗ IHM λIHM + A∗HM λAHM − (φ2 IHM + φ3 A∗HM )λRAM ∗ u2 = min 1, . 2E Due to the a priori boundedness of the state and adjoint functions and the resultingLipschitz structure of the ODE’s, we obtain the uniqueness of the optimal control for smalltf . The uniqueness of the optimal control pair follows from the uniqueness of the optimality system, which consists of (1), (14) and (15) with characterization (16). There is a restriction on the length of the time interval in order to guarantee the uniqueness of the optimality system. This smallness restriction of the length on the time is due to the opposite time orientations of (1), (14), and (15); the state problem has initial values and the adjoint problem has final values. This restriction is very common in control problems (see [10, 13, 16, 17, 18]). Next we discuss the numerical solutions of the optimality system and the corresponding optimal control pairs, the parameter choices, and the interpretations from various cases.

4.

Numerical Results

In this section, we study numerically an optimal control for the HIV/Malaria model. The optimal control is obtained by solving the optimality system, consisting of ODE’s from the state and adjoint equations. An iterative scheme is used for solving the optimality system. We start to solve the state equations with a guess for the controls over the simulated time using fourth order Runge-Kutta scheme. Because of the transversality conditions (15), the

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adjoint equations are solved by a backward fourth order Runge-Kutta scheme using the current iterations solutions of the state equation. Then, the controls are updated by using a convex combination of the previous controls and the value from the characterizations (16). This process is repeated and iterations is stopped if the values of the unknowns at the previous iterations are very close to the ones at the present iterations [19].

Total number of dually infected individuals

140 130 uncontrol

120 110

control 100 90 80 70 60

0

2

4

6

8

10 12 Time (days)

14

16

18

20

0.9 0.8

control u2

0.7 0.6 0.5 0.4 0.3 0.2

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0

5

10 Time (days)

15

20

Figure 1. Simulations of the HIV/Malaria model (1) showing total number of dually infected individuals. For the figures presented here, we assume that the weight factor C associated with AHM is greater than the weight factors B and A associated with IHM and IM . Also the weight factor E associated with control u2 is greater than D which is associated with control u1 . This assumption is based on the facts that the cost associated withu1 will include the cost of insecticide and insecticide treated bed nets, and the cost associated withu2 will include the cost of antimalaria and HIV antiretroviral drugs. Treating a dually infected individual takes longer than treating a singly infected individual. In the figures, the set of the weight factors, A = 50, B = 400, C = 500, D = 50 and E = 500 and initial state variables SH (0) = 40, EM (0) = 30, IM (0) = 40, IH (0) = 30, EHM (0) = 30, IHM (0) = 30, AH (0) = 30, EAM (0) = 30, AHM (0) = 30, RH (0) = 30, RAM (0) = 30, SV (0) = 90, EV (0) = 10, IV (0) = 50, RM V (0) = 20, are chosen to illustrate the optimal treatment response strategy. Other epidemiological and numerical parameters are presented in Table 1. Figure 1 shows the simulation for total number of dually infected individuals IHM + AHM , with improve drug response (control u2 at the upper bound for 21 days) and when

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F.B. Agusto

Total number of dually infected individuals recovered from malaria

drug response is not boosted. The total number of individuals IHM + AHM infected with malaria at time tf = 21 (days) is 137.1068 in the case with control and 148.8975 without control and the total cases of malaria prevented in dually infected individuals is 11.7907 (148.8975 − 137.1068). The simulation for total number of dually infected individuals who have recovered from malaria is depicted in Figure 2, in the case with control and without control at timetf = 14 (days). The total number recovered RAM with improved drug response is 33.2841 and 30.7629 without the improved response.

90

80

70

control

60

50

40

30

uncontrol

0

2

4

6 8 Time (days)

10

12

14

0

2

4

6 8 Time (days)

10

12

14

0.9 0.8

control u2

0.7 0.6 0.5 0.4

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0.3 0.2

Figure 2. Simulations of the HIV/Malaria model (1) showing total number of dually infected individuals recovered from malaria. A scenario with reducing different fraction of vector population is simulated in Figure 3. The result shows that the value of p = 0.2 (which may be achieve by a combination of insecticide and insecticide treated bed nets) gave the largest number of removed (RM V ) vector population, this is followed by p = 0.6, p = 0.85 and lastly by p = 1 (a case corresponding to no use or ineffective insecticide) as expected. This has the resultant effect (not depicted here) on total number of dually infected individuals exposed to malaria EHM + EAM . When p = 1, the total number of dually infected individuals exposed to malaria, EHM + EAM is 25.2637, when p = 0.85, EHM + EAM = 21.6372, when p = 0.6, EHM + EAM = 16.3878 and lastly when p = 0.2, the total number of dually infected individuals exposed to malaria, EHM + EAM is 13.4317. A similar resultant effect is observed for the total number of dually infected individuals IHM + AHM . When p = 1, the total number of dually infected individuals, IHM + AHM is 148.8975,

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120 p = 0.2

Total number of removed vectors

100 p = 0.6 80

60

p = 0.85

40

20

0

p=1

0

2

4

6 8 Time (days)

10

12

14

Figure 3. Simulations of the HIV/Malaria model (1) showing the removed vectors with various fraction (p = 1, 0.85, 0.6 and p = 0.2).

Total number of dually infected individuals

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140 130

u1 = 0, u2 ≠ 0

120 110

u1 ≠ 0, u2 ≠ 0

100 90 80 70 60

0

5

10 Time (days)

15

20

Figure 4. Simulations of the HIV/Malaria model (1) showing total number of dually infected individuals with u1 = 0 and u1 6= 0. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

F.B. Agusto Total number of dually infected individuals recovered from malaria

156

33

32.5 u = 0, u ≠ 0 1

2

32

31.5

u ≠ 0, u ≠ 0 1

2

31

30.5

30

0

5

10 Time (days)

15

20

Total number of dually infected individuals recovered from malaria

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Figure 5. Simulations of the HIV/Malaria model (1) showing total number of dually infected individuals recovered from malaria with u1 = 0 and u1 6= 0.

32.5 u ≠ 0, u ≠ 0 1

2

32

31.5

31

30.5

u1 ≠ 0, u2 = 0

30 0

5

10 Time (days)

15

20

Figure 6. Simulations of the HIV/Malaria model (1) showing total number of dually infected individuals recovered from malaria with u2 = 0 and u2 6= 0. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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157

when p = 0.85, IHM + AHM = 137.1068, when p = 0.6, IHM + AHM = 125.5911 and lastly when p = 0.2, the total number of dually infected individuals, IHM + AHM is 117.3512. A similar resultant effect is observed for the recovered dually infected individuals, RAM , with RAM = 33.2841 for p = 0.85 giving the largest recovered individuals and RAM = 32.7834 for p = 0.2 giving the least recovered individuals and these numbers stem from the amount of individuals infected with malaria due to the population of the vectors.

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Table 1. Description of Variables and Parameters of the HIV-Malaria Model (1). Parameter

Description

Baseline value

Reference

ΛH

Recruitment rate for humans

5 × 10−2 per day

[24]

ΛV

Recruitment rate for mosquitoes

6 per day

[8]

µH

Natural mortality rate for humans

3.9 × 10−5 5 per day

[5]

µV

Natural mortality rate for vectors

0.1429 per day

assumed

δH

HIV-induced mortality rate

9.13 × 10−4 per day

[25]

δM

Malaria-induced mortality rate

3.454 × 10−4 per day

[7]

βH

Transmission probabilities for HIV

0.00358 per day

variable

βM

Transmission probabilities for malaria in humans

0.8333 per day

[7]

βV

Transmission probabilities for malaria in vectors

0.00358 per day

variable

bM

Biting rate of mosquitoes

0.125 − 0.5 per day

[24]

ηH , ηHM , ξ

Modification parameters

1.4, 1.5, 1.002

[24]

θHM , στ

Modification parameters

1.002, 1.00, 1.001

[24]

, ϑ, ψ

Modification parameters

1.02, 1.002, 1.002

[24]

ηV , θV

Modification parameters

1.5, 1.5

[24]

φ 1 φ 2 , φ3

Recovery rate of humans from malaria

0.00556, 0.002, 0.0005

[24]

κ

Progression rate to AIDS class

0.000548 per day

[24]

γH

Progression rate from exposed to infectious state for humans

0.08333 per day

[7]

γV

Progression rate from exposed to infectious state for vectors

0.1

[7]

αM

Loss of immunity by individuals with malaria only

0.00556

assumed

αH

Loss of immunity by individuals with HIV

0.002

assumed

αA

Loss of immunity by individuals with AIDS

0.0005

assumed

p

Fraction of vectors reduced

variable

assumed

Another scenario is depicted in Figure 4 where controlu1 = 0 and u1 6= 0 with u2 6= 0. It is observed that the dually infected individuals whenu1 = 0 exceeds the dually infected Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

158

F.B. Agusto

individuals when u1 6= 0. A fact indicating the effect of controlling the population of the vectors. A similar observation is made in Figure 5 for the dually infected individuals who have recovered from malaria. The high number of individuals infected results to a high number of recovered individuals with improved treatment response (u2 6= 0). In Figure 6, control u2 = 0 and u2 6= 0 with u1 6= 0 and it is observed that the dually infected individuals who have recovered from malaria when u2 6= 0 exceeds the dually infected individuals who have recovered from malaria when u2 = 0. A fact indicating the effect of improving treatment response. In conclusion, our optimal control results shows how improving drug response in individuals dually infected with HIV and malaria increases the number of recovered persons dually infected. While reducing the vector population using a combination of insecticide and insecticide treated bed nets reduces malaria infection in individuals with HIV/AIDS.

Appendix In this appendix, we provide details of the MATLAB codes [19] used to obtain the results and plots in this paper. T=21;

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SH0 = 40; AH0 = 30; EV0 = 10;

EM0 = 30; EAM0 = 30; IV0 = 50;

IM0 = 40; AHM0 = 30; RMV0 = 20;

IH0 = 30; RH0 = 30;

EHM0 = 30; RAM0 = 30;

IHM0 = 30; SV0 = 90;

A =50 ; B = 200; C =300; D=100; E=500; betaH = 0.0007; betaV = 1.5; betaM = 0.8333; alphaM = 0.00556; alphaH = 0.002; alphaA = 0.0005; phi1 = 0.00556; phi2= 0.002; phi3=0.0005; vartheta = 10; gammaH=0.08333; gammaV = 0.1; muV = 0.1429; muH = 3.9e-5; epsilon = 1.02; deltaM = 3.454e-4; deltaH= 9.13e-4; kappa = 0.000548; sigma= 1.002; tau = 1.001; xi= 1.002; psi= 1.002; pp = 0.85; etaM = 0.05; etaHM = 1.5; etaA = 1.4; etaV=1.5; thetaHM = 1.002; thetaV = 1.5; bM = 0.25; LambdaH = 5e-2; LambdaV = 6; M = 1000; test = -1; count=0; h=T/M; h2 = h/2; t = linspace(0,T,M+1); STATE VARIABLES SH = zeros(1,M+1); IH = zeros(1,M+1); AH = zeros(1,M+1); RH = zeros(1,M+1); EV = zeros(1,M+1);

EM = zeros(1,M+1); EHM = zeros(1,M+1); EAM = zeros(1,M+1); RAM = zeros(1,M+1); IV = zeros(1,M+1);

IM = zeros(1,M+1); IHM = zeros(1,M+1); AHM = zeros(1,M+1); SV = zeros(1,M+1); RMV = zeros(1,M+1);

INITIAL STATE VALUES SH(1) = SH0; IH(1) = IH0; AH(1) = AH0; RH(1) = RH0; EV(1) = EV0;

EM(1) = EM0; EHM(1) = EHM0; EAM(1) = EAM0; RAM(1) = RAM0; IV(1) = IV0;

IM(1) = IM0; IHM(1) = IHM0; AHM(1) = AHM0; SV(1) = SV0; RMV(1) = RMV0;

Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Application of Optimal Control... ADJOINT VARIABLES lambdaSH = ones(1,M+1); lambdaIH = ones(1,M+1); lambdaAH = ones(1,M+1); lambdaRH = ones(1,M+1); lambdaEV = ones(1,M+1);

lambdaEM = ones(1,M+1); lambdaEHM = ones(1,M+1); lambdaEAM = ones(1,M+1); lambdaRAM = ones(1,M+1); lambdaIV = ones(1,M+1);

159

lambdaIM = ones(1,M+1); lambdaIHM = ones(1,M+1); lambdaAHM = ones(1,M+1); lambdaSV = ones(1,M+1); lambdaRMV = ones(1,M+1) ;

CONTROL VARIABLE u1 = zeros(1,M+1); u2 = zeros(1,M+1); count = 0; test = −1; UPDATE OF STATES, ADJOINTS, CONTROL while(test < 0) count = count +1; if count > 500 break end oldu1 = u1; oldSH = SH; oldIH = IH; oldAH = AH; oldRH = RH; oldEV = EV; oldlambdaSH = lambdaSH; oldlambdaIH = lambdaIH; oldlambdaAH = lambdaAH; oldlambdaRH = lambdaRH; oldlambdaEV = lambdaEV;

oldu2 = u2; oldEM = EM; oldEHM = EHM; oldEAM = EAM; oldRAM = RAM; oldIV = IV; oldlambdaEM = lambdaEM; oldlambdaEHM = lambdaEHM; oldlambdaEAM = lambdaEAM; oldlambdaRAM = lambdaRAM; oldlambdaIV = lambdaIV;

oldIM = IM; oldIHM = IHM; oldAHM = AHM; oldSV = SV; oldRMV = RMV; oldlambdaIM = lambdaIM; oldlambdaIHM = lambdaIHM; oldlambdaAHM = lambdaAHM; oldlambdaSV = lambdaSV; oldlambdaRMV = lambdaRMV;

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

RUNGE-KUTTA 4 FOR STATE VARIABLES for i = 1:M lambdaH

=

lambdaM lambdaV

= =

(betaH*(IH(i)+ etaHM*(EHM(i) + thetaHM* IHM(i)) + etaA*(AH(i) + etaHM*(EAM(i) + thetaHM *AHM(i))) ))/(NH(i)); (betaM *bM *IV(i))/(NH(i)); (betaV *bM *(IM(i) + etaV *(IHM(i) +thetaV *AHM(i))) )/(NH(i));

m11 m12 m13

= = =

LambdaH + alphaM* RH(i) - lambdaH *SH(i) -lambdaM* SH(i) - muH * SH(i); lambdaM* SH(i) - lambdaH* EM(i) - (gammaH + muH )*EM(i); gammaH *EM(i) - sigma *lambdaH *IM(i) - ( phi1 + deltaM + muH )*IM(i);

m14

=

m15

=

m16

=

lambdaH* SH(i) + alphaH* RAM(i) - vartheta*lambdaM *IH(i) - ( kappa + muH )*IH(i); lambdaH *EM(i) + vartheta *lambdaM *IH(i) - ( epsilon* gammaH + kappa + muH )*EHM(i); sigma* lambdaH *IH(i) + epsilon *gammaH *EHM(i) - ( (1+u2(i))*phi2 + xi* kappa + muH + tau *deltaM )*IHM(i);

m17

=

m18

=

m19

=

kappa *IH(i) + alphaA* RAM(i) - vartheta *lambdaM *AH(i) - ( muH + deltaH)* AH(i); vartheta *lambdaM *AH(i) + kappa* EHM(i) - (epsilon* gammaH+ muH + deltaH)*EAM(i); epsilon *gammaH* EAM(i) + xi* kappa* IHM(i) - ( (1+u2(i) )*phi3 + muH + psi *deltaH + tau *deltaM )*AHM(i);

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

160

F.B. Agusto

m110 m111

= =

m113 m114 m115 m116

= = = =

phi1 *IM(i) - ( alphaM + muH )*RH(i); (1+u2(i) )*phi2* IHM(i) + (1+u2(i) )*phi3* AHM(i) - ( alphaH + alphaA + muH + deltaH )*RAM(i); LambdaV - lambdaV* SV(i) - u1(i)*(1-pp)*SV(i) - muV *SV(i); lambdaV* SV(i) - u1(i)*(1-pp)*EV(i) - (gammaV +muV)* EV(i); gammaV *EV(i) - u1(i)*(1-pp)*IV(i) - muV* IV(i); u1(i)*(1-pp)*(SV(i) +EV(i) + IV(i) ) - muV *RMV(i) ;

lambdaH

=

lambdaM lambdaV

= =

m21

=

m22 m23

= =

LambdaH + alphaM*( RH(i)+h2*m110) - lambdaH *(SH(i)+h2*m11) - lambdaM* (SH(i)+h2*m11) - muH * (SH(i)+h2*m11); lambdaM* (SH(i)+h2*m11) - lambdaH*( EM(i)+h2*m12) - (gammaH + muH )*(EM(i)+h2*m12); gammaH *(EM(i)+h2*m12) - sigma *lambdaH *(IM(i)+h2*m13) - ( phi1 + deltaM + muH )*(IM(i)+h2*m13);

m24

=

m25

=

m26

=

m27

=

m28

=

m29

=

m210 m211

= =

m213

=

m214

=

m215

=

m216

=

(betaH*( (IH(i)+h2*m14)+ etaHM*( (EHM(i)+h2*m15) + thetaHM* (IHM(i)+h2*m16) )... + etaA*( (AH(i)+h2*m17) + etaHM*( (EAM(i)+h2*m18) + thetaHM *(AHM(i)+h2*m19) )) ))/( NH(i) ); (betaM *bM *(IV(i)+h2*m115) )/( NH(i) ); (betaV *bM *((IM(i)+h2*m13) + etaV *( (IHM(i)+h2*m16) +thetaV *(AHM(i)+h2*m19) )) )/( NH(i) );

lambdaH* (SH(i)+h2*m11) + alphaH* (RAM(i)+h2*m111) - vartheta*lambdaM *(IH(i)+h2*m14) - ( kappa + muH )*(IH(i)+h2*m14); lambdaH *(EM(i)+h2*m12) + vartheta *lambdaM *(IH(i)+h2*m14) -( epsilon* gammaH + kappa + muH )*(EHM(i)+h2*m15); sigma* lambdaH *(IH(i)+h2*m14) + epsilon *gammaH *(EHM(i)+h2*m15) -( (1+0.5*(u2(i)+u2(i)) )*phi2 + xi* kappa + muH + tau *deltaM )*(IHM(i)+h2*m16); kappa *(IH(i)+h2*m14) + alphaA* (RAM(i)+h2*m111)-vartheta *lambdaM *(AH(i)+h2*m17) - ( muH + deltaH )* (AH(i)+h2*m17); vartheta *lambdaM *(AH(i)+h2*m17) + kappa* (EHM(i)+h2*m15) - (epsilon* gammaH+ muH + deltaH)*(EAM(i)+h2*m18); epsilon *gammaH* (EAM(i)+h2*m18) + xi* kappa* (IHM(i)+h2*m16) - ( (1+0.5*(u2(i)+u2(i+1)) )*phi3 +muH + psi *deltaH + tau *deltaM )*(AHM(i)+h2*m19); phi1 *(IM(i)+h2*m13) - ( alphaM + muH )*(RH(i)+h2*m110); (1+0.5*(u2(i)+u2(i+1)) )*phi2* (IHM(i)+h2*m16) + (1+0.5*(u2(i)+u2(i+1)) )*phi3 * (AHM(i)+h2*m19) - ( alphaH + alphaA + muH + deltaH )*(RAM(i)+h2*m111); LambdaV - lambdaV* (SV(i)+h2*m113) - 0.5*(u1(i)+u1(i+1))*(1-pp)*(SV(i)+h2*m113) - muV *(SV(i)+h2*m113); lambdaV* (SV(i)+h2*m113) - 0.5*(u1(i)+u1(i+1))*(1-pp)*(EV(i)+h2*m114) - (gammaV +muV)* (EV(i)+h2*m114); gammaV *(EV(i)+h2*m114) - 0.5*(u1(i)+u1(i+1))*(1-pp)*( IV(i)+h2*m115) - muV* (IV(i)+h2*m115); 0.5*(u1(i)+u1(i+1))*(1-pp)*((SV(i)+h2*m113) + (EV(i)+h2*m114) + (IV(i)+h2*m115) ) - muV *(RMV(i)+h2*m116) ;

lambdaH

=

(betaH*( (IH(i)+h2*m24)+ etaHM*( (EHM(i)+h2*m25) + thetaHM* (IHM(i)+h2*m26) )... + etaA*( (AH(i)+h2*m27) + etaHM*( (EAM(i)+h2*m28) + thetaHM *(AHM(i)+h2*m29) )) ))/( NH(i) ); (betaM *bM *(IV(i)+h2*m215) )/( NH(i) ); (betaV *bM *((IM(i)+h2*m23) + etaV *( (IHM(i)+h2*m26) +thetaV *(AHM(i)+h2*m29) )) )/( NH(i) );

lambdaM lambdaV

= =

m31

=

m32

=

m33

=

LambdaH + alphaM*( RH(i)+h2*m210) - lambdaH *(SH(i)+h2*m21) -lambdaM* (SH(i)+h2*m21)- muH * (SH(i)+h2*m21); lambdaM* (SH(i)+h2*m21) - lambdaH*( EM(i)+h2*m22) - (gammaH + muH )*(EM(i)+h2*m22); gammaH *(EM(i)+h2*m22) - sigma *lambdaH *(IM(i)+h2*m23) - ( phi1 + deltaM + muH ) *(IM(i)+h2*m23);

Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Application of Optimal Control... m34

=

m35

=

m36

=

m37

=

m38

=

m39

=

m310 m311

= =

m313

=

m314

lambdaH* (SH(i)+h2*m21) + alphaH* (RAM(i)+h2*m211) - vartheta*lambdaM *(IH(i)+h2*m24) - ( kappa + muH )*(IH(i)+h2*m24); lambdaH *(EM(i)+h2*m22) + vartheta *lambdaM *(IH(i)+h2*m24) -( epsilon* gammaH + kappa + muH )*(EHM(i)+h2*m25); sigma* lambdaH *(IH(i)+h2*m24) + epsilon *gammaH *(EHM(i)+h2*m25) - ( (1+0.5*(u2(i)+u2(i+1)) )*phi2 + xi* kappa + muH + tau *deltaM )*(IHM(i)+h2*m26); kappa *(IH(i)+h2*m24) + alphaA* (RAM(i)+h2*m211) - vartheta *lambdaM *(AH(i)+h2*m27) - ( muH + deltaH )* (AH(i)+h2*m27); vartheta *lambdaM *(AH(i)+h2*m27) + kappa* (EHM(i)+h2*m25) - ( epsilon* gammaH + muH + deltaH )*(EAM(i)+h2*m28); epsilon *gammaH* (EAM(i)+h2*m28) + xi* kappa* (IHM(i)+h2*m26) - ( (1+0.5*(u2(i)+u2(i+1)) )*phi3 +muH + psi *deltaH + tau *deltaM )*(AHM(i)+h2*m29); phi1 *(IM(i)+h2*m23) - ( alphaM + muH )*(RH(i)+h2*m110); (1+0.5*(u2(i)+u2(i+1)) )*phi2* (IHM(i)+h2*m26) + (1+0.5*(u2(i)+u2(i+1)) )*phi3 * (AHM(i)+h2*m29) - ( alphaH + alphaA + muH + deltaH )*(RAM(i)+h2*m211); LambdaV - lambdaV* (SV(i)+h2*m213) - 0.5*(u1(i)+u1(i+1))*(1-pp)*(SV(i)+h2*m213) - muV *(SV(i)+h2*m213); = lambdaV* (SV(i)+h2*m213) - 0.5*(u1(i)+u1(i+1))*(1-pp)*(EV(i)+h2*m214) - (gammaV +muV)* (EV(i)+h2*m214); = gammaV *(EV(i)+h2*m214) - 0.5*(u1(i)+u1(i+1))*(1-pp)*( IV(i)+h2*m215) - muV* (IV(i)+h2*m215); = 0.5*(u1(i)+u1(i+1))*(1-pp)*((SV(i)+h2*m213) + (EV(i)+h2*m214) + (IV(i)+h2*m215) ) - muV *(RMV(i)+h2*m216) ;

m315 m316

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161

lambdaH

=

lambdaM lambdaV

= =

m41

=

m42 m43

= =

LambdaH + alphaM*( RH(i)+h*m310) - lambdaH *(SH(i)+h*m31) -lambdaM* (SH(i)+h*m31) - muH * (SH(i)+h*m31); lambdaM* (SH(i)+h*m31) - lambdaH*( EM(i)+h*m32) - (gammaH + muH )*(EM(i)+h*m32); gammaH *(EM(i)+h*m32) - sigma *lambdaH *(IM(i)+h*m33)... - ( phi1 + deltaM + muH )*(IM(i)+h*m33);

m44

(betaH*( (IH(i)+h*m34)+ etaHM*( (EHM(i)+h*m35) + thetaHM* (IHM(i)+h*m36) )... + etaA*( (AH(i)+h*m37) + etaHM*( (EAM(i)+h*m38) + thetaHM *(AHM(i)+h*m39) )) ))/( NH(i)); (betaM *bM *(IV(i)+h*m315) )/(NH(i)); (betaV *bM *((IM(i)+h*m33) + etaV *( (IHM(i)+h*m36) +thetaV *(AHM(i)+h*m39) )) )/(NH(i));

= lambdaH* (SH(i)+h*m31) + alphaH* (RAM(i)+h*m311) - vartheta*lambdaM *(IH(i)+h*m34) - ( kappa + muH )*(IH(i)+h*m34); = lambdaH *(EM(i)+h*m32) + vartheta *lambdaM *(IH(i)+h*m34) -( epsilon* gammaH + kappa + muH )*(EHM(i)+h*m35); = sigma* lambdaH *(IH(i)+h*m34) + epsilon *gammaH *(EHM(i)+h*m35) - ( (1+u2(i+1) )*phi2 + xi* kappa + muH + tau *deltaM )*(IHM(i)+h*m36);

m45 m46

m47

=

m48

=

m49

=

m410 m411

= =

m413

=

kappa *(IH(i)+h*m34) + alphaA* (RAM(i)+h*m311) - vartheta *lambdaM *(AH(i)+h*m37) - ( muH + deltaH)* (AH(i)+h*m37); vartheta *lambdaM *(AH(i)+h*m37) + kappa* (EHM(i)+h*m35) - (epsilon* gammaH+ muH + deltaH)*(EAM(i)+h*m38); epsilon *gammaH* (EAM(i)+h*m38) + xi* kappa* (IHM(i)+h*m36) - ( (1+ u2(i+1) )*phi3 +muH + psi *deltaH + tau *deltaM )*(AHM(i)+h*m39); phi1 *(IM(i)+h*m23) - ( alphaM + muH )*(RH(i)+h*m110); (1+ u2(i+1) )*phi2* (IHM(i)+h2*m26) + (1+ u2(i+1) )*phi3* (AHM(i)+h*m29) - ( alphaH + alphaA + muH + deltaH )*(RAM(i)+h*m211); LambdaV - lambdaV* (SV(i)+h*m213) - u1(i+1)*(1-pp)*(SV(i)+h*m213)... - muV *(SV(i)+h*m213);

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162

F.B. Agusto

m414

=

m415 m416

= =

SH(i+1) EM(i+1) IM(i+1)

lambdaV* (SV(i)+h*m213) - u1(i+1)*(1-pp)*(EV(i)+h*m214)... - (gammaV +muV)* (EV(i)+h*m214); gammaV *(EV(i)+h*m214) - u1(i+1)*(1-pp)*( IV(i)+h*m215) - muV* (IV(i)+h*m215); u1(i+1)*(1-pp)*((SV(i)+h*m213) + (EV(i)+h*m214) + (IV(i)+h*m215) )... - muV *(RMV(i)+h*m216) ;

= = =

SH(i) + (h/6)*(m11 + 2*m21 + 2*m31 + m41); EM(i) + (h/6)*(m12 + 2*m22 + 2*m32 + m42); IM(i) + (h/6)*(m13 + 2*m23 + 2*m33 + m43);

IH(i+1) EHM(i+1) IHM(i+1)

= = =

IH(i) + (h/6)*(m14 + 2*m24 + 2*m34 + m44); EHM(i) + (h/6)*(m15 + 2*m25 + 2*m35 + m45); IHM(i) + (h/6)*(m16 + 2*m26 + 2*m36 + m46);

AH(i+1) EAM(i+1) AHM(i+1)

= = =

AH(i) + (h/6)*(m17 + 2*m27 + 2*m37 + m47); EAM(i) + (h/6)*(m18 + 2*m28 + 2*m38 + m48); AHM(i) + (h/6)*(m19 + 2*m29 + 2*m39 + m49);

RH(i+1) RAM(i+1) SV(i+1)

= = =

RH(i) + (h/6)*(m110 + 2*m210 + 2*m310 + m410); RAM(i) + (h/6)*(m111 + 2*m211 + 2*m311 + m411); SV(i) + (h/6)*(m113 + 2*m213 + 2*m313 + m413);

EV(i+1) IV(i+1) RMV(i+1) end

= = =

EV(i) + (h/6)*(m114 + 2*m214 + 2*m314 + m414); IV(i) + (h/6)*(m115 + 2*m215 + 2*m315 + m415); RMV(i) + (h/6)*(m116 + 2*m216 + 2*m316 + m416);

