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FLORIDA 2007 S2007 Fernanda Oliveira Melo To my husband, who through his support and encouragement, made this all possible ACKENOWLED GMENTS I thank my advisor Dr. Maia Martcheva for her valuable inputs, her constructive feedback, and especially for ahrl . believing I could do this. I thank my supervisory committee, Dr. David Wilson and Dr. Robert Holt, for the continuous support and constructive criticism and discussions on the topic. Especial thanks go to Dr. Holt, for allowing me the opportunity to work in his laboratory. I thank my husband, who endured my absence during long nights of research, who assumed his responsibilities and mine in the household, who listened to my long It lI;nkl ining~! ahrlns being supportive and caring. I thank my son, Lucas, my strength and joy at the end of the long, and many times, frustrating d 7i< of research. I extend my gratitude to my parents, without whom I would never have made it this far. They ah .1< supported my efforts and praised my successes. They introduced me to the passion for science, research, curiosity, knowledge. I thank all who have given input and feedback into this thesis. Especial thanks to Dr. Michael Barfield, Gustavo Oliveira and Manojit Roy for valuable review of this thesis, and Dr. Rosangfela Xavier for her passionate discussions on the topic that came to inspire me to pursue this into a thesis. TABLE OF CONTENTS page ACK(NOWLEDGMENTS .......... . .. .. 4 LIST OF TABLES ......... ..... .. 6 LIST OF FIGURES ......... .... .. 7 LIST OF TERMS ............ ........... 8 CHAPTER ABSTRACT ......... ...... 10 1 INTRODUCTION ......... ... .. 11 1.1 Motivation ......... .... .. 11 1.2 Goals of Research ......... . .. 12 1.3 The Disease .... .. ...... .. 13 1.4 The Epidemic: Origin, History and Treatment ... . .. 14 1.5 Highly Active Antiretroviral Therapy .... .. 15 1.6 Transmission and Prevention . .... .. 16 1.7 Mathematical Models of STDs and the Basic Reproduction Number .. 18 2 FORMULATION OF THREE MODELS ...... .. 21 2.1 The Model Without Treatment .. .. .. ... .. 24 2.2 The Model With Treatment but Without C'!s lIl, in Behavior .. .. .. 25 2.3 The Model With Treatment and ChI .Is,,. in Behavior .. .. 26 3 ANALYSIS OF THE MODEL WITHOUT TREATMENT .. .. .. 28 3.1 Existence and Local Stability of the DFE .... ... .. 28 3.2 The Basic Reproduction Number and Its Interpretation .. .. .. .. 31 3.3 Existence of an Endemic Equilibrium ...... .... 34 4 SIMULATIONS ON THE THREE MODELS ..... ... .. 39 4.1 The Model Without Treatment .. .. .. .... .. 39 4.2 The Model With Treatment but Without C'!s lIly, in Behavior .. .. .. 45 4.3 The Model With Treatment and ChI .Is,,. in Behavior .. .. .. 47 5 CONCLUSION ......... . ... .. 50 REFERENCES ............. ............. 53 BIOGRAPHICAL SK(ETCH ......... . 56 LIST OF TABLES Table page 21 List of variables and parameters and their meaning ... .. .. 23 41 Simulation parameters and their value ranges .... ... .. 40 42 Specific parameter values used in simulations .... .. . 42 43 Reduced infectivity in individuals receiving HAART treatment .. .. .. .. 47 44 Prevalence of HIV infection with increasing prevention ... .. .. 48 45 Incidence of HIV infections with increasing prevention ... .. .. 48 LIST OF FIGURES Figure page 21 Diagram of the model with treatment . ..... .. 25 31 The model without treatment: diagram . ... 28 41 Prevention focused on decreasing number of partners only. .. .. .. .. 43 42 Prevention focus on increasing correct and consistent condom use only. .. .. 44 43 Prevention focused on changing recruitment only. .. .. .. .. 45 LIST OF TERMS N' *1.1 n An abnormal growth of tissue, a tumor. Immune deficiency Immunosuppression Seropositive Retrovirus Lentivirus Reverse Transcriptase Protease Integfrase CD4+ helper T cells Macrophages Dendritic cells Receptor Fusin Opportunistic infection Inability of the immune system to function properly. Results in greater susceptibility to disease. Reduced immune system response to pathogens, such as virus, bacteria or fungi. Individual infected with HIV. RNA virus that synthesizes DNA through reverse transcriptase. Type of retrovirus characterized by presenting a long interval between infection and the onset of symptoms in hosts. Enzyme responsible for transcription of singlestranded RNA into doublestranded DNA. HIV enzyme needed for assembly of an infectious virus particle. HIV enzyme used by the virus to incorporate its genetic material into that of the host cell. White blood cells that coordinate immune response. Large immune system cells responsible for removing invading pathogens by enveloping them. These cells move freely throughout the body. Immune system cells with long, tentacle branches. Function as specialized cells at the mucosa. A molecule on the surface of a cell that serves as a recognition or finding site for antigfens, antibodies, or other cellular or immunologfical components. A receptor present in all cells types that can he infected hv HIV. Fusin is necessary for HIV invasion Infection by an organism that does not ordinarily cause disease but which, under certain circumstances (impaired immune responses), becomes pathogenic. STD HAART Prevalence Sexually transmitted disease. Highly active antiretroviral treatment: a combination of drugs aimed at fighting HIV in the body. Proportion of individuals in a population who are infected. Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics THE NEW FACE OF AIDS: A MATHEMATICAL MODELING APPROACH TO NEW TRENDS IN HIV TREATMENT AND PREVENTION By Fernanda Oliveira Melo August 2007 C'I I!r: Maia Martcheva Major: Mathematics My study develops niathentatical models to investigate prevention strategies aimed at a reduction of HIV prevalence in a population. It focuses primarily on the impact of coupling prevention effort with treatment delivery. It gives an overview of possible prevention strategies for the entire population as well as for the subgroup of individuals who are undergoing treatment and/or receiving HIVrelated medical attention. My study focuses on exploring the effects of the most recent trend having a 1!! I r~~ impact in the HIV/AIDS epidemic: HAART treatment. Specifically, through simulation, it demonstrates the impact of integrating prevention efforts with the distribution of treatment and HIVrelated medical support. It shows that educating HIV patients about risks of HIV transmission, reliability of condoms, and safer sexual behavior may effectively decrease the prevalence of HIV in the population as effectively as aiming these prevention progranines at the entire population, provided that availability of testing and HIVrelated medical attention is adequate. CHAPTER 1 INTRODUCTION 1.1 Motivation Acquired immunodeficiency syndrome (AIDS) was first recognized in 1982 following a report published June 5, 1981 [1] of five cases of Pi,.;, i...u;,;is carinii pneumonia (PCP) among previously healthy young men in Los Angeles, California. These incidents were latter attributed to a retrovirus that came to be known as human immunodeficiency virus (HIV). By August 1981, 70 new cases were recorded of PCP and K~aposi's sarcoma (K(S), a rare, malignant neoplasm (Table 1) previously seen before in elderly man [2]. Within a few months it was clear that the world was facing a dangerous new epidemic, but, no one could have imagined the extent of what became the AIDS pandemic. Today, the Joint United N li;. ens Programme on HIV/AIDS (UNAIDS) and the World Health Organization (WHO) estimate that AIDS has killed more than :35 million people [:3]. In 2006, there were :39.5 million people worldwide living with HIV, including 2.3 million children. An estimated 4.3 million people were newly infected that same year [4]. In these 25 years the disease has spread to pandemic proportions. HIV infection and AIDS remain leading causes of illness and death in many countries in the world, including the United States [5]. The disease that once was viewed as restricted to men who have sex with men has now become prevalent among men who are heterosexual, women and children [6]. With the advance of this dangerous disease, the world has become involved in a frenetic search for vwsi~ to treat or cure HIV infection. Over the last 25 years there has been enormous progress in the fight against AIDS that might II r that there is hope for the future [:3]. A cure or vaccine is not yet available, but there has been a significant advance in treatment, resulting in an increase in the life expectancy and quality of life of those diagnosed with HIV infection [7]. Tod w highly active antiretroviral therapy (HAART) is used to slow the progression of HIV infection and thus postpone death from AIDS. HAART is credited as all in Hr factor in significantly reducing annual deaths by AIDS in the U.S. [7]. However, even though treatment can increase life expectancy and health prospects in HIV infected individuals, treatment does not provide a cure for the infection. This and other factors influence the need for continued efforts to reduce further spread of the disease. With globalization and increased nlovenient of people, the AIDS pandemic needs to be addressed as a global issue, especially given high cost of treatment both for individuals and for governments. Further, HIV is a highly mutable virus, so the risk of new niultiresistant strains emerging is ahrl . present [3]. Therefore, efforts to prevent the spread of HIV are still very important. There are, however, many issues that make it especially difficult to intplenient effective prevention progranines that reduce the prevalence of HIV infection in a population. Among these issues are difficulties in reaching a significant portion of the population with progranines for education about the disease and its transmission, the cost of prevention progranines, compliance of the population, and the fact that HIV infection is primarily a sexually transmitted disease and thus prevention strategies have to address many delicate issues of sexuality, intimacy and privacy [3, 6]. 1.2 Goals of Research I investigated the dynamics of HIV/AIDS in this new era of the epidemic marked by the introduction of HAART treatment. The clear need for improved, cost effective prevention and the difficulties in delivering prevention education to the entire population, or even to the group at higher risk of acquiringf the disease, motivates me to seek a different focus for prevention. I elo 1 that, given sufficient availability of testing and medical support, aiming prevention progranines (education about condom use and reduction in number of sexual partners) towards persons receiving HIV treatment and medical attention can effectively reduce the prevalence of the disease in the overall population. It is important to study the trends in the dynamics of the HIV epidemic. To effectively create and implement strategies of prevention it is necessary to focus on the driving forces of the epidemic and the impact of various epidemic parameters on the distribution of the pathogen in the population. Here I aim to explore the effects of a new and important dimension of HIV infection dynamics: treatment. Through analysis and simulation I illustrate and discuss some of the most relevant issues in tod~i's fight against HIV. I investigate the effects of different prevention strategies for HIV transmission in a population and possible countereffects that these strategies may cause. I demonstrate the impact of effectively integrating prevention efforts with delivery of treatment to HIV patients, and I conclude that, in countries such as the United States, where testing and treatment are widely available, it is sufficient to focus prevention programmes on HIV patients rather than on the entire population. I also demonstrate the risk of unmodified sexual behavior in HIV patients and II r prevention strategies that do not rely on reduced infectivity due to treatment. 1.3 The Disease Human immunodeficiency virus (HIV) was first isolated in 1983 by Luc Montagnier at the the Pasteur Institute in France. It has since been recognized as the pathogen responsible for the acquired immunodeficiency syndrome (AIDS) [6]. HIV is a lentivirus of the family Retroviridae (Table 1). It contains two copies of positive ribonucleic acid (RNA) that code for the virus' nine genes. The RNA is bound to proteins and enzymes necessary for viral development, including reverse transcriptase, protease, and integrase, and is enveloped in two l~i;rs of phospholipids [8] (Table 1). HIV primarily affects the human immune system by attacking helper T cells (more specifically, CD4+ cells), macrophages, and dendritic cells (Table 1) [9]. HIV attacks cells using protein receptors that are part of the normal immune response. The main receptor is the CD4 molecule on helper T cells. A secondary coreceptor is also necessary for the virus to invade a cell. The main coreceptor, present in all cell types that can be infected by HIV, is called fusin [8]. As the infection develops, the virus causes depletion and increasing disruption of the immune system opening many doors for the onset of opportunistic (Table 1), lifethreatening infections [810] There is usually an .imptomatic period, during which no visible symptoms of the infection occur. Even during this time, however, the virus is actively multiplying, infecting and killing cells [7]. Viral particles are also, during this time, moving into secretions and body fluids, including blood, semen and vaginal secretions. Treatment regiments todor focused on a combination of drugs that repress HIV reproduction and cell invasion, including: reverse transcriptase inhibitors and protease inhibitors (which impede viral reproduction), and fusion inhibitors (which block virus from entering cells) [11]. Other drugs aim to help prevent a number of opportunistic infections including PCP, toxoplasmosis, cryptococcus and cytomegalovirus infection [11]. 1.4 The Epidemic: Origin, History and Treatment It is believed that HIV originated in human populations in Africa, possibly from a spillover from primate populations, which could have occurred as a result of human exposure to primate blood during hunting expeditions [12]. HIV was isolated and recognized as the pathogen responsible for AIDS in 1983. No cure, vaccine or treatment existed, and treatments to combat the opportunistic infections caused by the immune deficiency were very few [7]. In 1987 the first treatment emerged. The drug was called AZT, a reverse transcriptase inhibitor. By 1992 combinations of drugs were introduced to improve treatment. In 1996 evidence of the efficacy of a new treatment, called highly active antiretroviral therapy (HAART), was presented for the first time at the 11th International AIDS Conference in Vancouver [6]. Since 2000 UNAIDS and WHO have been trying to increase access to HIV treatment in developing countries. By the end of 2005 1.3 million people in low and middleincome countries have received access to antiretroviral therapy [6]. Todor, successful treatment has changed the face of AIDS from a disease that killed in just a few years into a sustainable, chronic condition. By 2001, however, there was growing concern with both medication toxicity and effectiveness [11]. Although HAART has clear benefits, it also has its shortcomings and risks. HAART is not a cure: the drugs only control HIV, they cannot eliminate the virus from the body [7, 13]. This means that HIV can still be transmitted. The drugs have a strict schedule and adherence is difficult [6]. Understanding the ultimate impact of HAART upon HIV prevalence requires a quantitative approach. 1.5 Highly Active Antiretroviral Therapy A complicated dynamic exists between HIV prevention and treatment. As treatment access expands in resourcelimited countries, the health, longevity and quality of life for people with HIV will improve [6, 14], potentially increasing opportunities for sexual transmission. At the same time, optimism about the treatment or misperceptions about the effects of antiretroviral drugs may also cause some people to increase their risk behavior [1517]. On the other hand, and a goal of this thesis, the introduction of treatment in the population provides a clear target group for prevention efforts and a clear path through which to deliver such prevention programmes by integrating prevention efforts with delivery of treatment to HIV positive individuals. Since the introduction of HAART, increases in the quality of life and life expectancy for HIVpositive individuals under treatment has reshaped the face of the epidemic. HAART has been able to increase life expectancy and significantly reduce virus load in HIV positive patients, sometimes to levels below those that can he detected hv HIV tests [7]. In women, low blood plasma virus load has been correlated with lower vaginal virus load, indicating the possibility of a lower risk of perinatal and femaletomale heterosexual transmission from women under treatment [18]. Perinatal transmission rates have been reduced with screening of pregnant women, combined with prophylactic administration of HAART drugs [5]. However, since 2005 long term side effects of HIV medication use have become evident. Drugs for HAART require a strict schedule and requirements. Adherence to the treatment is difficult, and tod .r many patients fail to follow through with treatment because of difficulties with drug schedules, cost of drugs, and side effects [3, 9]. Another ill linr~ issue with HAART is the possibility of new resistant strains of HIV arising and increasing in frequency. For these reasons, some health experts recommend delaying drug treatment, while maintaining regular medical checkups [19]. 1.6 Transmission and Prevention HIV can he transmitted either through sexual contact where one partner is infected or through needles or syringes containing infected blood. Before blood was screened for HIV, the virus was also transmitted through contaminated blood transfusions [7]; however, after the introduction of screening of plasma for HIV in 1985 [6], the risk of transmission of HIV through such a path has become extremely small. HIV can also be transmitted from mother to child during pregnancy, birth or through nursing [7]. Prenatal HIV testing and HAART have reduced this transmission significantly [5]. There is no evidence that HIV can he transmitted by casual household or social contact or by insects [7, 20]. HIV is not transmitted through sweat, tears, kissing, or sharing common household items, such as eating utensils [20]. Today, the ul .In il ~ry of HIV infections are acquired through sexual relations between partners, one of whom has HIV [4, 7]. This makes understanding the dynamics regulating sexual transmission of HIV very important. Parameters that affect the sexual transmission of HIV include the correct use of harrier prevention methods such as male or female condoms, the number of different sexual partners that an individual acquires, and the presence of other sexually transmitted diseases (STDs) that can enhance transmission [20]. Goals of behavior change for HIV prevention include abstinence and d. 1 li. I1 sexual debuts for young people, monogamy within relationships, reduction in the number of partners, and correct and consistent use of condoms [3, 4]. Various studies II__ r that these strategies can reduce the prevalence of the disease and the risk of transmission [7, 20, 21]. It has been shown that even though HAART may reduce viral load in infected individuals significantly, eradication of the virus within a patient cannot yet be achieved. Levels of plasma virus load below detection do not necessarily reflect low levels of virus in other secretions such as semen [13]. Still, some studies have II__ r. II that there might be a correlation between low viral load in the blood and reduced infectivity [5, 22]. It so, treatment can be a facet of a prevention strategy. However, ]rn Ilny issues arise when treatment is used as a prevention strategy. The first is the high cost of treatment, both for individuals and for governments. Another issue is that, even if there is a correlation between low viral load and reduced infectivity, lower viral loads can only be achieved with adherence to the treatment, which is difficult to maintain [3] and may not even be the best immediate course of action for the patient early in infection [19]. But, de l .v 4 start of drug treatment means del lie II reduction in infectivity. Finally, widespread use of treatment increases the risk of producing multiresistant strains of HIV. However, d. 1.v liA start of drug treatment means de1 li, I reduction in infectivity. It has also been confirmed that there have been increases in bacterial STDs and risk behaviors correlated with the introduction of HAART [15]. The population as a whole perceives a reduced danger in HIV/AIDS, but the most significant issue might be the false overconfidence that treatment prevents transmission. Individuals who believe that treatment effectively reduces infectivity might increase risky behaviors, for instance, reducing compliance with the use of condoms and increasing the number of new sexual partners . Based on a study using selfreported sure vi the Centers for Disease Control and Prevention (CDC) found that infected individuals who know of their HIVpositive status have a significant reduction in highrisk behavior from before they learn about their status [5]. However, other studies [16, 17] have shown that HIVpositive patients are getting infected with hacterial STDs at rates comparable to or higher than those of HIVnegative patients. In a study in Brazil, the incidence of new gonorrhea infections (an acute hacterial STD) in women with known HIV infection was 12.9' .~ while the incidence among women who are HIVnegfative was 8.t:: in promiscuous women and "' in nonpromiscuous women for the same cohort [16]. A study in Nigeria found that 1 1' . of HIVpositive individuals, but only 2.(1' of HIVnegative individuals, tested positive for syphilis [17]. The Brazilian and Nigerian results raise the question of the effectiveness of changes in behavior due to the knowledge that one is seropositive (see Table 1). Even if persons that become aware of their HIV infection believe that they are substantially increasing precautions to prevent the spread of HIV [5], the fact that they are acquiring hacterial STDs II r that the prevention measures they are taking have not been sufficient. It is clear then that assuming that treatment will prevent the spread of HIV through reduced infectivity or through change in behavior without counseling is not ideal. Here I will argue that, if enough testing and medical counseling are available, it is possible to achieve comparable prevention results by focusing traditional, costeffective prevention programmes of education and support directly on the subpopulation receiving HIVrelated medical care. These programmes prevent new infections due to increased compliance with correct and consistent condom use and reduced number of sexual partners. I also illustrate that incorporating education programmes into HIVrelated medical attention can he substantially more effective than relying in reduced infectivity alone. Thus it is desirable to incorporate preventive changes in behavior for individuals who become aware of their seropositive status and receive treatment. 1.7 Mathematical Models of STDs and the Basic Reproduction Number In the early 1900's, Sir Ronald Ross, who received a Nobel Prize in 1902 for his work on malaria, laid the foundation of the field of mathematical epidemiology. In his efforts to establish that it is not necessary to eradicate the vector (mosquito) population in order to eliminate the disease in humans, he introduced the first concepts and models of mathematical epidemiology. In a mathematical model of malaria from 1911, he showed that bringing the mosquito population below a certain threshold was sufficient to eliminate malaria [23]. Such threshold phenomena have been central in mathematical epidemiology ever since [23, 25]. Traditionally, the threshold that determines the ability of an infectious disease to invade a stable susceptible population has been called the basic reproductive number, and denoted 7to. The basic reproductive number should reflect the reproductive success of a pathogen in a host population [25]. In classical mathematical epidemiology the reproductive number of the disease has been defined as the number of secondary cases of the disease that one ';id.:.rl infective individual will produce over his or her infective lifetime in an entirely susceptible stable population [23, 25]. A value greater than one is needed for disease persistence. In models that are homogeneous with respect to infectivity of individuals within a population, it is possible to construct the basic reproductive number in a straightforward and intuitive manner from this definition. With the appearance and spread of AIDS worldwide, more attention has been drawn to the dynamics of sexually transmitted diseases (STDs) [23, 24]. One of the most important and distinguishing aspects of modeling STD transmission versus classical modeling of infectious disease dynamics lies in the substantial heterogeneity of transmission within the population. Number of contacts ranks high in defining this heterogeneity, together with issues of social and interpersonal relationships [23]. It is known that sexual partner acquisition rates vary enormously among communities and among individuals, possibly ranging from less than 1 to 100 partners per year [24]. A consideration of contact processes is central to the understanding of threshold phenomena such as the basic reproduction number [23]. Mathematical models of STDs, and specifically of HIV/AIDS, that aim to incorporate heterogeneity of susceptibility and infectivity (see [24], [25], [26], [27]), or nonhomogenous mixing of individuals (see [28]) face am 1! r ~ problem in the computation of Ro because the mathematical description of what is a ';id.:.rl infectious individual is difficult to achieve in populations with high degrees of heterogeneity [23]. M1 I.ny researches have sought to resolve this issue either by assuming that the population in question is homogeneous in its infectivity and susceptibility using an average value for each individual or, probably more appropriately, by incorporating the heterogeneity of the population in the model and then letting Ro be the appropriate weighted average of the heterogeneous trait (such as number of sexual partners) in the population (see [23], [24], [29]). The number of new sexual partners constitutes an important aspect of heterogeneity within a population when we consider epidemics of STDs. The distribution of this number has been used for heterogeneous model formulations of HIV/AIDS [23, 29]. Here I plan to incorporate a similar approach to the issues of heterogeneity in number of sexual partners in a population with a more explicit rendering of Ro where the effects of heterogeneity in acquisition of new sexual partners can be explicitly treated, but still maintaining a general fidelity to the definition of Ro. A similar approach was used in [24], but there the heterogeneity in question was in the intrinsic viral infectivity variability between individuals, based on the assumption that there was a correlation between plasma viral load and infectivity. CHAPTER 2 FORMULATION OF THREE MODELS I consider a compartmental model of a population that consists of susceptible individuals (S), HIVinfected individuals who have not developed AIDS and are not receiving treatment or HIVrelated medical support (U), HIVinfected individuals under treatment and/or other HIVrelated medical support who have not developed AIDS (T) and infected individuals who have developed AIDS (A). The term AIDS will apply here only to the more advanced stage of HIV infection categorized by a CD4+ T cell count below 200 cells per cubic millimeter of blood, coupled with severe clinical conditions, most of which are opportunistic infections [7]. For this reason, I assume here that individuals in class A are removed from the sexually active population, given that they have developed severe opportunistic infections and/or cancers and are assumed to be severely debilitated or hospitalized [7]. I only consider the transmission of HIV via sexual intercourse and therefore the population under consideration is sexually active. I take into account that susceptible individuals and infected individuals (both with and without medical support) may differ in the number of sexual partners per unit of time. Thus, I subdivide the susceptible and infected classes into several subclasses (Si, Ui, Ti, i = 1, ..., m) with i corresponding to the number of sexual partners that individuals in each subclass will have per unit of time. It is assumed that formation of sexual pairs is random (does not depend on subclass). Let As, i = 1, ..., m, be the recruitment rates of individuals to the sexually active susceptible classes (Si), p be the natural death rate and removal rate and d be the disease induced death rate. Upon a sexual encounter, without protection, with an infected individual from subclasses Ui or Ti, a susceptible individual will become infected with probability rli and r15, respectively. To account for the effects of condom use, we assume condoms have an efficacy E and compliance ui. So the product pi = Eui represents the condom protection. To account for the fact that often individuals who are infected and not under treatment are not aware of their status and thus cannot seek treatment, I introduce the parameter vi, i = 1, ..., m, where vi is the proportion of infected individuals in class Ui who undergo testing and subsequently know their status. I assume tested individuals begin receiving medical support at a rate 6 and thereafter develop AIDS at a rate yi. I assume untreated individuals develop AIDS at a rate asi. Variable and parameter descriptions are summarized in Table 21. The total population size is given by NV = E Si + C Ui + Eg, Ti + A, and it satisfies the equation dN m di = As pN dA.