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

RUNGE-KUTTA 4 FOR ADJOINTS for i = 1:M j = M+2-i; lambdaH

=

lambdaM lambdaV

= =

(betaH*(IH(j)+ etaHM*(EHM(j) + thetaHM* IHM(j)) + etaA*(AH(j)+ etaHM*(EAM(j) + thetaHM *AHM(j))) ))/(NH(j)); (betaM *bM *IV(j))/(NH(j)); (betaV *bM *(IM(j) + etaV *(IHM(j) +thetaV *AHM(j))) )/(NH(j));

m11 m12 m13

= = =

(lambdaH + lambdaM - muH )*lambdaSH(j) - lambdaM*lambdaEM(j) - lambdaM*lambdaIH(j); (lambdaH + gammaH + muH )*lambdaEM(j) - gammaH*lambdaIM(j) - lambdaH*lambdaEHM(j) ; - A + ( sigma*lambdaH + phi1 + muH + deltaM )*lambdaIM(j)... - sigma*lambdaH*lambdaIHM(j) - ( phi1)*lambdaRH(j)... + ( (betaV*bM)/(NH(j)) )* ( SV(j)*lambdaSV(j) - SV(j)*lambdaEV(j) );

m14

=

m15

=

m16

=

- ( ( (betaH)/(NH(j)) )*SH(j) - vartheta*lambdaM - kappa - muH )*lambdaIH(j)... +( (betaH)/(NH(j)) )* ( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j) - EM(j)*lambdaEHM(j) )... + sigma*( (betaH)/(NH(j)) )* ( IM(j)*lambdaIM(j) - IM(j)*lambdaIHM(j) ) ; - epsilon*gammaH*lambdaIHM(j) - kappa*lambdaEAM(j) + ( epsilon*gammaH + kappa + muH)*lambdaEHM(j)... + ( (betaH*etaHM)/(NH(j)) )*( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j) - SH(j)*lambdaIH(j)... - EM(j)*lambdaEHM(j) ) + sigma*( (betaH*etaHM)/(NH(j)) )*( IM(j)*lambdaIM(j) - IM(j)*lambdaIHM(j) ); - B + ( epsilon*kappa + muH + tau*deltaM + (1+u2(j))*phi2 )*lambdaIHM(j) - xi*kappa*lambdaAHM(j)... - ( (1+u2(j) )* phi2)*lambdaRAM(j) + ( (betaV*bM*etaV)/(NH(j)) )*( SV(j)*lambdaSV(j) - SV(j)*lambdaEV(j) )... + ( (betaH*etaHM*thetaHM)/(NH(j)) )*( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j) - SH(j)*lambdaIH(j)... - EM(j)*lambdaEHM(j) )+ sigma*( (betaH*etaHM*thetaHM)/(NH(j) ) )*( IM(j)*lambdaIM(j)... - IM(j)*lambdaIHM(j) ) ;

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Application of Optimal Control... m17

=

m18

=

m19

=

163

( muH + deltaH )*lambdaAH(j) - vartheta*lambdaM*( lambdaAH(j) - lambdaEAM(j) )... + ( (betaH*etaA)/(NH(j)) )*( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j) - SH(j)*lambdaIH(j)... - EM(j)*lambdaEHM(j) ) + sigma*( (betaH*etaA)/(NH(j)) )*( IM(j)*lambdaIM(j) - IM(j)*lambdaIHM(j) ) ; (epsilon*gammaH + muH + deltaH)*lambdaEAM(j) - epsilon*gammaH*lambdaAHM(j)... + ( (betaH*etaA*etaHM)/(NH(j)) )*( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j) - SH(j)*lambdaIH(j)... - EM(j)*lambdaEHM(j) )+ sigma*( (betaH*etaA*etaHM)/(NH(j)) )*( IM(j)*lambdaIM(j) - IM(j)*lambdaIHM(j) ); - C + ( muH + psi*deltaH + tau*deltaM + (1 + u2(j))*phi3 )*lambdaIHM(j) - ( (1+u2(j) )*phi3)*lambdaRAM(j)... + ( (betaV*bM*etaV*thetaV)/( NH(j) ) )*( SV(j)*lambdaSV(j)-SV(j)*lambdaEV(j) )... + ( (betaH*etaHM*thetaHM)/( NH(j) ) )*( SH(j)*lambdaSH(j) + EM(j)*lambdaEM(j)-SH(j)*lambdaIH(j)... - EM(j)*lambdaEHM(j) ) + sigma*( (betaH*etaHM*thetaHM)/( NH(j) ) )*( IM(j)*lambdaIM(j)... - IM(j)*lambdaIHM(j) );

m110 m111 m113

= = =

- alphaM*lambdaSH(j) + (muH + alphaM)*lambdaRH(j) ; - alphaH*lambdaIH(j) - alphaA*lambdaAH(j) + ( alphaH + alphaA + muH + deltaH )*lambdaRAM(j); ( lambdaV + muV + (1-pp)*u1(j))*lambdaSV(j) - lambdaV*lambdaEV(j) - (1-pp)*u1(j)*lambdaRMV(j);

m114 m115

= =

m116

=

( (1-pp)*u1(j) + (gammaV*muV) )*lambdaEV(j) - (1-pp)*u1(j)*lambdaRMV(j) - gammaV*lambdaIV(j); ( muV + (1-pp)*u1(j))*lambdaIV(j) - (1-pp)*u1(j)*lambdaRMV(j)... + ( (betaH*bM)/(NH(j)) )*( SH(j)*lambdaSH(j) + SH(j)*lambdaEM(j) )... + vartheta*( (betaH*bM)/(NH(j)) )*( IM(j)*lambdaIM(j) - IM(j)*lambdaEHM(j) - AH(j)*lambdaAH(j)... - AH(j)*lambdaEAM(j) ); muV*lambdaRMV(j);

lambdaH

=

(betaH*(0.5*( IH(j) + IH(j-1))+ etaHM*( 0.5*( EHM(j) + EHM(j-1))... + thetaHM* 0.5*( IHM(j) + IHM(j-1)) ) + etaA*( 0.5*( AH(j) + AH(j-1) )+ etaHM*( 0.5*( EAM(j)... + EAM(j-1)) + thetaHM *0.5*( AHM(j) + AHM(j-1)))) ))/( 0.5*( NH(j) + NH(j-1)) ); (betaM *bM *0.5*( IV(j) + IV(j-1)) )/( 0.5*( NH(j) + NH(j-1)) ); (betaV *bM *( 0.5*( IM(j) + IM(j-1)) + etaV *( 0.5*( IHM(j) + IHM(j-1))... + thetaV *0.5*( AHM(j) + AHM(j-1)) )) )/( 0.5*( NH(j) + NH(j-1)) );

lambdaM lambdaV

= =

m21

=

m22

=

m23

=

(lambdaH + lambdaM - muH )*(lambdaSH(j)-h2*m11) - lambdaM*(lambdaEM(j)-h2*m12)... - lambdaM*(lambdaIH(j)-h2*m14); (lambdaH + gammaH + muH )*(lambdaEM(j)-h2*m12) - gammaH*(lambdaIM(j)-h2*m13)... - lambdaH*(lambdaEHM(j)-h2*m15); -A + ( sigma*lambdaH + phi1 + muH + deltaM )*(lambdaIM(j)-h2*m13)... - sigma*lambdaH*(lambdaIHM(j)-h2*m16) - ( phi1)*(lambdaRH(j)-h2*m110)... + ( (betaV*bM)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*(SV(j) + SV(j-1))*(lambdaSV(j)-h2*m113)... - 0.5*(SV(j) + SV(j-1))*(lambdaEV(j)-h2*m114) );

m24

=

- ( ( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )*0.5*(SH(j) + SH(j-1))... - vartheta*lambdaM - kappa - muH )*(lambdaIH(j)-h2*m14)... +( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*(EM(j) + EM(j-1))*(lambdaEM(j)-h2*m12) - 0.5*(EM(j) + EM(j-1))*(lambdaEHM(j)-h2*m15) )... + sigma*( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*(IM(j) + IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*(IM(j) + IM(j-1))*(lambdaIHM(j)-h2*m16) ) ;

m25

=

- epsilon*gammaH*(lambdaIHM(j)-h2*m16) - kappa*(lambdaEAM(j)-h2*m18)... + ( epsilon*gammaH + kappa + muH)*(lambdaEHM(j)-h2*m15)... + ( (betaH*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*(EM(j) + EM(j-1))*(lambdaEM(j)-h2*m12) - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m14)... - 0.5*( EM(j) + EM(j-1))*(lambdaEHM(j)-h2*m15) )... + sigma*( (betaH*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m16) ) ;

m26

=

- B + (epsilon*kappa + muH + tau*deltaM + (1+0.5*( u2(j)+ u2(j-1)) )*phi2 )*(lambdaIHM(j)-h2*m16)... - xi*kappa*(lambdaAHM(j)-h2*m19) - ( (1+0.5*( u2(j)+ u2(j-1)) )*phi2)*(lambdaRAM(j)-h2*m111)... + ( (betaV*bM*etaV)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SV(j)+ SV(j-1))*(lambdaSV(j)-h2*m113)... - 0.5*( SV(j)+ SV(j-1))*(lambdaEV(j)-h2*m114) )+ ( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )... *( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11) + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m12)... - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m14) - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m15) )... + sigma*( (betaH*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m16) ) ;

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164

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m27

=

( muH + deltaH )*(lambdaAH(j)-h2*m17) - vartheta*lambdaM*( (lambdaAH(j)-h2*m17) - (lambdaEAM(j)-h2*m18) )... + ( (betaH*etaA)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m12)- 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m14)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m15) )... + sigma*( (betaH*etaA)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m16) ) ;

m28

=

(epsilon*gammaH + muH + deltaH)*(lambdaEAM(j)-h2*m18) - epsilon*gammaH*(lambdaAHM(j)-h2*m19)... + ( (betaH*etaA*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m12) - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m14)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m15) )... +sigma*( (betaH*etaA*etaHM)/( 0.5*(NH(j)+NH(j-1)) ) )*( 0.5*( IM(j)+IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m16) );

m29

=

- C + ( muH + psi*deltaH + tau*deltaM + (1 + 0.5*( u2(j)+ u2(j-1)) )*phi3 )*(lambdaIHM(j)-h2*m16)... - ( (1+0.5*(u2(j) + u2(j-1)) )*phi3)*(lambdaRAM(j)-h2*m111)... + ( (betaV*bM*etaV*thetaV)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SV(j) + SV(j-1))*(lambdaSV(j)-h2*m113)... - 0.5*( SV(j)+ SV(j-1))*(lambdaEV(j)-h2*m114) )... + ( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m12)... - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m14) - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m15) )... + sigma*( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j) + IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m16) );

m210 m211

= =

m213

=

m214

=

m215

=

m216

=

-alphaM*(lambdaSH(j)-h2*m11) + (muH + alphaM)*(lambdaRH(j)-h2*m110) ; - alphaH*(lambdaIH(j)-h2*m14) - alphaA*(lambdaAH(j)-h2*m17)... + (alphaH + alphaA + muH + deltaH )*(lambdaRAM(j)-h2*m111); ( lambdaV + muV + (1-pp)*0.5*( u1(j)+ u1(j-1)) )*(lambdaSV(j)-h2*m113) - lambdaV*(lambdaEV(j)-h2*m114)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m116); ( (1-pp)*0.5*( u1(j)+ u1(j-1)) + (gammaV*muV) )*(lambdaEV(j)-h2*m114)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m116) - gammaV*(lambdaIV(j)-h2*m115); ( muV + (1-pp)*0.5*( u1(j)+ u1(j-1)) )*(lambdaIV(j)-h2*m115)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m116)... + ( (betaH*bM)/( 0.5*(NH(j) + NH(j-1)) ) )*(0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m11)... + 0.5*( SH(j)+ SH(j-1))*(lambdaEM(j)-h2*m12) )... + vartheta*( (betaH*bM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m13)... - 0.5*( IM(j)+ IM(j-1))*(lambdaEHM(j)-h2*m15) - 0.5*( AH(j)+ AH(j-1))*(lambdaAH(j)-h2*m17)... - 0.5*( AH(j)+ AH(j-1))*(lambdaEAM(j)-h2*m18) ); muV*(lambdaRMV(j)-h2*m116) ;

lambdaH

=

(betaH*(0.5*( IH(j) + IH(j-1))+ etaHM*( 0.5*( EHM(j) + EHM(j-1))... + thetaHM* 0.5*( IHM(j) + IHM(j-1)) ) + etaA*( 0.5*( AH(j) + AH(j-1) )... + etaHM*( 0.5*( EAM(j) + EAM(j-1)) + thetaHM *0.5*( AHM(j) + AHM(j-1)))) ))/( 0.5*( NH(j) + NH(j-1)) ); (betaM *bM *0.5*( IV(j) + IV(j-1)) )/( 0.5*( NH(j) + NH(j-1)) );

lambdaM

=

lambdaV

=

m31

=

m32

=

m33

=

(lambdaH + lambdaM - muH )*(lambdaSH(j)-h2*m21) - lambdaM*(lambdaEM(j)-h2*m22)... - lambdaM*(lambdaIH(j)-h2*m24); (lambdaH + gammaH + muH )*(lambdaEM(j)-h2*m22) - gammaH*(lambdaIM(j)-h2*m23) - lambdaH*(lambdaEHM(j)-h2*m25); -A + ( sigma*lambdaH + phi1 + muH + deltaM )*(lambdaIM(j)-h2*m23)... - sigma*lambdaH*(lambdaIHM(j)-h2*m26) - ( phi1)*(lambdaRH(j)-h2*m210)... + ( (betaV*bM)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*(SV(j) + SV(j-1))*(lambdaSV(j)-h2*m213)... - 0.5*(SV(j) + SV(j-1))*(lambdaEV(j)-h2*m214) );

(betaV *bM *( 0.5*( IM(j) + IM(j-1)) + etaV *( 0.5*( IHM(j) + IHM(j-1))... + thetaV *0.5*( AHM(j) + AHM(j-1)) )) )/( 0.5*( NH(j) + NH(j-1)) );

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Application of Optimal Control...

165

m34

=

- ( ( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )*0.5*(SH(j) + SH(j-1)) - vartheta*lambdaM - kappa - muH )... *(lambdaIH(j)-h2*m24)+ ( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*( SH(j) + SH(j-1))*(lambdaSH(j)-h2*m21)... + 0.5*(EM(j) + EM(j-1))*(lambdaEM(j)-h2*m22) - 0.5*(EM(j) + EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH)/( 0.5*(NH(j) + NH(j-1)) ) )* ( 0.5*(IM(j) + IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*(IM(j) + IM(j-1))*(lambdaIHM(j)-h2*m26) ) ;

m35

=

- epsilon*gammaH*(lambdaIHM(j)-h2*m26) - kappa*(lambdaEAM(j)-h2*m28) + ( epsilon*gammaH + kappa... + muH)*(lambdaEHM(j)-h2*m25) + ( (betaH*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))... * (lambdaSH(j)-h2*m21) + 0.5*(EM(j) + EM(j-1))*(lambdaEM(j)-h2*m22) - 0.5*( SH(j)+ SH(j-1))... * (lambdaIH(j)-h2*m24) - 0.5*( EM(j) + EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m26) );

m36

=

- B + ( epsilon*kappa + muH + tau*deltaM + (1 + 0.5*( u2(j)+ u2(j-1)) )*phi2 )*(lambdaIHM(j)-h2*m26)... - xi*kappa*(lambdaAHM(j)-h2*m29) - ( (1+0.5*( u2(j)+ u2(j-1)) )*phi2)*(lambdaRAM(j)-h2*m211)... + ( (betaV*bM*etaV)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SV(j)+ SV(j-1))*(lambdaSV(j)-h2*m213)... - 0.5*( SV(j)+ SV(j-1))*(lambdaEV(j)-h2*m214) )... + ( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m21) ... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m22)- 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m24)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m26) ) ;

m37

=

( muH + deltaH )*(lambdaAH(j)-h2*m27) - vartheta*lambdaM*( (lambdaAH(j)-h2*m27) - (lambdaEAM(j)-h2*m28) )... + ( (betaH*etaA)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m21)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m22) - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m24)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH*etaA)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m26) ) ;

m38

=

(epsilon*gammaH + muH + deltaH)*(lambdaEAM(j)-h2*m28) - epsilon*gammaH*(lambdaAHM(j)-h2*m29)... + ( (betaH*etaA*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m21)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m22) - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m24)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH*etaA*etaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m26) );

m39

=

- C + ( muH + psi*deltaH + tau*deltaM + (1 + 0.5*( u2(j)+ u2(j-1)) )*phi3 )*(lambdaIHM(j)-h2*m26)... - ( (1+0.5*(u2(j) + u2(j-1)) )* phi3)*(lambdaRAM(j)-h2*m211)... + ( (betaV*bM*etaV*thetaV)/(0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SV(j)+ SV(j-1))*(lambdaSV(j)-h2*m213)... - 0.5*( SV(j)+ SV(j-1))*(lambdaEV(j)-h2*m214) )... + ( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m21)... + 0.5*( EM(j)+ EM(j-1))*(lambdaEM(j)-h2*m22) - 0.5*( SH(j)+ SH(j-1))*(lambdaIH(j)-h2*m24)... - 0.5*( EM(j)+ EM(j-1))*(lambdaEHM(j)-h2*m25) )... + sigma*( (betaH*etaHM*thetaHM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j) + IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaIHM(j)-h2*m26) );

m310 m311

= =

-alphaM*(lambdaSH(j)-h2*m21) + (muH + alphaM)*(lambdaRH(j)-h2*m210); - alphaH*(lambdaIH(j)-h2*m24) - alphaA*(lambdaAH(j)-h2*m27)... + ( alphaH + alphaA + muH + deltaH )*(lambdaRAM(j)-h2*m211);

m313

=

( lambdaV + muV + (1-pp)*0.5*( u1(j)+ u1(j-1)) )*(lambdaSV(j)-h2*m213) - lambdaV*(lambdaEV(j)-h2*m214)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m216);

m314

=

( (1-pp)*0.5*( u1(j)+ u1(j-1)) + (gammaV*muV) )*(lambdaEV(j)-h2*m214)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m216) - gammaV*(lambdaIV(j)-h2*m215);

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m315

=

( muV + (1-pp)*0.5*( u1(j)+ u1(j-1)) )*(lambdaIV(j)-h2*m215)... - (1-pp)*0.5*( u1(j)+ u1(j-1))*(lambdaRMV(j)-h2*m216)... + ( (betaH*bM)/( 0.5*(NH(j) + NH(j-1)) ) )*(0.5*( SH(j)+ SH(j-1))*(lambdaSH(j)-h2*m21)... + 0.5*( SH(j)+ SH(j-1))*(lambdaEM(j)-h2*m22) )... + vartheta*( (betaH*bM)/( 0.5*(NH(j) + NH(j-1)) ) )*( 0.5*( IM(j)+ IM(j-1))*(lambdaIM(j)-h2*m23)... - 0.5*( IM(j)+ IM(j-1))*(lambdaEHM(j)-h2*m25) - 0.5*( AH(j)+ AH(j-1))*(lambdaAH(j)-h2*m27)... - 0.5*( AH(j)+ AH(j-1))*(lambdaEAM(j)-h2*m28) );

m316

=

muV*(lambdaRMV(j)-h2*m216);

lambdaH

=

lambdaM lambdaV

= =

m41

=

m42

=

m43

=

(lambdaH + lambdaM - muH )*(lambdaSH(j)-h*m31) - lambdaM*(lambdaEM(j)-h*m32)... - lambdaM*(lambdaIH(j)-h*m34); (lambdaH + gammaH + muH )*(lambdaEM(j)-h*m32) - gammaH*(lambdaIM(j)-h*m33)... - lambdaH*(lambdaEHM(j)-h*m35); -A + ( sigma*lambdaH + phi1 + muH + deltaM )*(lambdaIM(j)-h*m33)... - sigma*lambdaH*(lambdaIHM(j)-h*m36) - ( phi1)*(lambdaRH(j)-h*m310)... + ( (betaV*bM)/(NH(j-1)) )* ( SV(j-1)*(lambdaSV(j)-h*m313) - SV(j-1)*(lambdaEV(j)-h*m314) );

m44

=

- ( ( (betaH)/( (NH(j-1)) ) )* SH(j-1) - vartheta*lambdaM - kappa - muH )*(lambdaIH(j)-h*m34)... + ( (betaH)/(NH(j-1)) )* ( SH(j-1)*(lambdaSH(j)-h*m31) + EM(j-1)*(lambdaEM(j)-h*m32)... - EM(j-1)*(lambdaEHM(j)-h*m35) ) + sigma*( (betaH)/(NH(j-1)) )* ( IM(j-1)*(lambdaIM(j)-h*m33)... - IM(j-1)*(lambdaIHM(j)-h*m36) );

m45

=

- epsilon*gammaH*(lambdaIHM(j)-h*m36) - kappa*(lambdaEAM(j)-h*m38) + ( epsilon*gammaH... + kappa + muH)*(lambdaEHM(j)-h*m35 + ( (betaH*etaHM)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31)... + EM(j-1)*(lambdaEM(j)-h*m32))- SH(j-1)*(lambdaIH(j)-h*m34) - EM(j-1)*(lambdaEHM(j)-h*m35) )... + sigma*( (betaH*etaHM)/(NH(j-1)) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaIHM(j)-h*m36) );

m46

=

- B + ( epsilon*kappa + muH + tau*deltaM + (1+u2(j-1) )*phi2 )*(lambdaIHM(j)-h*m36)... - xi*kappa*(lambdaAHM(j)-h*m39) - ( (1+u2(j-1) )*phi2)*(lambdaRAM(j)-h*m311)... + ( (betaV*bM*etaV)/(NH(j-1)) )*( SV(j-1)*(lambdaSV(j)-h*m313) - SV(j-1)*(lambdaEV(j)-h*m314) )... + ( (betaH*etaHM*thetaHM)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31) + EM(j-1)*(lambdaEM(j)-h*m32)... - SH(j-1)*(lambdaIH(j)-h*m34) - EM(j-1)*(lambdaEHM(j)-h*m35) )... + sigma*( (betaH*etaHM*thetaHM)/( (NH(j-1))) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaIHM(j)-h*m36) );

m47

=

( muH + deltaH )*(lambdaAH(j)-h*m37) - vartheta*lambdaM*( (lambdaAH(j)-h*m37)... - (lambdaEAM(j)-h*m38) )+ ( (betaH*etaA)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31)... + EM(j-1)*(lambdaEM(j)-h*m32) - SH(j-1)*(lambdaIH(j)-h*m34) -EM(j-1)*(lambdaEHM(j)-h*m35) )... + sigma*( (betaH*etaA)/(NH(j-1)) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaIHM(j)-h*m36) );

m48

=

(epsilon*gammaH + muH + deltaH)*(lambdaEAM(j)-h*m38) - epsilon*gammaH*(lambdaAHM(j)-h*m39)... + ( (betaH*etaA*etaHM)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31) + EM(j-1)*(lambdaEM(j)-h*m32)... - SH(j-1)*(lambdaIH(j)-h*m34) - EM(j-1)*(lambdaEHM(j)-h*m35) )... + sigma*( (betaH*etaA*etaHM)/(NH(j-1)) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaIHM(j)-h*m36) );

m49

=

- C + ( muH + psi*deltaH + tau*deltaM + (1+u2(j-1) )*phi3 )*(lambdaIHM(j)-h*m36)... - ( (1+u2(j-1) )*phi3)*(lambdaRAM(j)-h*m311)... + ( (betaV*bM*etaV*thetaV)/(NH(j-1)) )*( SV(j-1)*(lambdaSV(j)-h*m313) - SV(j-1)*(lambdaEV(j)-h*m314) )... + ( (betaH*etaHM*thetaHM)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31) + EM(j-1)*(lambdaEM(j)-h*m32)... - SH(j-1)*(lambdaIH(j)-h*m34) - EM(j-1)*(lambdaEHM(j)-h*m35) )... + sigma*( (betaH*etaA*etaHM)/(NH(j-1)) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaIHM(j)-h*m36) );

m410 m411

= =

m413

=

(betaH*(IH(j-1)+ etaHM*( EHM(j-1) + thetaHM*IHM(j-1) ) + etaA*( AH(j-1)+ etaHM*( EAM(j-1)... + thetaHM *AHM(j-1) ) ) ))/( NH(j-1) ); (betaM *bM *IV(j-1) )/( NH(j-1) ); (betaV *bM *( IM(j-1) + etaV *( IHM(j-1) +thetaV * AHM(j-1) )) )/( NH(j-1) );

-alphaM*(lambdaSH(j)-h*m31) + (muH + alphaM)*(lambdaRH(j)-h*m310); - alphaH*(lambdaIH(j)-h*m34) - alphaA*(lambdaAH(j)-h*m37)... + ( alphaH + alphaA + muH + deltaH )*(lambdaRAM(j)-h*m311); ( lambdaV + muV + (1-pp)*u1(j-1) )*(lambdaSV(j)-h*m313) - lambdaV*(lambdaEV(j)-h*m314)... - (1-pp)* u1(j-1)*(lambdaRMV(j)-h*m316);

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Application of Optimal Control... m414

=

m415

=

m416

=

167

( (1-pp)*u1(j-1) + (gammaV*muV) )*(lambdaEV(j)-h*m314)... - (1-pp)*u1(j-1)*(lambdaRMV(j)-h*m316) - gammaV*(lambdaIV(j)-h*m315); ( muV + (1-pp)*u1(j-1) )*(lambdaIV(j)-h*m315) - (1-pp)*u1(j-1)*(lambdaRMV(j)-h*m316)... + ( (betaH*bM)/(NH(j-1)) )*( SH(j-1)*(lambdaSH(j)-h*m31) + SH(j-1)*(lambdaEM(j)-h*m32) )... + vartheta*( (betaH*bM)/( (NH(j-1) )) )*( IM(j-1)*(lambdaIM(j)-h*m33) - IM(j-1)*(lambdaEHM(j)-h*m35)... - AH(j-1)*(lambdaAH(j)-h*m37) -AH(j-1)*(lambdaEAM(j)-h*m38) ); muV*(lambdaRMV(j)-h*m316);

lambdaSH(j-1) lambdaEM(j-1) lambdaIM(j-1) lambdaIH(j-1) lambdaEHM(j-1) lambdaIHM(j-1) lambdaAH(j-1) lambdaEAM(j-1) lambdaAHM(j-1) lambdaRH(j-1) lambdaRAM(j-1) lambdaSV(j-1) lambdaEV(j-1) lambdaIV(j-1) lambdaRMV(j-1) end

= = = = = = = = = = = = = = =

lambdaSH(j) - (h/6)*(m11 +2*m21 +2*m31 +m41); lambdaEM(j) - (h/6)*(m12 +2*m22 +2*m32 +m42); lambdaIM(j) - (h/6)*(m13 +2*m23 +2*m33 +m43); lambdaIH(j) - (h/6)*(m14 +2*m24 +2*m34 +m44); lambdaEHM(j) - (h/6)*(m15 +2*m25 +2*m35 +m45); lambdaIHM(j) - (h/6)*(m16 +2*m26 +2*m36 +m46); lambdaAH(j) - (h/6)*(m17 +2*m27 +2*m37 +m47); lambdaEAM(j) - (h/6)*(m18 +2*m28 +2*m38 +m48); lambdaAHM(j) - (h/6)*(m19 +2*m29 +2*m39 +m49); lambdaRH(j) - (h/6)*(m110 +2*m210 +2*m310 +m410); lambdaRAM(j) - (h/6)*(m111 +2*m211 +2*m311 +m411); lambdaSV(j) - (h/6)*(m113 +2*m213 +2*m313 +m413); lambdaEV(j) - (h/6)*(m114 +2*m214 +2*m314 +m414); lambdaIV(j) - (h/6)*(m115 +2*m215 +2*m315 +m415); lambdaRMV(j) - (h/6)*(m116 +2*m216 +2*m316 +m416);

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UPDATE CONTROL tempu1 u11 u1

= = =

( (1-pp)*(SV.*lambdaSV + EV.*lambdaEV + IV.*lambdaIV ) - (1-pp)*(SV+EV+IV).*lambdaRMV)./2*D; min(1,max(tempu1 ,0)); 0.5*(u11 + oldu1);

tempu2 u21 u2

= = =

( lambdaIHM.*IHM + lambdaAHM.*AHM - (phi2*IHM + phi3*AHM).*lambdaRAM)./(2*E); min(1,max(tempu2,0)); 0.5*(u21 + oldu2);

temp1 temp2 temp3 temp4 temp5 temp6 temp7 temp8 temp9 temp10 temp11 temp13 temp14 temp15 temp16 temp17 temp18 temp19 temp20 temp21 temp22 temp23 temp24 temp25 temp26 temp27

= = = = = = = = = = = = = = = = = = = = = = = = = =

0.001*sum(abs(SH))-sum(abs(oldSH - SH)); 0.001*sum(abs(EM))-sum(abs(oldEM - EM)); 0.001*sum(abs(IM))-sum(abs(oldIM - IM)); 0.001*sum(abs(IH))-sum(abs(oldIH - IH)); 0.001*sum(abs(EHM))-sum(abs(oldEHM - EHM)); 0.001*sum(abs(IHM))-sum(abs(oldIHM - IHM)); 0.001*sum(abs(AH))-sum(abs(oldAH - AH)); 0.001*sum(abs(EAM))-sum(abs(oldEAM - EAM)); 0.001*sum(abs(AHM))-sum(abs(oldAHM - AHM)); 0.001*sum(abs(RH))-sum(abs(oldRH - RH)); 0.001*sum(abs(RAM))-sum(abs(oldRAM -RAM )); 0.001*sum(abs(SV))-sum(abs(oldSV - SV)); 0.001*sum(abs(EV))-sum(abs(oldEV - EV)); 0.001*sum(abs(IV))-sum(abs(oldIV - IV)); 0.001*sum(abs(RMV))-sum(abs(oldRMV - RMV)); 0.001*sum(abs(lambdaSH))-sum(abs(oldlambdaSH - lambdaSH)); 0.001*sum(abs(lambdaEM))-sum(abs(oldlambdaEM - lambdaEM)); 0.001*sum(abs(lambdaIM))-sum(abs(oldlambdaIM - lambdaIM)); 0.001*sum(abs(lambdaIH))-sum(abs(oldlambdaIH - lambdaIH)); 0.001*sum(abs(lambdaEHM))-sum(abs(oldlambdaEHM - lambdaEHM)); 0.001*sum(abs(lambdaIHM))-sum(abs(oldlambdaIHM - lambdaIHM)); 0.001*sum(abs(lambdaAH))-sum(abs(oldlambdaAH - lambdaAH)); 0.001*sum(abs(lambdaEAM))-sum(abs(oldlambdaEAM - lambdaEAM)); 0.001*sum(abs(lambdaAHM))-sum(abs(oldlambdaAHM - lambdaAHM)); 0.001*sum(abs(lambdaRH))-sum(abs(oldlambdaRH - lambdaRH)); 0.001*sum(abs(lambdaRAM))-sum(abs(oldlambdaRAM - lambdaRAM));

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168 temp29 temp30 temp31 temp32 temp33 temp34

F.B. Agusto = = = = = =

0.001*sum(abs(lambdaSV))-sum(abs(oldlambdaSV - lambdaSV)); 0.001*sum(abs(lambdaEV))-sum(abs(oldlambdaEV - lambdaEV)); 0.001*sum(abs(lambdaIV))-sum(abs(oldlambdaIV - lambdaIV)); 0.001*sum(abs(lambdaRMV))-sum(abs(oldlambdaRMV - lambdaRMV)); 0.001*sum(abs(u1)) - sum(abs(oldu1 - u1)); 0.001*sum(abs(u2)) - sum(abs(oldu2 - u2));

test = min(temp1, min(temp2,min( temp3,min(temp4,min( temp5,min( temp6,min(temp7,min( temp8,min( temp9,min (temp10,min (temp11,min (temp13,min (temp14,min (temp15,min (temp16,min (temp17,min (temp18,min (temp19,min (temp20,min (temp21,min (temp22,min (temp23,min (temp24,min (temp25,min (temp26,min (temp27,min (temp29,min (temp30,min (temp31,temp32)))))))))))))))))))) ) )) ) ) )))); end y(1,:) = t; y(5,:) = IH; y(9,:) = EAM; y(13,:) = SV; y(17,:) = u1;

y(2,:) = SH; y(6,:) = EHM; y(10,:) = AHM; y(14,:) = EV; y(18,:) = u2;

y(3,:) = EM; y(7,:) = IHM; y(11,:) = RH; y(15,:) = IV;

y(4,:) = IM; y(8,:) = AH; y(12,:) = RAM; y(16,:) = RMV;

plot(y(1,:),y(12,:), ’b’ ) xlabel(’Time (days)’);axis tight ylabel(’Total number of dually infected individuals recovered from malaria’)

References [1] Abdu-Raddad, L.J., Patnaik P., and Kublin J.G. (2006). Dual infection with HIV and Malaria fuels the spread of both diseases in sub-saharan Africa.Science. 314(5805): 1603-1606. [2] Anderson, R.M. and May, R.M. (1991). Infectious Diseases of Humans. Oxford University Press, Oxford,

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[3] Blower, S., and McLean, A. (1994). Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco. Science 265: 1451-1454. [4] Blower, S. and McLean, A. (1995). AIDS: Modeling epidemic control. Science 267: 1250-1253. [5] Bowman, C., Gumel A.B., van den Driessche P., Wu J. and Zhu H. (2005). A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67: 1107-1133. [6] CDC factsheet on HIV/AIDS and Malaria: www.cdc.gov/malaria/features/malaria− hiv.htm. Accessed October 30, 2008. [7] Chitnis, N., Cushing, J.M. and Hyman, J.M. (2006). Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 67(1): 24-45. [8] Chiyaka, C., Garira W. and Dube S. Mathematical modelling of the impact of vaccination on malaria epidemiology. Int. J. Qual. Theor. Diff. Eqn. Appl 1 (1) (2007): 28-58. [9] Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.P. (1990). On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28: 503-522. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[10] Felippe de Souza, J.A.M., Marco Antonio Leonel Caetano and Takashi Yoneyama (2000). Optimal control theory applied to the anti-viral treatment of AIDS.Proceedings of the 39th IEEE Conference on Decision and Control Sydney, Australia, Vol. 5: 4839-4844. [11] Fleming, W.H. and Rishel, R.W. (1975). Deterministic and Stochastic Optimal Control. Springer Verlag, New York. [12] Hethcote, H.W. (2000). The mathematics of infectious diseases, SIAM Rev. 42(4): 599-653. [13] Joshi, H.R. (2002). Optimal control of an HIV immunology model, Optim. Control Appl. Math, 23: 199-213. [14] Jung, E., Lenhart, S. and Feng, Z. (2002). Optimal control of treatments in a twostrain tuberculosis Model Discrete and Continuous Dynamical Systems-Series B. 2(4): 473-482. [15] Kern, D., Lenhart, S., Miller, R. and Yong, J. (2007). Optimal control applied to native-invasive population dynamics. J. Biol. Dyn. 1(4): 413-426. [16] Kirschner, D., Lenhart, S. and Serbin, S. (1997). Optimal Control of the Chemotherapy of HIV, J. Math. Biol., 35: 775-792.