(21) i=1 We also define the total population within each class as Nsi where Nsi = Si + Ui + Ti. Standard incidence is appropriate for large populations [? ]and is used here with the result that interactions are made independent of total population size. The idea behind this being that an increased population size does not necessarily make individuals more promiscuous. Thus we get the following system of nonlinear differential equations for i = 1, 2, ..., m, = sp, ( p~yp+ ( y:jy N (22) d~dT = (b ( + p)T+c~)U (24) dTi di = v6U, + pT, ( + d)Ai (25) j= 1 j= 1 Table 21: List of variables and parameters and their meaning Variable Description Si Susceptible individuals with i average number of new sexual partners per unit of time Ui HIVinfected individuals, with i average number of new sexual partners per unit of time, who have not developed AIDS and are not under treatment or HIVrelated medical support Ti HIVinfected individuals, with i average number of new sexual partners per unit of time, who have not developed AIDS but are receiving treatment and/or HIVrelated medical support A HIVinfected individuals who have developed AIDS NV Total population size 1V 1V = si + Ui + Ti Parameter Description n, a Average number of new sexual partners that an individual acquires per unit of time As Recruitment rate for individuals entering the sexually active population in the subclass Si p Natural death rate d AIDS induced death rate rli Infectivity from individuals in the subclass Ui of Infectivity from individuals in the subclass Ti E Condom efficacy (intrinsic) ui Compliance with the use of condom for individuals with i average number of new sexual partners per unit of time pi Condom induced protection (pi = Eui) vi proportion of HIVinfected individuals with i average number of new sexual partners per unit of time who have undergone testing and know their status 6 rate at which HIVinfected individuals who know their status receive treatment and/or HIVrelated medical support asi rate at which HIVinfected individuals with i average number of new sexual partners per unit of time develop AIDS without treatment Yi rate at which HIVinfected individuals with i average number of new sexual partners per unit of time develop AIDS with the treatment 2.1 The Model Without Treatment I will consider the above model ((22)(25)) for two discrete subclasses of susceptible individuals, infected individuals not on treatment or HIVrelated medical support, and infected individuals undergoing treatment and/or receiving HIVrelated medical support. I let S,, U, and T, be the subclasses of individuals in the susceptible, infected but not under treatment or medical support, and infected under treatment and/or medical support classes, respectively, who have relatively small number of new sexual partners per unit of time. These will henceforth be collectively referred to as the moderately sexually active class. Similarly I let (S,), (U,) and (T,) be the subclasses of individuals in the susceptible, infected not under treatment or medical support, and infected under treatment and/or medical support classes, respectively, who have relatively large number of new sexual partners per unit of time. These will be henceforth collectively referred to as the highly sexually active class. I let a be the average number of new sexual partners that individuals in S,, U, and T, acquire per unit of time and a be the average number of sexual partners that individuals in S,, U, and T, acquire per unit of time. I then consider the general model above for i = n, s. I will focus first on a special case of this model assuming no individuals get treatment (vi = 0). I will, in this case, let Ii be the total number of infective individuals with i number of new sexual partners per unit of time. Then the model ((22)(25)) simplifies to the following: dS, nS, = s pS, (n1 ps~qI, s( p~al) "(26) d t n N, +s N, dS, sS, = As pS n(1 psal, s(1 p~qI) *(27) d t n N, +s N, dl, nS, =(n( ps~,I + (1 p~as) (p + a~n)I, (28) d t n N, + s N, dl, sS, = (n1 ps~qI, s( p~al) (p+ a)Is(29) dt nNV, +sNV, dA = 0,l, + asl, (p + d)A (210) A, + As pNV dA (211) 2.2 The Model With Treatment but Without Change in Behavior I will also consider the above model ((22)(25)) including medical intervention and treatment. The model is described by the flow chart diagram in Figure 21. Figure 21. Diagfram of the model with treatment For A(t) = (n1 ps U +M s(1 ps/lU + mT / n( psmqT, + s(1 p s)(T,), the resulting model is given by the following system of nonlinear differential equations: d S, a S, = As pS, A~t)(212) dt ` nNV, + slV, dS, sS, = A, s , A(t) dt nNV, + slV, (213) d t dt dA dt a S, A(t) (p + vab n N, + s N, s S, x(t) (p + usb nNV, + slV, dT, "= vnGUn (p l+ 7,)T, + a~n)U. + cas)Us (214) (215) (216) (217) (218) (219) td dT" dt usbU, (p + y,)Ts asU, + casU, + y,T, + yT,~ (p + d)A dNV = + As pNV dA dt 2.3 The Model With Treatment and Change in Behavior Now I introduce behavior change in the population under treatment. We will allow for individuals who are under treatment, and thus assumed to know their serologfical status, to undergo a behavioral change. I introduce the parameters no and so defined as the average number of new partners that individuals in T, and T, acquire per unit of time, respectively. Similarly, I also introduce the parameter po and ps, for the condom related protection of individuals in subclass T, and T,, respectively. The generalized model becomes the following. For A(t) = n(1 p,)rkL,U+ s(1 p,)rl,U + no(1 po)rnT, + so(1 Pso)rsls,T, dS " dt dS " dt nS, As pS, A(t) nNV, + (no n)T, + slV, + (so s)T, sS, As pS, A(t) * nNV,+ (n n)T, + sl,+ (s s)T, nS, (p + v,6 + an,)U, i,+ (no n)T, + slV, + (so s)T, sS, (p + usb + as,)Us i, + (no n)T, + slV, + (so s)T, dT, =vnGUn (p l+ q,)T, dT, = usU, (p + y,)Ts (220) (221) (222) (223) (224) (225) dU " dt dU " dt A(t) n X)nN , dA = anU, + CsUs + YnTn + YsTs (p + d)A (226) dtV = An + As pNV dA (227) CHAPTER 3 ANALYSIS OF THE MODEL WITHOUT TREATMENT 3.1 Existence and Local Stability of the DFE The following result concerns the existence of a unique diseasefree equilibrium (DFE). Theorem 1. The system (26) (211) has a unique diseasefree equilibrium which is given by Eo ,0 ) Proof Consider the model in the absence of infection. That is, let 1, = I, A = 0. At equilibrium, setting the right hand side of (26) (211) equal to zero, we get S, =" and S, = So there exists a unique diseasefree equilibrium given by Eo = ( 0, 0, 0). The following results concern with the local stability of the diseasefree equilibrium (DFE). I now linearize the system (26) (211) around the DFE. The Jacobian matrix for this system, which I will call J from here on, at the DFE is given as follows: ^n S I ac nl n at Figure 31. The model without treatment: diagram T ii r ~y c 5 r c c c F+ a a a v, a r c a a 5 i y % % fi fi r + fi r a a a c r ici a c F1 Let J(DFE) be the Jacobian matrix for this system at the disease free equilibrium given above. It is clear that three of the eigenvalues of J(DFE) are given by A1,2  and A3 = (p + d). The remaining eigenvalues are eigenvalues of the 2x2 matrix I now compute the determinant of Jo nA, sn,  Jo = n(1 s% p s psl, (p + a~n) + (p 1+ a~n)(p + Cas) nA, + sn, nA, + sn, It is clear that when nA, sn, n(1 pn)rlz (p+ s) +s(1 ps)rls (p + a~n) < (p 1+ a~n)(p + Cas) (31) nnA sn, nA,+sn, then the determinant of Jo is positive. nA, sn, n( s)% (p+ a) s(1 ps)rls (p + a~n) + (p 1+ a~n)(p + Cas) > 0 nnA sn, nA,+sn, then we have nA, sn, (p + a~s)[n(1 pn)rl (p + a~n)] s(1 ps)rls (p + a~n) > 0. nA, + sn nA, + sn But since sn, s(1 p~q,(p + a~n) nA, sn, is clearly negative, we must have that nA, (p + a~s)[n(1 pn)rl (p + a~n)] nA, sn, is positive. Thus, an, n(1 pn)q, (p + a~n) n~n + sn, is negative. Analogfous considerations lead to the fact that sn, nA, + sn, is also negative. Therefore all diagonal entries of Jo are negative. That is, assuming that the determinant of Jo is positive, the trace of Jo is negative. Therefore I reach the condition that determines the local stability of our system at the DFE. If det(Jo) > 0, then I showed that the trace of Jo < 0 and Theorem 5.4 [30] implies that the eigenvalues of Jo have negative real part. Then all eigenvalues of J(DFE) have negative real part. Therefore the DFE is locally .Iiangind' ;cally stable. If condition 31 is not satisfied, det(Jo) < 0, then Jo has an eigenvalue with a positive real part. Thus the DFE is unstable. I can now write the above threshold condition as Re > 1 for Rc "(1Pn)lln nh, (pL+a,) nh,+sh, s(1ps)s sA" This leads to the following result: Theorem 2. The DFE for the system (26) (211) is '.. a~~ll;i icr;l,''l.:H..;ll;i stable if Rc < 1 and unstable if R, > 1, for n( p,nA, s1 sq sn, Re=+ c (p + n)nA+, + (p + as) nA, + sn In section 3.2 I show that the value R, can actually be understood in terms of basic reproduction number, Ro, which epidemiologically describes the reproductive success of the pathogen in a susceptible host population. Biologically, stability of the DFE means that the infection, if initially rare, will fade away. 3.2 The Basic Reproduction Number and Its Interpretation The quantity Ro reflects the reproductive success of a pathogen in a host population. In classical epidemiology the reproductive number of the disease can be constructed from the number of secondary cases of the disease that one infective individual will produce over its infective lifetime in an entirely susceptible stable population. This number then determines if an initial surge of an epidemic will be possible [23, 31]. However, because the models studied here deal with a heterogeneous host population where individuals in the host population differ in the number of contacts they make, this simple definition poses a particular problem for these models. Namely, the number of secondary cases that one infective individual in I,z will produce will be different than the number of secondary cases that one infective individual in I., will produce, and, more importantly, the longternt behavior of the disease is dependent on the original structure of the susceptible population in terms of subclasses S,z and Ss. Therefore I propose that this definition needs to be further specified for this model. Such specifications have been studied in more complex epidemiological models for which heterogeneity was introduced [29, 31]. Here I make a distinction between the basic reproductive number of the disease in the population, the usual 2,,, or population 2,,, and the intrinsic 2,, of an individual in the population. The population 2,, is the measurement of reproductive success of the pathogen in the host population. The intrinsic 2,, is the number of secondary cases of the disease that one infective individual will produce over its infective lifetime in an entirely susceptible stable population. The latter is coninonly used in the construction of the population 2,, in homogeneous populations [31]. 2,, of the model with two distinct subclasses without treatment. First, notice that, as mentioned above, the total host population is assumed to be subdivided into two classes: a moderately sexually active subpopulation and a highly sexually active subpopulation, namely individuals who respectively have n or s number of sexual partners per unit of time. If there are only susceptible individuals, the total population becomes A,+,+A I consider the number of secondary cases that one individual in the moderately active subclass (I,z) will produce over its entire infective lifetime. The average rate at which an infective individual in I, leaves the infective class is no, + p, and therefore the average time that he or she remains infective is given by ~F. The number of new contacts that suchl an individual has per unit of time is n. The probability that each of these contacts will be with a. suscecptible is .. The probabhility of transmissions? per partner is (1 p11)17,. Therefore, the average number of transmissions by one infective individual in I, per unit of timne is given by n1(1 pSnjl .~i~ In an entirely susceptible population, the expected number of secondary cases that one infective individual in I, will produce over its infective lifetime is "(")~". This can be thought of as an intrinsic reproductive number for an individual in I,. I will refer to this as R". Thus I have R'" = "")"", and similarly by the same construction as above for one infective individual in I, I obtain R" "(P" One can see now that the overall population Ro (the measurement of reproductive success of the pathogen in the entire population) that was derived by the analysis of the local stability of the DFE is given by a weighted sum of the intrinsic reproductive number of hosts in echcl of the two host c~la~sses. MLore specifically, Ro = R, (,of;~k I +R3(,asfsy ) at the disease free equilibrium. Since the entirely susceptible population has S, = An and S, = A it folloWS that Ro = s s')FO (1s"!"' and p~ ~ nns nns o a,+pL R = """"" A similar result canl be found in [23]. This can be understood the following way. I consider a typical infective individual to be an individual chosen at random from the original susceptible population and then made infective. An individual is chosen through a sexual contact. The probability of a random sexual contact being with an individual of type Sr, is ;1 for i =n, s. This accounts for the structure of the susceptible population in which the pathogen is being introduced. Thus, I arrive at the value of Ro: the sum of the products of the intrinsic reproductive number of hosts in each of the two host classes and the probability that an individual chosen at random (through a sexual contact) to become infected is from each of the classes. Thus, Ro is an appropriately weighted sum of the Ros that would result if the population was made up of just one (or the other) of the two classes. It is interesting to notice that this construction of Ro could be generalized to any number of host population classes defined by average number of sexual partners. If the total population were subdivided into m classes, I could construct Ro as above and get It is possible to rewrite the basic reproduction number in a more intuitive way as the product of the mean duration of infection, mean probability of transmission per partner, and an appropriately weighted average number of partners per unit time. This allows Ro to be interpreted as the average number of individuals that an individual infected at random will infect in a completely susceptible population. 3.3 Existence of an Endemic Equilibrium First, let R" =" l"j" and R" ")j". Now, I proceed to prove the existence of an endemic equilibrium. Theorem 3. If Ro > 1 then the system (26) (211) has at least one endemic equilibrium which is given by E* = (S* > 0, S* > 0, I ,> 0, I* > 0, A* > 0) . Proof We again consider the system (26) (211) at equilibria and we get the followingf equations: nS, sS, nlV, + s Ns a S, O =(n1 psq,, +s( ps~ls (p + a~n)I, (34) n(s + s~ ~ l s5 0 = c0,l, + adsl (p + d) A (36) 0 = A, +A,pNV dA (37) Now I proceed to find an endemic equilibrium and therefore assume I,~ O or I, / 0. Notice that if either I, = 0 or I, = 0, then both I, = I, = 0 and A = 0. Therefore, system (26)(211) can only have two kinds of equilibria: the diseasefree equilibriumn given by Eo = ( 0, 0, 0) and possible endemric~ equilibria given as E* = (S* > 0, ~SI > 0, I ,> 0, I* > 0, A > 0). From equations (34) and (35) I get the following n S, (n( ps~l,+ s1 psqs)= (p + a~n)I, (38) sS, (n( psq,, s( psal) =(p + cas)Is (39) nlV, + s Ns Let A = n(1 p,)rl,, + s(1 p,)rl,Is. Multiplying both sides of equation (38) by "Op'" ""77 and both? sides of eqluations (39) by "il"j")" anld th~en adding (38) and (39) I get n(1 p~q, nS, S(1 prs~q S A + (310) p1 cs nNV,+sN, p1+ s nNV,+N Then I can write equation (310) as nS s R"A R"A (311) Thus I get the following relation (assuming A / 0) R~nS, + R~sS, = nNV, + slV, (312) Now I rewrite equations (32) and (33) using (312): R"ns, O = R"As R~IpS, A (313) R" sS 0 = R"As R~pS, A OUU (314) RM S, + Es S, Now adding equations (313) and (314) I get the following 0 =R"As R"As "pS "S (315) which can be written in the form p(R~"S, + R"S,) =R"As + R"As A (316) Equation (314) can be rewritten as 0 = As pS, A (317) RM S, + gs S, N ow,, I multiply, equation (31 7)\ byT R nSn RSsS to get the following \ = (As p~ )(RnS, + RS o) AsS (318) Now rewriting (318) using (316) as follows 0 = (s ps)(R s+ n( O"As + gs \ A R"S,)) AsS, (319) 0 A p(s ),+ n(RA + U"" A) AsS, (320) Multiplying everything out in (320) I get the following quadratic equation in S, O =( n)R, ca + (mnR", nRASh (8 M)R"Sh + (s n)A)Ss n~h,(321' *(A,R"+ AsR" A) Since s > a there is a unique positive solution to equation (321) for all 0 < A < I can write S, and S, as functions of A. Thus let S, = f,(A) and S, = f,(A) for 0 A* = R"As + Rgn,. I also have S,(A*) = f,(A*) = 0 and S,(A*) = f,(A*) = 0. Now I go back to equations (34) and (35) where the denominator has been replaced with the left hand side of equation (312). nS, X (p + a~n)I, (322) nR gS, +sR gS, X = (p + a~s)Is (323) nn"S, sR:Ss A nS, In (324) p + annR"S, +sRgS, A s S, Is (325) p + a nR"S,+ sRgS, Recalling (312) I have R~a, +R"sS, = n(S, + I,) + s(S, + Is) (326) R"a f (A) + R"s f() =~ nf,(A) + s f,(A)+ A n2 nX 82 s"(x (3 27) pI + /v nlRnf f()r + slRSf f()r p + Cas nRn f (A) + s'RSf f()r rU ~ ~ (R 1)nvon\ f ,(A)+(R 1)s fs(A) = os\ A 8 nzX 82 sasX (328) p a nlRnf f()r +sRS f() pr + a n'Rnf f()r +sRS f f(A) Let F() (" 1n\l f,() (" 1S fs(A (329) A n2 nX 82 sfsx G(A) = +(330) iThe funlctionls F(A) andu G(A) are continuous functions of A for 0 < A < A*. AtA G(0 =V 0 and, F(0 =V (RO , 1)ls_)A"s > 0 since Ro > 1. Therefore, F(0) > G(0). At A = A*, G(A) is not defined. I define G(A*) = lim . G(A), provided this limit exists. To see that this limit exists, consider equations (32) and (33). n S, A =ci As pS, (331) R~nS, + Rss sS, A n A pS, (332) RMnS, + RgsS, So the quotient s" (AnpS"). ThuS f"(A) (Anpf"(A)) Ss n(AspLS,) fs(A) n(An pfs(A)) Now let A A * .fn(A) sA, him (333) Tha]t is, thle limit limr** . is finite and nonlzero. Tlhererfore limn~h C (A) exists: and is positive. In addition, lim . F(A) = 0. Consequently, G(A*) > F(A*). The Intermediate Value Theorem implies that there exists at least one endemic equilibrium for the system (26)(211). There is at least one A with 0 < A < A* for which equation (321) is satisfied which gives a positive value for S,. If S, = fs(A) > 0 for 0 < A < A* and f,(A*) = 0, then I want to show S, = f,(A) > 0 in the interval 0 < A < A*. f,(0) = ~ > 0. Assume there exists a A** in 0 < A** < A* such that Sn = f,(A**) = 0 and S, > 0 f,(A**) > 0. At A** from equation (316) I have pRgS, = R"As + Rgn, A** At A** from (318) we have 0 = (As pS,)R~sS, A**sS, and thus 0 = AsRg pR"S, A**. So I have pR"S, = AR" A** So I get a contradiction: R"As + AsR" A** = AR" A** since R"As / 0. So S, = f,(A) / 0 for all A with 0 < A < A*, and the proof is completed. . CHAPTER 4 SIMULATIONS ON THE THREE MODELS 4.1 The Model Without Treatment To understand the possible outcomes of different prevention strategies in the absence of medical intervention or treatment, I first simulate the baseline model without treatment given by system (26)(211). Parameters of the models were estimated from the literature and are summarized in Table 41. Values for the simulations where chosen within these ranges. Below, I discuss briefly these estimates. Partner acquisition rates vary greatly within populations of nonmonogamous sexually active individuals, ranging from 1 per year to 100 partners per year [24]. The removal rate p, that tracks natural death rate and removal from the sexually active population by changes in sexual behavior was taken from [24]. The individuals in question are assumed to be sexually active young adults who are expected to live an average of 50 years and engage in nonmonogamous sexual relations for 20 years [24]. The parameters an, and as,, that track the duration of infectivity, or the time it takes for an infected individual to develop the terminal symptoms of AIDS and be thus removed to the class of AIDS patients (who are not sexually active), are taken close to values found in [32], with variations introduced here correlating with level of sexual activity. The study in [32] was chosen for these parameters because this study was performed before the introduction of HAART therapy and many other antiretroviral treatments and therefore does not incorporate the impact of treatments in the average time to the development of AIDS. The values for rl,, rl,, rls, rls, the probability of transmission per partner, depend on the number of contacts per partner and the probability of transmission per contact [24], as well as infectivity of seropositive individuals. Estimates on the transmission probability per sexual contact range from 0.0003, for transmission from female to male, to 0.08, for transmission from male to male [24]. The number of contacts per partner is taken from [24] to be 2 Table 41: Simulation parameters and their value ranges Parameter Value Description reference n, a 0.0)8 10 partner month Average nrlumber of new sexual partners [24] acquired per unit of time An .90Are Proportion of individuals entering the [33] moderately sexually active population As .10Are Proportion of individuals entering the [33] highly sexually active population p 0.01 monthl Natural death rate and removal from [24] sexually active class d 0.125 monthl AIDS induced death rate [24] ps, f ,Dio0.0003 0.64 Infectivity (Probability of transmission [24, 34] per susceptible partner without preventive strategies) E >1'.9'.Condom efficacy (intrinsic) [11] Un 35' 0 Compliance with the use of condom for [35] individuals in steady relationships Us 9 n: Compliance with the use of condom for [35] individuals engaging in casual sexual relationships pi Condom induced protection (pi EUi) vi 0.008 0.0667month1 Rate at which HIVinfected individuals [3, 7] undergo testing and know their status 6 1.11'(ma I') Proportion of HIVinfected individuals [3] who know their status and receive treatment and/or other medical support asi 0.01 monthl Rate at which HIVinfected individuals [32] develop AIDS without treatment or HIVrelated medical support Yi 0.005 0.006 monthl Rate at which HIVinfected individuals [19] develop AIDS with the treatment and/or HIVrelated medical support * AT : Total recruitment into sexually active population per week for individuals with few sexual partners, and 1 per partner for individuals with many partners. The probability of transmission per partner are given by the probability of transmission per contact multiplied by the number of contacts. I chose parameters for the simulations within the ranges in Table 41 and suninarize them in Table 42. I investigate two 1!! ri ~ prevention focal points: reducing number of partners and condom use. I further investigate possible counter effects of these focused prevention efforts. For the case of an effort to reduce the number of new sexual partners per unit of time, I explore the changes in the value of 2,, when there is no other change in behavior, and when individuals have a reduced compliance with the correct and consistent use of condoms because they feel more secure with fewer partners. Similarly, I consider the strategy of prevention that focuses on increasing the correct and consistent use of condoms. I simulate the scenario where there is no other change in behavior and the possible effects of increased risk behavior in increased number of new partners per unit of time that can result front overconfidence in condom protection. Results of the simulation are given in Figures 41 and 42. It can he noted that given enough compliance with either of the focused prevention strategies, 2,, decreases even given some counter effect to the progranine. However, it is also clear that when there are secondary changes in behavior, for example an increase in number of sexual partners due to overconfidence in condom protection, the prevention strategy becomes less efficient. The results seem to ell 1 that there should be some concern with preventing secondary counter effective changes in behavior when intplenienting any prevention program that focuses mostly in one area of prevention, but also that as long as such counter effective changes are kept at manageable levels, the focused prevention program is still better than no prevention at all. The explicit division of the population into two discrete classes based on the number of sexual partners that individuals acquire, allows for the investigation of the effect of reducing the proportion of the population that becomes highly sexually active. Efforts Table 42: Specific parameter values Parameter Value n 0.08 partner month' s 10 partner month' r7; 0.08 rls 0.02 e U1 . U, 1'. usPn'. p, t'. used in simulations Description Average number of new sexual partners per month for individuals who are moderately sexually active Average number of new sexual partners per month for individuals who are highly sexually active Probability of transmission from moderately sexually active infective individual per susceptible partner without preventive strategies Probability of transmission from highly sexually active infective individual per susceptible partner without preventive strategies Condom efficacy (intrinsic) Compliance with the use of condom for individuals in steady relationships Compliance with the use of condom for individuals engaging in casual sexual relationships Condom induced protection (pi = emi) Condom induced protection (pi = emi) Rate at which HIVinfected individuals undergo testing and know their status Rate at which of HIVinfected highly active individuals undergo testing and know their status Proportion of HIVinfected individuals who know their status and receive treatment and/or other medical support Rate at which moderately sexually active HIVinfected individuals develop AIDS without intervention of treatment or HIVrelated medical support Rate at which highly sexually active HIVinfected individuals develop AIDS without intervention of treatment or HIVrelated medical support Rate at which moderately sexually active HIVinfected individuals develop AIDS with the intervention of treatment and/or HIVrelated medical support Rate at which highly sexually active HIVinfected individuals develop AIDS with the intervention of treatment and/or HIVrelated medical support 0.036month 0.03675month 1 0.09 month 0.012 month 0.005 month 0.006 month Focus on reducing number of partners only 45 3.5  25 I 5 10 15 20 25 30 35 40 45 percentage reduction i n number of new partners per unit of tirne Figure 41. Prevention focused on decreasing number of partners only. Let p be the percentage decrease in average number of new sexual partners per unit of time. Thus, average number of new sexual partners per unit of time of moderately sexually active class here is given by n(1 p) and average number of new sexual partners per unit of time of highly sexually active class is given by s(1 p). Graph shows 7 o versus p. Let q be the percentage decrease in correct and consistent condom use. Then condom use protection is given by 1 (u,(1 q)e) for moderately active group and 1 (u,(1 q)e) for highly active group. I show curves for q = p, q = 0.5p and q = 2p. The line S1 is the baseline case where there is a reduction in number of partners and no other change in behavior. Curves S2 S4 illustrate a reduction in number of partners coupled with an increasing reduction in the correct and consistent use of condoms. Focus on Increasing correct and consistent condom use S3 S2 a=0 5p 0 10 20 30 40 50 percentage increase in correct and consistent condom use Figure 42. Prevention focus on increasing correct and consistent condom use only.Let p be the percentage increase in correct and consistent condom use. Thus, condom use protection is given by 1 (u,(1 + p)e) for moderately sexually active individuals and 1 (u,(1 + p)e) for highly sexually active individuals. Graph shows Ro versus p. Let q be the percentage increase in average number of new sexual partners per unit of time. Then average number of new sexual partners per unit of time of moderately sexually active class here is given by n(1 + q) and average number of new sexual partners per unit of time of highly sexually active class is given by s(1 + q). I show curves for q = p, q = 0.5p and q = 2p. The line S1 is the base line case where there is increase in correct and consistent condom use and no other change in behavior. Curves S2 S4 illustrate an increase in condom use coupled with an increase in number of new sexual partners per unit of time as a result of possible overconfidence that the use of condoms will protect against the transmission of HIV. Increasing recruitment into moderately sexually active class 5.5 4.5 3.5 2.5 1.5 90 91 92 93 94 95 96 97 98 99 100 percentage of population being recruited into moderately sexual active group Figure 43. Prevention fowused on changing recruitment only. OsI Iuse; in recruitment proportions can he used as a prevention strategy with low possible counter effects. to educate young people who are about to become sexually active about safer sexual practices involving a low number of different sexual partners has a clear impact in disease prevention as illustrated in Figure 43. 4.2 The Model With Treatment but Without Change in Behavior: Treatment as a Prevention Strategy Now I fowus on examining the model which incorporates the new trends in treatment. The parameters v,z and v, refer to the rate at which sexually active individuals get tested. According to the IT.S. Department of Health and Human Services [7] and the ITNAIDS Report on the Global AIDS Epidemic [4], in the IT.S. onequarter of those individuals who are infected with HIV do not get tested and are unaware of their status. The parameter 6 reflects the proportion of the population made up of seropositive individuals who know their status that actually receive treatment and medical support. Also according to UNAIDS, in the world todwi, >II' of seropositive individuals requiring treatment and medical support do not get it. In some countries less than 1 of known HIVpositive individuals have access to treatment and medical support, while there are countries that have achieved 1011'. coverage, the mean world value being >II'.