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[17] Lenhart, S. and Bhat, M.G. (1992). Application of distributed parameter control model in wildlife damage management, Math. Models & Methods in Appl. Sci. 2(4): 423-439. [18] Lenhart, S.M. and Yong, J. (1995). Optimal control for degenerate parabolic equations with logistic growth, Nonl. Anal., Vol. 25(7), 681-698. [19] Lenhart, S. and Workman, J.T. (2007). Optimal Control Applied to Biological Models. Chapman and Hall. [20] Lakshmikantham, V., Leela, S. and Martynyuk, A.A. (1989). Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., New York and Basel. [21] Lukes, D.L. (1982). Differential Equations: Classical to Control, Mathematics in Science and Engineering, Academic Press, New York. [22] Macdonald, G. (1957). The Epidemiology and Control of Malaria. Oxford University Press. [23] Morgan D et al., (2002). Progression to symptomatic disease in people infected with HIV-1 in rural Uganda: prospective cohort study. British Medical Journal,324: 193196. [24] Mukandavire, Z., Gumel, A.B., Garira, W. and Tchuenche, J.M. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences and Engineering. To appear. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

170

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[25] Mukandavire, Z. and Garira, W. (2007). Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe. J. Math. Biol. 54(5): 669699. [26] Pontryagin,L. S., Boltyanskii, V.G., Gamkrelidze, R. V. and Mishchenko,E. F. (1962). The Mathematical Theory of Optimal Processes. Wiley, New York. [27] Roseberry, W., Seale A. and Mphuka S. (2005). Assessing the application of the three ones principle in Zambia. DFID Health Resources System Centre, London. [28] ScienceDaily (2006). Malaria treatment efficacy compromised in certain HIVpositive patients. http://www.sciencedaily.com/releases/2006/09/060908193928.htm. Accessed on November 4, 2008. [29] UCSF Malaria and HIV (2006). http://hivinsite.ucsf.edu/InSite?page=kb-05-04-04 [30] van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180: 29-48. [31] UNAIDS/WHO (2005). AIDS epidemic update, Geneva. [32] WHO (2004). Malaria and HIV/AIDS interactions and implications: Conclusions of a technical consultation. Accessed on October 29, 2008.

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[33] WHO (2004). Malaria and HIV interactions and their implications for public health policy. Report of technical consultation. Accessed on October 29, 2008. [34] Xiefei Yan, Yun Zou1 and Jianliang Li. (2007). Optimal quarantine and isolation strategies in epidemics control. World Journal of Modelling and Simulation. 33: 202211.

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 171-194

ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 6

T WO S TRAIN HIV/AIDS M ODEL AND THE E FFECTS OF S UPERINFECTION

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

N. J. Malunguzaa∗, S. Dubeb , J.M. Tchuenchec , S.D. Hove-Musekwaa and Z. Mukandavirea a Department of Applied Mathematics, National University of Science and Technology, Box AC 939 Ascot, Bulawayo, Zimbabwe b Department of Applied Biology/Biochemistry, National University of Science and Technology, Box AC 939, Bulawayo, Zimbabwe c Mathematics Department, University of Dar es Salaam, P.O.Box 35062, Dar es Salaam, Tanzania

Abstract Two retroviruses HIV-1 and HIV-2 are the cause of HIV/AIDS around the world. HIV-1 virus is spread world wide whereas HIV-2 is concentrated to the West African countries. Individuals who engage in HIV predisposing risk behaviour such as truck drivers across nations, commercial sex workers, drug abusers and others can find themselves infected with both types of strains. Dual infection with HIV-1 and HIV-2 can occur in locales where these viruses co-circulate, most commonly in West Africa. This paper presents a deterministic mathematical model to compare the dynamics of HIV/AIDS when there is no superinfection and when there is superinfection. The mathematical features including the epidemic threshold, equilibria and stabilities are determined. It is shown using comprehensive analytic and numerical analysis that in the absence of superinfection, the strain with the highest reproduction number above unity forces the other strain to die off in a process of competitive exclusion. For an enhancement factor greater than unity, superinfection is shown to promote coexistence between strains even if one strain has its reproduction number being less than unity.

Keywords: HIV/AIDS, equilibria, epidemic threshold, stability, invasion reproduction numbers, persistence, superinfection. ∗

E-mail addresses: [email protected], [email protected]. (Corresponding author.)

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1.

N.J. Malunguzam S. Dube, J.M. Tchuenche et al.

Introduction

Two retroviruses HIV-1 and HIV-2 are the cause of HIV/AIDS around the world. HIV-1 virus is spread world wide whereas HIV-2 is concentrated to the West African countries where it was first discovered in 1986. In Europe, it was isolated in Sweden from a woman who had an infected Senegalese boyfriend. Each type has many sub-types whose envelope protein mutate very fast. Both types enter the CD4+T cells by attaching to the CD4+T receptor and the chemokine (CCR5) receptor. The viruses slowly multiply and in the first three months the antibodies against it are not detectable. It enters the lysogenic state for a period of time. The human body mounts a defense mechanism by producing interferons, interleukins and antibodies. These produce no perceivable effect on the virus. The antibodies produced by each type are different so that a kit specifically made for detecting one may not detect the other type. Nowadays, the kits are constructed to be able to detect both types. People who engage in HIV predisposing risk behaviour such as truck drivers across nations, commercial sex workers, drug abusers, can find themselves infected with both type at the same time [20]. The virus variants isolated from HIV-infected persons worldwide share remarkable diversity, especially in the envelope glycoprotein, gp120. Phylo-genetic studies have clustered HIV-1 isolates into eight subtypes (A-H). Nevertheless, even within a single infected person, HIV is present as a quasi-species, or a swarm of closely related variants. This genetic diversity, which in the case of HIV-1 accumulates at a rate of approximately one nucleotide substitution per genome per replication cycle, gives the virus an enormous flexibility to respond to a wide array of in vivo selection pressures. As a consequence, drugresistant and immunologic escape mutants are rapidly generated in infected persons through all stages of infection. On a global scale, the HIV pandemic is recognized as consisting of many separate epidemics, each with characteristic geography, affected populations, and predominant viral strain types [11]. HIV-2 has also been found in India and analysis of the genetic divergence of HIV-2 isolates has led to their classification into seven phylogenetic subtypes, designated A to G [20]. Unlike HIV-1, for which data on strain diversity and molecular epidemiology are abundant, little is known about the variation and geographic distribution of HIV-2 subtypes. Only subtypes A and B appear to be prevalent. Subtype A, which contains most of the HIV2 strains characterized to date, has been identified in almost all western African countries. Subtype B viruses seem to be much more limited geographically and have been reported mainly in Cˆ ote d’Ivoire and Ghana. Subtypes C, D, E, and F are represented only by partial sequences of single viral genomes; they have been identified in Liberia (subtype C and D) and Sierra Leone (subtype E and F). The new subtype, G, is represented by the fulllength genomic sequence of a strain collected in Cˆ ote d’Ivoire from an asymptomatic blood donor [7]. While HIV-2 is related to HIV-1 by its morphology, its tropism and its in vitro cytopathic effect on CD4+T (T4) positive cell lines and lymphocytes, HIV-2 differs from previously described human retroviruses known to be responsible for AIDS. Moreover, the proteins of HIV-1 and HIV-2 have different sizes and their serological cross-reactivity is restricted mostly to the major core protein, as the envelope glycoproteins of HIV-2 are not immune precipitated by HIV-1-positive sera except in some cases where very faint cross-reactivity

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Two Strain HIV/AIDS Model and the Effects of Superinfection

173

can be detected. Since a significant proportion of the HIV infected patients lack antibodies to the major core protein of their infecting virus, it is important to include antigens to both HIV-1 and HIV-2 in an effective serum test for the diagnosis of the infection by these viruses. HIV-2 was first discovered in the course of serological research on patients native to Guinea-Bissau who exhibited clinical and immunological symptoms of AIDS and from whom sero-negative or weakly sero-positive reactions to tests using an HIV-1 lysate were obtained. Due to the sero-negative or weak sero-positive results obtained when using kits designed to identify HIV-1 infections in the diagnosis of these new patients with HIV-2 infection, it has been necessary to devise a new diagnostic kit capable of detecting HIV-2 infection, either by itself or in combination with an HIV-1 infection. It was observed that HIV-2 is related more closely to the Simian Immunodeficiency Virus (SIV) than it is to HIV-1 [10]. Dual infection with HIV-1 and HIV-2 can occur in locales where these viruses cocirculate, most commonly in West Africa. Although dual seropositivity is common in this region, the true rate of dual infection remains unclear. In addition, whether unique HIV-1 subtypes are circulating in dually infected individuals is unknown. A cohort of 47 HIV-1 and HIV-2 dually seropositive individuals from Senegal, West Africa, was screened for the presence of HIV-1 and HIV-2 gag and env peripheral blood mononuclear cell (PBMC) viral DNA sequences using polymerase chain reaction (PCR). Of the 47 dual HIV-1/ HIV-2 seropositive individuals tested, 19 (40.4%) had infection with both HIV-1 and HIV-2 confirmed by genetic sequence analysis, whereas only HIV-1 or HIV-2 was confirmed in 17 (36.2%) or 9 (19.1%), respectively. The majority of HIV-1 subtypes found were CRF-02 and A, although subtypes D, C, G, J and B were also found, reflecting the subtypes known to be circulating in Senegal. There was no significant difference in HIV-1 subtype distribution between individuals with confirmed dual infection and patients in this study with dual seropositivity but lacking HIV-2, or with HIV-1 infected patients within the general population in Senegal, although the study was underpowered to detect anything but large differences. The prevalence of HIV-1/ HIV-2 dual infection appears to be significantly less than that of dually seropositive individuals and this likely reflects cross-reactive serology. The common HIV-1 subtypes prevalent in West Africa (CRF-02 and subtype A) have a similar distribution to those found in the cohort of dually infected and dually seropositive subjects [4]. HIV-1 is more prevalent and has three sub-groups M, N and O. Group O is restricted to West Central Africa and group N, discovered in 1998 in Cameroon is a rarity. Ninety percent of the world’s infections belong to the M subgroup of HIV-1. Within group M there are 9 genetically distinct subtypes of HIV-1. These are subtypes A, B, C, D, F, G, H, J, K and Circulating Recombinant Forms (CRFs). In certain instances, two viruses of different subtype can meet in the cell of an infected individual and combine their genetic material to create a new hybrid virus in a process called “viral” sex [2]. These new strains often die off very quickly but where they infect more than one person, they are called Circulating Recombinant Forms. For example the CRF A/B is a mixture of subtypes A and B. A study presented in 2006 showed that Ugandans infected with subtype D or CRFs incorporating subtype D developed AIDS and died much sooner than those infected with subtype A. The study suggested that subtype D is more effective at binding to immune receptor cells [8]. Recent evidence suggested that these different strains may be transmitted at different rates

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N.J. Malunguzam S. Dube, J.M. Tchuenche et al.

(Soto-Ramirez [12]). It has been observed that certain subtypes/CRFs are predominantly associated with specific modes of transmission. In particular, subtype B is spread mostly by homosexual contact and intravenous drug use (essentially via blood), while subtype C and CRF A/E tend to fuel heterosexual epidemics [6]. In this chapter we seek to bring out to the fore the effect on the usual and known mathematical and epidemiological properties of the presence of two viruses co-circulating in the same community. The world is a global village now and it is easy to envisage a situation where quite soon HIV-2 and sub-types of HIV-1 are not going to be limited to West Africa alone. As mathematical epidemiologists, we need to be well equipped to understand the implication of this dual circulation of variant HIV strains in a community in order to advice the public health policy decision-makers on the possible outcome of the co-dynamics as well as control strategies that will yield optimal results.

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2.

Two Strain HIV/AIDS Model without Superinfection

The model divides the sexually active population into four classes that are, S(t) the sexually mature susceptibles in the population who have not had effective contact with the virus at time t. Maturity in sexuality increases this compartment at a constant rate φ. The sexually mature individuals who are infected with the virus but have not yet developed symptomatic AIDS are denoted by I(t). This compartment increases by way of graduands from the susceptible class who have had effective contact with the virus and decreases by way of natural death and development to clinical AIDS. In this model, I(t) is made up of I1 (t) and I2 (t) representing classes afflicted with strain 1 and strain 2, respectively. The probability that a new sex partner will carry a given strain is Ii /N , the fraction of the sexually active population afflicted by strain i. Let the rate of transmitting infection from an infective partner to a susceptible partner be βi which measures the transmissibility of strain i. Then, the hazard rate for strain i (rate of infection with strain i) per unit time is βi Ii /N . Infected individuals leave the infected class by way of natural mortality µ and by progression to indistinguishable clinical AIDS at constant rates α1 and α2 for both strain 1 and strain 2, respectively. Thus, αi measures the virulence of strain i. Once an individual has developed AIDS, they either die naturally at a constant rate µ or they die at a disease induced death rate ν. The per capita life expectancy is given by 1/µ while 1/(µ + αi ) is the death adjusted average infectious life of an individual infected with the ith strain. We assume transmission through sexual contact only and that AIDS cases do not acquire new sexual partners and therefore do not give rise to new transmissions. Thus, N = S + I1 + I2 is assumed to be the sexually interacting population. Babies born with the virus die before they reach sexual maturity. Homogeneous mixing is assumed so that the standard incidence is of the form βi SIi /N . No latency is assumed for HIV because the latent period is negligible compared to the average infectious period. It is a consequence of our model here that infection by strain i confers cross immunity from other strains to the newly infected and therefore we do not have a superinfected class. The model flow diagram is shown in Figure 1.

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Two Strain HIV/AIDS Model and the Effects of Superinfection

175

Figure 1. Model flow diagram.

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The assumptions lead us to the following model structure, S′

=φ−

I1′

=

I2′

=

β1 SI1 β2 SI2 − − µS, N N

β1 SI1 − (µ + α1 )I1 , N

(1)

β2 SI2 − (µ + α2 )I2 , N

A′ = α1 I1 + α2 I2 − (µ + ν)A, with initial condition S(t) = S0 (t), I(t) = I0 (t) and A(t) = A0 (t), and NT (t) = S(t) + I1 (t) + I2 (t) + A(t),

(2)

where NT (t) is the total human population at time t, NT′ (t) = S ′ (t) + I1′ (t) + I2′ (t) + A′ (t).

(3)

Adding the differential equations in model system (1) gives, NT′ (t) = φ − νA(t) − µNT , Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(4)

176

N.J. Malunguzam S. Dube, J.M. Tchuenche et al.

Direct integration of (4) gives, NT (t) = NT (0) + φt − (ν

Z

t

A(τ )dτ + µ

0

Z

t

NT (τ )dτ ),

(5)

0

with initial conditions S(0) = S0 , I1 (0) = I0 , I2(0) =, I0 , A(0) = A0 and defined over the domain, Ω given by n Ω = (S, I1 , I2 , A) ∈ R4+ |S ≥ 0, I1 ≥ 0, I2 ≥ 0, A ≥ 0, S + I1 + I2 ≤ 1}. (6) Theorem 1. Any solutions (S(t), I1 (t), I2 (t), A(t)) of model system (1) are positive for all t ≥ 0. Proof. From model system (1), I1′

= ζ1 − (µ + α1 )I1 ,

I2′

= ζ2 − (µ + α2 )I2 ,

(7)

A′ = ζ3 − (µ + ν)A, where ζ1 = β1NSI1 , ζ2 = be written as,

β2 SI2 N , ζ3

= α1 I1 + α2 I2 . The equations in model system (7) can

I1′ ≥ −(µ + α1 )I1 , I2′ ≥ −(µ + α2 )I2 , and

(8)

A′ ≥ −(µ + ν)A.

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Integration of equations in (8) gives, I1 ≥ I1 (0)e(µ+α1 )t , I2 ≥ I2 (0)e(µ+α2 )t , and

(9)

A ≥ A(0)e(µ+ν)t . The non-negativity of I1 , I2 and A, implies the non-negativity of S since these compartments all come from a biologically feasible region Ω.

2.1.

Existence and Uniqueness of Solutions

Theorem 2. The model (1) is dissipative, that is, all feasible solutions are uniformly bounded in a proper subset of Ω ⊂ R4+ . Proof. Let W = S + I1 + I2 + A(t) and (S(t), I1 (t), I2 (t), A(t)) ∈ R4+ be any solution of the model system (1) with non-negative initial conditions. Taking with respect to t, derivatives of W gives W ′ = S ′ + I1′ + I2′ + A′ (10) = φ − µW − νA.

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Applying Rota and Birkhoff’s Theorem [1] on differential inequality, equation (10) now reads 0 ≤ W ≤ φ/µ + W (0)e−µt , (11) where W (0) represents the value of W evaluated at the initial values of the respective variables. Thus, as t → ∞ (11) becomes 0 ≤ W ≤ φ/µ.

(12)

Therefore, all feasible solutions of the model enter the region n o Ω1 = (S, I1 , I2 , A) ∈ R4+ |W ≤ φ/µ + δ .

(13)

Hence, the region Ω of biological interest is a compact set which is positively invariant and attracting. We illustrate the phase plane portraits of model system (1) in Figures (2), (a), (b) and (c) which also illustrate the existence and uniqueness of a solution set (S(t), I1 (t), I2 (t), A(t)) ◦

◦

of model system (1) in Ω, the interior of Ω. In this region Ω, basic results such as existence, uniqueness and continuation of solutions are valid for system (1). Thus, system (1) is mathematically and epidemiologically well-posed.

2.2.

Disease-Free Equilibrium and Stability

Model system (1) has the disease-free equilibrium (DFE),

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ξ0 = {S =

φ , I1 = 0, I2 = 0, A = 0}. µ

(14)

The basic reproduction number R0 gives the average number of secondary cases produced by an infected individual in a totally susceptible population during the entire infectious period. If R0 < 1, then, the infective produces less than one new case throughout its infectious period and the disease cannot invade the population. If R0 > 1, then, one new infective produces on average more than one new infection and the disease can invade the population. To find R0 , we use van den Driessche and Watmough’s technique [17] so that,     β1 0 µ + α1 0 . (15) and F = V= 0 β2 0 µ + α2 Thus, R0 = ρ(FV −1 ) where ρ(FV −1 ) is the spectral radius of the next generation matrix (FV −1 ), giving n β β2 o 1 R0 = max = max{R1 , R2 }, (16) , (µ + α1 ) (µ + α2 ) β2 β1 , and R2 = (µ+α . The partial reproductive number Ri , gives the where R1 = (µ+α 1) 2) number of secondary infectives produced by an individual infected with strain i during his/her infectious period when introduced into a wholly susceptible population. Equilibria analysis in the absence of disease gives conditions under which a disease will establish itself or be eradicated in the population. Using Theorem 2 in [17], the following result is established.

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(a)

(b)

(c) Figure 2. A phase plane portrait for model (1) in the SI phase plane for (a) β1 = β2 =

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0.3314, α1 = α2 = 0.125, φ = µ = 0.029, ν = 0.333, (R1 = R2 = 2.15), (b) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.6, µ = 0.029, ν = 0.333, (R1 = 2.15, R2 = 0.53), (c) β1 = β2 = 0.3314, α1 = 0.5, α2 = 0.8, µ = 0.029, ν = 0.333, (R1 = 0.63, R2 = 0.4), for τ = 10 varying initial conditions.

Lemma 1. The model system (1) has the disease-free equilibrium ξ0 . If R0 < 1, then, the disease-free equilibrium is locally asymptotically stable. If R0 > 1, then, the disease-free equilibrium is unstable.

2.3.

Persistence

The disease (for both strains) will die out if R0 < 1. The disease may persist if R0 > 1 for at least any one strain. Persistence (or permanence) theory has developed into a fascinating area with important applications in mathematical ecology and epidemiology. It addresses the long-term survival of certain (if not all) components in ecological (or other) systems using and further developing the concepts and tools of dynamic system theory [14, 15]. In the epidemiology of infectious diseases, persistence has two faces, persistence (or endemicity) of the disease and survival of the host population [14]. The transmission rate βi measures the transmissibility of the disease while αi measures the virulence (level of harm) of the disease. These are the two most important parameters in the spread and persistence of the disease. Strains that are highly virulent, that kill their host before transmission to another host, reduce their chance of survival and ultimately the persistence of that strain

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may not be possible. The reproduction number is the product of the transmission rate βi , 1 . From the definition of and the average death adjusted length of the infectious period µ+α i R0 , if the strain is highly virulent as measured by αi , such that R0 < 1, then, in this case the strain and the disease cannot persist. If αi is reduced, then, R0 > 1 and the disease may persist in the population. We consider persistence next. Lemma 2. Any positive solution of system (1) is uniformly persistent for any time t > 0 whenever R0 is greater than unity. Proof. For uniform persistence, Lasalle invariance principle [9] guarantees that ∃ c > 0, such that c < lim inf (S(t), I1 (t), I2 (t), A(t)), t→∞

provided ξ0 ∈ Ω. Taking the limits of the equations in (8) as t → ∞, we have: lim inf I1 (t) ≥ I1 (0) = c2 = 0, t→∞

lim inf I2 (t) ≥ I2 (0) = c3 = 0. t→∞

Similarly, from the positivity of the first and last equations of model (1) we have φ , t→∞ µ lim inf A(t) ≥ A(0) = c4 = 0.

lim inf S(t) ≥ S(0) = c1 = t→∞

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Thus, we can find a constants c = min(c1 , c2 , c3 , c4 ) = 0 such that 0 ≤ lim inf (S(t), I1 (t), I2 (t), A(t)), t→∞

and since the only invariant subsets on ∂Ω is ξ0 = {( µφ , 0, 0, 0)} which is isolated, while the ◦

only largest compact invariant set in Ω, the interior of Ω is any of the endemic steady states ξ1 , ξ2 for any of the exclusive strains which is absorbing (because Ω is positively invariant and attracting), it follows from a result of Hofbauer and So [5] that system (1) is strongly uniformly persistent. That is, the disease will persist at an endemic equilibrium level if it initially exists, with R0 > 1 and ξ0 unstable, and all solutions starting in Ω and sufficiently close to ξ0 move away from ξ0 . We note here that the uniform persistence of the system is independent of the model strains 1 (R1 > R2 ) and 2 (R2 > R1 ), but only on R0 > 1, and lim sup(S(t), I1 (t), I2 (t), A(t)) = t→∞

φ . µ

The behaviour of the local dynamics near the disease-free steady state ξ0 for any of the exclusive strain implies that system (1) is permanent provided R0 > 1. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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2.4.

Global Stability of Disease-Free Equilibrium

Theorem 3 (Castillo-Chavez et al., [3]). If system (1) can be written in the form dX = F (x, Z), dt

(17) dZ = G(X, Z), G(x, 0) = 0, dt where X ∈ Rm denotes (its components) the number of uninfected individuals and Z ∈ Rn denotes (its components) the number of infected individuals including latent, infectious, etc. U0 = (x∗ , 0) denotes the disease-free equilibrium of the system. And assume that (i) ∗ For dX dt = F (X, 0), X is globally asymptotically stable (GAS), (ii) G(X, Z) = AZ − ˆ ˆ G(X, Z), G(X, Z) ≥ 0 for (X, Z) ∈ Ω, where A = DZ G(X ∗ , 0) is an M -matrix (the off diagonal elements of A are nonnegative) and Ω is the region where the model makes biological sense. Then the fixed point U0 = (x∗ , 0) is a globally asymptotic stable (GAS) equilibrium of model system (1) provided that R0 < 1. Applying Theorem 3 to model system (1) gives    S β1 I1 1 − S + I1 + I2     ˆ  ≥ 0. S G(X, Z) =  β2 I2 1 −   S + I1 + I2  0

(18)

ˆ Since 0 ≤ S ≤ S + I1 + I2 , it is clear that G(X, Z) ≥ 0. Hence, by Theorem 3, ξ0 is GAS. We summarise the result in Lemma 3. Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Lemma 3. If R0 < 1, the DFE is globally asymptotically stable and unstable if R0 > 1.

2.5.

Exclusive Endemic Equilibrium and Stability

The strain 1 exclusive endemic equilibrium point (ξ1 ) is given by n o φ+ (α µφ φα1 (µ+α1 −β1 ) 1 −β1 ) S = (−α1φ+β1 ) , I1 = µ+α , I = 0, A = 2 (µ+ν)(µ+α1 )(α1 −β1 ) , 1 which can be expressed in terms of the reproductive number as o n φ(1−R1 ) φα1 (1−R1 ) φ , I = . , I = 0, A = S = α1 (R1 −1)+µR 1 2 α1 −β1 (µ+ν)(α1 −β1 ) 1

(19)

(20)

The exclusive endemic equilibrium ξ1 exists if R1 > 1. The strain 2 exclusive endemic equilibrium point (ξ2 ) is given by, n o φ+ (α µφ φα2 (µ+α2 −β2 ) 2 −β2 ) (21) S = (−α2φ+β2 ) , I1 = µ+α , I = 0, A = 1 (µ+ν)(µ+α2 )(α2 −β2 ) , 2 which can be expressed in terms of the reproductive number as o n φ(1−R2 ) φα2 (1−R2 ) φ , I = 0, I = . , A = S = α2 (R2 −1)+µR 1 2 α2 −β2 (µ+ν)(α2 −β2 ) 2

(22)

The exclusive endemic equilibrium ξ2 exists if R2 > 1. We establish the following Lemma; Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Lemma 4. The model system (1) has at least one unique exclusive endemic boundary equilibrium if and only if Ri > 1 for i = 1, 2.

Model system (1) has a Jacobian matrix given by   µ+α1 +R21 (µ+α1 )−R1 (µ+2α1 ) (µ+α1 )(1−R1 )+R2 (µ+α2 ) µ+α1 − − 0 − R1 R1 R1   (R1 −1)(µ+α1 ) (R1 −1)(µ+α1 ) (R1 −1)2 (µ+α1 )   − − 0 ,  R1 R1 R1 J =  (R1 −1)(µ+α2 )  0 0 − 0  R1 0 α1 α2 −µ−ν (23) It follows from the Jacobian matrix (23) that the eigenvalues are given by the roots of the characteristic equation ∆(z) = z 4 + a1 z 3 + a2 z 2 + a3 z + a4 = 0.