~ [:3]. I focus on the current scenario in U.S. and let treatment availability be at 1011' According to the CDC, HAART treatment can just about double life expectancy of seropositive individuals [19]. First, I explore the effects of treatment in the absence of any further prevention strategy. Some literature II ; that even though treatment lowers the plasma virus load of seropositive patients, this reduction does not correlate with a reduction in virus load in other body fluids, such as semen [1:3]. However, some studies of couples with discordant serostatus, individuals with lower virus load have a lower probability of transmitting HIV [5], while it is not clear by how much. I show the effects of reduced infectivity of individuals undergoing treatment (Table 4:3). The incidence without treatment was of 146 individuals in 100,000. Without reduction of infectivity or change in risk behavior the introduction of treatment increases incidence of HIV infections. With a decrease in infectivity, introducing treatment reduces the incidence of HIV infection. It is important to recall that not all individuals who receive HIV related medical attention are actually taking HAART drugs. Some health professionals delay the start of HAART but still provide medical attention in the form of viral load screening, hacterial STDs treatment, and treatment of opportunistic infections. For this simulation, however, I assume individuals receiving medical intervention are actually receiving HAART drugs. Table 43: Reduced infectivity in individuals receiving HAART treatment Infectivity reduction Incidence in 100,000 O' 151 10I' 140 21 1' 135 SII'.129 11 I' .120 50' .112 I .I I' 103 711' ,92 H I' .78 911' .60 I recall that to effectively reduce viral load there needs to be adherence to treatment at all times (which has proved difficult), that deb i. I start of HAART drugs might be appropriate, and that treatment ahrl . caries the risk of producing multiresistant strains. 4.3 The Model With Treatment and Change in Behavior: Incorporating Prevention Programmes in HIVRelated Medical Intervention Now I allow for a behavior change in individuals who come to know of their HIVpositive status and receive treatment. Consider the model (220) (226). As a result of expanded treatment access, millions of people living with HIV are periodically visiting healthcare delivery sites to monitor their disease and treatment progress [3]. I believe this provides an important path through which to incorporate prevention programmes that focus on behavior change. In countries like the United States, where only about "1.' of HIV infections are unknown, many individuals who know of their seropositive status have the opportunity to receive medical support and treatment to improve their lifestyle and increase their longevity. I explore the outcome of integrating change of behavior prevention programmes focused on these individuals with the delivery of treatment and HIV medical support. Assuming effective prevention efforts implemented through integration of prevention and medical support, in general, assume that no < n and so < s as well as po > p, and ps. > Ps. Infectivity is assumed do be reduced by 501' for treated individuals. The values for parameters are taken as before but now I introduce behavior change in persons who know of their seropositive status and are undergoing treatment and/or medical counseling. Results of this simulation are given in Table 44 and Table 45. Table 44: Prevalence of HIV infection with increase in condom use and decrease in number of sexual partners per unit of time Increased prevention For the entire For the group under population treatment (5' of total population) 5'~ 6.5' 6.'7' . 10I' 6.0I' 6i. !' . 15' .5. !' 6.0I' . 21 1' 4.'7' 5. ' 25' 3.4' 5. !' . Table 45: Incidence (per 100 000) of HIV infections with increase in condom use and decrease in number of sexual partners per unit of time Increased prevention For the entire For the group under population treatment (5' of total population) O' 112 112 5' 104 109 10I' 96; 100 15'.85 98 21 1' 76 90 Observe that with increased effective prevention aimed only at the subgroup of the population consisting of seropositive individuals receiving medical support the values for the resulting prevalence of HIV infection in the population are very close to the values reached by increasing effective prevention on the entire population. The group under treatment corresponds to 5.25' of the entire population. To reach an incidence level of 90 new cases in 100 000 per unit of time, there needs to be an increase of 211' in prevention in the group under treatment, while for the same level of incidence there needs to be around 1"' increase in prevention in the entire population. Notice reaching 211' of the individuals in the treated group corresponds to effectively reaching 1000 individuals, while changing 12.5' of the entire population corresponds to effectively reaching 12500 individuals. In both cases, the resulting incidence is 90 individuals per 100 000. Here I reach my main result: given that testing and treatment availability are comparable to the United States, it is not necessary to effectively reach the entire population with increased prevention programmes to reduce the prevalence and incidence of HIV. It is sufficient to effectively change the behavior of the subgroup of the population composed of individuals of known seropositive status who are receiving HIV related medical support and counseling. But more than that, these results show that coupling prevention efforts with treatment delivery could be as as effective as targeting prevention to the entire population. This conclusion demonstrates how large an impact individuals of known seropositive status who receive medical intervention can have in the prevention of HIV. By actively incorporating and improving prevention strategies, individuals who are aware of their seropositive status can effectively help reduce the prevalence of HIV in the population and slow the epidemic. It is very important then not to neglect HIV prevention related education for these individuals. It is clear from reports of the CDC [5] that individuals who become aware of their seropositive status do actively try to prevent further transmission of the disease. Therefore, it is necessary for society to provide appropriate tools for these individuals to effectively help in the fight against HIV/AIDS. It is necessary to provide clear information, resources, and support to those individuals who are infective and who are receiving HIV related medical attention. CHAPTER 5 CONCLUSION The model with treatment and differentiated rate of partner acquisition, which to my knowledge has not been previously studied, accounts for heterogeneities among individuals in their behavior towards acquisition of new partners and use of condoms while exploring the effects of treatment and medical support, with and without prevention counseling, on the overall disease dynamics. The model without treatment accounts for heterogeneities among individuals in their behavior towards acquisition of new partners and use of condoms, but emphasizes these mechanisms explicitly, especially in the explicit rendering of Ro, and gives further insight on prevention strategies. For the model without treatment I derived the explicit formula for the basic reproduction number and proved existence of at least one endemic equilibrium when Ro > 1. I discuss the value of the basic reproduction number in view of other more traditional epidemiological papers and the meaning of this value of Ro U~sing simulations, I examined the transmission dynamics of the disease in a population where HAART is introduced, with and without focal prevention efforts. The main result I reach is that prevention programmes do not need to reach the entire population, but only need to focus on the group receiving HIVrelated medical attention. This approach to prevention is costeffective, since the target population is significantly smaller than the total population, since there is a clear channel for the delivery of treatment, and since this channel does not rely on the costly antiretroviral drugs for reduced infectivity. Coupling prevention programmes, such as counseling and education programmes, with the delivery of HIVrelated medical attention can liberate HAART from being used as a prevention strategy, so that the start of HAART drugs can be de 1lwed if necessary and the risk of producing multiresistant HIV strains can be reduced. This approach prevents seropositive patients from increasing risky behavior due to overconfidence in treatment and reduced infectivity. There also exists a clear path for delivery of prevention programmes to HIV patients, since contrary to the target population of individuals under high risk of acquiring the disease, HIV patients are a known, welldefined subgroup of the total population. With this model I advocate that health policies for prevention of HIV infection in a population should not rely solely on reduced infectivity due to treatment for the prevention of the transmission of HIV from seropositive patients, but rather it should take an active approach to prevention focusing on change of behavior of these individuals. Further, I advocate that it is more effective to focus change of behavior prevention on the subgroup of the population receiving treatment to achieve a significant reduction in the prevalence of the disease, given rates of testing comparable to the United States, than are indiscriminate measures. In the absence of a cure or vaccine, the world must rely on effective implementation of prevention strategies all of which involve behavior change of the population as a whole or, and maybe especially, of the individuals of known seropositive status. Ideally, all infected individuals should be tested and provided with treatment [24], as well as information and support to incorporate changes in risk behavior and prevention techniques in everyd1 life. Also, ideally all individuals who are aware of having seropositive status would have the will, the resources and the support to actively engage in prevention. However, changes in behavior, especially sexual behavior, are akals a challenge. Relatively few studies have been undertaken to measure the effectiveness of behavioral interventions for prevention for people living with HIV, but emerging evidence indicates that such programmes are effective in reducing the likelihood that people with HIV will engage in sexual activity that might expose others to the virus [36]. Thus, it is very useful to focus prevention on this group. 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The New England Journal of Medicine (N Engl J Med). 339 (1998) 1803. [14] Lisa R. Metsch, Margaret Pereyra, Carlos del Rio, Lytt Gardner, Wayne A. Duffus, Gordon Dickinson, Peter K~erndt, Pamela AndersonMahoney, Steffanie A. Strathdee, Alan E. Greenberg, for the Antiretroviral Treatment and Access Study Group (ARTAS). Delivery of HIV Prevention Counseling by Physicians at HIV Medical Care Settings in 4 US Cities. Am J Public Health. 94 (7) (2004) 1186 [15] M.C. Boily, F.I. Bastos, K(. Desai, B. Masse. C'!s Ia ss in the transmission dynamics of the HIV epidemic after the widescale use of antiretroviral therapy could explain increases in sexually transmitted infections: results from mathematical models. Sex Transm Dis. 31(2004)100. [16] Brazil, Ministrio da Sade Secretaria de Vigilncia em Sade/Programa Nacional de DST e AIDS. Tabelas contend os dados de vaginose bacteriana, candidate, sfilis, HPV, HIV, HBV e HCV. (2004). [17] Uneke Jesse, Chigozie and Ogbu, Ogborn .T .i and Alo, Moses and Ariom, Thaddeus (2006) Syphilis serology in HIVpositive and HIVnegative Nigerians: The public health significance. Online Journal Of Health And Allied Sciences 5(2). [18] C.E. Hart, J.L. Lennox, M. PrattPalmore, T.C. Wright, R.F. Schinazi, T. EvansStrickfaden, T.J. Bush, C. Schnell, L.J. Conley, K(.A. Clancy, and T.V. Ellerbrock. Correlation of Human Immunodeficiency Virus Type 1 RNA Levels in Blood and the Female Genital Tract. J Infect Dis. 179 (1999) 871. [19] Panel on Clinical Practices for Treatment of HIV Infection (October 6, 2005). Guidelines for the Use of Antiretroviral Agents in HIV1Infected Adults and Adolescents. Department of Health and Human Services. (2005). [20] Centers for Disease Control and Prevention (CDC). HIV Prevention: HIV and its transmission. (1999). [21] Centers for Disease Control and Prevention (CDC). HIV and its transmission. (1999). [22] T.C. Quinn, M.J. Wawer, N. Sewankambo. Viral load and heterosexual transmission of human immunodeficiency virus type 1. Rakai Project Study Group. N Engl J Med 342 (2000)921. [23] C. CastilloC'!s l.. ., Z. F in_ W. Huang, On the computation of 7 o and its role on global stability, in: C. Castillo'lei.;. with S. Blower, P. van den Driessche, D. K~irschner, A.A. Yakubu (eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, SpringerVerlag, (2002)229. [24] J.M. Hyman, J. Li, E.A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci. 155(1999)77. [25] J.M. Hynian, J Li. The reproductive number for an HIV model with differential infectivity and staged progression. Linear Algebra and its applications. :398 (2005) 101. [26] J.M. Hynian, J. Li. Differential susceptibility epidemic models. J. Math. Biol. 50 (2005) 626. [27] J.M. Hynian, J. Li. Differential susceptibility and infectivity epidemic models. Math. Biosci. and Eng. :3 (2006) 89. [28] J.M Hynian, J. Li. Behavior changes in SIS STD models with selective mixing. SIAM J. Appl. Math. 57 (1997) 1082. [29] O. Dieknmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Disease; Model Building, Analysis and Interpretation, Wiley, New York, (1999). [:30] L.J.S. Allen, An Introduction to Mathematical Biology, Pearson Prentice Hall, I~pper Saddle River, NJ, (2007). [:31] P. van den Driessche, J. Watnlough, Reproduction numbers and subthreshold endemic equilibria for compartniental models of disease transmission, Math. Biosci. 180 (2002) 29. [:32] J.M. Logini, W.S. Clark, 31. Haber, R. Horsburgh. The stages of HIV infection: Waiting times and infection transmission probabilities, in: CastilloChl li. ;, Levin, Shoemaker (Eds.), Mathematical Approaches to AIDS Epidemiology Lecture Notes in Biontathentatics, vol. 8:3. Springer. (1989) 111. [:33] R.M. Anderson, R.M. May. Infectious diseases of humans: Dynamics and control. Oxford, New York. (1991). [:34] M.D. Guiniaraes, A. Munoz, C. BoschiPinto, E.A. Castilho. HIV infection among female partners of seropositive men in Brazil. Rio de Janeiro Heterosexual Study Group. Am J Epidenmiol 1995;142(5):5:3847. [:35] J.E. Anderson, R. Wilson, T.S. Jones, P. Barker. Condom use and HIV risk behaviors among ITS adult: data front a national survey. Fant. Plan. Perspect. :31 (1999) 2428. [:36] G. Marks, N. Crepaz, J.W. Senterfitt, R.S. Janssen. Metaanalysis of highrisk sexual behavior in persons aware and unaware they are infected with HIV in the United States: Implications for HIV prevention programs. J Acquir Ininune Defic Syndr. :39(4) (2005) 446. [:37] Y. Huang, W.A. Paxton, S.M. Wolinsky, A.IT. Neumann, L. Zhang, T. He, S. K~ang, D. Ceradini, Z. Jin, K(. Yazdanhakhsh, K(. K~unstnman, D. Erickson, E. Dragon, N.R. Landau, J. Phair, D.D. Ho, R.A. K~oup. The role of a mutant CCR5 allele in HIV1 transmission and disease progression. 