(24)

when evaluated at ξ1 and ξ2 .

a1 = (1 −

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a2 =

R2 ) + α1 (R1 − 1) + µ(R1 + 1) + ν, R1

1 [α1 (µ + α1 ) − R2 (µ + ν − α1 )(µ + α2 ) + R21 (µ + α1 )(3µ + ν + α1 + α2 ) R1 +R1 (µν − α1 (5µ + ν + 2α1 )

= +(µ + ν − α1 )α2 − R2 (µ + α1 )(µ + α2 ))], a3 =

1 [(−α1 + R1 (µ + α1 ))(R2 (µ + α1 )(µ + α2 ) + R21 (µ(3µ + 2ν) R1 +(2µ + ν)α2 + α1 (2µ + ν + α2 ))

= −R1 (R2 (2µ + ν + α1 )(µ + α2 ) + (µ + α1 )(2µ + ν + α2 )))] a4 =

and

(µ + ν)(R1 − 1)(R1 − R2 )(µR1 + α1 (R1 − 1))(µ + α2 ) . R21

(25) By the Routh Hurwitz criterion it follows that all eigenvalues of the characteristic equation (24) have negative real parts if a1 > 0, a1 (a2 + a3 ) > 0 and a4 > 0. From (25), it can be seen that a1 > 0, a4 > 0 if R1 > 1 and R1 > R2 .

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The expression a1 (a2 + a3 ) > 0 can be expressed as

1 A (A2 R31 1

+ A3 ) > 0, where

A1 = R21 (µ + α1 ) − R2 (µ + α2 ) + R1 (2µ + ν − α1 + α2 )), A2 = R1 α1 (µ + α1 ) − R2 (µ + ν − α1 )(µ + α2 ) + R21 (µ + α1 )(3µ + ν + α1 + α2 ) +R1 (µν − α1 (5µ + ν + 2α1 ) + µ + ν − α1 )α2 − R2 (µ + α1 )(µ + α2 )), A3 = (−α1 + R1 (µ + α1 ))(R2 (µ + α1 )(µ + α2 ) + R21 (µ(3µ + 2ν) + (2µ + ν)α2 +α1 (2µ + ν + α2 ) − R1 (R2 (2µ + ν + α1 )(µ + α2 ) + µ + α1 )(2µ + µ + α2 )), (27) A1 > 0, A2 + A3 > 0. (28) Theorem 4. The exclusive strain 1 endemic equilibrium ξ1 , for the model system (1) exists when R1 > 1 and R1 > R2 and is locally asymptotically stable if the inequalities (26) and (28) are satisfied. Similarly we have the following Theorem: Theorem 5. The exclusive strain 2 endemic equilibrium ξ2 for the model system (1) exists when R2 > 1 and R2 > R1 and is locally asymptotically stable. The proof for Theorem 5 is the same as that for Theorem 4.

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3.

Two Strain HIV/AIDS Model with Superinfection

Model system (1) is extended to incorporate the idea of a population afflicted with two HIV strains by incorporating a situation where the class of infectives includes those individuals infected by both strains 1 and 2 and this new superinfected class is denoted by J. The susceptible population S can be infected by strain 1 at a transmission rate β1 and move to the class I1 . Those infected with strain 1, I1 individuals in turn progress to clinical AIDS at a constant rate α1 . Conversely, the susceptible population can be infected by strain 2 at a transmission rate β2 and move to join the class of infectives I2 . Individuals infected with the strain 2 virus will progress to clinical AIDS at a constant rate α2 . Individuals infected by one strain can either get partial or total immunity from other strains or they can become more susceptible to reinfection with an enhancement or reduction of infection factor ǫ. In the extreme case, ǫ = 0 and infection by one strain confers immunity to all other strains. For ǫ = 1, an infected individual is as vulnerable as an uninfected individual. If ǫ > 1, then reinfection is more likely than the regular infection. If 0 < ǫ < 1, then, reinfection is less likely than the regular infection but immunity is not total. Individuals infected with strain 1 can come into effective contact with individuals infected with strain 2 only and be infected at a rate ǫ2 β2 and move to the superinfected class J. Strain 2 only infectives can come into effective contact with strain 1 only infected individuals and get infected by strain 1 at a rate ǫ1 β1 and move into the superinfected class J. Superinfected infectives either transmit strain 1 or strain 2 at transmission rates γ1 and γ2 , respectively,

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where γ2 = 1 − γ1 . Susceptible individuals who have not had any contact with any of the strains may come into contact with the superinfected and be infected with strain 1 only at rate γ1 β3 and move to the class I1 or be infected with strain 2 at a rate γ2 β3 in which case they move to class I2 . The superinfected progress to symptomatic AIDS at a rate α3 . Strain 1 only infectives can come into contact with the super-infectives and be inoculated with strain 2 at a rate ǫ2 γ2 β3 and move to the superinfected class J. Conversely, strain 2 only infectives can come into contact with the super-infectives and be inoculated with strain 1 at a rate ǫ1 γ1 β3 and move to the superinfected class J. For convenience, the probability of simultaneous infection by both strains is assumed to be negligible. The model flow diagram is shown in Figure 3 where the dotted lines denote indirect interaction. Thus, from the above assumptions, the proposed model takes the form; β1 SI1 β2 SI2 β3 SJ − − , N N N

S′

= φ − µS −

I1′

=

ǫ2 β2 I1 I2 ǫ2 γ2 β3 I1 J β1 SI1 γ1 β3 SJ + − (µ + α1 )I1 − − , N N N N

I2′

=

β2 SI2 γ2 β3 SJ ǫ1 β1 I2 I1 ǫ1 γ1 β3 I2 J + − (µ + α2 )I2 − − , N N N N

J′

=

ǫ1 γ1 β3 I2 J ǫ2 β2 I1 I2 ǫ1 β1 I2 I1 ǫ2 γ2 β3 I1 J + + + − (µ + α3 )J, N N N N

(29)

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A′ = α1 I1 + α2 I2 + α3 J − (µ + ν)A, with initial conditions S(t) = S0 (t), I(t) = I0 (t), J(t) = J0 (t) and A(t) = A0 (t) with NT (t) = S(t) + I1 (t) + I2 (t) + J(t) + A(t),

(30)

where NT (t) is the total human population. Differentiating (30) and substituting (29) gives NT′ (t) = S ′ (t) + I1′ (t) + I2′ (t) + J ′ + A′ (t) = φ − (ν + µ)A(t).

(31)

Integration of (31) gives

NT (t) = NT (0) + φt − (ν + µ)

and thus NT (t) is a non constant population. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Z

t

A(τ )dτ, 0

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Figure 3. Flow diagram for model with super-infection.

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Theorem 6. Any solutions (S(t), I1 (t), I2 (t), J(t), A(t)) of model system (29) are positive and bounded for all t ≥ 0. Proof. The second equation in model system (29) can be written as, I1′ ≥ −(µ + α1 + ǫ2 β2 I2 + ǫ2 γ2 β3 J)I1 .

(33)

Integration of (33) gives, n

I1 (t) ≥ I1 (0) exp − [(µ + α1 )t +

Z

0

t

o (ǫ2 β2 I2 (τ ) + ǫ2 γ2 β3 J(τ ))dτ ] ,

(34)

as t → ∞, the exponential term in (34) decays to 0 and I1 (t) ≥ 0, similarly I2 ≥ 0. The fourth and fifth equations in model system (29) can be written as J ′ ≥ −(µ + α3 )J, A′ ≥ −(µ + ν)A.

(35)

J(t) ≥ J(0) exp{−(µ + α3 )t}, A(t) ≥ A(0) exp{−(µ + ν)t}.

(36)

By direct integration of (35) we get

as t → ∞, equation (36) gives J(t) ≥ 0 and A(t) ≥ 0. The non-negativity of I1 , I2 , J and A, implies the non-negativity of S, since these compartments all come from the biologically feasible region Φ.

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3.1.

185

Existence and Uniqueness of Solutions

Theorem 7. Model system (29) is dissipative, that is, all feasible solutions are uniformly bounded in a proper subset of Φ ⊂ R5+ . n o Proof. Let W = S + I1 + I2 + J and S(t), I1 (t), I2 (t), J(t), A(t) ∈ R5+ be any solution of the system with non negative initial conditions. Taking derivatives of W gives, W ′ = S ′ + I1′ + I2′ + J ′ , A′ (37) = φ − µW − νA. Equation (37) can be written as W ′ ≤ φ − µW.

(38)

Applying Birkhoff and Rota’s Theorem [1] on differential inequality, (38) results in 0 ≤ W ≤ φ/µ + W (0)e−µt ,

(39)

where W (0) represents the value of equation W evaluated at the initial values of the respective variables. Thus, as t → ∞ in (39) 0 ≤ W ≤ φ/µ. Therefore, all feasible solutions of the model system (29) enter the region n o Φ = (S, I1 , I2 , J, A) ∈ R5+ |W ≤ φ/µ + δ .

(40)

(41)

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Hence, the region Φ of biological interest is positively invariant.

3.2.

Reproduction Numbers and Equilibria

n o At the disease-free equilibrium ξ0 = S = φ/µ, I1 = 0, I2 = 0, J = 0, A = 0 and using van den Driessche and Watmough’s technique [17] gives the spectral radius n R0 = ρ(FV −1 ) = max

n o β1 β2 o , = max R1 , R2 , µ + α1 µ + α2

and the following Theorem follows from [17]. Theorem 8. The model system (29) has the disease-free equilibrium ξ0 and if R0 < 1 the disease-free equilibrium is locally asymptotically stable. If R0 > 1 the disease-free equilibrium is unstable. The stability of the disease free equilibrium for model (29) can be explored using similar techniques to those used in the analysis for model system (1) but are omitted here to avoid repetition. We note that besides the disease-free equilibrium, the model system (29) has two one strain exclusive equilibria coinciding with equilibria for model system (1) and a co-existence equilibrium given by

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n o ¯ I¯1 , 0, 0, A¯ ; S, n φ φ(1 − R1 ) ¯ φα1 (1 − R1 ) o S¯ = , I¯1 = , I2 = 0, J¯ = 0, A¯ = , α1 (R1 − 1) + µR1 α1 − β1 (µ + ν)(α1 − β1 ) (42)

if R1 > 1, n o ¯ 0, I¯2 , 0, A¯ ; S, n S¯ =

φ(1 − R2 ) ¯ φα2 (1 − R2 ) o φ , I¯1 = 0, I¯2 = , J = 0, A¯ = , α2 (R2 − 1) + µR2 α2 − β2 (µ + ν)(α2 − β2 ) (43)

if and only if R2 > 1 and ξp = (S ∗ , I1∗ , I2∗ , J ∗ , A∗ ),

(44)

this is the co-existence equilibrium whose dynamics is governed by the invasion reproduction numbers R12 and R21 (see next section).

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3.3.

Invasion Reproduction Numbers

The invasion reproduction number Rik gives the number of secondary infections that one infected individual with strain i will produce in a population in which strain k is at equilibrium. The invasion reproduction number Rik is found by using the method in [17] but linearizing at ξk instead. The strain 2, exclusive endemic equilibrium is given by o n ¯ 0, I¯2 , 0, A¯ ; S, n φ φ(1 − R2 ) ¯ φα2 (1 − R2 ) o S¯ = , I¯1 = 0, I¯2 = , J = 0, A¯ = . α2 (R2 − 1) + µR2 α2 − β2 (µ + ν)(α2 − β2 ) (45)

Linearizing about this equilibrium gives the dominant eigenvalue which in this case is R12 , the invasion reproduction number near the strain 2 equilibrium. It gives conditions under which strain 1 will invade a population in which strain 2 has reached endemicity and is at equilibrium. Resultantly, R12 =

R1 (µ + α1 )(µ + α3 ) + (−1 + R2 )(µ + α2 )β3 γ1 ǫ2 . β3 γ1 ǫ1 + R2 (µ + α3 − β3 γ1 ǫ1 ))(µ + α1 + (−1 + R2 )(µ + α2 )ǫ2

(46)

If R2 > 1 > R1 , then, strain 2 can invade the disease-free equilibrium but strain 1 cannot. We note the superinfection effect which does not make a requirement for R1 to be greater than 1 in order to be able to invade the strain 2 equilibrium as long as R12 > 1. Similarly, the strain 1 exclusive endemic equilibrium ξ2 is given by

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n o ¯ I¯1 , 0, 0, A¯ ; S, n S¯ =

φ φ(1 − R1 ) ¯ φα1 (1 − R1 ) o , I¯1 = , I2 = 0, J¯ = 0, A¯ = . α1 (R1 − 1) + µR1 α1 − β1 (µ + ν)(α1 − β1 ) (47)

Linearizing about this equilibrium gives R12 , the invasion reproduction number near the strain 1 equilibrium. It gives conditions under which strain 2 will invade a population in which strain 1 has reached endemicity and is at equilibrium. Resultantly

R21 =

R2 (µ + α2 )(µ + α3 ) + (−1 + R1 )(µ + α1 )β3 γ2 ǫ1 . β3 γ2 ǫ2 + R1 (µ + α3 − β3 γ2 ǫ2 ))(µ + α2 + (−1 + R1 )(µ + α1 )ǫ1

(48)

If R1 > 1 > R2 , then, strain 1 can invade the disease-free equilibrium but strain 2 cannot. Strain 2 can invade ξ1 if R21 > 1. That is, the presence of strain 1 increases the susceptibility of the the population to strain 2. Model system (29) has the following endemic equilibria; • the strain 1 exclusive boundary equilibrium given by (42), • the strain 2 exclusive boundary equilibrium given by (43) and

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• co-existence equilibrium given by, S∗

=

φ β1 F1 + β2 F2 + β3 F3 − , µ µN ∗

I1∗

=

β1 F1 − β2 ǫ2 F4 + β3 (γ1 F3 − ǫ2 F5 ) , N ∗ (µ + α1 )

I2∗

=

β2 F2 − β1 ǫ1 F4 + β3 (γ2 F3 − ǫ1 F5 ) , N ∗ (µ + α2 )

J∗

=

A∗ = +

F4 (β1 ǫ1 + β2 ǫ2 ) + β3 (ǫ2 F2 + ǫ1 F6 ) , N ∗ (µ + α3 ) α1 (β1 F1 −β2 ǫ2 F4 +β3 (γ1 F3 −ǫ2 F5 )) 4 +β3 (γ2 F3 −ǫ1 F6 )) + α2 (β2 F2 −β1 ǫ1 Fµ+α µ+α1 2 N ∗ (µ + ν) α3 (β1 ǫ1 F4 +β2 ǫ2 F4 +β3 (ǫ2 F5 +ǫ1 F6 )) µ+α3 N ∗ (µ+ν)

.

where F1 = S ∗ I1∗ , F2 = S ∗ I2∗ , F3 = S ∗ J ∗ , F4 = I1∗ I2∗ , F5 = I1∗ J ∗ , F6 = I2∗ J ∗ , and N ∗ = S ∗ + I1∗ + I2∗ + J ∗ .

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Lemma 2 also holds for the two strain model with superinfection. Further, asymptotic properties of the equilibria will also be explored using numerical simulation. It can be shown however that • The strain 1 exclusive equilibrium ξ1 exists if and only if R1 > 1 in (42), and is locally asymptotically stable for R21 < 1 in (46) otherwise ξ1 is unstable. • The strain 2 exclusive equilibrium ξ2 exists if and only if R2 > 1 in (43), and is locally asymptotically stable for R12 < 1 in (48) otherwise ξ2 is unstable.

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4.

Numerical Simulations

In this section, numerical simulations are used to comprehensively analyse and compare the models in Sections 2 and 3. A Runge-Kutta algorithms of order four is employed using a code in C++ to obtain numerical solutions to models (1) and (29). In our numerical simulation, we use some parameter values for Zimbabwe obtained from the Central Statistical Office of Zimbabwe (CSOZ) while some are reasonable estimates. Zimbabwe is a country in Sub-Saharan Africa and it is estimated that one fifth [16] of its adult population is living with HIV and that at least 565 [18] new adults and children infections take place every day (roughly one person every three minutes). During the years 2002 to 2006, the population is estimated to have decreased by four million people [13], while infant mortality has doubled since 1990 [13]. Average life expectancy for women, who are particularly affected by Zimbabwe’s AIDS epidemic, is 34 years and is one of the the lowest in the world [19]. The simulations are conducted using absolute numbers and then converted back to proportions for the figures. Table 1. Model parameters and their interpretations Parameter µ ν β1

Interpretation Natural death rate AIDS induced death rate Transmission rate of strain 1

Value 0.02 0.333 0.3314

β2

Transmission rate of strain 2

0.3314

α1

Rate of progression to AIDS for strain 1 infectives Rate of progression to AIDS for strain 2 infectives

0.125

α2

4.1.

0.125

Source CSOZ CSOZ deduced from doubling time of 3.5 years deduced from doubling time of 3.5 years Average incubation period of 8 years Average incubation period of 8 years

The Model without Superinfection

Here, we present the numerical simulations for model (1) in Figure 4. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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(a)

(b)

(c)

(d)

Figure 4. Numerical simulations for model (1) for (a) β1 = β2 = 0.3314, α1 = α2 = 0.125, φ = µ = 0.029, ν = 0.333 ( R1 = R2 = 2.15 > 1), (b) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.6, µ = 0.029, ν = 0.333 (R1 = 2.15 > 1) and (R2 = 0.53 < 1), (c) β1 = β2 = 0.3314, α1 = 0.5, α2 = 0.8, µ = 0.029, ν = 0.333 (R1 = 0.63, R2 = 0.4), (d) β1 = β2 = 0.3314, α1 = 0.1, α2 = 0.2, µ = 0.029, ν = 0.333 (R1 = 2.57 > 1, R2 = 1.45 > 1) for τ = 10 varying initial conditions. Figure 4 illustrates the following: 4(a) The two strains have equal virulence and transmissibility and R1 = R2 = 2.15 > 1. The two strains only differ in subtype, thus, we have the same reproduction number being greater than unity. No single strain dominates and the two strains co-exist at their exclusive equilibria and the disease persists. They therefore trace the same trajectory paths in phase space. Thus, the graph for I1 (t) is superimposed on the graph for I2 (t) since we use the same initial conditions and parameter values. 4(b) The basic reproduction number due to strain 1 is greater than unity R1 = 2.15 > 1 and the disease persists. The reproduction number due to strain 2 is less than unity (R2 = 0.53 < 1) and strain 2 dies off. 4(c) The reproduction numbers for both strains are less than unity (R1 = 0.63, R2 = 0.4) and their trajectories in phase space enter the disease-free equilibrium and HIV/AIDS is eradicated.

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4(d) Both reproduction numbers are greater than unity (R1 = 2.57 > 1, R2 = 1.45 > 1), but strain 2 is forced to die off in a process of competitive exclusion.)

4.2.

The Progression of the Infective Population with Time for Model (29)

In Figure 5 we illustrate the numerical simulation of model (29) showing the progression of the infective sub-populations for varying values of R0 . In Figure 5 above we illustrate numerical simulations for model (29) with superinfection 5(a) Strains 1 and 2 have the same parameters and both reproduction numbers are greater than unity (R1 = 2.15 = R2 = 2.15 > 1). The graph illustrates that the trajectories for strain 1 are superimposed on the trajectories for strain 2 in phase space since we used the same parameters and initial conditions. However, there is no interaction among infectives carrying differing strains as well as the joint strain hosts and the populaton of joint infectives J is not replenished and therefore quickly dies off.

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5(b) Strains I1 and I2 have the same parameters and both reproduction numbers are greater than unity (R1 = R2 = 2.15 > 1). Here, the enhancement factor at reinfection is ǫ = 1. The graph illustrates that the trajectories for strain 1 are superimposed on the trajectories for strain 2 in phase space. There is interaction among infectives carrying differing strains and a single strain infected individual is equally susceptible to infection as a non-infected (that is, the presence of an infection does not influence the rate of infection by other strains). There exists a sub-populaton of joint infectives J whose proportion remains less than that of the single infectives. 5(c) Strains 1 and 2 have the same parameters and both reproduction numbers are greater than unity (R1 = R2 = 2.15 > 1). The enhancement factor at reinfection (ǫ = 4) is greater than unity and infectives are more susceptible to re-infection by subsequent strains. A class of joint infectives exists and converges to an endemic equilibrium solution. The graph of J lies above that of I1 and I2 . Thus, the proportion of joint infectives is now higher than the proportion of single strain infectives. 5(d) The basic reproduction number for strain 1, R1 = 2.15 > R2 = 1.01 > 1 and I1 settles at the exclusive endemic equilibrium. The coefficient of enhancement (ǫ = 0.9) is less than unity. Infection by one strain confers partial immunity to infection by other strains. Even though R2 > 1, strain 2 dies off in a process of competitive exclusion. 5(e) Reproduction numbers for both strains are greater than unity (R1 = 2.15 > R2 = 1.01 > 1). The coefficient of enhancement is greater than unity (ǫ = 4) and both co-exist (compare with (d) above). The subpopulations infected with strain 2 alone remain near but above the disease-free equilibrium. 5(f) Strain 2 cannot invade the disease-free equilibrium on its own because its reproduction number, (R2 = 0.95 < 1) is less than unity but superinfection allows it to invade the strain 1 endemic equilibrium since (R21 = 4.23 > 1) and therefore maintain the existence of joint infectives J(t). Since ξ1 can be invaded by strain 2 (R21 = 4.23 > 1), this causes ξ1 to be unstable.

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(a)

(b)

(c)

(d)

(e)

(f )

Figure 5. Numerical simulation of the infected sub-populations I1 , I2 , J for model (29) for (a) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.125, µ = 0.029, ν = 0.333, γ1 = γ2 = 0.5, ǫ = 0 (R1 = 2.15 = R2 = 2.15 > 1), (b) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.125, α3 = 0.125, µ = 0.029, ν = 0.333, γ1 = γ2 = 0.5, ǫ = 1 (R1 = 2.15 = R2 = 2.15 > 1), (c) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.125, α3 = 0.125, µ = 0.029, ν = 0.333, γ1 = γ2 = 0.5, ǫ = 4 (R1 = R2 = 2.15 > 1), (d) β1 = β2 = 0.3314, α1 = 0.125, α2 = 0.3, α3 = 0.3, µ = 0.029, ν = 0.333, γ1 = 0.3, γ2 = 0.7, ǫ = 0.9 R1 = 2.15 > R2 = 1.01 > 1, (e) β1 = β2 = β3 = 0.3314, α1 = 0.125, α2 = 0.3, α3 = 0.3, µ = 0.029, ν = 0.333, γ1 = 0.3, γ2 = 0.7, ǫ = 7 ((R1 = 2.15 > R2 = 1.01 > 1)), (f ) β1 = β2 = β3 = 0.3314, α1 = 0.1, α2 = 0.32, α3 = 0.32, µ = 0.029, ν = 0.333, γ1 = 0, γ2 = 1, ǫ = 7 (R2 = 0.95 < 1, R21 = 4.23 > 1) for τ = 10 varying initial conditions.

5.

Summary and Concluding Remarks

In model (1) we presented a deterministic compartmental model for the transmission dynamics of HIV in a community where two subtypes are co-circulating with no cases of Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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super-infection. Infection by one strain confers on the infected total immunity to other strains. It was shown that the disease-free equilibrium is locally and globally asymptotically stable whenever both reproduction numbers R1 and R2 are less than unity. Also, it was shown through numerical simulation that: • If the two strains have the same parameters and therefore the same reproduction numbers greater than unity with cases of superinfection negligible, then, the two strains co-exist and co-circulate in the community as shown in Figure 4(a). • If both strains have reproduction numbers higher than unity, the strain with the higher reproduction will drive the strain with the lower reproduction number into extinction in a process of competitive exclusion. Thus, competition for susceptibles kills off strains with higher virulence as shown in Figure 4(d). In model (29), we presented a deterministic compartmental model of HIV in a community where two sub-types are co-circulating and where a jointly infected population is in existence. The existence of this class was found to be dependent on the size of ǫ, the enhancement factor at re-infection. The obtained results are as follows.

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• If R1 = R2 > 1 and the coefficient of enhancement ǫ = 0, the joint infected population J is forced to extinction as there is no interaction between infectives carrying the two different strains. In this case the long term asymptotic dynamics of model (29) are the same as those for model (1) as shown in Figure 5(b). The two exclusive boundary equilibria still exist and converge to an equilibrium. • If R1 > R2 > 1 and ǫ < 1, then, infection by one strain confers on the infected partial immunity to other strains. The strain with the highest reproduction number dominates and stabilizes at its endemic equilibrium. The strain with the lower reproduction number is suffocated (fails to get susceptibles) and dies off even though its reproduction number is greater than unity. The jointly infected cases J is replenished at a decreasing rate and dies off as shown in Figure 5(c). • If R1 and R2 are greater than unity and ǫ is sufficiently greater than unity, then, the two strains co-exist and there is a stable co-existence equilibrium. The size of the jointly infected population J is larger than the ith strain infected population as shown in Figure 5. • If R1 > 1 but R2 < 1, with ǫ sufficiently larger than unity, strain 2 will approach but will not get to disease-free equilibrium and the strain will not die off. It sustains itself on the strain 1 infectives since these are more susceptible to infection than those who have not come into contact with any virus. This invasion of the strain 1 boundary equilibrium causes it to be unstable as shown in Figure 5(d). We conclude from the study that in cases of superinfection, co-existence of both strains is feasible and that superinfection may cause the persistence of strains with high levels of virulence. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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References [1] Birkhoff G. and G.C. Rota, Ordinary differential equations, Ginn (1982). [2] Burke D. S., Recombination of HIV; An important viral evolutionary strategy, Emerg. Infect. Dis., 3(3), 253-259 (1997). [3] Castillo-Chavez C., Z. Feng and W. Huang, On the computation of R0 and its role on global stability (2002) ( math.la.asu.edu/˜chavez/2002/JB276.pdf). [4] Gottlieb G. S., P. S. Sow, et al., Molecular Epidemiology of Dual HIV-1/HIV-2 Seropositive Adults from Senegal, West Africa, AIDS Research and Human Retroviruses, 19(7), 575-584 (2003). [5] Hofbauer J. and J. W. H. So, Uniform persistence and repellers for maps. Proc. AMS, 107, 1137-1142 (1989). [6] Introduction to HIV types, groups and subtypes, ( http://www.avert.org/hivtypes.htm), (Accessed, 2008) [7] Kassim S., A. Ackah, L. Abouya, et al., Maurice Prospective cohort study of HIV1/HIV-2 dual infection among HIV-infected tuberculosis patients in Abidjan, Cote d’Ivoire, Int. Conf. AIDS, Jul 9-14, 13: abstract no. TuPeC3355 (2000).

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[8] Laeyendecker O., X. Li, M. Arroyo, F. McCutchan, et al. The effect of HIV subtype on rapid disease progression in Rakai, Uganda. 13th CROI, Denver, Abstract 44LB (2006). [9] LaSalle J.P. The stability of dynamical systems. CBMS-NSF Regional Conf. Ser. in Appl. Math., 25, SIAM, Philadelphia (1976). [10] Marc A., M. Luc, G. Denise, et al., Human immunodeficiency virus type 2 (HIV-2) env polypeptide and diagnostic assays ( http://www.freepatentsonline.com/6979535.html) (Acessed, 2008). [11] Pieniazek D., L. M. Janini, A. Ramos, et al., HIV-1 Patients May Harbor Viruses of Different Phylogenetic Subtypes: Implications for the Evolution of the HIV/AIDS Pandemic, Emerg. Infect. Dis., 1(3), 86-88, (1995). [12] Soto-Ramirez L. E., B. Renjifo, M. T. Mclane, et al., HIV-1 Langerhans cell tropism associated with heterosexual transmission of HIV, Science, 71, 1291–1293 (1996). [13] The Independent, How AIDS and Starvation Condemn Zimbabwe’s Women to Early Grave, (November, 2006). [14] Thieme H. R., Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166, 173–201 (2000). [15] Thieme H. R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (2), 405–435 (1993). Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[16] UNAIDS, Report on the Global AIDS epidemic 2006, Annex 2: HIV/AIDS estimates and data (2006). [17] van den Driessche P. and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180, 29–48 (2002). [18] World Health Organisation, Zimbabwe country profile for HIV/AIDS treatment scale up (December 2005), (http://www.who.int/hiv/HIVCP ZWE.pdf), (Accessed 2008). [19] World Health Organisation, The World Health Report (2006).

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[20] Zhang J., J. K. Maniar, D. G. Saple, Tsuchie H, Kageyama S, Wakamiya N, Kurimura T. Int Conf AIDS. Study of HIV-1/HIV-2 dual infection in Bombay., 11; 169 (abstract no. Mo.C. 1668) (1996).

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ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 7

M ODELLING THE T RANSMISSION OF M ULTIDRUG - RESISTANT AND E XTENSIVELY D RUG - RESISTANT T UBERCULOSIS C.P. Bhunua,∗and W. Gariraa,b a Modelling Biomedical Systems Research Group, Department of Applied Mathematics, National University of Science and Technology, P. O. Box AC 939 Ascot, Bulawayo, Zimbabwe b Department of Mathematics and Applied Mathematics, University of Venda, South Africa

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Abstract In recent years, bacteria have become resistant to antibiotics, leading to a decline in the effectiveness of antibiotics in treating infectious diseases. This chapter presents a mathematical model for multi-strain tuberculosis transmission dynamics that is used to determine the burden of drug-sensitive, multidrug-resistant and extensively drugresistant tuberculosis. The reproduction numbers for the model are determined and stabilities analysed. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and the unique endemic equilibrium when the associated reproduction number is greater than unity. From the study, we conclude that effective isoniazid preventive therapy and active tuberculosis treatment results in a reduction of both drug sensitive and multidrugresistant tuberculosis as most cases of multidrug resistant tuberculosis are caused by misuse of first line tuberculosis drugs. Furthermore, effective use of second line tuberculosis results in a decrease of the multidrug-resistant and extensively drug resistant tuberculosis as most cases of extensively drug resistant tuberculosis are a result of inappropriate treatment of multidrug resistant tuberculosis. Thus, the model is used to establish a set of epidemiological conditions for the control of these tuberculosis strains in poor-resource settings.

Keywords: Multidrug-resistant tuberculosis, extensively drug-resistant tuberculosis, first line tubercusis drugs, second line tuberculosis drugs. ∗

E-mail addresses: [email protected], [email protected]. Corresponding author.

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1.