2(11) Nat Med. (1996) 1240. BIOGRAPHICAL SKETCH Fernanda Oliveira Melo was born on April 29, 1981 in Brasilia, Brazil. The daughter of two medical doctors, she developed a passion for the sciences early on. Her early school years were spent mostly in a small bilingual school, Es cola das Na coes, in her hometown. In high school she enrolled in the American School of Brasilia. In 1998, she moved to the United States with her parents. She lived in Orlando, Florida for 3 years. She completed high school at Dr. Phillips High School and received an associate in arts degree from Valencia Community College. Upon receiving her A.A. she moved to Gainesville, Florida to begin her undergraduate studies in mathematics at the University of Florida. In August 2003, at the end of her junior year, she married Edgar Melo. Fernanda was awarded a bachelor of science degree in mathematics with a minor in education in August 2004. She graduated with magnet cum laude. She specialized in applied mathematics and produced, under the sponsorship of Dr. Serguei Pilyugin, an undergraduate thesis entitled An SIR model with discrete .;;;;;;;; i 01;; subcles~sse~s. In April 2005, her first son, Lucas, was born in K~issimmee, Florida. She started graduate school at the Mathematics Department in the University of Florida in 2005. During her two years as a graduate student there, she had the opportunity to take individual research courses under biomathematicians such as Dr. Alaia Martcheva, Dr. Serguei Pilyugin and Dr. Patrick DeLenher. Upon graduating with her M.S. in applied mathematics, Fernanda will be moving to the Zoology Department at the University of Florida to complete a doctorate degree under the mentorship of theoretical biologists Dr. Be ni l...ilr 31. Bolker and Dr. Robert Holt. There she hopes to further her research in the area of infectious disease dynamics, combining theoretical biology with her knowledge of biomathematics. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IthankmyadvisorDr.MaiaMartchevaforhervaluableinputs,herconstructivefeedback,andespeciallyforalwaysbelievingIcoulddothis.Ithankmysupervisorycommittee,Dr.DavidWilsonandDr.RobertHolt,forthecontinuoussupportandconstructivecriticismanddiscussionsonthetopic.EspecialthanksgotoDr.Holt,forallowingmetheopportunitytoworkinhislaboratory.Ithankmyhusband,whoenduredmyabsenceduringlongnightsofresearch,whoassumedhisresponsibilitiesandmineinthehousehold,wholistenedtomylongbrainstorming,alwaysbeingsupportiveandcaring.Ithankmyson,Lucas,mystrengthandjoyattheendofthelong,andmanytimes,frustratingdaysofresearch.Iextendmygratitudetomyparents,withoutwhomIwouldneverhavemadeitthisfar.Theyalwayssupportedmyeortsandpraisedmysuccesses.Theyintroducedmetothepassionforscience,research,curiosity,knowledge.Ithankallwhohavegiveninputandfeedbackintothisthesis.EspecialthankstoDr.MichaelBareld,GustavoOliveiraandManojitRoyforvaluablereviewofthisthesis,andDr.RosangelaXavierforherpassionatediscussionsonthetopicthatcametoinspiremetopursuethisintoathesis. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 LISTOFTERMS ..................................... 8 CHAPTER ABSTRACT ........................................ 10 1INTRODUCTION .................................. 11 1.1Motivation .................................... 11 1.2GoalsofResearch ................................ 12 1.3TheDisease ................................... 13 1.4TheEpidemic:Origin,HistoryandTreatment ................ 14 1.5HighlyActiveAntiretroviralTherapy ..................... 15 1.6TransmissionandPrevention .......................... 16 1.7MathematicalModelsofSTDsandtheBasicReproductionNumber .... 18 2FORMULATIONOFTHREEMODELS ...................... 21 2.1TheModelWithoutTreatment ........................ 24 2.2TheModelWithTreatmentbutWithoutChangeinBehavior ....... 25 2.3TheModelWithTreatmentandChangeinBehavior ............ 26 3ANALYSISOFTHEMODELWITHOUTTREATMENT ............ 28 3.1ExistenceandLocalStabilityoftheDFE ................... 28 3.2TheBasicReproductionNumberandItsInterpretation ........... 31 3.3ExistenceofanEndemicEquilibrium ..................... 34 4SIMULATIONSONTHETHREEMODELS ................... 39 4.1TheModelWithoutTreatment ........................ 39 4.2TheModelWithTreatmentbutWithoutChangeinBehavior ....... 45 4.3TheModelWithTreatmentandChangeinBehavior ............ 47 5CONCLUSION .................................... 50 REFERENCES ....................................... 53 BIOGRAPHICALSKETCH ................................ 56 5 PAGE 6 Table page 21Listofvariablesandparametersandtheirmeaning ................ 23 41Simulationparametersandtheirvalueranges ................... 40 42Specicparametervaluesusedinsimulations .................... 42 43ReducedinfectivityinindividualsreceivingHAARTtreatment .......... 47 44PrevalenceofHIVinfectionwithincreasingprevention .............. 48 45IncidenceofHIVinfectionswithincreasingprevention .............. 48 6 PAGE 7 Figure page 21Diagramofthemodelwithtreatment ........................ 25 31Themodelwithouttreatment:diagram ....................... 28 41Preventionfocusedondecreasingnumberofpartnersonly. ............ 43 42Preventionfocusonincreasingcorrectandconsistentcondomuseonly. ..... 44 43Preventionfocusedonchangingrecruitmentonly. ................. 45 7 PAGE 8 Neoplasm Anabnormalgrowthoftissue,atumor.Immunedeciency Inabilityoftheimmunesystemtofunctionproperly.Resultsingreatersusceptibilitytodisease.Immunosuppression Reducedimmunesystemresponsetopathogens,suchasvirus,bacteriaorfungi.Seropositive IndividualinfectedwithHIV.Retrovirus RNAvirusthatsynthesizesDNAthroughreversetranscriptase.Lentivirus Typeofretroviruscharacterizedbypresentingalongintervalbetweeninfectionandtheonsetofsymptomsinhosts.ReverseTranscriptase EnzymeresponsiblefortranscriptionofsinglestrandedRNAintodoublestrandedDNA.Protease HIVenzymeneededforassemblyofaninfectiousvirusparticle.Integrase HIVenzymeusedbythevirustoincorporateitsgeneticmaterialintothatofthehostcell.CD4+helperTcells Whitebloodcellsthatcoordinateimmuneresponse.Macrophages Largeimmunesystemcellsresponsibleforremovinginvadingpathogensbyenvelopingthem.Thesecellsmovefreelythroughoutthebody.Dendriticcells Immunesystemcellswithlong,tentaclebranches.Functionasspecializedcellsatthemucosa.Receptor Amoleculeonthesurfaceofacellthatservesasarecognitionorbindingsiteforantigens,antibodies,orothercellularorimmunologicalcomponents.Fusin AreceptorpresentinallcellstypesthatcanbeinfectedbyHIV.FusinisnecessaryforHIVinvasionOpportunisticinfection Infectionbyanorganismthatdoesnotordinarilycausediseasebutwhich,undercertaincircumstances(impairedimmuneresponses),becomespathogenic. 8 PAGE 9 Sexuallytransmitteddisease.HAART Highlyactiveantiretroviraltreatment:acombinationofdrugsaimedatghtingHIVinthebody.Prevalence Proportionofindividualsinapopulationwhoareinfected. 9 PAGE 10 MystudydevelopsmathematicalmodelstoinvestigatepreventionstrategiesaimedatareductionofHIVprevalenceinapopulation.Itfocusesprimarilyontheimpactofcouplingpreventioneortwithtreatmentdelivery.Itgivesanoverviewofpossiblepreventionstrategiesfortheentirepopulationaswellasforthesubgroupofindividualswhoareundergoingtreatmentand/orreceivingHIVrelatedmedicalattention.MystudyfocusesonexploringtheeectsofthemostrecenttrendhavingamajorimpactintheHIV/AIDSepidemic:HAARTtreatment.Specically,throughsimulation,itdemonstratestheimpactofintegratingpreventioneortswiththedistributionoftreatmentandHIVrelatedmedicalsupport.ItshowsthateducatingHIVpatientsaboutrisksofHIVtransmission,reliabilityofcondoms,andsafersexualbehaviormayeectivelydecreasetheprevalenceofHIVinthepopulationaseectivelyasaimingthesepreventionprogrammesattheentirepopulation,providedthatavailabilityoftestingandHIVrelatedmedicalattentionisadequate. 10 PAGE 11 1 ]ofvecasesofPneumocystiscariniipneumonia(PCP)amongpreviouslyhealthyyoungmeninLosAngeles,California.Theseincidentswerelatterattributedtoaretrovirusthatcametobeknownashumanimmunodeciencyvirus(HIV).ByAugust1981,70newcaseswererecordedofPCPandKaposi'ssarcoma(KS),arare,malignantneoplasm(Table 1 )previouslyseenbeforeinelderlyman[ 2 ].Withinafewmonthsitwasclearthattheworldwasfacingadangerousnewepidemic,but,noonecouldhaveimaginedtheextentofwhatbecametheAIDSpandemic. Today,theJointUnitedNationsProgrammeonHIV/AIDS(UNAIDS)andtheWorldHealthOrganization(WHO)estimatethatAIDShaskilledmorethan35millionpeople[ 3 ].In2006,therewere39.5millionpeopleworldwidelivingwithHIV,including2.3millionchildren.Anestimated4.3millionpeoplewerenewlyinfectedthatsameyear[ 4 ].Inthese25yearsthediseasehasspreadtopandemicproportions.HIVinfectionandAIDSremainleadingcausesofillnessanddeathinmanycountriesintheworld,includingtheUnitedStates[ 5 ].Thediseasethatoncewasviewedasrestrictedtomenwhohavesexwithmenhasnowbecomeprevalentamongmenwhoareheterosexual,womenandchildren[ 6 ]. Withtheadvanceofthisdangerousdisease,theworldhasbecomeinvolvedinafreneticsearchforwaystotreatorcureHIVinfection.Overthelast25yearstherehasbeenenormousprogressintheghtagainstAIDSthatmightsuggestthatthereishopeforthefuture[ 3 ].Acureorvaccineisnotyetavailable,buttherehasbeenasignicantadvanceintreatment,resultinginanincreaseinthelifeexpectancyandqualityoflifeofthosediagnosedwithHIVinfection[ 7 ].Todayhighlyactiveantiretroviraltherapy(HAART)isusedtoslowtheprogressionofHIVinfectionandthuspostponedeathfrom 11 PAGE 12 7 ]. However,eventhoughtreatmentcanincreaselifeexpectancyandhealthprospectsinHIVinfectedindividuals,treatmentdoesnotprovideacurefortheinfection.Thisandotherfactorsinuencetheneedforcontinuedeortstoreducefurtherspreadofthedisease.Withglobalizationandincreasedmovementofpeople,theAIDSpandemicneedstobeaddressedasaglobalissue,especiallygivenhighcostoftreatmentbothforindividualsandforgovernments.Further,HIVisahighlymutablevirus,sotheriskofnewmultiresistantstrainsemergingisalwayspresent[ 3 ].Therefore,eortstopreventthespreadofHIVarestillveryimportant. Thereare,however,manyissuesthatmakeitespeciallydiculttoimplementeectivepreventionprogrammesthatreducetheprevalenceofHIVinfectioninapopulation.Amongtheseissuesaredicultiesinreachingasignicantportionofthepopulationwithprogrammesforeducationaboutthediseaseanditstransmission,thecostofpreventionprogrammes,complianceofthepopulation,andthefactthatHIVinfectionisprimarilyasexuallytransmitteddiseaseandthuspreventionstrategieshavetoaddressmanydelicateissuesofsexuality,intimacyandprivacy[ 3 6 ]. 12 PAGE 13 6 ].HIVisalentivirusofthefamilyRetroviridae(Table 1 ).Itcontainstwocopiesofpositiveribonucleicacid(RNA)thatcodeforthevirus'ninegenes.TheRNAisboundtoproteinsandenzymesnecessaryforviraldevelopment,includingreversetranscriptase,protease,andintegrase,andisenvelopedintwolayersofphospholipids[ 8 ](Table 1 ). HIVprimarilyaectsthehumanimmunesystembyattackinghelperTcells(morespecically,CD4+cells),macrophages,anddendriticcells(Table 1 )[ 9 ].HIVattackscellsusingproteinreceptorsthatarepartofthenormalimmuneresponse.ThemainreceptoristheCD4moleculeonhelperTcells.Asecondarycoreceptorisalsonecessaryforthevirustoinvadeacell.Themaincoreceptor,presentinallcelltypesthatcanbeinfectedby 13 PAGE 14 8 ].Astheinfectiondevelops,theviruscausesdepletionandincreasingdisruptionoftheimmunesystemopeningmanydoorsfortheonsetofopportunistic(Table 1 ),lifethreateninginfections[ 8 { 10 ].Thereisusuallyanasymptomaticperiod,duringwhichnovisiblesymptomsoftheinfectionoccur.Evenduringthistime,however,thevirusisactivelymultiplying,infectingandkillingcells[ 7 ].Viralparticlesarealso,duringthistime,movingintosecretionsandbodyuids,includingblood,semenandvaginalsecretions. TreatmentregimentstodayfocuseonacombinationofdrugsthatrepressHIVreproductionandcellinvasion,including:reversetranscriptaseinhibitorsandproteaseinhibitors(whichimpedeviralreproduction),andfusioninhibitors(whichblockvirusfromenteringcells)[ 11 ].OtherdrugsaimtohelppreventanumberofopportunisticinfectionsincludingPCP,toxoplasmosis,cryptococcusandcytomegalovirusinfection[ 11 ]. 12 ].HIVwasisolatedandrecognizedasthepathogenresponsibleforAIDSin1983.Nocure,vaccineortreatmentexisted,andtreatmentstocombattheopportunisticinfectionscausedbytheimmunedeciencywereveryfew[ 7 ].In1987thersttreatmentemerged.ThedrugwascalledAZT,areversetranscriptaseinhibitor.By1992combinationsofdrugswereintroducedtoimprovetreatment. In1996evidenceoftheecacyofanewtreatment,calledhighlyactiveantiretroviraltherapy(HAART),waspresentedforthersttimeatthe11thInternationalAIDSConferenceinVancouver[ 6 ].Since2000UNAIDSandWHOhavebeentryingtoincreaseaccesstoHIVtreatmentindevelopingcountries.Bytheendof20051.3millionpeopleinlowandmiddleincomecountrieshavereceivedaccesstoantiretroviraltherapy[ 6 ].Today, 14 PAGE 15 By2001,however,therewasgrowingconcernwithbothmedicationtoxicityandeectiveness[ 11 ].AlthoughHAARThasclearbenets,italsohasitsshortcomingsandrisks.HAARTisnotacure:thedrugsonlycontrolHIV,theycannoteliminatethevirusfromthebody[ 7 13 ].ThismeansthatHIVcanstillbetransmitted.Thedrugshaveastrictscheduleandadherenceisdicult[ 6 ].UnderstandingtheultimateimpactofHAARTuponHIVprevalencerequiresaquantitativeapproach. 6 14 ],potentiallyincreasingopportunitiesforsexualtransmission.Atthesametime,optimismaboutthetreatmentormisperceptionsabouttheeectsofantiretroviraldrugsmayalsocausesomepeopletoincreasetheirriskbehavior[ 15 { 17 ].Ontheotherhand,andagoalofthisthesis,theintroductionoftreatmentinthepopulationprovidesacleartargetgroupforpreventioneortsandaclearpaththroughwhichtodeliversuchpreventionprogrammesbyintegratingpreventioneortswithdeliveryoftreatmenttoHIVpositiveindividuals. SincetheintroductionofHAART,increasesinthequalityoflifeandlifeexpectancyforHIVpositiveindividualsundertreatmenthasreshapedthefaceoftheepidemic.HAARThasbeenabletoincreaselifeexpectancyandsignicantlyreducevirusloadinHIVpositivepatients,sometimestolevelsbelowthosethatcanbedetectedbyHIVtests[ 7 ].Inwomen,lowbloodplasmavirusloadhasbeencorrelatedwithlowervaginalvirusload,indicatingthepossibilityofalowerriskofperinatalandfemaletomaleheterosexualtransmissionfromwomenundertreatment[ 18 ].Perinataltransmissionrateshavebeenreducedwithscreeningofpregnantwomen,combinedwithprophylacticadministrationofHAARTdrugs[ 5 ]. 15 PAGE 16 3 9 ].AnothermajorissuewithHAARTisthepossibilityofnewresistantstrainsofHIVarisingandincreasinginfrequency.Forthesereasons,somehealthexpertsrecommenddelayingdrugtreatment,whilemaintainingregularmedicalcheckups[ 19 ]. 