C.P. Bhunu and W. Garira

Introduction

Tuberculosis (TB) is a bacterial disease with about an estimated one third of the world’s population as its reservoir (Castillo-Chavez and Feng, 1998). TB is the second largest cause of death from an infectious agent after HIV/AIDS and in developing countries, Mycobacterium tuberculosis (Mtb) is one of the most common infections associated with HIV (Dye et al., 1999). The emergence of multidrug-resistant tuberculosis (MDR-TB) and extensively drug-resistant tuberculosis (XDR-TB) will likely impact TB treatment and control strategies (Reichman and Tanne, 2002). MDR-TB is defined as resistance to isoniazid and rifampcin (major two first line drugs) with/ without resistance to other first line drugs (Davies, 1999). Cases of MDR-TB will continue to increase as long as drug susceptible and other drug resistant cases persist anywhere and as long as some patients fail treatment (Dye and William, 2000). MDR-TB generated by inadequate treatment can transmit the infection to others, suggesting that interrupting the transmission cycle is the limiting factor in MDR-TB control because cure rates of MDR strains are significantly lower than those of drug sensitive or drug resistant strains (Espinal et al., 2000). Second line tuberculosis drugs are drugs used to treat MDR-TB. Failure to effectively treat MDR-TB leads to the creation of XDR-TB which is resistant to first and second line drugs. XDR-TB has been defined by the WHO Global Task Force as resistance to at least rifampicin and isoniazid in addition to any fluoroquinolone, and at least one of the three following injectable drugs: capreomycin, kanamycin and amikacin (WHO, 2006). XDRTB has been found in all regions of the world but is rare. MDR-TB usually has to occur before XDR-TB arises. Wherever second-line drugs to treat MDR-TB are being misused, there is a risk of XDR-TB. Countries in southern Africa have moved quickly to draw up a regional strategy for managing and preventing extensively drug-resistant TB. This follows an outbreak in South Africa that demonstrated the high mortality of XDR-TB when associated with HIV infection (McGregor, 2006). Worldwide attention was focused on South Africa when a research project publicized a deadly outbreak of XDR-TB in the small town of Tugela Ferry in KwaZulu-Natal. XDR-TB has now been reported in all provinces of South Africa, yet so far there have been no confirmed reports of cases in other countries in the region (Gandhi et al., 2006). It makes economic sense to treat TB properly in the first place. It costs US$ 52 to treat each patient with ordinary TB. If a patient develops multidrug-resistant TB, the cost of treatment dramatically increases to US$3168, which includes hospitalization and more expensive drugs. Global surveillance of anti-TB drug resistance indicates that drug-resistant TB is present everywhere and that it is especially severe in parts of China and in countries of the former Soviet Union (Frieden et al., 1993). The presence of XDR-TB in developing countries in the absence of its treatment only point to quarantine as a necessary solution to prevent the further spread of this strain of TB (Singh et al., 2007). Several groups have developed models of the spread of TB drug resistance (Blower et al., 1996; Castillo-Chavez and Feng, 1997; Gumel and Song, 2008). Our work differs from these studies in various ways: we are modelling a three strain TB model while all these groups were looking at a two strain model. This work further differs from Gumel and Song (2008) in that we have incorporated the likely benefits of quarantining individuals detected to have XDR-TB. The analysis of a three strain TB model incorporating drug sensitive,

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multidrug-resistant and extensively drug-resistant strains have not yet been explored to the best of our knowledge. This chapter is organised as follows. Section 2 presents the model description and analysis. In Section 3 there are numerical simulations for the model and finally we present a summary and concluding remarks.

2.

Model Description

We develop a mathematical model of the transmission of TB drug-sensitive, multidrugresistant and extensively drug-resistant strains between the following mutually exclusive compartments: susceptibles S(t), who have never encountered the Mtb; those exposed to TB or latently infected Ei (t), representing those that were infected and have not yet developed active TB; those latently infected with drug sensitive Mtb but under isoniazid chemoprophylaxis T (t); Infectives Ii (t), containing individuals with active TB and are infectious; Infectives under TB treatment Ji (t), and the recovered R(t), who were previously infected and successfully treated or self-cured, for i = s, m, x denote drug-sensitive strain, multidrug-resistant strain and extensively drug-resistant strain of Mtb, respectively. We assume that drug-resistant strains were initially generated through inadequate treatment of drug-sensitive TB, and that these strains could subsequently be transmitted to other individuals. Here, the term drug-resistant TB refers to multidrug-resistant and/or extensively drug-resistant TB. The total population size is N (t) and is given by N (t) = S(t) + Es (t) + Em (t) + Ex (t) + Is (t) + Im (t) (1)

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+ Ix (t) + T (t) + Js (t) + Jm (t) + Jx (t) + R(t). In the model, individuals enter the susceptible class at a constant rate Λ through birth or migration and they may be infected with a circulating Mtb strain at a time-dependent rate λi (t) =

βi cIi (t) + αi cJi (t) , N (t)

(2)

where βi is the probability of one individual being infected with Mtb strain i (i = s, m, x) by an untreated individual and αi is the probability of one individual being infected with Mtb strain i = s, m, x by an individual under TB treatment and c is the per capita contact rate. It is assumed that individuals who have active TB and not on treatment are more infectious than the corresponding active TB cases on treatment or quarantine (βi > αi ). Individuals in the different compartments suffer from natural death at a constant rate µ. A proportion fi of the Mtb-infected individuals move to the latently infected state Ei (t), whereas the remaining proportion of the infecteds (1 − fi ) move fast to the active TB disease state Ii (t). Those with latent infections progress to active TB as a result of endogenous reactivation of the latent bacilli and exogenous reinfection at rates ki and δi λi , respectively with δi ∈ (0, 1) since latently infected individuals have acquired partial immunity, which reduces the risk of subsequent infection, but does not completely prevent it (Gomes et al., 2004). Individuals latently infected with drug sensitive Mtb strain are detected at a rate ρ and move to a state T (t). Latently infected individuals in T (t) are given isoniazid preventive therapy and cure at rate η and move into the recovered class, R(t). Individuals with

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active TB, Ii (t) will eventually be detected at a rate ri to move into infectives detected class, Ji (t). They also experience disease induced death at rates ds1 , dm1 , dx for individuals in Is (t), Im (t) and Ix (t), respectively. Individuals in Js (t) and Jm (t) are treated at rates ωs and ωm , respectively and proportions ps and pm recover and move to the recovered state, R(t) and those in Jx (t) are all quarantined given the absence of XDR-TB treatment program in developing countries (Sidley, 2006). Here pm < ps because those with drug-resistant TB are cured at reduced rate. The complementary proportions experience treatment failure and develop drug resistance. Individuals in Js , Jm , Jx classes experience the following disease-induced death rates ds2 , dm2 , and dx , respectively. Recovered individuals are infected with Mtb at rate λi , with i = s, m, x since recovery from TB does not offer permanent immunity against the disease. A proportion fi develops latent TB to move into Ei (t) and the complementary (1 − fi ) of those in R(t) will develop fast TB and move into Ii (t). The model flow diagram is given in Figure 1

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The assumptions result in the following model equations: dS(t) = Λ − λS(t) − µS(t), dt dEs (t) = fs λs (S(t) + R(t)) − (ρ + µ + ks )Es (t) − (δs λs + λm + λx )Es (t), dt dEm (t) = fm λm (S(t) + R(t)) + λm Es (t) − (km + µ)Em (t) dt −(δm λm + λx )Em (t) + (1 − ps )ωs Js (t),

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dEx (t) = fx λx (S(t) + R(t)) + λx Es (t) + λx Em (t) − (kx + µ)Ex (t) dt −δx λx Ex (t) + (1 − pm )ωm Jm (t), dT (t) = ρEs (t) − (η + µ)T (t), dt dIs (t) = (ks + δs λs )Es (t) + (1 − fs )λs (S(t) + R(t)) − (rs + µ + ds1 )Is (t), dt dIm (t) = (km + δm λm )Em (t) + (1 − fm )λm (S(t) + R(t)) − (rm + µ + dm1 )Im (t), dt dIx (t) = (kx + δx λx )Ex (t) + (1 − fx )λx (S(t) + R(t)) − (rx + µ + dx )Ix (t), dt dJs (t) = rs Is (t) − (ωs + µ + ds2 )Js (t), dt dJm (t) = rm Im (t) − (ωm + µ + dm2 )Jm (t), dt dJx (t) = rx Ix (t) − (µ + dx )Jx (t), dt dR(t) = ωs ps Js (t) + ωm pm Jm (t) + ηT (t) − (λ + µ)R(t). dt (3) where λ = λs + λm + λx . (4)

2.1.

Basic Properties

In this section, we study some basic results of solutions of model system (3) which will be useful in the sequel in the proofs of stability and persistence results. Let Rn+ = (0, ∞) denote the set x = (x1 , x2 , · · · , xn ) with xj > 0 for j = 1, 2, · · · , n. We will use the following results in Appendix A of Thieme (2003): Lemma 1. Let F : Rn+ → Rn , F (x) = (F1 (x), F2 (x), · · · , Fn (x)), x = (x1 , x2 , · · · , xn ) ∂Fj which exist and are continuous in Rn+ for be continuous and have partial derivatives ∂xk j, k = 1, 2, · · · , n. Then, F is locally Lipschitz continuous in Rn+ . Theorem 2.1. Let F : Rn+ → Rn be locally Lipschitz continuous and for each j = 1, 2, · · · , n satisfy Fj (x) ≤ 0 whenever x ∈ Rn+ , xj = 0. Then, for every x0 ∈ Rn+ ,

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there exists a unique solution of x′ = F (x), x(0) = x0 with values in Rn+ which is defined n X in some interval (0, b] with b ∈ (0, ∞]. If b < ∞, then sup xj (t) = ∞. j=1

0 , E 0 , T 0 , I 0 , I 0 , I 0 , J 0 , J 0 , J 0 , R0 > 0, there exists Theorem 2.2. For all S 0 , Es0 , Em m x s m x s x S, Es , Em , Ex , T, Is , Im , Ix , Js , Jm , Jx , R : (0, ∞) → (0, ∞) which solve model system (3) with 0 , E = E0, T = T 0, I = I 0, I = I 0 , initial conditions S = S 0 , Es = Es0 , Em = Em x s m x s m 0 , J = J 0 , R = R0 . Ix = Ix0 , Js = Js0 , Jm = Jm x x

Proof. Applying Theorem 2.1, we define ′ (t), F (x) = E ′ (t), F1 (x) = S ′ (t), F2 (x) = Es′ (t), F3 (x) = Em 4 x ′ (t), F (x) = I ′ (t), F5 (x) = T ′ (t), F6 (x) = Is′ (t), F7 (x) = Im 8 x ′ (t), F (x) = J ′ (t), F (x) = R′ (t) F9 (x) = Js′ (t), F10 (x) = Jm 11 12 x

(5)

where x = (S, Es , Em , Ex , T, Is , Im , Ix , Js , Jm , Jx , R).

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By the properties of continuity over operations, we have continuity of Fl for all l = 1, 2, · · · , 12. Further ∂F1 N −S = −µ − c(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) , ∂x1 N2 ∂F1 Sc(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) = , ∂x2 N2 ∂F1 Sc(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) = , ∂x3 N2 ∂F1 Sc(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) = , ∂x4 N2 ∂F1 Sc(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) = , ∂x5 N2 ∂F1 βs cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) =− , ∂x6 N2 ∂F1 βm cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) =− , ∂x7 N2 βx cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) ∂F1 =− , ∂x8 N2 αs cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) ∂F1 =− , ∂x9 N2 αm cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) ∂F1 =− , ∂x10 N2 αx cN S − S(βs cIs + αs cJs + βm cIm + αm cJm + βx cIx + αx cJx ) ∂F1 =− , ∂x11 N2 ∂F1 Sc(βs Is + αs Js + βm Im + αm Jm + βx Ix αx Jx ) = , ∂x12 N2

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and these partial derivatives exist and are continuous. As a result, by Lemma 1, F is locally Lipschitz continuous. Let x1 = S = 0 and x2 = Es > 0, x3 = Em > 0, x4 = Ex > 0, x5 = T > 0, x6 = Is > 0, x7 = Im > 0, x8 = Ix > 0, x9 = Js > 0, x10 = Jm > 0, x11 = Jx > 0, x12 = R > 0 then F1 (x) = Λ > 0.

(7)

Now, let x2 = Es = 0 and x1 = S > 0, x3 = Em > 0, x4 = Ex > 0, x5 = T > 0, x6 = Is > 0, x7 = Im > 0, x8 = Ix > 0, x9 = Js > 0, x10 = Jm > 0, x11 = Jx > 0, x12 = R > 0, then, fs (βs cIs + αs cJs )(S + R) (8) > 0. N This is further done up to the case when x12 = R = 0 and the rest x1 = S > 0, x2 = Es > 0, x3 = Em > 0, x4 = Ex > 0, x5 = T > 0, x6 = Is > 0, x7 = Im > 0, x8 = Ix > 0, x9 = Js > 0, x10 = Jm > 0, x11 = Jx > 0 then F2 (x) =

F12 (x) = ωs ps Js (t) + ωm pm Jm (t) + ηT (t) > 0.

(9)

0 , E 0 , T 0 , I 0 , I 0 , I 0 , J 0 , J 0 , J 0 , R0 ) ∈ R12 , By Theorem 2.1, for every x0 = (S 0 , Es0 , Em x s m x s m x + ′ there exists a unique solution of x = F (x), x(0) = x0 with values in R12 which is defined + in some interval (0, b] with b ∈ (0, ∞]. If b < ∞, then,

sup N (t) = ∞. 0≤t≤b

(10)

Thus, N ′ = Λ − µN − ds1 Is − ds2 Js − dm1 Im − dm2 Jm − dx (Ix + Jx ) ≤ Λ − µN. So that using Birkhoff and Rota (1982) Theorem on differential inequality,

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0≤N ≤

Λ + N (0)e−µt , µ

(11)

where N (0) represents the value of (1) evaluated at the initial values of the respective variables. As t → ∞ Λ 0≤N ≤ , (12) µ so N (t) is bounded, a contradiction to Theorem 2.1. As a result b = ∞, implying that solutions of model system (3) are positive and are defined on (0, ∞). For the boundedness of solutions, the following Theorem 2.3 is established. Theorem 2.3. All solutions of model system (3) are bounded. Proof. Using model system (3) we have N ′ = Λ−µN −ds1 Is −ds2 Js −dm1 Im −dm2 Jm − Λ dx (Ix + Jx ) ≤ Λ − µN. Assume N (t) ≤ M for all t ≥ 0 where M = + 1. Suppose the µ assumption is not true then there exists a t1 > 0 such that N (t1 ) =

Λ Λ + 1, N (t) < + 1, t < t1 , N ′ (t1 ) ≥ 0, µ µ

(13)

N ′ (t1 ) ≤ Λ − µN (t1 ) = −µ < 0, which is a contradiction, implying that the assumption is true. This means N (t) ≤ M for all t ≥ 0.

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C.P. Bhunu and W. Garira Therefore, all feasible solutions of model system (3) enter the region   Λ . Γ = (S, Es , Em , Ex , T, Is , Im , Ix , Js , Jm , Jx , R) ∈ R12 : N ≤ + µ

(14)

Thus, Γ is positively invariant and it is sufficient to consider solutions in Γ. Existence, uniqueness and continuation results for system (3) hold in this region and all solutions of system (3) starting in Γ remain in Γ for all t ≥ 0. All parameters and state variables for model system (3) are assumed to be non-negative for t ≥ 0 since it monitors human population.

2.2.

Disease-Free Equilibrium and Stability Analysis

Disease-free equilibrium points are steady-state solutions where there is no disease. We define the diseased classes as the human populations that are either latently infected or 12 infectious. The positive orthant in R12 is denoted by R12 + , and the boundary of R+ by ∂R12 + . The positive equilibrium human population value in the absence of disease for (3) is Λ N = S = . The disease-free equilibrium of model system (3), E 0 is given by µ
0 0 0 E 0 = S 0 , Es0 , Em , Ex0 , T 0 , Is0 , Im , Ix0 , Js0 , Jm , Ix0 , R0   (15) Λ = , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 . µ

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Theorem 2.4. Model system (3) has exactly one equilibrium point with no disease in the population (on Γ ∩ ∂R12 + ). Proof. We need to show that E 0 is an equilibrium point of model system (3) and that there 0 is no other equilibrium point on Γ ∩ ∂R12 + . Substituting E into (3) shows all derivatives 0 equal to zero, so E is an equilibrium point. We now have to use Lemma 2 stated below. Lemma 2. For all equilibrium points on Γ ∩ ∂R12 + , Es = Em = Ex = T = Is = Im = Ix = Js = Jm = Jx = R = 0. We know from Lemma 2 that in Γ ∩ ∂R12 + , Es = Em = Ex = T = Is = Im = Ix = Js = Jm = Jx = R = 0 and from the equation of S ′ (t) in model system (3), the only Λ 0 equilibrium point is S = . Thus the only equilibrium point on Γ ∩ ∂R12 + is E . µ The reproduction number for model system (3) is given by RSM X = max{RS , RM , RX }, with (ks + (ρ + µ)(1 − fs )) (αs crs + (ωs + µ + ds2 )βs c) , (ρ + µ + ks )(rs + µ + ds1 )(ωs + µ + ds2 ) (km + (1 − fm )µ) (αm crm + (ωm + µ + dm2 )βm c) RM = , (km + µ)(rm + µ + dm1 )(ωm + µ + dm2 ) (kx + (1 − fx )µ) (αx crx + (µ + dx )βx c) RX = , (kx + µ)(rx + µ + dx )(µ + dx ) RS =

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being the reproduction numbers for the drug sensitive TB only model, multidrug resistant TB only model and extensively drug resistant TB only model, respectively. Mathematically, the reproduction number is defined as the spectral radius (Diekman et al., 1990; van den Driessche and Watmough, 2002). The spectral radius defines the number of new infections generated by a single infected individual in a completely susceptible population (May and Anderson, 1998). Theorem 2.5 follows from van den Driessche and Watmough (2002). Theorem 2.5. The disease-free equilibrium E 0 is locally asympotically stable when RSM X < 1 and unstable otherwise.

2.3.

Identification and Analysis of the Endemic Equilibria

We have the following endemic equilibria: (i) the extensively drug-resistant TB endemic equilibrium only, (ii) multidrug-resistant TB endemic equilibrium only, (iii) drug sensitive TB endemic equilibrium only, (iv) co-existence of the drug sensitive and multidrug-resistant TB endemic equilibrium only, (v) co-existence of the drug sensitive and extensively drugresistant TB endemic equilibrium only, (vi) co-existence of the extensively drug-resistant and multidrug-resistant TB endemic equilibrium only, and (vii) the endemic equilibrium where all the strains (drug sensitive, multidrug resistant, extensively drug resistant) exist. 2.3.1.

Extensively Drug Resistant Equilibrium Only

This occurs when Es = Em = T = Is = Im = Js = Jm = R = 0. In terms of the force of infection λ = λx , this equilibrium value is denoted by, Ex∗ = (Sx∗ , 0, 0, Ex∗ , 0, 0, 0, Ix∗ , 0, 0, Jx∗ , 0)

(17)

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where, Sx∗ =

Λ , µ + λ∗x

Ex∗ =

Λfx λ∗x , (µ + λ∗x )(kx + µ + δx λ∗x )

Ix∗ =

Λλ∗x (kx + µ(1 − fx ) + δx λ∗x ) , (µ + dx + rx )(µ + λ∗x )(kx + µ + δx λ∗x )

Jx∗ =

Λrx λ∗x (kx + µ(1 − fx ) + δx λ∗x )

(µ + dx )(µ + dx + rx )(µ + λ∗x )(kx + µ + δx λ∗x )

,

Λ(µ + dx + rx ) [(µ + dx )(µ + kx + δx λ∗x + fx λ∗x ) + λ∗x (kx + µ(1 − fx ) + δx λ∗x )] . (µ + dx )(µ + dx + rx )(µ + λ∗x )(kx + µ + δx λ∗x ) (18) From Theorem 2.5 we have that if the disease-free equilibrium exists and is locally asymptotically stable if and only if RSM X < 1. However, the disease-free equilibrium may not be globally asymptotically stable even if RX < 1. There is a possibility of backward bifurcation (bistability), where a stable endemic state co-exists with the disease-free equilibrium N∗ =

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when RX < 1 (Alexander et al., 2004; Arino et al., 2004; Dushoff et al., 1998; Kgosimore and Lungu, 2004; Kribs-Zaleta and Velasco-Hernandez, 2000; Mukandavire and Garira, 2006). Substituting equation (18) into the force of infection λ = λx to investigate bistability we find that the endemic equilibrium Ex∗ satisfy the following polynomial, λx (Aλ2x + Bλx + C) = 0,

(19)

where, A=

δx , (µ + dx )(µ + kx ) (20)

δx , C = 1 − Rx . B= (µ + dx )(µ + kx )(µ + dx + rx ) We have λx = 0 corresponding to the disease-free equilibrium and f (λx ) = Aλ2x + Bλx + C = 0

(21)

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for the existence of multiple endemic equilibria. If δx = 0, that is if exposure to TB prevents C reinfection, A = 0, B > 0 and the quadratic equation becomes linear and λx = − . In B this case, model system (3) with Es = Em = T = Is = Im = Js = Jm = R = 0 has a unique endemic equilibrium if and only if C < 0 that is Rx > 0, discarding the possibility of backward bifurcation for this case. By examining the quadratic f (λx ), it is noted that there is a unique endemic equilibrium if B < 0, and C = 0 or B 2 − 4AC = 0, there are two if C > 0, B < 0 and B 2 − 4AC > 0 and there is none otherwise. These conditions are therefore summarise in Lemma 3. Lemma 3. Model system (3)when Es = Em = T = Is = Im = Js = Jm = R = 0 has (i) precisely one unique endemic equilibrium if C < 1 ⇔ Rx > 1. (ii) precisely one unique endemic equilibrium if B < 0, and C = 0 or B 2 − 4AC = 0. (iii) precisely two endemic equilibria if C > 0, B < 0 and B 2 − 4AC > 0. (iv) otherwise there are none To find the backward bifurcation point, we set the descriminant B 2 − 4AC = 0. Solving this equation for Rx yields B2 (22) , Rcx = 1 − 4A from which it can be shown that backward bifurcation occurs for values of Rx in the range Rcx < Rx < 1. To determine the local asymptotic stability of this endemic equilibrium we make use of the centre manifold theory (Carr, 1981) as described in Theorem 4.1 of Castillo-Chavez and Song (2004). To apply the centre manifold theory for model system (3) when Es = Em = T = Is = Im = Js = Jm = R = 0 we make use of the change of 4 X variables S = x1 , Ex = x2 , Ix = x3 , Jx = x4 so that N = xn . We use the vector n=1

notation X = (x1 , x2 , x3 , x4 )T , then model system (3) when Es = Em = T = Is = Im = Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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dX Js = Jm = R = 0 can now be written in the form = F = (f1 , f2 , f3 , f4 )T such that dt model system (3) when Es = Em = T = Is = Im = Js = Jm = R = 0 is given as, x′1 (t) = f1 = Λ −

x′2 (t) = f2 =

x′3 (t) = f3 =

(βx cx3 + αx cx4 )x1 − µx1 , P4 x n n=1

fx (βx cx3 + αx cx4 )x1 δx (βx cx3 + αx cx4 )x2 − (µ + kx )x2 − , P4 P4 n=1 xn n=1 xn

(1 − fx )(βx cx3 + αx cx4 )x1 + kx x2 P4 n=1 xn

+

(23)

δx (βx cx3 + αx cx4 )x2 − (µ + rx + dx )x3 , P4 n=1 xn

x′4 (t) = f4 = rx x3 − (µ + dx )x4 .

The Jacobian matrix of system (23) at the disease-free equilibrium is given by  −µ 0 −βx c −αx c  0 −(µ + kx ) fx βx c fx αx c J(E 0 ) =   0 kx (1 − fx )βx c − (µ + dx + rx ) (1 − fx )αx c 0 0 rx −(µ + dx )



 , 

(24)

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from which it can be shown that the reproduction number is RX . If βx = αx is taken as bifurcation point, consider the case where RX = 1 and solve for βx (αx ), we obtain βx = βx∗ =

(kx + µ)(µ + dx ) (kx + (1 − fx )µ)c

(25)

Note that the linearised system of the transformed system 23 with βx = αx = βx∗ has a simple zero eigenvalue, thus allowing the use of the Center Manifold theory. It can be shown that the Jacobian of (23) at βx = βx∗ has a right eigenvector associated with the zero eigenvalue given by u = [u1 , u2 , u3 , u4 ]T where, u1 = −

fx βx∗ c(µ + dx + rx )u3 βx∗ c(µ + dx + rx )u3 , u2 = , µ(µ + dx ) (µ + kx )(µ + dx )

(26)

rx u 3 u3 = u3 > 0, u4 = . µ + dx The left eigenvector of J(E 0 ) associated with the zero eigenvalue at βx = βx∗ is given by v = [v1 , v2 , v3 , v4 ]T where, v1 = 0, v2 =

kx v3 , v3 = v3 > 0, v4 = v3 . µ + kx

(27)

In order to establish the conditions for the existence of backward bifurcations, we use Theorem 2.6 of Castillo-Chavez and Song (2004). Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Theorem 2.6. Consider the following general system of ordinary differential equations with a parameter φ dx (28) = f (x, φ), f : Rn × R → R and f ∈ C2 (Rn × R), dt where 0 is an equilibrium of the system that is f (0, φ) = 0 for all φ and assume,   ∂fi 1. A = Dx f (0, 0) = (0, 0) is the linearisation of system (28) around the equi∂xj librium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts; 2. Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue. Let fk be the k th component of f and a=

n X

vk u i u j

k,i,j=1

∂ 2 fk (0, 0), ∂xi ∂xj (29)

b=

n X

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k,i=1

vk u i

∂ 2 fk ∂xi ∂φ

(0, 0).

The local dynamics of (28) around 0 are totally governed by a and b. i. a > 0, b > 0. When φ < 0 with |φ| 0, then, model system (3) has a backward bifurcation at RX = 1, otherwise a < 0 and a unique endemic equilibrium Ex∗ guaranteed by Theorem 2.6 is locally asymptotically stable for RX > 1 but close to 1. Backward bifurcation occurs when the disease-free and the endemic equilibria co-exists for RX < 1. In this case making RX < 1 will not control the TB epidemic as the disease remains endemic. Backward bifurcation in this case is caused by exogenous re-infection of the latently infected individuals. In the absence of exogenous reinfection, model system (3) will have a unique endemic equilibrium which exists whenever RX > 1.

2.3.2.

Multidrug-Resistant TB Endemic Equilibrium Only

This endemic equilibrium occurs when Es = Ex = T = Is = Ix = Js = Jx = 0, pm = 1 and is given by ∗ , 0, E ∗ , 0, 0, 0, I ∗ , 0, 0, J ∗ , 0, R∗ ) , Em = (Sm m m m m

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∗ , E ∗ , I ∗ , J ∗ , R∗ defined in terms of λ∗ such that with Sm m m m m m

Λ , µ + λ∗m

∗ Sm =

∗ Em =

θ1 θ2 θ3 +δm λ∗2 m (θ2 θ3 −rm ωm )

+

Λλ∗m fm θ2 θ3 ∗ λm (θ2 θ3 µδm −km fm rm ωm +θ1 (θ2 θ3 −(1−fm )rm ωm ))

∗ = Im

Λλ∗m θ3 (θ1 (1−fm )+km fm +δm λ∗m ) ∗ θ1 θ2 θ3 +δm λ∗2 m (θ2 θ3 −rm ωm )+λm (θ2 θ3 µδm −km fm rm ωm +θ1 (θ2 θ3 −(1−fm )rm ωm ))

∗ = Jm

Λλ∗m rm (θ1 (1−fm )+km fm + δm λ∗m ) ∗2 θ1 θ2 θ3 +δm λm (θ2 θ3 −rm ωm )+λ∗m (θ2 θ3 µδm −km fm rm ωm +θ1 (θ2 θ3 −(1−fm )rm ωm ))

∗ Rm =

Λλ∗m rm g(λ∗m )(θ1 (1 − fm ) + km fm + δm λ∗m ) , ∗ θ1 θ2 θ3 +δm λ∗2 m (θ2 θ3 −rm ωm )+λm (θ2 θ3 µδm −km fm rm ωm +θ1 (θ2 θ3 −(1−fm )rm ωm ))

with θ1 = µ + km , θ2 = rm + µ + dm1 , θ3 = ωm + µ + dm2 , g(λ∗m ) =

ωm . λ∗m + µ

(35)

To investigate bistability, we use the centre manifold theory similar to the analysis ofEx in previous section. 2.3.3.

Drug Sensitive TB Endemic Equilibrium Only

The drug sensitive endemic equilibrium only occurs when Em = Ex = Im = Ix = Jm = Jx = 0, ps = 1 and is given by Es = (Ss∗ , Es∗ , 0, 0, T ∗ , Is∗ , 0, 0, Js∗ , 0, 0, Rs∗ ) ,

(36)

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with terms Ss∗ , Es∗ , Is∗ , Js∗ , Rs∗ given in terms of λ∗s such that Ss∗ =

Λ , µ + λ∗s

Es∗ =

Λλ∗s fs m2 m3 , ∗ ∗ m1 m2 m3 m4 µ + δs m2 λ∗2 s (m3 m4 − rs ωs ) + λs z(λs )

T∗ =

Λλ∗s fs m3 m4 ρ , ∗ ∗ m1 m2 m3 m4 µ + δs m2 λ∗2 s (m3 m4 − rs ωs ) + λs z(λs )

Is∗

Λλ∗s m2 m4 (m1 (1 − fs ) + ks fs + δs λ∗s ) , = ∗ ∗ m1 m2 m3 m4 µ + δs m2 λ∗2 s (m3 m4 − rs ωs ) + λs z(λs )

Js∗ =

Λλ∗s m2 rs (m1 (1 − fs ) + ks fs + δs λ∗s ) , ∗ ∗ m1 m2 m3 m4 µ + δs m2 λ∗2 s (m3 m4 − rs ωs ) + λs z(λs )

Rs∗ =

Λλ∗s h(λ∗s )(m3 m4 ηρfs + m2 rs (m1 (1 − fs ) + ks fs + δs λ∗s )ωs ) , ∗ ∗ m1 m2 m3 m4 µ + δs m2 λ∗2 s (m3 m4 − rs ωs ) + λs z(λs )

with h(λ∗s ) =

(37)

1 , m1 = µ + ρ + ks , m2 = η + µ, m3 = µ + ks + ds1 , m4 = µ + ωs + ds2 , λ∗s + µ

z(λ∗s ) = −m3 m4 ηρfs + m2 m3 m4 µδs − ks m2 fs rs ωs + m1 m2 (m3 m4 − (1 − fs )rs ωs ).

To investigate bistability we use the centre manifold theory similar to the analysis ofEx in previous section.

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z(λ∗s ) = −m3 m4 ηρfs + m2 m3 m4 µδs − ks m2 fs rs ωs + m1 m2 (m3 m4 − (1 − fs )rs ωs ). To investigate bistability we use the centre manifold theory similar to the analysis of Ex in previous section. 2.3.4.

Co-existence of the Drug Sensitive and Multidrug-Resistant TB Endemic Equilibrium Only

This equilibrium point exists when Ex = Ix = Jx = 0, pm = 1 and is given by sm , 0, T sm , I sm , I sm , 0, J sm , J sm , 0, Rsm ), Esm = (S sm , Essm , Em m s m s

(38)

sm with components of Esm being defined in terms of λsm s and λm .

2.3.5.

Co-existence of the Drug Sensitive and Extensively Drug-Resistant TB Endemic Equilibrium Only

This equilibrium point exists when Em = Im = Jm = 0, ps = 1 and is given by Esx = (S sx , Essx , 0, Exsx , T sx , Issx , 0, Ixsx , Jssx , 0, Jxsx , Rsx ),

(39)

sx with components of Esx being terms in λsx s and λx .

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2.3.6.

Co-existence of the Extensively Drug-Resistant and Multidrug-Resistant TB Endemic Equilibrium Only

This equilibrium point exists when Em = T = Is = Js = 0 and is given by mx , E mx , 0, 0, I mx , I mx , 0, J mx , J mx , Rmx ), Emx = (S mx , 0, Em x m x m x

(40)

mx with components of Emx being terms in λmx m and λx .

2.3.7.

Coexistence of the Drug Sensitive, Multidrug Resistant and Extensively Drug Resistant Strains Endemic Equilibrium

This endemic equilibrium point is denoted by ∗ smx , E smx , T smx , I smx , I smx , I smx , J smx , J smx , J smx , Rsmx ) , Esmx = (S smx , Essmx , Em x s m x s m x (41) smx smx smx smx smx smx smx smx ∗smx smx smx with terms S , Es , Em , Ex , T , Is , Im , Ix , Js , Jm , Jx , Rsmx smx being expressions in λi , i = (s, m, x). The local asymptotic stability analysis of this equilibrium point can also be analysed using the centre manifold theory similar to the analysis of Ex∗ and is not done here to avoid repetition.

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Analysis of the Reproduction Number RSM X

2.4.

From RSM X , we consider the following situations: Case 1: No intervention and case detection for all TB strains In the absence of any intervention strategy and case detection we have, lim

(ri ,ωi ,ρ)→(0,0,0)

RSM X = max

(ki + (1 − fi )µ)βi c = R0 , i = (s, m, x), (µ + ki )(µ + di1 )

(42)

which is the pre-detection and pre-treatment reproduction number for model system (3). Case 2: Only latent and active forms of drug sensitive TB cases are detected and treated In this case, rm = rx = ωm = 0. Thus, lim

RSM X =

(rm ,rx ,ωm ,ρ)→(0,0,0)

lim

max{RS , RM , RX } = max{RS , R0M , R0X },

(rm ,rx ,ωm ,ρ)→(0,0,0)

where R0M =

(km + (1 − fm )µ)βm c , (µ + km )(µ + dm1 )

R0X =

(kx + (1 − fx )µ)βx c , (µ + kx )(µ + dx1 )

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(43) are the pre-detection and pre-treatment reproduction numbers for the multidrug-resistant and extensively drug-resistant TB only models, respectively. Rewriting RM as RM = K1 R0M where K1 =

(αm rm + (ωm + µ + dm2 )βm )(µ + dm1 ) < 1 for αm (µ + dm1 ) (ωm + µ + dm2 )(µ + dm1 + rm )βm

< βm (ωm + µ + dm2 ), ∆K1 = R0M − RM = R0M (1 − K1 ) = βm rm (ωm + µ + dm2 ) − αm rm (µ + dm1 )(ωm + µ + dm2 )(µ + dm1 + rm )βm > 0 for αm (µ + dm1 ) < βm (ωm + µ + dm2 ). (44) Whenever K1 < 1 (∆K1 > 0), increasing detection and treatment effort for MDR-TB will increase its benefit. It is evident from the definition of RM that lim RM =

rm →∞

(km + (1 − fm )αm c > 0, (km + µ)(ωm + µ + dm2 ) (45)

lim RM

ωm →∞

(km + (1 − fm )βm c = > 0. (km + µ)(rm + µ + dm1 )

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Thus, a sufficiently effective MDR-TB treatment program that focuses on detecting (at a rate, rm → ∞) and treating (at a rate, ωm → ∞) can lead to effective MDR-TB control if it results in making the right hand sides of equation (45) less than unity. Further sensitivity analysis on detection and treatment parameters is carried out by differentiating RM partially with respect to rm and ωm giving, ∂RM (αm (µ + dm1 ) − βm (ωm + µ + dm2 ))(µ + dm1 )R0M rm m < ωm , a lot of MDR-TB cases are detected although not all of the detected cases are treated, thus, in this c and case MDR-TB cases continue to spread in the community. However, when rm < rm c ωm > ωm , then, despite successful treatment for those detected being available, MDR-TB continue to spread in the community as not all cases are being detected. Thus, to effectively c and ω > ω c allowing detection and treatment control MDR-TB we should have rm > rm m m of most MDR-TB cases in the community. Now we have to rewrite RX as RX = K2 R0X where, αx rx + (µ + dx )βx < 1 for αx < βx , K2 = βx (µ + dx + rx ) (48) (βx − αx )rx R0X ∆K2 = R0X (1 − K2 ) = > 0 for αx < βx . βx (µ + dx + rx )

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c ωm =

Whenever K2 < 1 (∆K2 > 0), then, detecting individual cases with XDR-TB, in the absence of treatment will have a positive impact on its control. It is evident from the definition of RX that (kx + (1 − fx )µ)αx c lim RX = > 0. (49) rx →∞ (kx + µ)(µ + dx )

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Thus, a sufficiently effective XDR-TB detection program focusing on detecting XDR-TB active cases can lead to effective disease control if it succeeds in making the right hand side of (49) less than unity. Sensitivity analysis on the detection parameter carried out by partial differentiation of RX with respect to rx gives ∂RX (µ + dx )(αx − βx )R0X < 0 for βx > αx . = (50) ∂rx βx (µ + dx + rx )2   ∂RX Whenever βx > αx < 0 , increasing XDR-TB detection effort will increase ben∂rx efit. But, when βx < αx , increasing XDR-TB detection effort will be detrimental to the community as this will result in an increase of RX . Setting RX = 1 and solving for the critical detection rate gives rxc =

βx (µ + dx )(R0X − 1) βx − αx RX

(51)

If detection can slow the spread of XDR-TB, it will be able to effectively do so, when rx > rxc and it will not be able to effectively control when rx < rxc . Case 3: Only intervention for drug resistant cases (MDR and XDR) In this case rs = ωs = ρ = 0 thus, lim

(rs ,ωs ,ρ)→(0,0,0)

R0S

RSM X = max{R0S , RM , RX } where,

(ks + (1 − fs )µ)βs c = . (µ + ks )(µ + ds1 )

(52)

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Expressing RS in terms of R0S we have, RS = K30 R0S with, K30 =

(ks + (ρ + µ)(1 − fs ))(αs rs + (ωs + µ + ds2 )βs )(µ + ks )(µ + ds1 ) < 1 for (ρ + µ + ks )(rs + µ + ds1 )(ωs + µ + ds2 )(ks + (1 − fs )µ)βs

(µ + ks )(µ + ds1 ) [αs rs (ks + (µ + ρ)(1 − fs )) + (1 − fs )(ωs + µ + ds2 )ρβs ] < 1, (ρ(rs + µ + ds1 ) + rs (µ + ks )) (ks + (1 − fs )µ) (ωs + µ + ds2 )βs ∆K30 = R0S (1 − K30 ) > 0 for (µ + ks )(µ + ds1 ) [αs rs (ks + (µ + ρ)(1 − fs )) + (1 − fs )(ps + µ + ds2 )ρβs ] < 1. (ρ(rs + µ + ds1 ) + rs (µ + ks )) (ks + (1 − fs )µ) (ps + µ + ds2 )βs (53) Whenever K30 (∆K30 > 0), isoniazid preventive therapy for those latently infected coupled with treatment of TB will have a positive impact on the control of TB. If there is no detection and preventive isoniazid therapy for those latently infected, then, lim RS = RSn1 =

ρ→0

(ks + (1 − fs )µ)(αs crs + (ωs + µ + ds2 )βs c) . (µ + ks )(rs + µ + ds1 )(ωs + µ + ds2 )

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(54)

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Expressing RSn1 in terms of R0S we have RSn1 = K31 R0S with K31 =

(µ + ds1 )(αs rs + (ωs + µ + ds2 )βs ) < 1 for αs (µ + ds1 ) < βs (µ + ωs + ds2 ), (µ + rs + ds1 )(ωs + µ + ds2 )

∆K31 = R0S (1 − K31 ) =

((βs (ωs + µ + ds2 ) − αs (µ + ds1 ))rs R0S >0 (µ + rs + ds1 )(ωs + µ + ds2 )βs

for αs (µ + ds1 ) < βs (µ + ωs + ds2 ). (55) Thus, whenever K31 < 1 (∆K31 > 0), detecting and treating TB will be beneficial to the community as this results in decrease in the spread of the epidemic. If there is no detection and treatment of active TB, then, lim

(rs ,ωs )→(0,0)

RS = RSn2 =

(ks + (µ + ρ)(1 − fs ))βs c . (ρ + µ + ks )(µ + ds1 )

(56)

Expressing RSn2 in terms of R0S gives, RSn2 = K32 RSn2 ,

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K32 =

(ks + (µ + ρ)(1 − fs ))(µ + ks ) < 1, (ks + µ(1 − fs ))(ρ + µ + ks )

∆K32 = R0S (1 − K32 ) =

(57)

ks ρfs R0S > 0. (ks + µ(1 − fs ))(ρ + µ + ks )

Thus, K32 < 1 (∆K32 > 0), suggesting that isoniazid preventive therapy will have a positive impact in controlling TB. It is evident from the definition of RS that, lim RS =

ρ→∞

(1 − fs )(αs crs + (ωs + µ + ds2 )βs c) > 0, ((ωs + µ + ds2 )(rs + µ + ds1 )

lim RS =

(ks + (µ + ρ)(1 − fs ))αs c > 0, (ωs + µ + ds2 )(ρ + µ + ks )

lim RS =

(ks + (µ + ρ)(1 − fs ))βs c > 0. (rs + µ + ds1 )(ρ + µ + ks )

rs →∞

ωs →∞

(58)

A sufficiently effective isoniazid preventive therapy and treatment program that focuses on detecting latent TB (at rate ρ → ∞), detecting active TB (at rate rs → ∞) and treatment of active TB (at a rate ωs → ∞) can lead to effective TB control if it succeeds in making the right hand sides of equation (58) less than unity. Sensitivity analysis on latent TB detection, active TB detection and active TB treatment parameters is done by partial differentiation of

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C.P. Bhunu and W. Garira

RS with respect ρ, rs and ωs , respectively giving, ∂RS fs ks (αs crs + (ωs + µ + ds2 )βs c) < 0, =− ∂ρ (rs + µ + ds1 )(ωs + µ + ds2 )(ρ + µ + ks )2 ∂RS (ks + (ρ + µ)(1 − fs ))(βs c(ωs + µ + ds2 ) − αs c(µ + ds1 )) =− < 0, ∂rs (rs + µ + ds1 )2 (ωs + µ + ds2 )(ρ + µ + ks )

(59)

for βs c(ωs + µ + ds2 ) > αs c(µ + ds1 ), ∂RS (ks + (ρ + µ)(1 − fs ))αs crs =− < 0. ∂ωs (rs + µ + ds1 )(ωs + µ + ds2 )2 (ρ + µ + ks ) ∂RS ∂RS < 0 and < 0, implying that increasing both ∂ρ ∂ωs case detection of latent TB and treatment of active TB effort its benefit in the   will increase ∂RS < 0 , suggesting that case community. Whenever βs (ωs +µ+dm2 ) > αs (µ+ds1 ) ∂rs detection of active drug sensitive TB cases will also have a positive impact on TB control. Setting RS = 1 and solving for the critical latent TB detection, active TB detection and active TB treatment rates give, From equation (59), we have

ρc =

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rsc =

(ks +(1 − fs )µ)(αs crs +(ωs +µ+ds2 )βs c)−(µ+ks )(rs +µ+ds1 )(ωs +µ+ds2 ) , (rs + µ + ds1 )(ωs + µ + ds2 ) − (1 − fs )(αs crs + (ωs + µ + ds2 )βs c) (ωs +µ+ds2 )[(ks +(µ+ρ)(1−fs ))βs c−(ρ+µ+ks )(µ+ds1 )] , (ωs + µ + ds2 )(ρ + µ + ks ) − αs c(ks + (µ + ρ)(1 − fs ))

(ks +(µ+ρ)(1−fs ))(αs crs +(µ+ds2 )βs c)−(ρ+µ+ks )(rs +µ+ds1 )(µ+ds2 ) . (ρ + µ + ks )(rs + µ + ds1 ) − (ks + (µ + ρ)(1 − fs ))βs c (60) If rs < rsc and ωs < ωsc , then, not all cases of drug sensitive active TB are detected and treated, implying they will continue spreading slightly unabated. When rs > rsc and ωs < ωsc , a lot of drug sensitive active TB cases are detected although not all of the detected cases are treated, thus, in this case, drug sensitive active TB cases continue to spread in the community. However, when rs < rsc and ωs > ωsc , then, despite successful treatment for those detected being available, drug sensitive active TB continue to spread in the community as not all cases are being detected. Thus, to effectively control drug sensitive active TB, we should have rs > rsc and ps > pcs , allowing detection and treatment of most drug sensitive cases in the community. It is worth noting that for ρ > ρc , detecting and giving isoniazid preventive therapy will be able to keep TB at bay. ωsc =

3.

Numerical Simulations

In this section, we carry out numerical simulations for model system (3) using the fourth order Runge-Kutta numerical scheme coded in C++ programming language. The parameter values that we use for numerical simulations are in Table 1. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Table 1. Model parameters and their interpretations.

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Definition Recruitment rate Natural mortality rate Contact rate Drug sensitive TB induced death rate MDR-TB induced death rate XDR-TB induced death rate Transmission rate Endogeneous reactivation rates Probability of successful treatment Treatment rate for the drug sensitive latently infected Treatment rate for the drug sensitive infectives Treatment rate for the MDR-TB Protective factor for individuals in Ei Rate of developing active TB among susceptibles and the recovered

Symbol Λ µ c d s1 , d s2

Estimate(Range) 0.029yr−1 0.01yr−1 3yr−1 0.3yr−1

Source CSOZ a* d* a*

dm1 , dm2 dx βi , αi ki ps , pm

0.3yr−1 0.3yr−1 0.35 (0.1-0.6)yr−1 0.00013 (0.0001-0.0003)yr−1 0.9999yr−1

a* Estimate a* a* Estimate

η

0.7yr−1

b*

ωs

0.88yr−1

c*

ωm

0.8yr−1

Estimate

δi

0.7yr−1

Estimate

1 − fi

0.1yr−1

d*

In Table 1, CSOZ means Central Statistics Office of Zimbabwe, a* denotes parameter values and ranges from Dye et al., (1998), Dye and William (2000), b* denotes parameter values from Bhunu et al., (2008)(a), c* denotes parameter values from Qing-Song Bao et al., (2007) and d* denotes parameter values from Bhunu et al., (2008)(b). For the numerical results in Figure 2, we assume that, S(0) = 8000000, Es (0) = 5000000, Em (0) = 100000, Ex (0) = 50000, T (0) = 950000, Is (0) = 500000, Im (0) = 30000, Ix (0) = 10000, Js (0) = 340000, Jm (0) = 16000, Jx (0) = 4000, RT (0) = 0 to be the initial population sizes in each compartment. Parameters used are as given in Table 1. Figure 2 is a graphical representation showing the effects of preventive isoniazid preventive therapy, first line TB drugs and second line TB drugs on the susceptible population, drug sensitive latently infected population, MDR latently infected population, XDR latently infected population, drug sensitive active TB population, MDR active TB population and XDR active TB population. Figure 2(a) shows that there is more drug sensitive latently infected individuals when there is no intervention strategy than when there is some form of intervention. When isoniazid preventive therapy and TB treatment with first line drugs are implemented, drug sensitive latent TB cases decline to low levels. It is worth noting here that the rate of decline is higher when isoniazid preventive therapy is coupled with drug

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216

C.P. Bhunu and W. Garira 900000

N Drug sensitive active TB cases

14000000

Latently infected with drug sensitive strain

12000000 10000000 8000000 6000000 T1

4000000

CT1 CT2

2000000 0

0

800000

N

700000 600000 500000 400000 300000 200000 T1

100000

20

40 60 Time (years)

80

100

CT1

0

20

40 60 Time (years)

80

100

( a)

( b) Multi-drug resistant active TB cases

16000000

drug resistant strain

12000000 10000000 8000000 CT1

6000000 4000000 CT2

2000000

N

Extensively drug resistant active TB cases

0

( c)

T1

20

40 60 Time (years)

80

700000 600000 CT1

500000 400000 300000 200000 100000 0

450000

N 40 60 Time (years)

20

80

100

( d)

20000000

400000 350000

CT2

300000 250000 200000 150000 100000 50000 0

800000

100

Latently infected with extensively drug resistant strain

Latently infected with

14000000

900000

CT1 0

20

T1 40 60 Time (years)

( e)

80

18000000 CT2

16000000 14000000 12000000 10000000 8000000 6000000 4000000 N

2000000

100

0

CT1 20

T1 40 60 Time (years)

22000000 20000000

80

100

(f )

CT2

Suceptible population

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18000000 CT1

16000000 14000000 12000000

T1

10000000 8000000 6000000 N

4000000 2000000 0

20

40 Time (years)

60

80

100

( g) Figure 2. Simulation results showing the effect of preventive isoniazid preventive therapy, first line drugs and second line drugs on the susceptible population, drug sensitive latently infected population, MDR latently infected population, XDR latently infected population, drug sensitive active TB population, MDR active TB population and XDR active TB population. N denotes no intervention strategy, T1 denotes drug sensitive active TB treatment only, CT1 denotes drug sensitive active TB treatment and isoniazid preventive therapy for those latently infected with drug sensitive strain and CT2 denotes isoniazid preventive therapy for those latently infected with drug sensitive strain and active TB treatment (drug sensitive and MDR).

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sensitive TB treatment than when TB treatment is the only intervention strategy. Figure 2(b) is a direct result of Figure 2(a) as an increase (decrease) in the number of drug sensitive latently infected individuals translate to an increase (decrease) in the number of drug sensitive active TB cases and vice-versa. In Figure 2(c), MDR latently infected cases are decreasing to low levels when there is no intervention due to the fact that MDR strains may be at a comparative disadvantage for survival in an environment free of isonianid and rifampcin. Also, MDR latently infected are also decreasing to low levels, when there is treatment of MDR-TB in addition to isoniazid preventive for therapy and TB treatment for latently infected and TB infectives with drug sensitive strain. Latently infected individuals with MDR strains are found to be increasing and reaching its asymptotic states when there is only treatment of drug sensitive TB with or without isoniazid preventive therapy for those latently infected with drug sensitive strain. This is in total support of the argument that drug resistance increases with increase in drug use (misuse) as bacterial resistance develops as a result of selective pressure on non-resistant strains due to antibiotic use (Levy, 1992; McGowan, 1983). Figure 2(d) follows from Figure 2(c) and vice-versa. Figures 2(e) and (f) shows the XDR active TB and latent TB, respectively. They both show that XDR forms of TB when second line TB drugs are in use. In both cases, when there is no treatment for MDR-TB, XDR cases are decreasing to low levels as XDR strains are at a comparative disadvantage in an environment free of second line TB drugs. It is worth noting that Figure 2(e) follows from Figure 2(f) and vice-versa. In Figure 2(g), how the susceptible population fares under different conditions is shown. In the presence of isoniazid preventive therapy for the latently infected with drug sensitive strain, treatment for drug sensitive active TB and treatment for MDR-TB the susceptible population increases and reaches its peak and then decreases asymptotically to a steady state. The decrease may be due to MDR strains becoming resistant to second line drugs and thus continue infecting the susceptible population uncontrolled. When there is treatment for drug sensitive active TB with and/ or without preventive isoniazid preventive therapy, the susceptible population starts increasing to its peak and then declines asymptotically to a steday level state, due to drug sensitive strains becoming resistant to isoniazid and rifampcin and thus continue to infect the susceptibles unabated. In the absence of any intervention, the susceptible population simply decrease to its asymptotic steady state. It is worth mentioning here that the peak of the susceptibles in the presence of isoniazid preventive therapy, drug sensitive TB treatment and MDR-TB treatment is the highest followed by isoniazid preventive therapy and drug sensitive TB treatment which is then followed by drug sensitive TB treatment alone and the last is when there is no treatment. Basing on the results of this figure, we can safely conclude that the effects of XDR is in the long run going to be a threat to human life as seen on the low level of susceptible population.

4.

Summary and Concluding Remarks

A mathematical model was presented and studied to assess the impact of isoniazid preventive therapy on the latently infected individuals and active TB treatment (drug sensitive and MDR) in controlling the spread of TB. The centre manifold theory was used to establish the local asymptotic stability of the endemic equilibrium and it showed the existence of

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backward bifurcation when the associated reproduction number is less than unity and the unique endemic equilibrium when the associated reproduction number is greater than unity. From the analysis of the reproduction numbers obtained it is concluded that (i) case detection, isoniazid preventive therapy and TB treatment of drug sensitive Mtb strains will help reduce the spread of drug sensitive TB , (ii) case detection and treatment of MDR-TB help in reducing the spread of MDR-TB, (iii) detecting individuals with XDR-TB have a positive impact in the control XRTB even in the absence of treatment by reducing the rate at which individuals detected to XDR-TB interact with other non-infected people in the community. From numerical simulations carried out, it can be concluded that, (i) case detection, isoniazid preventive therapy and TB treatment of drug sensitive Mtb strains while help reduce the spread of drug sensitive TB also results in an increase of MDR-TB cases supporting the argument that bacterial resistance develops as a result of selective pressure on non-resistant strains due to antibiotic use (Levy, 1992; McGowan, 1983), (ii) case detection and treatment of MDR-TB while reducing the spread of MDR-TB also results in an increase of XDR-TB cases again supporting the previous argument that bacterial resistance develops as a result of selective pressure on non-resistant strains due to antibiotic use (Levy, 1992; McGowan, 1983).

Acknowledgements C. P. Bhunu would like to acknowledge the financial support given to him by the International Clinical Operational Health Services Research Training Award (ICOHRTA) through the Biomedical Research Training Institute, Zimbabwe.

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References [1] Alexender ME, Bowman C, Gumel AB et al., A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dynam. Syst (2004), 3:503–524. [2] Andrews JR et al., Multidrug-resistant and extensively drug-resistant tuberculosis: Implications for the HIV epidemic and antiretroviral therapy rollout in South Africa, J. Inf. Dis. (2007), 196:482–90. [3] Arino J, Cooke KL, van den Driessche P, Velasco-Hernandez J, An epidemiology model that includes a leaky vaccine with general waning function, Discr. Cont. Dynam. Syst. B. (2004), 4:479–495. [4] Blower SM, Gerberding JL, Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework, J. Mol. Med. (1998), 76:624– 636. [5] Blower SM, Small PM, Hopewell PC, Control strategies for tuberculosis epidemics, Sci. (1996), 273:497–500. [6] Bhunu C, Garira W, Mukandavire Z, Zimba M,Tuberculosis transimission model with chemoprophylaxis and treatment, Bull. Math. Biol. (2008), 70:1163–1191. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[7] Bhunu C, Garira W, Mukandavire Z, Magomebedze G, Modelling the effects of preexposure and post-exposure vaccines in tuberculosis control, J. Theor. Biol.(2008), doi:10.1016/j.jtbi:2008.06.023 [8] Birkhoff G, Rota GC, Ordinary differential equations, Ginn,1982. [9] Carr J, Applications Centre Manifold theory, Springer-Verlag, New York, 1981. [10] Castillo-Chavez C, Feng Z, To treat or not to treat: The case of tuberculosis, J. Math. Biol. (1997), 35:629–656. [11] Castillo-Chavez C, Feng Z, Mathematical models for the disease dynamics of tuberculosis, In Advances in Mathematical Population Dynamics - Molecules, Cells and Man (Arino O., Axelrod D. and Kimmel M., Eds) World Scientific (1996), 629–656. [12] Castillo-Chavez C, Song B, Dynamical models of tuberculosis and their applications, Math. Biosci. Engrg. (2004), 1(2): 361–404. [13] Davies PDO. Multi-drug resistant tuberculosis.Priory Lodge Education Ltd, (1999). [14] Diekman 0, Heesterbeek JAP, Metz JAP, On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases, J. Math. Biol. (1990), 28:365–382.

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[15] Dushoff J, Huang W, Castillo-Chavez C, Backward bifurcations and catastrophe in simple models of fatal diseases, J.Math. Biol. (1998), 36:227–248. [16] Dye C, Schele S, Dolin P, Pathania V, Raviglione M, Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Directly observed short-course therapy, Lancet (1998), 352:1886–1891. [17] Dye C, Schele S, Dolin P, Pathania V, Raviglione M, For the WHO global surveillance and monitoring project. Global burden of tuberculosis estimated incidence, prevalence and mortality by country, J.A.M.A. (1999), 282:677–686. [18] Dye C, Williams BR, Criteria for the control of drug-resistant tuberculosis, Proc. Natl. Acad. Sci. (2000), 1–6. [19] Espinal MA, Kim SJ, Suarez PG, Kam KM, Khomenko AG, Migliori GB, Baz J, Kochi A, Dye C, Raviglione MC, Standard short-course chemotherapy for drugresistant tuberculosis: treatment outcomes in 6 countries, J. Am. Med. Assoc. (2000), 283:2537–2545. [20] Frieden TR, Sterling T, Pablos-Mendez A, et al., The emergence of drug-resistant tuberculosis in New York City, N. Engl. J. Med. (1993), 328(8):521–556. [21] Gandhi NR, Moll A, Sturm AW, et al., Extensively drug-resistant tuberculosis as a cause of death in patients co-infected with tuberculosis and HIV in a rural area of South Africa, Lancet (2006), 368:1575–1580. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[22] Gomes, MGM, Franco, AO, Gomes, M.C and Medley, G.F. The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy, Proc. R. Soc. Lond. B (2004), 271:617–623. [23] Gumel AB, Song B, Existence of multiple-stable endemic equilibria for a multidrugresistant model of Mycobacterium tuberculosis, Math. Biosci. Engrg. (2008), 5(3):437–455. [24] Hallinan JS, Wiles J, Modelling the spread of antibiotic resistance. I. E. E. (2000), 1152–1159. [25] Kgosimore M, Lungu EM, The effects of vaccination and treatment on the spread of HIV/AIDS, J. Biol. Sys. (2004), 12:399–417. [26] Kribs-Zaleta CM, Velasco-Hernandez J, A simple vaccination model with multiple endemic states, Math. Biosci. (2000), 164:183–201. [27] Levy SB, The antibiotic paradox: How miracle drugs are destroying the miracle, New York: Plenum Press, 1992. [28] May RM, Anderson RM, The transmission dynamics of Human Immunodeficiency Virus(HIV), Philos. R. S. Lond. B. Biol. Sci. (1998), 321:565–607. [29] McGowan JE, Antimicrobial resistance in hospital organisms and its relation to antibiotic use, Rev. Inf. Dis. (1983), 5(6):1033–1048.

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[30] McGregor S, New TB strain could fuel South Africa AIDS toll, http://go.reuters.com/newsArticle.jhtml?type=healthNews&StoryID=13448940. [31] Qing-Song Bao, Yu-Hua Du, Ci-Yong Lu, Treatment outcome of new pulmonary tuberculosis in Guangzhou, China 1993–2002: a register-based cohort study, BMC Public Health 7:344, 2007. [32] Mukandavire Z, Garira W, HIV/AIDS model for assessing effects of prophylactic sterilizing vaccines, J. Biol. Sys. (2006), 14(3):323–355. [33] Shah, NS, et al., Worldwide emergence of extensively drug-resistant tuberculosis. Emerg. Inf. Dis. (2007), 13(3):380–387. [34] Sidley P, South Africa acts to curb spread of lethal strain of TB, Brit. Med. J. (2006), 333:825. [35] Thieme HR, Mathematics in population biology, Princeton Univ. Press, Princeton and Oxford, 2003. [36] van den Driessche P, Watmough J, Reproduction numbers and sub-threshold endemic equilibria for the compartmental models of disease transmission, Math. Biosci. 180: 29–48, 2005. [37] World Health Organization, WHO global task force outlines measures to combat XDR-TB worldwide (http://www.who.int/mediacentre/news/notes/2006/np29/en/index.html).

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 221-241

ISBN 978-1-60741-452-0 c 2009 Nova Science Publishers, Inc.

Chapter 8

HIV/AIDS AND THE U SE OF M ATHEMATICAL M ODELS IN THE T HEORETICAL A SSESSMENT OF I NTERVENTION S TRATEGIES : A R EVIEW

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Zindoga Mukandavirea ,∗ Jean M. Tchuencheb , Christinah Chiyakaa and Godfrey Musukac a Department of Applied Mathematics, National University of Science and Technology Box AC 939 Ascot, Bulawayo, Zimbabwe b Mathematics Department, University of Dar es Salaam, Box 35062, Dar es Salaam, Tanzania c African Comprehensive HIV/AIDS Partnerships, Private Bag X033 Gaborone, Botswana

Abstract Mathematical models have long provided basic insights in the theoretical assessment of HIV/AIDS intervention strategies. This chapter gives a global history of the HIV/AIDS epidemic and various intervention strategies in place. Mathematical modelling of epidemics, its history and role in controlling infectious diseases are discussed. An in-depth literature review on the mathematical modelling and theoretical assessment of HIV/AIDS intervention strategies including the epidemiologic and demographic effects are presented.