7 ];however,aftertheintroductionofscreeningofplasmaforHIVin1985[ 6 ],theriskoftransmissionofHIVthroughsuchapathhasbecomeextremelysmall.HIVcanalsobetransmittedfrommothertochildduringpregnancy,birthorthroughnursing[ 7 ].PrenatalHIVtestingandHAARThavereducedthistransmissionsignicantly[ 5 ].ThereisnoevidencethatHIVcanbetransmittedbycasualhouseholdorsocialcontactorbyinsects[ 7 20 ].HIVisnottransmittedthroughsweat,tears,kissing,orsharingcommonhouseholditems,suchaseatingutensils[ 20 ]. Today,themajorityofHIVinfectionsareacquiredthroughsexualrelationsbetweenpartners,oneofwhomhasHIV[ 4 7 ].ThismakesunderstandingthedynamicsregulatingsexualtransmissionofHIVveryimportant.ParametersthataectthesexualtransmissionofHIVincludethecorrectuseofbarrierpreventionmethodssuchasmaleorfemalecondoms,thenumberofdierentsexualpartnersthatanindividualacquires,andthepresenceofothersexuallytransmitteddiseases(STDs)thatcanenhancetransmission[ 20 ]. GoalsofbehaviorchangeforHIVpreventionincludeabstinenceanddelayedsexualdebutsforyoungpeople,monogamywithinrelationships,reductioninthenumberofpartners,andcorrectandconsistentuseofcondoms[ 3 4 ].Variousstudiessuggestthat 16 PAGE 17 7 20 21 ]. IthasbeenshownthateventhoughHAARTmayreduceviralloadininfectedindividualssignicantly,eradicationoftheviruswithinapatientcannotyetbeachieved.Levelsofplasmavirusloadbelowdetectiondonotnecessarilyreectlowlevelsofvirusinothersecretionssuchassemen[ 13 ].Still,somestudieshavesuggestedthattheremightbeacorrelationbetweenlowviralloadinthebloodandreducedinfectivity[ 5 22 ].Itso,treatmentcanbeafacetofapreventionstrategy. However,manyissuesarisewhentreatmentisusedasapreventionstrategy.Therstisthehighcostoftreatment,bothforindividualsandforgovernments.Anotherissueisthat,evenifthereisacorrelationbetweenlowviralloadandreducedinfectivity,lowerviralloadscanonlybeachievedwithadherencetothetreatment,whichisdiculttomaintain[ 3 ]andmaynotevenbethebestimmediatecourseofactionforthepatientearlyininfection[ 19 ].But,delayedstartofdrugtreatmentmeansdelayedreductionininfectivity.Finally,widespreaduseoftreatmentincreasestheriskofproducingmultiresistantstrainsofHIV.However,delayedstartofdrugtreatmentmeansdelayedreductionininfectivity. IthasalsobeenconrmedthattherehavebeenincreasesinbacterialSTDsandriskbehaviorscorrelatedwiththeintroductionofHAART[ 15 ].ThepopulationasawholeperceivesareduceddangerinHIV/AIDS,butthemostsignicantissuemightbethefalseovercondencethattreatmentpreventstransmission.Individualswhobelievethattreatmenteectivelyreducesinfectivitymightincreaseriskybehaviors,forinstance,reducingcompliancewiththeuseofcondomsandincreasingthenumberofnewsexualpartners. Basedonastudyusingselfreportedsurveys,theCentersforDiseaseControlandPrevention(CDC)foundthatinfectedindividualswhoknowoftheirHIVpositivestatushaveasignicantreductioninhighriskbehaviorfrombeforetheylearnabouttheir 17 PAGE 18 5 ].However,otherstudies[ 16 17 ]haveshownthatHIVpositivepatientsaregettinginfectedwithbacterialSTDsatratescomparabletoorhigherthanthoseofHIVnegativepatients.InastudyinBrazil,theincidenceofnewgonorrheainfections(anacutebacterialSTD)inwomenwithknownHIVinfectionwas12.9%,whiletheincidenceamongwomenwhoareHIVnegativewas8.33%inpromiscuouswomenand2%innonpromiscuouswomenforthesamecohort[ 16 ].AstudyinNigeriafoundthat14%ofHIVpositiveindividuals,butonly2.0%ofHIVnegativeindividuals,testedpositiveforsyphilis[ 17 ]. TheBrazilianandNigerianresultsraisethequestionoftheeectivenessofchangesinbehaviorduetotheknowledgethatoneisseropositive(seeTable 1 ).EvenifpersonsthatbecomeawareoftheirHIVinfectionbelievethattheyaresubstantiallyincreasingprecautionstopreventthespreadofHIV[ 5 ],thefactthattheyareacquiringbacterialSTDssuggestthatthepreventionmeasurestheyaretakinghavenotbeensucient. ItisclearthenthatassumingthattreatmentwillpreventthespreadofHIVthroughreducedinfectivityorthroughchangeinbehaviorwithoutcounselingisnotideal.HereIwillarguethat,ifenoughtestingandmedicalcounselingareavailable,itispossibletoachievecomparablepreventionresultsbyfocusingtraditional,costeectivepreventionprogrammesofeducationandsupportdirectlyonthesubpopulationreceivingHIVrelatedmedicalcare.Theseprogrammespreventnewinfectionsduetoincreasedcompliancewithcorrectandconsistentcondomuseandreducednumberofsexualpartners.IalsoillustratethatincorporatingeducationprogrammesintoHIVrelatedmedicalattentioncanbesubstantiallymoreeectivethanrelyinginreducedinfectivityalone.Thusitisdesirabletoincorporatepreventivechangesinbehaviorforindividualswhobecomeawareoftheirseropositivestatusandreceivetreatment. 18 PAGE 19 23 ]. Suchthresholdphenomenahavebeencentralinmathematicalepidemiologyeversince[ 23 25 ].Traditionally,thethresholdthatdeterminestheabilityofaninfectiousdiseasetoinvadeastablesusceptiblepopulationhasbeencalledthebasicreproductivenumber,anddenotedRo.Thebasicreproductivenumbershouldreectthereproductivesuccessofapathogeninahostpopulation[ 25 ].Inclassicalmathematicalepidemiologythereproductivenumberofthediseasehasbeendenedasthenumberofsecondarycasesofthediseasethatonetypicalinfectiveindividualwillproduceoverhisorherinfectivelifetimeinanentirelysusceptiblestablepopulation[ 23 25 ].Avaluegreaterthanoneisneededfordiseasepersistence. Inmodelsthatarehomogeneouswithrespecttoinfectivityofindividualswithinapopulation,itispossibletoconstructthebasicreproductivenumberinastraightforwardandintuitivemannerfromthisdenition.WiththeappearanceandspreadofAIDSworldwide,moreattentionhasbeendrawntothedynamicsofsexuallytransmitteddiseases(STDs)[ 23 24 ].OneofthemostimportantanddistinguishingaspectsofmodelingSTDtransmissionversusclassicalmodelingofinfectiousdiseasedynamicsliesinthesubstantialheterogeneityoftransmissionwithinthepopulation.Numberofcontactsrankshighindeningthisheterogeneity,togetherwithissuesofsocialandinterpersonalrelationships[ 23 ].Itisknownthatsexualpartneracquisitionratesvaryenormouslyamongcommunitiesandamongindividuals,possiblyrangingfromlessthan1to100partnersperyear[ 24 ].Aconsiderationofcontactprocessesiscentraltotheunderstandingofthresholdphenomenasuchasthebasicreproductionnumber[ 23 ]. 19 PAGE 20 24 ],[ 25 ],[ 26 ],[ 27 ]),ornonhomogenousmixingofindividuals(see[ 28 ])faceamajorprobleminthecomputationofRobecausethemathematicaldescriptionofwhatisatypicalinfectiousindividualisdiculttoachieveinpopulationswithhighdegreesofheterogeneity[ 23 ].Manyresearcheshavesoughttoresolvethisissueeitherbyassumingthatthepopulationinquestionishomogeneousinitsinfectivityandsusceptibilityusinganaveragevalueforeachindividualor,probablymoreappropriately,byincorporatingtheheterogeneityofthepopulationinthemodelandthenlettingRobetheappropriateweightedaverageoftheheterogeneoustrait(suchasnumberofsexualpartners)inthepopulation(see[ 23 ],[ 24 ],[ 29 ]). ThenumberofnewsexualpartnersconstitutesanimportantaspectofheterogeneitywithinapopulationwhenweconsiderepidemicsofSTDs.ThedistributionofthisnumberhasbeenusedforheterogeneousmodelformulationsofHIV/AIDS[ 23 29 ].HereIplantoincorporateasimilarapproachtotheissuesofheterogeneityinnumberofsexualpartnersinapopulationwithamoreexplicitrenderingofRowheretheeectsofheterogeneityinacquisitionofnewsexualpartnerscanbeexplicitlytreated,butstillmaintainingageneraldelitytothedenitionofRo.Asimilarapproachwasusedin[ 24 ],buttheretheheterogeneityinquestionwasintheintrinsicviralinfectivityvariabilitybetweenindividuals,basedontheassumptionthattherewasacorrelationbetweenplasmaviralloadandinfectivity. 20 PAGE 21 Iconsideracompartmentalmodelofapopulationthatconsistsofsusceptibleindividuals(S),HIVinfectedindividualswhohavenotdevelopedAIDSandarenotreceivingtreatmentorHIVrelatedmedicalsupport(U),HIVinfectedindividualsundertreatmentand/orotherHIVrelatedmedicalsupportwhohavenotdevelopedAIDS(T)andinfectedindividualswhohavedevelopedAIDS(A).ThetermAIDSwillapplyhereonlytothemoreadvancedstageofHIVinfectioncategorizedbyaCD4+Tcellcountbelow200cellspercubicmillimeterofblood,coupledwithsevereclinicalconditions,mostofwhichareopportunisticinfections[ 7 ].Forthisreason,IassumeherethatindividualsinclassAareremovedfromthesexuallyactivepopulation,giventhattheyhavedevelopedsevereopportunisticinfectionsand/orcancersandareassumedtobeseverelydebilitatedorhospitalized[ 7 ].IonlyconsiderthetransmissionofHIVviasexualintercourseandthereforethepopulationunderconsiderationissexuallyactive.Itakeintoaccountthatsusceptibleindividualsandinfectedindividuals(bothwithandwithoutmedicalsupport)maydierinthenumberofsexualpartnersperunitoftime.Thus,Isubdividethesusceptibleandinfectedclassesintoseveralsubclasses(Si;Ui;Ti;i=1;:::;m)withicorrespondingtothenumberofsexualpartnersthatindividualsineachsubclasswillhaveperunitoftime.Itisassumedthatformationofsexualpairsisrandom(doesnotdependonsubclass). Leti;i=1;:::;m,betherecruitmentratesofindividualstothesexuallyactivesusceptibleclasses(Si),bethenaturaldeathrateandremovalrateanddbethediseaseinduceddeathrate.Uponasexualencounter,withoutprotection,withaninfectedindividualfromsubclassesUiorTi,asusceptibleindividualwillbecomeinfectedwithprobabilityiandi,respectively.Toaccountfortheeectsofcondomuse,weassumecondomshaveanecacy"andcomplianceui.Sotheproducti="uirepresentsthecondomprotection. 21 PAGE 22 21 ThetotalpopulationsizeisgivenbyN=Pmi=1Si+Pmi=1Ui+Pmi=1Ti+A,anditsatisestheequation dt=mXi=1iNdA:(2{1) WealsodenethetotalpopulationwithineachclassasNiwhereNi=Si+Ui+Ti. Standardincidenceisappropriateforlargepopulations[?]andisusedherewiththeresultthatinteractionsaremadeindependentoftotalpopulationsize.Theideabehindthisbeingthatanincreasedpopulationsizedoesnotnecessarilymakeindividualsmorepromiscuous. Thuswegetthefollowingsystemofnonlineardierentialequationsfori=1;2;:::;m, dt=mXj=1jUj+mXj=1jTj(+d)A(2{5) 22 PAGE 23 Listofvariablesandparametersandtheirmeaning Variable Description Description 23 PAGE 24 2{2 )( 2{5 ))fortwodiscretesubclassesofsusceptibleindividuals,infectedindividualsnotontreatmentorHIVrelatedmedicalsupport,andinfectedindividualsundergoingtreatmentand/orreceivingHIVrelatedmedicalsupport.IletSn,UnandTnbethesubclassesofindividualsinthesusceptible,infectedbutnotundertreatmentormedicalsupport,andinfectedundertreatmentand/ormedicalsupportclasses,respectively,whohaverelativelysmallnumberofnewsexualpartnersperunitoftime.Thesewillhenceforthbecollectivelyreferredtoasthemoderatelysexuallyactiveclass.SimilarlyIlet(Ss),(Us)and(Ts)bethesubclassesofindividualsinthesusceptible,infectednotundertreatmentormedicalsupport,andinfectedundertreatmentand/ormedicalsupportclasses,respectively,whohaverelativelylargenumberofnewsexualpartnersperunitoftime.Thesewillbehenceforthcollectivelyreferredtoasthehighlysexuallyactiveclass.IletnbetheaveragenumberofnewsexualpartnersthatindividualsinSn,UnandTnacquireperunitoftimeandsbetheaveragenumberofsexualpartnersthatindividualsinSs,UsandTsacquireperunitoftime.Ithenconsiderthegeneralmodelabovefori=n;s. Iwillfocusrstonaspecialcaseofthismodelassumingnoindividualsgettreatment(i=0).Iwill,inthiscase,letIibethetotalnumberofinfectiveindividualswithinumberofnewsexualpartnersperunitoftime. Thenthemodel(( 2{2 )( 2{5 ))simpliestothefollowing: dt=nIn+sIs(+d)A(2{10) 24 PAGE 25 dt=n+sNdA(2{11) 2{2 )( 2{5 ))includingmedicalinterventionandtreatment.ThemodelisdescribedbytheowchartdiagraminFigure 21 Figure21. Diagramofthemodelwithtreatment For(t)=(n(1n)nUn+s(1s)sUs+n(1n)nTn+s(1s)sTs),theresultingmodelisgivenbythefollowingsystemofnonlineardierentialequations: 25 PAGE 26 dt=nUn+sUs+nTn+sTs(+d)A(2{18) dt=n+sNdA(2{19) 26 PAGE 27 dt=nUn+sUs+nTn+sTs(+d)A(2{26) dt=n+sNdA(2{27) 27 PAGE 28 2{6 )( 2{11 )hasauniquediseasefreeequilibriumwhichisgivenby"o=(n 2{6 )( 2{11 )equaltozero,wegetSn=n Thefollowingresultsconcernwiththelocalstabilityofthediseasefreeequilibrium(DFE). Inowlinearizethesystem( 2{6 )( 2{11 )aroundtheDFE.TheJacobianmatrixforthissystem,whichIwillcallJfromhereon,attheDFEisgivenasfollows: Figure31. Themodelwithouttreatment:diagram 28 PAGE 30 thenthedeterminantofJoispositive. Ifn(1n)nnn 30 PAGE 31 ThereforeIreachtheconditionthatdeterminesthelocalstabilityofoursystemattheDFE.Ifdet(Jo)>0,thenIshowedthatthetraceofJo<0andTheorem5.4[ 30 ]impliesthattheeigenvaluesofJohavenegativerealpart.ThenalleigenvaluesofJ(DFE)havenegativerealpart.ThereforetheDFEislocallyasymptoticallystable.Ifcondition 3{1 isnotsatised,det(Jo)<0,thenJohasaneigenvaluewithapositiverealpart.ThustheDFEisunstable. IcannowwritetheabovethresholdconditionasRc>1forRc=n(1n)n 2{6 )( 2{11 )islocallyasymptoticallystableifRc<1andunstableifRc>1,forRc=n(1n)n 3.2 IshowthatthevalueRccanactuallybeunderstoodintermsofbasicreproductionnumber,Ro,whichepidemiologicallydescribesthereproductivesuccessofthepathogeninasusceptiblehostpopulation.Biologically,stabilityoftheDFEmeansthattheinfection,ifinitiallyrare,willfadeaway. 23 31 ]. 31 PAGE 32 29 31 ]. HereImakeadistinctionbetweenthebasicreproductivenumberofthediseaseinthepopulation,theusualRo,orpopulationRo,andtheintrinsicRoofanindividualinthepopulation.ThepopulationRoisthemeasurementofreproductivesuccessofthepathogeninthehostpopulation.TheintrinsicRoisthenumberofsecondarycasesofthediseasethatoneinfectiveindividualwillproduceoveritsinfectivelifetimeinanentirelysusceptiblestablepopulation.ThelatteriscommonlyusedintheconstructionofthepopulationRoinhomogeneouspopulations[ 31 ]. Iconsiderthenumberofsecondarycasesthatoneindividualinthemoderatelyactivesubclass(In)willproduceoveritsentireinfectivelifetime.TheaveragerateatwhichaninfectiveindividualinInleavestheinfectiveclassisn+,andthereforetheaveragetimethatheorsheremainsinfectiveisgivenby1 32 PAGE 33 ThusIhaveRno=n(1n)n OnecanseenowthattheoverallpopulationRo(themeasurementofreproductivesuccessofthepathogenintheentirepopulation)thatwasderivedbytheanalysisofthelocalstabilityoftheDFEisgivenbyaweightedsumoftheintrinsicreproductivenumberofhostsineachofthetwohostclasses.Morespecically,Ro=Rno(nSn 23 ]. Thiscanbeunderstoodthefollowingway.Iconsideratypicalinfectiveindividualtobeanindividualchosenatrandomfromtheoriginalsusceptiblepopulationandthenmadeinfective.Anindividualischosenthroughasexualcontact.TheprobabilityofarandomsexualcontactbeingwithanindividualoftypeSiisii 33 PAGE 34 Itispossibletorewritethebasicreproductionnumberinamoreintuitivewayastheproductofthemeandurationofinfection,meanprobabilityoftransmissionperpartner,andanappropriatelyweightedaveragenumberofpartnersperunittime.ThisallowsRotobeinterpretedastheaveragenumberofindividualsthatanindividualinfectedatrandomwillinfectinacompletelysusceptiblepopulation. 2{6 )( 2{11 )hasatleastoneendemicequilibriumwhichisgivenby"=(Sn>0;Ss>0;In>0;Is>0;A>0). 2{6 )( 2{11 )atequilibriaandwegetthefollowingequations: 0=nSn(n(1n)nIn+s(1s)sIs)nSn 0=sSs(n(1n)nIn+s(1s)sIs)sSs 0=(n(1n)nIn+s(1s)sIs)nSn 0=(n(1n)nIn+s(1s)sIs)sSs 0=nIn+sIs(+d)A(3{6) 0=n+sNdA(3{7) NowIproceedtondanendemicequilibriumandthereforeassumeIn6=0orIs6=0. 