Keywords: HIV/AIDS, Intervention strategies, Mathematical modelling, Theoretical assessment.

1.

Introduction

The extensive spread of human immunodeficiency virus (HIV) and the associated acquired immune deficiency syndrome (AIDS) continues around the world since its recognition in the early 1980s. In 2003, almost five million people became newly infected with HIV, the ∗

E-mail addresses: [email protected], [email protected]. Corresponding author.

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greatest number in any one year since the beginning of the epidemic. At global level, the number of people living with HIV/AIDS continued to grow from 35 million in 2001 to 38 million in 2003 and in the same year, almost three million were killed by AIDS (UNAIDS, 2004). HIV/AIDS had killed more than 25 million people by 2005, making it one of the most destructive epidemics recorded in history. Despite recent, improved access to antiretroviral treatment and care in many regions of the world, the HIV/AIDS epidemic claimed 3.1 million [2.8-3.6 million] lives in 2005, of which more than half a million (570 000) were children (UNAIDS/WHO, 2005). Globally, the HIV incidence rate is believed to have peaked in the late 1990s and to have stabilized subsequently, notwithstanding increasing incidence in a number of countries. In some countries, favourable trends in incidence associated with changes in behaviour and prevention programmes have been observed. Changes in incidence along with rising AIDS mortality have caused global HIV prevalence to level off but the number of people living with HIV has continued to rise, due to population growth and, more recently, the life prolonging effects of antiretroviral therapy (ART). Among the notable new trends are the recent declines in national HIV prevalence in Kenya, Zimbabwe, urban areas of Burkina Faso and similarly Haiti, Tamil Nadu, Cambodia and Thailand (UNAIDS, 2006; UNAIDS/WHO, 2005). However, HIV/AIDS prevalence is increasing in some countries, notably China, Indonesia, Papua New Guinea, and Vietnam, and there are signs of HIV outbreaks in Bangladesh and Pakistan. In eastern and central Asia, the majority of people living with HIV/AIDS are in Ukraine, where the annual number of new HIV diagnoses keeps rising, and the Russian Federation has the biggest AIDS epidemic in all of Europe. Meanwhile, evidence continues to emerge of resurgent epidemics in the United States of America and in some countries in Europe among homosexuals, and of largely hidden epidemics among their counterparts in Latin America and Asia (UNAIDS, 2006). The HIV/AIDS epidemic has remained one of the leading causes of death in the world and has been destructive in Africa with Sub-Saharan Africa remaining the epidemiological locus of the epidemic. For a long time, it appeared that most severe HIV/AIDS epidemics might be restricted to countries in east and central Africa but subsequently the pandemic has spread to southern Africa and now countries in Sub-Saharan Africa are experiencing more devastating national epidemics (Garnett et al., 2001). Since the very start of the epidemic in Sub-Saharan Africa, heterosexuality has remained the principal mode of transmission (May and Anderson, 2004; UNAIDS/ WHO, 1998; Tallis, 2002; UNAIDS, 2004; Mufune, 2004). Over time and due to the fact that women are inordinately affected, vertical transmission from mother-to-child is of increasing importance in transmission and this is in contrast to other areas of the globe where the principal mode of transmission includes homosexuals and intravenous drug users (Mufune, 2004). Sub-Saharan Africa is currently the home to 25.8 million [23.8-28.9 million] people living with HIV, almost one million more than in 2003 (UNAIDS, 2004). Two thirds of all people living with HIV are in Sub-Saharan Africa, as are 77% of all women with HIV. An estimated 2.4 million [2.1-2.7 million] people died of HIV-related illnesses in this region in 2005, while a further 3.2 million [2.8-3.9 million] became infected with HIV (UNAIDS/WHO, 2005). The detrimental impact of the HIV/AIDS epidemic is more strongly felt in developing countries. In Sub-Saharan Africa countries, declines in national HIV prevalence associated with changes in behaviour and prevention programmes have been observed in two countries that are Kenya and Zimbabwe. In all

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affected countries with either high or low HIV/AIDS prevalence, HIV/AIDS hinders development, exacting a devastating toll on individuals and families. In the hardest-hit countries, it is erasing decades of health, economic and social progress, reducing life expectancy by years, deepening poverty, and contributing to and exacerbating food shortages (UNAIDS, 2004).

2.

HIV/AIDS Interventions

To ensure a comprehensive response to HIV/AIDS, a number of intervention strategies have been adopted across the globe. In recent years, international consensus on the need for comprehensive response to HIV/AIDS comprising prevention, treatment and care has strengthened. International and national funding available to control the spread of HIV/AIDS has also increased. These advances present an important opportunity to further intensify efforts and increase the momentum towards gaining universal access to prevention, treatment and care for all countries affected by HIV/AIDS. The key elements in comprehensive HIV/AIDS prevention, treatment and care include the following: • behaviour change programmes especially for young people and populations at higher risk of HIV exposure, as well as for people living with HIV/AIDS, • prevent sexual transmission of HIV by promoting the use of male and female condoms as a protective option along with abstinence, fidelity and reducing the number of sexual partners,

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• prevent mother-to-child transmission (MTCT) of HIV, • prevent the transmission of HIV through injecting drug use, including harm-reduction measures, • ensure safety of blood supply, • prevent HIV transmission in health care settings, • promote greater access to voluntary HIV/AIDS counselling and testing while promoting principles of confidentiality and consent, • integrate HIV/AIDS prevention into HIV/AIDS treatment services, • focus on HIV/AIDS prevention among young people, • provide HIV-related information and education to enable individuals to protect themselves from infection, • confront and mitigate HIV/AIDS-related stigma and discrimination, • prepare for access and use of vaccines and microbicides. Advances in Disease Epidemiology, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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In June 2005, the UNAIDS governing board comprising member states, co-sponsoring UN agencies and civil society endorsed a policy position paper for intensifying of HIV prevention with the ultimate aim of achieving universal access to HIV prevention, treatment and care (UNAIDS/WHO, 2005). This policy position paper included essential proven programmes and actions for HIV prevention, treatment and care (previously mentioned) that could be used to close the prevention, treatment and care gap as well as twelve essential policy actions that would be needed to ensure universal access which are as follows: • ensure that human rights are promoted, protected and respected and that measures are taken to eliminate discrimination and combat stigma, • build and maintain leadership from all sections of society, including governments, affected communities, nongovernmental organizations, faith-based organizations, the education sector, media, the private sector and trade unions, • involve people living with HIV/AIDS, in the design, implementation and evaluation of prevention strategies, addressing the distinct prevention needs, • address cultural norms and beliefs, recognizing both the key role they may play in supporting prevention efforts and the potential they have to fuel HIV transmission, • promote gender equality and address gender norms and relations to reduce the vulnerability of women and girls, involving men and boys in this effort, • promote widespread knowledge and awareness of how HIV is transmitted and how infection can be averted,

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• promote the links between HIV prevention and sexual reproductive health, • support the mobilization of community-based responses throughout the continuum of prevention, care and treatment, • promote programmes targeted at HIV prevention needs of key affected groups and populations, • mobilizing and strengthening financial, human and institutional capacity across all sectors, particularly in health and education, • review and reform legal frameworks to remove barriers to effective, evidence based HIV prevention, combat stigma and discrimination and protect the rights of people living with HIV or vulnerable or at risk to HIV, • ensure that sufficient investments are made in the research and development of, and advocacy for, new prevention technologies. There are also other new prevention methods for HIV/AIDS that are, vaccines which are still under research, male circumcision, pre-exposure prophylaxis and microbicides. Other HIV/AIDS control measures include, the use of antiretroviral drugs for life prolonging and reducing MTCT, condom use and public health educational campaigns. We briefly discuss each of these in the following paragraphs.

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The best long-term hope for control of the HIV epidemic may be a preventive HIV vaccine. There is an urgent need to develop new candidate vaccines, but also a need to plan the considerable programme requirements in introducing new vaccines and in fitting them into other prevention strategies (Nagelkerke and De Vlas, 2003). Several different HIV vaccination strategies have been proposed. These include therapeutic vaccination (administered to those who are already infected) and prophylactic vaccination (administered prior to infection) that either prevents infection (sterilizing vaccination) or ameliorates disease (disease-modifying vaccinen) (Davenport et al., 2004). A number of candidate vaccines are on trial in several places to obtain definite information about their efficacy in inducing protection against infection. But developing a vaccine remains an enormous challenge for reasons related to inadequate resources, clinical trial and regulatory capacity concerns, intellectual property issues and scientific challenges. There are now 17 vaccine candidates in phase I trials (trial is done in a small number (20-80) of healthy, low-risk and non-infected volunteers) and four vaccines in phase II (trial is conducted with larger numbers up to a few hundred; including the promising Merck adenovirus vector vaccine now in phase IIb, which may stimulate anti-HIV cell-mediated immunity) and there is only one in phase III (trial is done at a much larger-scale involving thousands of non-infected and high-risk volunteers), under way in Thailand (UNAIDS/WHO, 2005). Male circumcision is the surgical removal of all or part of the foreskin of the penis. The foreskin has a high density of Langerhans cells, which represent a possible source of initial cell contact for HIV infection (Soto-Ramirez et al., 1996). In addition, the foreskin may provide an environment for survival of bacterial and viral matter and may be susceptible to tears, scratches, and abrasions, which suggests that its presence may increase the likelihood of contracting HIV (Cameron et al., 1989). Male circumcision may be practised as part of a religious ritual performed shortly after birth, a traditional coming of age ritual practiced at or after puberty in certain cultures, or a medical procedure related to infections, injury, or anomalies of the foreskin. It is increasingly being considered as a preventive medical procedure to reduce the acquisition of sexually transmitted HIV infection. The earliest documentary evidence for male circumcision is from Egypt. Tomb artwork from the Sixth Dynasty [2345-2181 B.C.] shows circumcised men, and one belief from this period shows the rite being performed on a standing adult male (Auvert et al., 2005). Genesis (17:11) places the origin of the rite among the Jews in the age of Abraham, who lived around 2000 B.C. Hutchinson (1885) suggested that the removal of the foreskin reduces susceptibility of men to sexually transmitted infections. Fink (1986) published the first paper suggesting a protective effect of male circumcision against HIV. Since then, many observational studies have been published and some of which have observed that most men living in east and southern Africa, the regions with the highest prevalence of HIV/AIDS are uncircumcised (Bongaarts et al., 1989; Caldwell and Caldwell, 1996; Moses et al., 1990). In a recent article (Auvert et al., 2005) presented the results of their randomized controlled trial on male circumcision to prevent HIV transmission. They concluded that male circumcision reduces the risk of HIV infection by some 60% (95% confidence interval, 32%-76%). The randomised, controlled, blindly evaluated intervention trial was carried out in Orange Farm and surrounding areas, a semi-urban region close to the city of Johannesburg. Pre-exposure prophylaxis is a novel approach to HIV prevention in which antiretroviral drugs (ARVs) are used by an individual prior to potential HIV exposure to reduce the like-

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lihood of infection. Pre-exposure prophylaxis should be distinguished from post-exposure prophylaxis, in which an individual takes ARVs soon after a potential HIV exposure with the goal of reducing the likelihood of infection (Szekeres et al., 2004). Small-to-medium sized phase II trials are under way in Atlanta and San Francisco, with larger phase II/III studies under way or planned in Botswana, Ghana, and possibly Thailand. Some of these studies have been dogged by controversy. The main issues were the adequacy of pre-trial community consultation and informed consent, linkages to HIV treatment programmes for those found to be infected at baseline or in the course of the study, and in the case of Thailand, the lack of access to sterile needles in a study designed to examine HIV transmission among injecting drug users. Two pre-exposure prophylaxis studies were cancelled (Cambodia, Nigeria) and another (Cameroon) postponed. A consultation in Seattle and a series of consultations led by UNAIDS in two African regions, Asia and Geneva involving community activists, researchers, sponsors and others helped identify the problems in trial design in this promising research area. Trials have moved forward in six other sites (UNAIDS/WHO, 2005). Microbicides offer the best promise of a prevention tool women can control. They could have a substantial impact on the epidemic. Currently, the HIV microbicide field has four candidate microbicides entering or in phase III trials, five in phase II, and six in phase I. They include soaps, acid buffering agents, seaweed derivatives and anti-HIV compounds. Modelling indicates that even a 60%-efficacious microbicide could have considerable impact on HIV spread. If used regularly by just 20% of women in countries with substantial epidemics, hundreds of thousands of new infections could be averted over three years (Rockefeller, 2001). Currently, antiretroviral drugs are being used as a way of reducing the destruction caused by HIV/AIDS. Antiretroviral drugs are only given to AIDS individuals who are ill and have experienced AIDS-defining symptoms, or whose CD4 + T cell count is below 200/µl , which is the recommended AIDS defining stage guideline (WHO, 2002). Current combination antiretroviral drugs increase survival time of HIV-infected individuals, but do not lead to viral eradication within individuals and hence do not cure. These therapies are based upon three or more anti-HIV medications that typically combine a protease inhibitor (PI), or a non-nucleoside reverse transcriptase inhibitor (nnRTI), with at least two nucleoside reverse transcriptase inhibitors (nRTI) (Velasco-Hern´andez et al., 2002). Unfortunately, because of the high cost of antiretroviral drugs, very few individuals in developing countries can afford them. There is also considerable support for accelerating and intensifying efforts to prevent HIV infection in infants especially in countries most affected by the HIV/AIDS epidemic. Programmes using antiretroviral drugs to reduce HIV transmission from mother to child (MTCT) are being introduced or taken to scale in a number of countries. A two-dose regimen of nevirapine (single dose to the mother at onset of labour and single dose to the infant within 72 hours of birth) is increasingly being used because of its low cost and simplicity (WHO, 2001). The use of nevirapine among women of unknown serostatus at the time of labour and delivery has also been proposed, especially in settings with high HIV prevalence among pregnant women and limited availability or low uptake of programme activities for the prevention of MTCT, such as voluntary counselling and testing and antiretroviral prophylaxis.

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The use of condoms has remained the only feasible preventative strategy for HIV/AIDS. Since the recognition of the condom as a preventive tool for controlling the spread of HIV, numerous experimental and clinical studies have been conducted to ascertain their effectiveness. The most conclusive evidence of condom effectiveness in reducing HIV/AIDS transmission has come from studies of serodiscordant couples, in which one person is infected with HIV and one person is not (McNeill et al., 2001) and on a larger scale, evidence in favour of condom effectiveness is supported by the experience of the government of Thailand. The Thai government’s 100% condom policy, which required commercial sex workers and their clients to use condoms for every act of intercourse, led to an increase in the use of condoms from 14% in 1989 to 94% in 1994, and a decrease in cases of bacterial STIs from 410406 in 1987 to 29362 in 1994 (RHO, 2005). Countries such as Ethiopia, Uganda, and Vietnam have also achieved a dramatic increase in condom use through national programmes promoting condoms for HIV/STI prevention (WHO/UNAIDS, 2001). Public health educational campaigns defined as the counselling of individuals to have fewer sexual partners, abstain, and/or otherwise reduce risky behaviour have been widely adopted as a control measure for the spread of HIV/AIDS. Public health educational campaigns have resulted in declines of HIV/AIDS prevalence in countries such as Uganda and Senegal (UNAIDS, 2001). Theoretical studies of partially effective HIV vaccines and antiretroviral treatment and the potential changes in risky behaviour associated with a vaccination and treatment campaigns have also shown that the benefits offered by an only partly effective vaccination or treatment programme may be offset by rises in potentially infectious contacts unless an education campaign (defined in the context of counselling of individuals) accompanies it (Blower and Maclean, 1994; Del Valle et al., 2004; Maclean and Blower, 1993, 1995).

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3.

Mathematical Modelling of Epidemics

Mathematical modelling is the use of mathematical language to describe the behaviour of a system. Mathematical models are used particularly in the natural sciences and engineering disciplines. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables representing the properties of the system. A model is a set of functions that describe the relations between the different variables. Some of the basic groups of variables are, decision variables, input variables, state variables, exogenous variables, random variables, and output variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Mathematical models can be classified in several ways, some of which are described below: 1. Linear and nonlinear: Mathematical models are usually composed by variables, which are abstractions of quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions and differential operators. If all the operators in a mathematical model present linearity the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise.

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2. Deterministic and stochastic: A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.

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3. Static and dynamic: A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with difference or differential equations. An epidemic is generally defined as a widespread disease that affects many individuals in a population. It can be endemic or non-endemic depending on the time period it occurs. An epidemic is endemic if it persists in a population in a long term without needing to be reintroduced from outside, during which there is a renewal of susceptibles by birth or recovery from temporary or permanent immunity and is non-endemic if it occurs in a short period of time. It is called pandemic if it is widely distributed in space. Epidemics may arise from the introduction of a pathogen or strain to a previously unexposed population or as a result of the regrowth of susceptible numbers some time after a previous epidemic due to the same infectious agent (Swinton, 2003). HIV/AIDS epidemic has not only severe personal and social implications for affected individuals and families, but also socio-economic implications for the society. The most urgent public-health problem worldwide is to devise effective strategies to minimise the destruction caused by this HIV/AIDS epidemic. To understand, predict and control the epidemic, it is important to establish the transmission dynamics of the epidemic. HIV/AIDS transmission dynamics provides a large number of new problems to mathematicians, biologists and epidemiologists since it has many features different from traditional infectious diseases. Mathematical modelling of epidemics play a crucial role in understanding the transmission dynamics of infectious diseases. Models are particularly helpful as experimental tools with which to evaluate and compare control procedures and prevention strategies, and to investigate the relative effects of various sociological and biological factors on the spread of diseases. The capability of mathematical models in understanding and predicting epidemics goes back to Daniel Bernoulli, who in 1760 used mathematical methods to study techniques of protecting against smallpox. Although his work preceded the identification of the agent responsible for the transmission of smallpox by a century, he formulated and solved a differential equation describing the dynamics of the infection which is still of value today. The development of mathematical epidemiology was stalled by a lack of understanding of the mechanism of infection spread until the beginning of the nineteenth century (Bailey, 1975). Further concepts on mathematical modelling of epidemics were developed by Ronald Ross (1911) and his students Kermack and Mackendrick (Mackendrick, 1926; Kermack and Mackendrick, 1927, 1932, 1933). The 1927 SIR (susceptible-infectives-removed) model by Kermack and Mackendrick used to understand the cholera epidemic was and is still famous among researchers. The epidemic model has duration dependent infectivity, that is the infection rate depends on the duration in the infected and infectious status and the infection happens only one time in the life time of host individuals. Assuming constant

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infectivity, this structured SIR model is reduced to the well known ordinary differential equation model (Inaba, 2000). Kermack and Mackendrick’s structured SIR model has been re-examined by Diekmann (1977) and Metz (1978) (see also Metz and Diekmann, 1986). The importance of this kind of structured SIR is now widely recognised, since it provides a model for epidemics with long incubation period and variable infectivity such as HIV/AIDS epidemic (Thieme and Castillo-Chavez, 1993). In the past, the SIR-type models have been well studied and extended to various kinds of epidemic-demographic situations (Anderson and May, 1991). In their theoretical papers written in 1932 and 1933, Kermack and Mackendrick proposed a kind of duration-dependent epidemic model, where the transmission rate depends on both duration of infected host and duration of susceptible host. The model classified the total population into three classes, the never infected, infected and recovered. The host population is structured by duration variable in each status, but the chronological age is neglected. In this model, recovered individuals can be reinfected repeatedly, and their reinfection probability depends on how long it takes since the last infection. Kermack and McKendrick concentrated on the problem of endemicity of this model, thus they examined conditions under which existence and uniqueness of the endemic steady state can be established. It is the theoretical studies by Kermack and Mackendrick that formed the basis for mathematical modelling of infectious diseases and with the emergence of HIV/AIDS, this discipline has had a renaissance. Following Kermack and Mackendrick, various epidemic models have been proposed using differential equations which are, SI, SIS and SIR in which S(t) is the number of susceptibles at time t, I(t) is the number of infectious individuals at time t and R(t) are the removed or recovered class at time t. The removed or recovered group are those who may have gained temporary or permanent immunity, have died or have been isolated (or quarantined) depending on the disease being modelled. These models have been widely studied and applied in modelling parasitic diseases, host-vector-host type of diseases (for example malaria and sleeping sickness) and sexually transmitted diseases (e.g. gonorrhea and HIV/AIDS) (see Bailey 1975; Hethcote, 1980; Hoppensteadt, 1974, 2000; Hyman and Li, 1997; Murray 1989; Razan, 2001; Hraba and Dolezal, 1996). The SI model divides the population into two classes that are the susceptibles S and the infectives I. This model assumes that once an individual becomes infective, the individual remains so but is not removed. The SIS model also divides the population into two classes but unlike the SI model, the infectives have a chance to recover and become susceptibles. We can include another class or compartment to the SI and SIS models obtaining the SEI and SEIS models where E(t) is the exposed class, which is a class of infected individuals who are not yet infectious at time t. The SIR model divides the population into three classes which are the susceptibles S, the infectives I and the removed or recovered R as previously defined. This model allows removals or recoveries. The removed or recovered group includes the dead, the isolated and the immune. The individuals who recover from the disease are assumed to have become immune. The SIR model can also be extended by adding another class of the exposed E giving an SEIR model. In a case in which the recovered group looses immunity to become susceptibles, the model becomes an SIRS model. The exposed class E can also be included to obtain an SEIRS model. In the modelling of infectious diseases, deterministic and stochastic models have been adopted but deterministic models have perhaps received more attention in the literature. The

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main advantage of deterministic models is that they can be more complex, yet still possible to analyse, at least when numerical solutions are adequate and for a stochastic epidemic model to be mathematically manageable it has to be quite simple and thus not entirely realistic. Several reasons suggest that stochastic models are to be preferred when their analysis is possible. First, the most natural way to describe the spread of disease is stochastic; one defines the probability of disease transmission between two individuals, rather than stating certainly whether or not transmission will occur. Deterministic models describe the spread under the assumption of mass action law, relying on the law of large numbers (Grais et al., 2004). Secondly, there are phenomena which are genuinely stochastic and do not satisfy the law of large numbers. A third important advantage concerns estimation. Knowledge about uncertainty in estimates requires a stochastic model, and an estimate is not of much use without some knowledge of its uncertainty. Generally, stochastic models are preferred when their analysis is possible, otherwise deterministic models should be used. However, there are no conflicts between the types of models and both play an important role in better understanding the mechanisms of disease spread (Andersson and Britton, 2000).

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4.

Modelling HIV/AIDS and Interventions

Mathematical modelling and computer simulations of HIV/AIDS control efforts have become powerful tools for health policy evaluation, policy dialogue, and advocacy. A number of such mathematical and computer simulation models have been proposed in the literature. Indeed, the number of publications for infectious disease modelling has seen an exponential growth and as was pointed out by someone, it has become an epidemic itself. Nevertheless, it is not the purpose of this survey to list all relevant works related to HIV/AIDS. Hence, the need to review some previous studies on the effects of intervention strategies in curtailing the epidemic. In this section, we review some previous studies that contributed in the understanding of the effectiveness of HIV/AIDS intervention strategies up to date. The first model used for the explicit study of sexually transmitted disease, namely gonorrhea, was a one-sex model (Cooke and Yorke, 1973). A two-sex model developed specifically for gonorrhea was formulated by Lajmanovich and Yorke (1976). Dietz and Hadeler (1988) and Waldst´atter (1989) also studied epidemiological models of sexually transmitted diseases using a simple two-sex model. Concerns with the HIV/AIDS epidemic have generated extensive research activity on the models for the sexual transmission of HIV/AIDS. Other studies of heterosexual transmission models of HIV include Anderson et al., (1988), May et al., (1988), LePont and Blower (1991), Lin et al., (1993), and Bursenberg et al., (1995). However, most works have concentrated on the study of models for the homosexual transmission (see Anderson and May, 1991; Gabriel et al., 1990; Velasco-Hernandez and Hsieh, 1994). For some notable exceptions see the work of Castillo-Chavez and Busenberg (1991), CastilloChavez et al., (1991), Hadeler (1989), Hadeler et al., (1988) and Hoppensteadt (1974). Theoretical studies in the past on control measures for the transmission of HIV/AIDS using mathematical models include Scalia-Tomba (1991), on effects of behaviour change on spread of HIV, Hsieh (1991) on screening and removal of HIV positive, Anderson et al., (1991) and Hsieh and Velasco-Hern´andez (1995) on community-wide treatment of HIV, and Velasco-Hernandez and Hsieh (1994) on effects of treatment and behavioural change, with homosexual transmission accounted for.

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The epidemiological impact of widescale use of (highly active) antiretroviral therapy (HAART, or ART) among HIV/AIDS patients in industrialised countries have been analysed using a number of mathematical modelling studies (see (Blower et al., 2000; VelascoHern´andez, 2002) for a general reference). Successful ART use decreases plasma and seminal viral load and so is thought to reduce HIV infectiousness (Vernazza et al., 1997). Mathematical models can be used to assess the potential impact and effectiveness of various intervention strategies. In terms of ART use, they can be used to investigate, the efficient use of ART, the epidemiological consequences of ART and interaction with behavioural changes/interventions, the likely course of drug resistance evolution (within and between individuals), achievable levels of coverage and effectiveness, the effective and efficient use of second line treatments, and demographic/health care impact (Baggaley et al., 2005). Currently, UNAIDS/WHO is using the Goals model to produce national HIV/AIDS epidemic projections. The Goals model is a Microsoft Excel spreadsheet model using linear equations, designed to improve resource allocation for national HIV/AIDS programmes. It feeds into the dynamic epidemic projection package (EPP) and Spectrum, used by the UNAIDS/WHO to produce national HIV/AIDS estimates, to predict the impact of interventions (Ghys et al., 2004). A number of models have been used to investigate the impact of ART use in various settings. Blower et al., (2000) constructed a deterministic model to study the effects of ART among the homosexual population in San Francisco and concluded that, increasing ART usage would decrease the death rate and substantially reduce HIV incidence. Using the same model, Blower et al., (2001) noted that prevalence of ART resistance is already high in San Francisco and will continue to increase substantially through 2005. Transmitted drug resistance will remain low, only increasing gradually, with a doubling time of around 4 years and a predicted median 15.6% [range 0.05-73.21%] new infections resistant to ARVs by 2005. Law et al., (2001), used a mathematical deterministic model to explore the use of ART among homosexuals and noted that changes in risky behaviour were linearly associated with increases in incidence, while decreases in infectivity were nonlinearly associated with decreases in incidence. Decreases in infectivity of 2-, 5- and 10- fold would be counterbalanced (in terms of incidence) by increases in risky behaviour of 40, 60 and 70%, respectively. In the extension of their model, Law et al., (2002) noted that decreases in infectivity of 2-, 5- and 10- fold would be counterbalanced (in terms of incidence) by increases in risky behaviour of 30, 50 and 65%, respectively, that is, even more modest increases than in their previous publication (Law et al., 2001). They also noted that even small increases in Sexually Transmitted Infections (STIs) as a result of increased risky behaviour could have an important multiplicative effect increasing HIV incidence. Velasco-Hern´andez et al., (2002) studied effects of ART among homosexuals in San Francisco and showed that median R0 = 0.90 (R0 is the basic reproductive number or the epidemic threshold) if risky sex decreased, 1.0 if risky sex remained stable, and 1.16 if risky sex increased and R0 decreased as ART coverage increased. The probability of epidemic eradication is high (p = 0.85) if risky sex decreases (median 25% reduction), moderate (p = 0.5) if it remains stable, and low (p = 0.13) if it increases (median 50% increase). They concluded that ART can function as an effective HIV prevention tool, even with high levels of drug resistance and risky sex, and could eradicate a high HIV prevalence (30%). A large body of literature exists on the search for an HIV/AIDS vaccine and over the

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past decade, a number of researchers have looked at the potential impact of a vaccine on the HIV/AIDS epidemic (Anderson and Hanson, 2005; Blower and McLean, 1994; Blower et al., 2002; Blower et al., 2005; Garnett et al., 2001; McLean and Blower, 1993; Nagelkerke and De Vlas, 2003; Seitz, 2001; Smith and Blower, 2004; Stover et al., 2002, to name but a few). Most of the vaccination models studied in the literature have considered different types of vaccine action, a range of vaccine characteristics (varying levels of vaccine efficacy with regard to preventing infection, reducing infectiousness and delaying progression to the development of AIDS, duration of vaccine induced protection for example), the possibility of behavioural reversals due to perceived vaccine-related protection, application in different epidemic settings, and various coverage levels and targeting strategies. Most of these modelling investigations recognize that the first generation of AIDS vaccines is likely not to have the ideal vaccine characteristics of high preventative efficacy and lifelong protection, and so they specifically consider future vaccines with only low to moderate efficacy and a limited duration of protection. Several articles, particularly McLean and Blower (1993), Garnett et al., (2001), Seitz (2001), Stover et al., (2002), Blower et al., (2002), and Nagelkerke and De Vlas (2003) explored the effects of vaccines with take type protection (where 50% efficacy means that 50% of those vaccinated are completely protected while the other 50% receive no protection) versus those with degree type protection (where 50% efficacy means that the probability of infection per contact is reduced by 50% for everyone vaccinated). The type of protection seems to make little difference to the overall impact in the general population or when the average risk is low. But, in populations at very high risk, degree protection produces much less impact than take protection since even 50% reductions in risk may still result in high average risk in these populations. McLean and Blower (1993), Blower and McLean (1994) and Blower et al., (2005) use both analytic techniques and simulation models to examine the likelihood that an AIDS vaccine could eventually eradicate an epidemic. They conclude in general that eradication through vaccination programmes alone will be challenging without very effective vaccines or very low levels of risk and high vaccine coverage. The availability of a prophylactic vaccine might lead to riskier behavior. A number of papers address this issue, including Blower and McLean (1994), Garnett et al., (2001), Seitz (2001), Stover et al., (2002), Nagelkerke and De Vlas (2003), Smith and Blower (2004), and Anderson and Hanson (2005). These studies note that behavioural reversals could mitigate the gains from vaccination or even produce perverse results where the use of the vaccine results in more infections. The conclusion is that any program to implement vaccination should be accompanied by strong general prevention efforts. Anderson et al., (1991), Blower et al., (2003), Smith and Blower (2004), Davenport et al., (2004) and Anderson and Hanson (2005) also investigate the impact of vaccines that do not prevent infection but do affect viral load and disease progression. They find that while such vaccines could prevent many deaths, they could result either in more new infections if they only lengthen the period of infectiousness or they could result in fewer new infections if they also significantly reduce viral load and, thus, the level of infectiousness. Blower et al., (2003) and Gray et al., (2003) examine the individual and joint effects of vaccines and widespread treatment with antiretroviral therapy (ART). They observe that widespread ART that reduces infectiousness combined with a low efficacy vaccine could reduce new infections to very low levels as long as behavioral reversals do not overwhelm the reductions in the risk of transmission per sexual act.

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Although therapeutic treatment strategies appear promising for retarding the progression of HIV-related diseases, prevention remains the most effective strategy against the HIV/AIDS epidemic. The use of condoms as preventive strategy for HIV have been studied. Gordon (1989) estimated R0 based on data for condom efficacy to avoid pregnancy. Greenhalgh et al., (2001) analysed the dynamics of a two-group deterministic model for assessing the impact of condom use on the sexual transmission of HIV/AIDS within a homosexual population. Bistability, a phenomenon where a stable disease-free equilibrium and a stable endemic equilibrium co-exist when R0 < 1 was observed in their study. This phenomenon has been observed in a number of epidemiological studies (see (Kribs-Zaleta, 2000) for a general reference). Moghadas et al., (2003) studied the effect of condom use as a single-strategy approach in HIV prevention in the absence of any treatment. Two primary factors in the use of condoms to halt the HIV/AIDS epidemic, condom efficacy and compliance using deterministic mathematical model were considered. Stability and sensitivity analysis, based on a plausible range of parameter values were used to identify key parameters that govern the persistence or eradication of HIV/AIDS. They concluded from the study that for populations where the average number of HIV-infected partners is large, the associated preventability threshold is high and perhaps unattainable, suggesting that for such a population, HIV may not be controlled using condoms alone. On the other hand, for a population where the average number of HIV-infected partners is low, it is shown that preventability threshold is about 75%, suggesting that the epidemic could be stopped by using condoms. Thus, for such a population, public health measures that can bring preventability above the threshold and continuous quantitative monitoring to make sure it stays there, are what would be necessary. In other words, for populations with reasonable average numbers of HIV-infected partners, given the will and effort, it is within our means to halt this epidemic using condoms. Mukandavire and Garira (2006) also assessed the effectiveness of female and male condom use as a preventative strategy in a heterosexually active population following a similar approach by Moghadas et al., (2003). The use of vaginal microbicides as HIV transmission barriers has been advocated for some time, but development has been slow (Potts, 2000). A vaginal preparation inserted before intercourse may be more acceptable to a large number of women who know their sexual partners may be at risk of HIV infection but who cannot negotiate the use of condoms (Ramjee et al., 1999). In the role of reaching out and empowering women for whom condom use is not possible, microbicides could be a significant force in reducing new infections. Karmon et al., (2003) developed a mathematical model for assessing the effects of introducing a microbicide as an HIV infection protective method. From their study, they found that in general, if existing condom usage in a community is low, introducing a microbicide will most likely have a positive impact on HIV/AIDS incidence as abandonment of condom use in favour of microbicides will not play a significant role. If condom use in a community is high, though, attrition of condom users could play a role large enough to overwhelm any added risk reduction afforded new microbicide users. Their model illustrates the importance of knowing key behavioural parameters, such as the proportion of the population that uses condoms, before microbicides can be safely introduced. These key parameters include the proportion of condom users likely to maintain condom use and the proportion of condom nonusers likely to adopt microbicides, as well as the efficacy of the candidate microbicide.

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While access to life-saving anti-retroviral therapy (ART) for those infected with HIV/AIDS is increasing rapidly throughout the world, effective programmes to reduce HIV transmission are still needed (Salomon et al., 2005), especially in Africa. Many factors influence the risk of acquiring and transmitting HIV, but where measures intended to reduce transmission have been rigorously tested at a population level, the results have been mixed (Sangani et al., 2004; Siegfried et al., 2005). Furthermore, an effective vaccine will probably not be available for a decade or more (Harrington et al., 2005). There are indications that male circumcision (MC) significantly reduces female-to-male transmission of HIV. A randomized controlled trial (RCT) has shown that male circumcision (MC) reduces sexual transmission of HIV from women to men by 60% [32%-76%, 95% CI] offering an intervention of proven efficacy for reducing the sexual spread of HIV (Auvert et al., 2005). Recently, Williams et al., (2006) used dynamical simulation models to consider the impact of MC on the relative prevalence of HIV in men and women and in circumcised and uncircumcised men. Using country level data on HIV prevalence and MC, they estimated the impact of increasing MC coverage on HIV incidence, HIV prevalence, and HIV-related deaths over the next ten, twenty, and thirty years in Sub-Saharan Africa. Assuming that full coverage of MC is achieved over the next ten years, they considered three scenarios in which the reduction in transmission is given by the best estimate and the upper and lower 95% confidence limits of the reduction in transmission observed in the RCT. A recent study by Mukandavire et. al. (2007) considers a holistic approach of intervention strategies involving MC and condom use. Theoretical studies by Blower and Maclean (Blower and Maclean, 1994; Maclean and Blower, 1993, 1995) on partially effective HIV vaccines and the potential changes in risky behaviour associated with a vaccination campaign, showed that the benefits offered by an only partly effective vaccination program may be offset by rises in potentially infectious contacts unless accompanied by an educational campaign. Following the concepts of Blower and Maclean that partly effective vaccines should be accompanied by educational campaigns, Del Valle et al., (2004) studied the effects of education in a setup with vaccination and treatment on HIV transmission in homosexuals with heterogeneity. Mukandavire and Garira (2007) studied the effects of public health educational campaigns and the role of sex workers on the spread of HIV/AIDS among heterosexuals and these studies conclude that the presence of sex workers enlarges the epidemic threshold R0 , thus fuels the epidemic among the heterosexuals, and that public health educational campaigns among the high-risk heterosexual population reduces R0 , thus can help slow or eradicate the epidemic.

5.

Conclusion

In this commentary, we gave a global history of the HIV/AIDS epidemic and various intervention strategies in place. Mathematical modelling of epidemics, its history and role in controlling infectious diseases especially HIV/AIDS, are discussed. An in-depth literature review on the mathematical modelling and theoretical assessment of HIV/AIDS intervention strategies including epidemiological and demographic effects are presented. As HIV spreads, it interacts with other infectious diseases, facilitated by the increase in numbers of immunosuppressed individuals and because its own clinical course can be altered by other infections (UNICEF, 2003). Infectious diseases often synergize or nega-

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tively affect each other, and this is most noticeable with HIV and tuberculosis (Corbett et al., 2002). In fact, there is a synergistic interaction between HIV-1 transmission and genital herpes. Some examples of co-infections with HIV/AIDS are: Tuberculosis (TB) - Malaria Sexually Transmitted Infections (STIs) - Hepatitis B and C. Thus, UNICEF ESARO (Eastern and Southern Africa) has identified HIV/AIDS as the priority of priorities. In addition to enhancing access to HIV-1 prevention and care, public health surveillance and control programmes should be greatly intensified to cope with the new realities of infectious disease control in Africa (Corbett et al., 2002). So far from being a disease which is unstoppable in its epidemic consequences, AIDS has produced an epidemic which owes its present virulence to sociological configurations of rather recent existence (Thomson, 1989). It is the very low probability of transmission of AIDS which distinguishes its epidemiological properties from customary venereal diseases and probably gives the key to the fragility of the current AIDS epidemic.

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In: Advances in Disease Epidemiology Editors: J.M. Tchuenche et al, pp. 243-269

ISBN: 978-1-60741-452-0 © 2009 Nova Science Publishers, Inc.

Chapter 9

A MODEL FOR THE SPREAD OF HIV/AIDS IN A TWO SEX POPULATION Ram Naresh a, Agraj Tripathi b1 and Dileep Sharma a a

Department of Mathematics, Harcourt Butler Technological Institute Kanpur-208002, India b* Department of Mathematics, Bhabha Institute of Technology Kanpur-209204, India

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Abstract A nonlinear mathematical model is proposed and analyzed to study the spread of HIV in a two sex population with constant immigration. In the model, we have divided the population into two classes, the male population N1 and the female population N2. The male and female populations are further partitioned into three subclasses of susceptibles, infectives and AIDS patients. It is assumed that the male susceptibles exhibit bisexual character and become infected by homosexual as well as by heterosexual contacts whereas female susceptibles get infected by heterosexual interaction with male infectives only. The model has been studied qualitatively using stability theory of nonlinear differential equations and numerical simulation. The model exhibits two equilibria namely, a disease free and an endemic equilibrium. The existence of an endemic equilibrium is found to be dependent on the basic reproduction number R0. It is observed that disease persistence is higher if the population exhibits a bisexual character. A numerical study of the model is also performed to investigate the influence of some other key parameters on the spread of the disease and to support the analytical results.

Keywords: HIV/AIDS, epidemic, heterosexual, homosexual, bisexual character, reproduction number, stability.

*

E-mail address: [email protected], [email protected]. Corresponding author

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Ram Naresh, Agraj Tripathi and Dileep Sharma

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1. Introduction The Human Immuno-deficiency Virus (HIV) infection, which can lead to Acquired Immunodeficiency Syndrome (AIDS), is an infectious disease that remains a problem worldwide [6]. The most susceptible individuals at risk of acquiring infection include people having sexual contacts with HIV infecteds, homosexual and bisexual men, intravenous drug abusers and persons transfused with contaminated blood [19]. The HIV infection and AIDS are continuing to spread in both the developed and developing nations with alarming rate. AIDS has started getting attention as it has become a death sentence and a fear to a lot of people mainly because there is no cure available till date. It is now spreading rapidly in Asia, where new infections are increasing faster than anywhere else in the world. It is therefore, essential to study the transmission dynamics of HIV/AIDS and take necessary steps to prevent it from spreading [6, 17, 19]. Various modeling studies have been made to help improve our understanding of the major contributing factors of the AIDS epidemic. From the initial models of May and Anderson [2, 3, 16], several refinements have been added into modeling frameworks, and specific issues have been addressed by researchers [see for example,1, 6, 11, 15, 18-22]. In particular, Lin et al. [15] presented a model for HIV transmission for homosexual population of varying size with recruitment into the susceptible class proportional to the active population size and with stages of progression to AIDS. Greenhalgh et al. [11] studied the impact of condom use on the sexual transmission of HIV and AIDS amongst a homogeneously mixing male homosexual population. Aggarwala [1] developed a density dependent HIV transmission model for Canadian population by taking into account the vertical transmission and by using simple mass action type interaction. Srinivasa Rao [21] gave a framework for modeling the AIDS epidemic in India. Piqueira et al., [20] presented a model for transmission in homosexual populations by taking into account different attitudes, blood screening and effects of social networks. Naresh et al., [17] presented a model to study the transmission of HIV in a population with vertical transmission. Chen [7] presented an epidemic model of HIV transmission with self-protective behavior and preferred mixing. Heterogeneity in sexual activity and the different social/sexual mixing patterns influence an AIDS epidemic and were recognized early in the history of the pandemic. However, some of the above features in both homogeneous and heterogeneous populations have been taken in the AIDS modeling [4, 5, 9, 10, 12, 13]. Lin [14] proposed an HIV model for heterogeneous population and obtained the complicated steady patterns and multiple endemic equilibria. Doyle et al., [9] and Doyle and Greenhalgh [10] proposed a mathematical model for the spread of HIV by sexual transmission in a heterosexual population. Hsieh and Chen [12] developed a model for a community which has the structure of two classes of commercial sex workers and two classes of sexually active male customers with different levels of sexual activity. They observed that the change of behavior by the commercial sex workers has a more direct effect on the spread of HIV than that of male customers.

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It is pointed out that the above studies consider either the homosexual population only or the heterosexuals but very often the populations exhibit bisexual character which not only increases the chances of interactions but the associated risk of infection too. In this paper our objective is to model the spread of HIV infection in a population exhibiting bisexual character with constant immigration of susceptibles. Both the analytical and numerical studies of the model are conducted to obtain necessary information towards reducing the spread of the disease.

2. Mathematical Model

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The model considers a population with two subclasses of N1 males and N2 females. These subclasses are further partitioned into three compartments of susceptibles Si(t), infectives Ii(t) and AIDS patients Ai(t), respectively, (i = 1, 2). The total male population size is N1(t) = S1(t) + I1(t) + A1(t) and the total female population size is N2(t) = S2(t) + I2(t) + A2(t). For the model building, we make the following assumptions, (i)

The model considers only the sexual interaction responsible for the spread of HIV infection.

(ii)

The bisexual (homosexual and heterosexual) character is assumed to be exhibited by males only whereas females are considered to be only heterosexual.

(iii)

The spread of HIV infection via female to female sexual interaction is not taken into account due to lack of penetrative sex.

(iv)

The AIDS patients (male and female both) are assumed to be exposed and isolated and hence do not take part in sexual interaction.

(v)

It is also assumed that anti-HIV treatment is not available within the community so that persons once infected are bound to develop full blown AIDS at a later date.

2.1. Male Susceptible Individuals, S1(t) Male susceptibles are assumed to become infected following sexual contacts with male infectives I1 at a rate β1 and with female infectives I2 at a rate β2 with constant immigration at a rate Q1 and natural mortality at a rate d. It is further assumed that the rate of transmission of infection is in direct proportion to the susceptible population, and also to the ratio between the number of infected population and the total subpopulation. The equation governing the dynamics of susceptible male population is given by,

⎛β c S I dS1 β c SI = Q1 − ⎜⎜ 1 1 1 1 + 2 2 1 2 dt N2 ⎝ N1

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⎞ ⎟⎟ − dS1 ⎠

(1)

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where c1 and c2 are the number of sexual partners per unit time for male to male homosexual contacts and for male to female heterosexual contacts, respectively.

2.2. Male HIV Infected Individuals, I1(t) This population is generated by the HIV infection of male susceptibles either by homosexual contacts with male infectives or by heterosexual contacts with female infectives. It is diminished by natural deaths at a rate d and by the development of clinical AIDS at a rate δ. Thus, we have,

dI1 β1c1S1 I1 β 2 c 2 S1 I 2 = + − (δ + d )I1 dt N1 N2

(2)

2.3. Male Individuals with Clinical AIDS, A1(t) The population of individuals with clinical AIDS, A1(t), generates when male infective population I1(t) looses individuals with disease symptoms at a rate δ. This population suffers natural mortality at a rate d and by disease induced deaths at a rate α1. Thus,

dA 1 = δI1 − (α 1 + d )A 1 dt

(3)

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2.4. Female Susceptible Individuals, S2(t) The female susceptibles immigrating at a rate Q2 are considered to become infected following heterosexual contacts with male HIV infectives only. The female-to-female interaction is considered to be unlikely to transmit the infection due to lack of penetrative sex. The population in this class looses individuals following sexual contacts with male infectives I1 at a rate β3 and with natural mortality at a rate d. Thus,

β cS I dS 2 = Q 2 − 3 3 2 1 − dS 2 N1 dt

(4)

where c3 is the number of sexual partners per unit time for female to male heterosexual contacts.

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2.5. Female HIV Infected Individuals, I2(t) The female infective population is generated following heterosexual contacts of female susceptibles with the male infectives I1 at a rate β3. It is also diminished by natural deaths at a rate d and by the development of clinical AIDS at a rate δ. Thus, we have,

dI 2 β 3 c 3S 2 I1 = − (δ + d )I 2 dt N1

(5)

2.6. Female Individuals with Clinical AIDS, A2(t) The population of female individuals with clinical AIDS, A2(t), is generated following loss of individuals from female infective class at a rate δ who progress to exhibit the disease symptoms. This population is reduced by natural mortality rate d and by disease induced death at a rate α2. Thus,

dA 2 = δI 2 − (α 2 + d )A 2 dt

(6)

The schematic diagram of the disease progression is shown in Fig.1.

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Male

Q1

S1

β1c1S1I1/N1

β2c2S1I2/N2

Female

Q2

I1

δ

A1

β3c3S2I1/N1

S2

I2

δ

A2

Figure 1. Transfer diagram of mathematical model.

Since we assume that N1(t) = S1(t) + I1(t) + A1(t) and N2(t) = S2(t) + I2(t) + A2(t), the above equations can now be rewritten as,

dN 1 = Q1 − dN 1 − α 1 A 1 , dt

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dI1 β1c1 ( N 1 − I1 − A 1 )I1 β 2 c 2 ( N 1 − I1 − A 1 )I 2 = + − (δ + d )I1 , dt N1 N2

(8)

dA 1 = δI1 − (α 1 + d )A 1 , dt

(9)

dN 2 = Q 2 − dN 2 − α 2 A 2 , dt

(10)

dI 2 β 3 c 3 ( N 2 − I 2 − A 2 )I1 = − (δ + d )I 2 , dt N1

(11)

dA 2 = δI 2 − (α 2 + d )A 2 , dt

(12)

Continuity of the right hand side of system (7)-(12) and its derivative imply that the model is well posed for N1>0 and N2>0. As N1 and N2 tend to zero, Si, Ii and Ai (i =1, 2) also tend to zero. Hence, each of these subpopulations, (Si, Ii and Ai for i =1, 2) tends to zero as N1 and N2 does. It is therefore natural to interpret these terms as zero at N1 = 0 and N2 = 0.

3. Stability Analysis

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In the following, we present the existence of model equilibria and their local and global stability analysis.

3.1. Equilibria of the Model The model (7)-(12) has two non negative equilibria namely, (i)

E0 =(Q1/d, 0, 0, Q2/d, 0, 0), the disease free equilibrium

(ii)

E * = ( N 1 , I1 , A 1 , N 2 , I 2 , A 2 ) , the endemic equilibrium

*

*

*

*

*

*

*

*

*

where N 1 , I1 , A 1 and N 2 , I 2 , A 2

*

*

*

are positive solutions of the following system of

algebraic equations,

Q1 − dN 1 − α 1 A1 =0

(13)

β 1c1 ( N 1 − I1 − A 1 ) I1 β 2 c 2 ( N 1 − I1 − A 1 ) I 2 + − (δ + d )I1 = 0 N1 N2

(14)

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δI1 − (α1 + d )A 1 = 0

(15)

Q 2 − dN 2 − α 2 A 2 = 0

(16)

β 3 c 3 ( N 2 − I 2 − A 2 ) I1 − (δ + d )I 2 = 0 N1

(17)

δI 2 − (α 2 + d )A 2 = 0

(18)

On solving simultaneously the algebraic equations (13)-(18), we obtain,

(α 1 + d ) A1 δ

(19)

Q1 − dN 1 α1

(20)

(α 2 + d ) A2 δ

(21)

A2 =

Q 2 − dN 2 α2

(22)

N2 =

f 4 − f1 N1 f 2 − f 3 N1

(23)

I1 =

A1 =

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I2 =

N2 =

and

g 1 N 12 − g 2 N 1 g 3 − g 4 N 1 + g 5 N 12

where

f1 =

Q 2 {β 3 c 3 (α 2 + δ + d )d − α1 (δ + d )(α 2 + d )} , α 1α 2 δ f2 =

f3 =

Q1β 3 c 3 (α1 + d )(α 2 + d )(δ + d ) , α 1α 2 δ 2

Q Q β c (α + d )(α 2 + δ + d ) d (δ + d )(α 2 + d )[β 3 c 3 − α 1 ] , f4 = 1 2 3 3 1 , α 1α 2 δ α1α 2 δ 2

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Q 2 β 2 c 2 (α1 + d )(α 2 + d )(δ + d ) Q Q β c (α + δ + d )(α 2 + d ) , g2 = 1 2 2 2 1 , 2 α1α 2 δ α1α 2 δ 2

g1 =

g3 =

Q12 β1c1 (α1 + d )(α 1 + δ + d) , α12 δ 2

⎛ β c Q ⎞ ⎡ (δ + d)(α 1 + d ) 2 ⎛ (α + d ) (α 2 + d ) ⎞ ⎤ ⎟⎟⎥ , + d (α 1 + δ + d)⎜⎜ 1 + g 4 = ⎜⎜ 1 1 2 1 ⎟⎟ ⎢ α α α α δ 1 1 2 ⎝ ⎠⎦ ⎝ 1 ⎠⎣ ⎡⎛ β c (α + d ) β 2 c 2 (α 2 + d) ⎞ d(α1 + d)(δ + d) d(α1 + d)(δ + d) ⎤ ⎟⎟ − + g 5 = ⎢⎜⎜ 1 1 1 ⎥ 2 α α α 1δ α δ 1 2 ⎠ 1 ⎣⎝ ⎦

From equation (23) we note that as N1 → 0 then N2 →

f4 > 0 and f2

dN 2 = dN 1

f 3 f 4 − f 1f 2 > 0 at (0, N2) . f 22

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From equation (24) we note that as N2 → 0 then N1 → 0 or N1 → Also,

g2 > 0. g1

g 14 g 2 dN 2 dN 2 g = − 2 < 0 at (0, 0) and = at (N1, 0), dN 1 dN 1 g3 [g 22 g 5 + g 12 g 3 − g 4 g 2 g 1 ]

we see that

dN 2 > 0 at (N1, 0) if β 2 c 2 (α 2 + d ) > [β 1c1 (α 2 + d ) + α 2 δ] dN 1 *

*

Thus, we get positive values of N1 and N2 (say N 1 and N 2 respectively) if the following condition is satisfied,

β 2 c 2 (α 2 + d ) > [β 1c1 (α 2 + d ) + α 2 δ] *

*

*

*

*

*

Using N 1 and N 2 , the values of I1 , A 1 , I 2 and A 2 can be found from equations (19)(22) respectively (see Fig. 2).

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Figure 2. Intersection of two isoclines (23) and (24) to determine E*.

Figure 3. Numerical existence of non-negative equilibrium point E*.

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We have also shown numerically the existence of a unique positive endemic equilibrium using the following set of parameter values, Q1 = 3000, Q2 =2000, d = 0.02, β1 =0.1, β2 = 0.102, β3 = 0.223, c1 = 5, c2 = 10 = c3, α1 = 1, α2 = 0.8, δ = 0.2 (see Fig. 3).

3.2. Local Stability of the Equilibria *

To determine the local stability of E 0 and E , the variational matrix of the system (7)-(12) is computed corresponding to equilibrium E 0 as follows,

0 − α1 0 0 0 ⎤ ⎡− d ⎥ ⎢ Q 0 0 β 2c2 1 0 ⎥ ⎢ 0 [β1c1 − (δ + d )] Q2 ⎥ ⎢ δ − (α 1 + d ) 0 0 0 ⎥ ⎢ 0 M0 = ⎢ 0 0 0 −d 0 − α2 ⎥ ⎥ ⎢ Q2 ⎥ ⎢ 0 β 3c 3 0 0 − (δ + d ) 0 ⎥ ⎢ Q1 ⎢ 0 δ 0 0 0 − (α 2 + d )⎥⎦ ⎣ From M0 it is clear that all the eigenvalues will be negative if β 1c1 < (δ + d ) . We define the basic reproduction number for male homosexual population as,

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R00 =

β 1 c1 (δ + d )

(25)

Thus, if R00 < 1, the infection does not persist in the male homosexual population. We now define R01 and R02 respectively as the number of secondary infections male (female) 1 in an generated by a typical male (female) infective over his (her) infectious period (δ + d ) otherwise totally susceptible population at equilibrium [9]. They are referred to as the partial reproduction numbers for male population and female population respectively. R01 =

β2c2 and R02 = (δ + d )

β3c 3 (δ + d )

(26)

Thus, the basic reproduction number for the two sex population is the spectral radius of the next generation matrix given as [8, 13] R12 =

R 01 R 02

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Hence, we may define the basic reproduction number for total population as R0 = max{R00, R12}

(28)

Thus, for R0 < 1, the disease free equilibrium E0 is locally asymptotically stable, so that the infection does not persist in the population and under this condition the endemic equilibrium E* does not exist. It is unstable for R0>1 and then E* exists and the infection is maintained in the population. Thus, the epidemic will take off for R0 > 1and will die out for R0 < 1. The variational matrix corresponding to E* is given by

0 − α1 0 0 0 ⎤ ⎡− d * ⎥ ⎢ N s *2 −r −s 0 ⎥ ⎢ p − (q + δ + d ) I2 ⎥ ⎢ ⎥ ⎢ δ ( α d ) 0 − + 0 0 0 1 M* = ⎢ ⎥ 0 0 −d 0 − α2 ⎥ ⎢ 0 N 1* ⎥ ⎢ u u v − ( v + δ + d) − 0 −v ⎥ * ⎢ I1 ⎥ ⎢ δ 0 0 0 0 − (α 2 + d ) ⎦ ⎣

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where,

p=

β 2 c 2 I *2 β1c1 (I1* + A 1* )I1* β1c1 I1* β 2 c 2 I *2 β1c1 ( N 1* − I1* − A 1* ) , q = , + + + N *2 N 1*2 N 1* N *2 N 1*

r=

β 2 c 2 ( N 1* − I1* − A 1* )I *2 β1c1 I1* β 2 c 2 I *2 + , s = , 2 N 1* N *2 N *2

u=

β 3 c 3 ( N *2 − I *2 − A *2 )I1* N 1*

2

,v=

β 3 c 3 I1* N 1*

which are all positive. *

The characteristic equation corresponding to M is given by

f (λ ) = λ6 + a 1 λ5 + a 2 λ4 + a 3 λ3 + a 4 λ2 + a 5 λ + a 6 = 0

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where

a 1 = α 1 + α 2 + 2δ + 6d + v + q a 2 = (α 2 + v + δ + 4d )(α 1 + q + δ + 2d ) + (α1 + d )(q + d + δ) + (α 2 + 2d )( v + δ + 3d )

+ v(δ + d) + δ(r + d ) − su

N 1* N *2 I1* I *2

a 3 = δ(α1 p + α 2 v) + rδ(α 2 + v + δ + 4d) + 3d(α 2 + 2d)(v + δ) + 3vδd + d 2 (6α 2 + 10d)

+ (q + δ + d )[ (α 1 + d )(α 2 + v + δ + 4d ) + (α 2 + 2d )( v + δ + 3d ) + v(δ + d ) + δd ]

+ α 1 v(α 2 δ + 3d) + α1 (α 2 δ + 3α 2 d + 3δd + 6d 2 ] − su

N 1* N *2 [α 1 + α 2 + 4d ] I1* I *2

a 4 = α 1 pδ( v + δ + α 2 + 3d ) + ( v + δ + 2d )(α 2 rδ + 3rδd ) + α 2 rδd + 3d 2 (α 2 v + vδ + α 2 δ)

+ d 3 [4( v + δ + α 2 ) + 5d ] + α 1d[2(α 2 v + vδ + α 2 δ) + 3d( v + δ + α 2 ) + 4d 2 ] + α 2 vδ(q + δ + α 1 + 3d ) + (α 1 + d )(α 2 + 2d )(δ + 3d )(q + δ + d ) + dδ(α 1 + d )(q + δ + d )

+ v(α 1 + d )(α 2 + δ + 3d )(q + δ + d ) + d[2( vα 2 + δv + α 2 δ) + d (3v + 3δ + 3α 2 + 4d )]

⎛ N* N* ⎞ N* N* (q + δ + d ) − suδ⎜⎜ α 2 *1 + α1 *2 ⎟⎟ − su *1 *2 [α1 (α 2 + 3d ) + 3d (α 2 + 2d )] I1 I2 ⎠ I1 I 2 ⎝

a 5 = α 1 pδ(2vd + 2δd + 2α 2 d + 3d 2 + α 2 v + α 2 δ) + α 2 rδd(2v + 2δ + 3d) Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

+ α 2 vδ(α 1q + 2qd + 2δd + 2α 1d + α 1δ + 3d 2 ) + d 2 (α 1 + d )(α 2 v + vd + vδ + α 2 δ + δd )

+ d 3 (α1 + d)(α 2 + d ) + (q + δ + d)[ vd 2 (α 2 + δ + d) + d 2 (α 2 δ + δd + α 2 d + d 2 )] + (q + δ + d)(α1 + d)[ vd(2α 2 + 2δ + 2d ) + δd(2α 2 + 3d ) + d 2 (3α 2 + 4d)] ⎡ N 1* N *2 ⎤ + rδd (3v + 3δ + 4d ) − suδ ⎢α 2 (α 1 + 2d ) * + α 1 (α 2 + 2d ) * ⎥ I1 I2 ⎦ ⎣ N* N* − α1d(2α 2 + 3d ) + d 2 (3α 2 + 4d) su *1 *2 I1 I 2 2

[

]

a 6 = (α 1 + d )(q + δ + d)[d 2 ( vα 2 + vδ + vd + α 2 δ + α 2 d + δd + d 2 )] + (α1 pδd + α1α 2 pδ)(δv + vd + δd + d 2 ) + rδd 2 (α 2 v + α 2 δ + α 2 d + vd + δd + d 2 )

+ α 2 vδd (α1q + qd + α1δ + δd + α1d + d 2 ) − su

N 1* N *2 .(α1α 2 + α1d + α 2 d + d 2 )d 2 * * I1 I 2

⎡ N* N* ⎤ − suδd ⎢α 2 (α1 + d ) *1 + α1 (α 2 + d ) *2 ⎥ I1 I2 ⎦ ⎣

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It has been checked that all the ai (i = 1-6) are positive, thus by Routh-Hurwitz criteria, *

E is locally asymptotically stable if,

a1 a3

a1 1 > 0, a 3 a2 0

a1

1

0

0

a3

a2

a1

1

a4

a3

a2

0

0

a4

1

0

a2 0

a 1 > 0, 0 a3 0

a1

1

0

0

0

a3

a2

a1

1

0

> 0 and a 5

0

a4 a6

a3 a5

a2 a4

a 1 > 0. a3

0

0

0

a6

a5

3.3. Global Stability of the Endemic Equilibrium Now to show the globally stability behavior of E*, we need the bounds of the dependent variables involved. For this we find the region of attraction stated without proof in the form of the following lemma.

Lemma 1. The region Ω = {(N1, I1, A1, N2, I2, A2); 0 < N1 ≤ N 1 ;0