34 PAGE 35 2{6 )( 2{11 )canonlyhavetwokindsofequilibria:thediseasefreeequilibriumgivenby"o=(n Fromequations( 3{4 )and( 3{5 )Igetthefollowing (n(1n)nIn+s(1s)sIs)nSn (n(1n)nIn+s(1s)sIs)sSs Let=n(1n)nIn+s(1s)sIs.Multiplyingbothsidesofequation( 3{8 )byn(1n)n 3{9 )bys(1s)s 3{8 )and( 3{9 )Iget ThenIcanwriteequation( 3{10 )as ThusIgetthefollowingrelation(assuming6=0) NowIrewriteequations( 3{2 )and( 3{3 )using( 3{12 ): 0=RnonRnoSnRnonSn 0=RsosRsoSsRsosSs Nowaddingequations( 3{13 )and( 3{14 )Igetthefollowing 0=Rnon+RsosRnoSnRsoSs(3{15) 35 PAGE 36 Equation( 3{14 )canberewrittenas 0=sSssSs NowImultiplyequation( 3{17 )byRnonSn+RsosSstogetthefollowing 0=(sSs)(RnonSn+RsosSs)sSs(3{18) Nowrewriting( 3{18 )using( 3{16 )asfollows 0=(sSs)(RsosSs+n(Rnon+Rsos RsoSs))sSs(3{19) 0=(sSs)(Rso(sn)Ss+n(Rnon+Rsos ))sSs(3{20) Multiplyingeverythingoutin( 3{20 )IgetthefollowingquadraticequationinSs Sinces>nthereisauniquepositivesolutiontoequation( 3{21 )forall0 PAGE 37 +nnSn +ssSs Recalling( 3{12 )Ihave +nn2fn() +ss2fs() (Rno1)nfn()+(Rso1)sfs()= +nn2fn() +ss2fs() Let +nn2fn() +ss2fs() ThefunctionsF()andG()arecontinuousfunctionsoffor0<. At=0,G(0)=0andF(0)=(Rno1)nn Toseethatthislimitexists,considerequations( 3{2 )and( 3{3 ). SothequotientSn 37 PAGE 38 lim!fn() Thatis,thelimitlim!fn() Inaddition,lim!F()=0.Consequently,G()>F(). TheIntermediateValueTheoremimpliesthatthereexistsatleastoneendemicequilibriumforthesystem( 2{6 )( 2{11 ). Thereisatleastonewith0 PAGE 39 2{6 )( 2{11 ). ParametersofthemodelswereestimatedfromtheliteratureandaresummarizedinTable 41 .Valuesforthesimulationswherechosenwithintheseranges.Below,Idiscussbrieytheseestimates. Partneracquisitionratesvarygreatlywithinpopulationsofnonmonogamoussexuallyactiveindividuals,rangingfrom1peryearto100partnersperyear[ 24 ].Theremovalrate,thattracksnaturaldeathrateandremovalfromthesexuallyactivepopulationbychangesinsexualbehaviorwastakenfrom[ 24 ].Theindividualsinquestionareassumedtobesexuallyactiveyoungadultswhoareexpectedtoliveanaverageof50yearsandengageinnonmonogamoussexualrelationsfor20years[ 24 ].Theparametersnands,thattrackthedurationofinfectivity,orthetimeittakesforaninfectedindividualtodeveloptheterminalsymptomsofAIDSandbethusremovedtotheclassofAIDSpatients(whoarenotsexuallyactive),aretakenclosetovaluesfoundin[ 32 ],withvariationsintroducedherecorrelatingwithlevelofsexualactivity.Thestudyin[ 32 ]waschosenfortheseparametersbecausethisstudywasperformedbeforetheintroductionofHAARTtherapyandmanyotherantiretroviraltreatmentsandthereforedoesnotincorporatetheimpactoftreatmentsintheaveragetimetothedevelopmentofAIDS.Thevaluesforn,n,s,s,theprobabilityoftransmissionperpartner,dependonthenumberofcontactsperpartnerandtheprobabilityoftransmissionpercontact[ 24 ],aswellasinfectivityofseropositiveindividuals.Estimatesonthetransmissionprobabilitypersexualcontactrangefrom0.0003,fortransmissionfromfemaletomale,to0.08,fortransmissionfrommaletomale[ 24 ].Thenumberofcontactsperpartneristakenfrom[ 24 ]tobe2 39 PAGE 40 Simulationparametersandtheirvalueranges ParameterValue Descriptionreference 24 ]n:90T 33 ]s:10T 33 ]0.01month1 24 ]d0.125month1 24 ]i;i;io0.0003{0.64 Infectivity(Probabilityoftransmissionpersusceptiblepartnerwithoutpreventivestrategies)[ 24 34 ]"80%95% Condomecacy(intrinsic)[ 11 ]un35%50% Compliancewiththeuseofcondomforindividualsinsteadyrelationships[ 35 ]us59%63% Compliancewiththeuseofcondomforindividualsengagingincasualsexualrelationships[ 35 ]i 3 7 ]1%100%(mean80%) ProportionofHIVinfectedindividualswhoknowtheirstatusandreceivetreatmentand/orothermedicalsupport[ 3 ]i0.01month1 32 ]i0.0050.006month1 19 ] *T:Totalrecruitmentintosexuallyactivepopulation 40 PAGE 41 IchoseparametersforthesimulationswithintherangesinTable 41 andsummarizetheminTable 42 .Iinvestigatetwomajorpreventionfocalpoints:reducingnumberofpartnersandcondomuse.Ifurtherinvestigatepossiblecountereectsofthesefocusedpreventioneorts.Forthecaseofaneorttoreducethenumberofnewsexualpartnersperunitoftime,IexplorethechangesinthevalueofRowhenthereisnootherchangeinbehavior,andwhenindividualshaveareducedcompliancewiththecorrectandconsistentuseofcondomsbecausetheyfeelmoresecurewithfewerpartners.Similarly,Iconsiderthestrategyofpreventionthatfocusesonincreasingthecorrectandconsistentuseofcondoms.Isimulatethescenariowherethereisnootherchangeinbehaviorandthepossibleeectsofincreasedriskbehaviorinincreasednumberofnewpartnersperunitoftimethatcanresultfromovercondenceincondomprotection. ResultsofthesimulationaregiveninFigures 41 and 42 .Itcanbenotedthatgivenenoughcompliancewitheitherofthefocusedpreventionstrategies,Rodecreasesevengivensomecountereecttotheprogramme.However,itisalsoclearthatwhentherearesecondarychangesinbehavior,forexampleanincreaseinnumberofsexualpartnersduetoovercondenceincondomprotection,thepreventionstrategybecomeslessecient.Theresultsseemtosuggestthatthereshouldbesomeconcernwithpreventingsecondarycountereectivechangesinbehaviorwhenimplementinganypreventionprogramthatfocusesmostlyinoneareaofprevention,butalsothataslongassuchcountereectivechangesarekeptatmanageablelevels,thefocusedpreventionprogramisstillbetterthannopreventionatall. Theexplicitdivisionofthepopulationintotwodiscreteclassesbasedonthenumberofsexualpartnersthatindividualsacquire,allowsfortheinvestigationoftheeectofreducingtheproportionofthepopulationthatbecomeshighlysexuallyactive.Eorts 41 PAGE 42 Specicparametervaluesusedinsimulations ParameterValue Description Probabilityoftransmissionfrommoderatelysexuallyactiveinfectiveindividualpersusceptiblepartnerwithoutpreventivestrategiess0.02 Probabilityoftransmissionfromhighlysexuallyactiveinfectiveindividualpersusceptiblepartnerwithoutpreventivestrategies90% Condomecacy(intrinsic)un40% Compliancewiththeuseofcondomforindividualsinsteadyrelationshipsus60% Compliancewiththeuseofcondomforindividualsengagingincasualsexualrelationshipsn36% Condominducedprotection(i=ui)s54% Condominducedprotection(i=ui)n0:036month1 ProportionofHIVinfectedindividualswhoknowtheirstatusandreceivetreatmentand/orothermedicalsupportn0.09month1 42 PAGE 43 Preventionfocusedondecreasingnumberofpartnersonly.Letpbethepercentagedecreaseinaveragenumberofnewsexualpartnersperunitoftime.Thus,averagenumberofnewsexualpartnersperunitoftimeofmoderatelysexuallyactiveclasshereisgivenbyn(1p)andaveragenumberofnewsexualpartnersperunitoftimeofhighlysexuallyactiveclassisgivenbys(1p).GraphshowsRoversusp.Letqbethepercentagedecreaseincorrectandconsistentcondomuse.Thencondomuseprotectionisgivenby1(un(1q))formoderatelyactivegroupand1(us(1q))forhighlyactivegroup.Ishowcurvesforq=p,q=0:5pandq=2p.ThelineS1isthebaselinecasewherethereisareductioninnumberofpartnersandnootherchangeinbehavior.CurvesS2S4illustrateareductioninnumberofpartnerscoupledwithanincreasingreductioninthecorrectandconsistentuseofcondoms. 43 PAGE 44 Preventionfocusonincreasingcorrectandconsistentcondomuseonly.Letpbethepercentageincreaseincorrectandconsistentcondomuse.Thus,condomuseprotectionisgivenby1(un(1+p))formoderatelysexuallyactiveindividualsand1(us(1+p))forhighlysexuallyactiveindividuals.GraphshowsRoversusp.Letqbethepercentageincreaseinaveragenumberofnewsexualpartnersperunitoftime.Thenaveragenumberofnewsexualpartnersperunitoftimeofmoderatelysexuallyactiveclasshereisgivenbyn(1+q)andaveragenumberofnewsexualpartnersperunitoftimeofhighlysexuallyactiveclassisgivenbys(1+q).Ishowcurvesforq=p,q=0:5pandq=2p.ThelineS1isthebaselinecasewherethereisincreaseincorrectandconsistentcondomuseandnootherchangeinbehavior.CurvesS2S4illustrateanincreaseincondomusecoupledwithanincreaseinnumberofnewsexualpartnersperunitoftimeasaresultofpossibleovercondencethattheuseofcondomswillprotectagainstthetransmissionofHIV. 44 PAGE 45 Preventionfocusedonchangingrecruitmentonly.Changesinrecruitmentproportionscanbeusedasapreventionstrategywithlowpossiblecountereects. toeducateyoungpeoplewhoareabouttobecomesexuallyactiveaboutsafersexualpracticesinvolvingalownumberofdierentsexualpartnershasaclearimpactindiseasepreventionasillustratedinFigure 43 Theparametersnandsrefertotherateatwhichsexuallyactiveindividualsgettested.AccordingtotheU.S.DepartmentofHealthandHumanServices[ 7 ]andtheUNAIDSReportontheGlobalAIDSEpidemic[ 4 ],intheU.S.onequarterofthose 45 PAGE 46 3 ].IfocusonthecurrentscenarioinU.S.andlettreatmentavailabilitybeat100%.AccordingtotheCDC,HAARTtreatmentcanjustaboutdoublelifeexpectancyofseropositiveindividuals[ 19 ]. First,Iexploretheeectsoftreatmentintheabsenceofanyfurtherpreventionstrategy.Someliteraturesuggeststhateventhoughtreatmentlowerstheplasmavirusloadofseropositivepatients,thisreductiondoesnotcorrelatewithareductioninvirusloadinotherbodyuids,suchassemen[ 13 ].However,somestudiesofcoupleswithdiscordantserostatus,individualswithlowervirusloadhavealowerprobabilityoftransmittingHIV[ 5 ],whileitisnotclearbyhowmuch.Ishowtheeectsofreducedinfectivityofindividualsundergoingtreatment(Table 43 ).Theincidencewithouttreatmentwasof146individualsin100,000.WithoutreductionofinfectivityorchangeinriskbehaviortheintroductionoftreatmentincreasesincidenceofHIVinfections.Withadecreaseininfectivity,introducingtreatmentreducestheincidenceofHIVinfection.ItisimportanttorecallthatnotallindividualswhoreceiveHIVrelatedmedicalattentionareactuallytakingHAARTdrugs.SomehealthprofessionalsdelaythestartofHAARTbutstillprovidemedicalattentionintheformofviralloadscreening,bacterialSTDstreatment,andtreatmentofopportunisticinfections.Forthissimulation,however,IassumeindividualsreceivingmedicalinterventionareactuallyreceivingHAARTdrugs. 46 PAGE 47 ReducedinfectivityinindividualsreceivingHAARTtreatment InfectivityreductionIncidencein100,000 0%151 10%140 20%135 30%129 40%120 50%112 60%103 70%92 80%78 90%60 Irecallthattoeectivelyreduceviralloadthereneedstobeadherencetotreatmentatalltimes(whichhasproveddicult),thatdelayedstartofHAARTdrugsmightbeappropriate,andthattreatmentalwayscariestheriskofproducingmultiresistantstrains. 2{20 )( 2{26 ).Asaresultofexpandedtreatmentaccess,millionsofpeoplelivingwithHIVareperiodicallyvisitinghealthcaredeliverysitestomonitortheirdiseaseandtreatmentprogress[ 3 ].Ibelievethisprovidesanimportantpaththroughwhichtoincorporatepreventionprogrammesthatfocusonbehaviorchange.IncountriesliketheUnitedStates,whereonlyabout25%ofHIVinfectionsareunknown,manyindividualswhoknowoftheirseropositivestatushavetheopportunitytoreceivemedicalsupportandtreatmenttoimprovetheirlifestyleandincreasetheirlongevity.IexploretheoutcomeofintegratingchangeofbehaviorpreventionprogrammesfocusedontheseindividualswiththedeliveryoftreatmentandHIVmedicalsupport. Assumingeectivepreventioneortsimplementedthroughintegrationofpreventionandmedicalsupport,ingeneral,assumethatno PAGE 48 44 andTable 45 Table44: PrevalenceofHIVinfectionwithincreaseincondomuseanddecreaseinnumberofsexualpartnersperunitoftime IncreasedpreventionFortheentirepopulationForthegroupundertreatment(5%oftotalpopulation) 0%7.0%7.0% 5%6.5%6.7% 10%6.0%6.4% 15%5.4%6.0% 20%4.7%5.8% 25%3.9%5.4% Table45: Incidence(per100000)ofHIVinfectionswithincreaseincondomuseanddecreaseinnumberofsexualpartnersperunitoftime IncreasedpreventionFortheentirepopulationForthegroupundertreatment(5%oftotalpopulation) 0%112112 5%104109 10%96100 15%8598 20%7690 ObservethatwithincreasedeectivepreventionaimedonlyatthesubgroupofthepopulationconsistingofseropositiveindividualsreceivingmedicalsupportthevaluesfortheresultingprevalenceofHIVinfectioninthepopulationareveryclosetothevaluesreachedbyincreasingeectivepreventionontheentirepopulation.Thegroupundertreatmentcorrespondsto5:25%oftheentirepopulation. Toreachanincidencelevelof90newcasesin100000perunitoftime,thereneedstobeanincreaseof20%inpreventioninthegroupundertreatment,whileforthesamelevelofincidencethereneedstobearound12%increaseinpreventionintheentirepopulation.Noticereaching20%oftheindividualsinthetreatedgroupcorrespondstoeectivelyreaching1000individuals,whilechanging12.5%oftheentirepopulationcorresponds 48 PAGE 49 HereIreachmymainresult:giventhattestingandtreatmentavailabilityarecomparabletotheUnitedStates,itisnotnecessarytoeectivelyreachtheentirepopulationwithincreasedpreventionprogrammestoreducetheprevalenceandincidenceofHIV.ItissucienttoeectivelychangethebehaviorofthesubgroupofthepopulationcomposedofindividualsofknownseropositivestatuswhoarereceivingHIVrelatedmedicalsupportandcounseling.Butmorethanthat,theseresultsshowthatcouplingpreventioneortswithtreatmentdeliverycouldbeasaseectiveastargetingpreventiontotheentirepopulation. ThisconclusiondemonstrateshowlargeanimpactindividualsofknownseropositivestatuswhoreceivemedicalinterventioncanhaveinthepreventionofHIV.Byactivelyincorporatingandimprovingpreventionstrategies,individualswhoareawareoftheirseropositivestatuscaneectivelyhelpreducetheprevalenceofHIVinthepopulationandslowtheepidemic.ItisveryimportantthennottoneglectHIVpreventionrelatededucationfortheseindividuals.ItisclearfromreportsoftheCDC[ 5 ]thatindividualswhobecomeawareoftheirseropositivestatusdoactivelytrytopreventfurthertransmissionofthedisease.Therefore,itisnecessaryforsocietytoprovideappropriatetoolsfortheseindividualstoeectivelyhelpintheghtagainstHIV/AIDS.Itisnecessarytoprovideclearinformation,resources,andsupporttothoseindividualswhoareinfectiveandwhoarereceivingHIVrelatedmedicalattention. 49 PAGE 50 Themodelwithtreatmentanddierentiatedrateofpartneracquisition,whichtomyknowledgehasnotbeenpreviouslystudied,accountsforheterogeneitiesamongindividualsintheirbehaviortowardsacquisitionofnewpartnersanduseofcondomswhileexploringtheeectsoftreatmentandmedicalsupport,withandwithoutpreventioncounseling,ontheoveralldiseasedynamics.Themodelwithouttreatmentaccountsforheterogeneitiesamongindividualsintheirbehaviortowardsacquisitionofnewpartnersanduseofcondoms,butemphasizesthesemechanismsexplicitly,especiallyintheexplicitrenderingofRo,andgivesfurtherinsightonpreventionstrategies. ForthemodelwithouttreatmentIderivedtheexplicitformulaforthebasicreproductionnumberandprovedexistenceofatleastoneendemicequilibriumwhenRo>1.IdiscussthevalueofthebasicreproductionnumberinviewofothermoretraditionalepidemiologicalpapersandthemeaningofthisvalueofRo. Usingsimulations,IexaminedthetransmissiondynamicsofthediseaseinapopulationwhereHAARTisintroduced,withandwithoutfocalpreventioneorts.ThemainresultIreachisthatpreventionprogrammesdonotneedtoreachtheentirepopulation,butonlyneedtofocusonthegroupreceivingHIVrelatedmedicalattention.Thisapproachtopreventioniscosteective,sincethetargetpopulationissignicantlysmallerthanthetotalpopulation,sincethereisaclearchannelforthedeliveryoftreatment,andsincethischanneldoesnotrelyonthecostlyantiretroviraldrugsforreducedinfectivity.Couplingpreventionprogrammes,suchascounselingandeducationprogrammes,withthedeliveryofHIVrelatedmedicalattentioncanliberateHAARTfrombeingusedasapreventionstrategy,sothatthestartofHAARTdrugscanbedelayedifnecessaryandtheriskofproducingmultiresistantHIVstrainscanbereduced.Thisapproachpreventsseropositivepatientsfromincreasingriskybehaviorduetoovercondenceintreatmentandreducedinfectivity.Therealsoexistsaclear 50 PAGE 51 WiththismodelIadvocatethathealthpoliciesforpreventionofHIVinfectioninapopulationshouldnotrelysolelyonreducedinfectivityduetotreatmentforthepreventionofthetransmissionofHIVfromseropositivepatients,butratheritshouldtakeanactiveapproachtopreventionfocusingonchangeofbehavioroftheseindividuals.Further,Iadvocatethatitismoreeectivetofocuschangeofbehaviorpreventiononthesubgroupofthepopulationreceivingtreatmenttoachieveasignicantreductionintheprevalenceofthedisease,givenratesoftestingcomparabletotheUnitedStates,thanareindiscriminatemeasures. Intheabsenceofacureorvaccine,theworldmustrelyoneectiveimplementationofpreventionstrategiesallofwhichinvolvebehaviorchangeofthepopulationasawholeor,andmaybeespecially,oftheindividualsofknownseropositivestatus.Ideally,allinfectedindividualsshouldbetestedandprovidedwithtreatment[ 24 ],aswellasinformationandsupporttoincorporatechangesinriskbehaviorandpreventiontechniquesineverydaylife.Also,ideallyallindividualswhoareawareofhavingseropositivestatuswouldhavethewill,theresourcesandthesupporttoactivelyengageinprevention.However,changesinbehavior,especiallysexualbehavior,arealwaysachallenge. RelativelyfewstudieshavebeenundertakentomeasuretheeectivenessofbehavioralinterventionsforpreventionforpeoplelivingwithHIV,butemergingevidenceindicatesthatsuchprogrammesareeectiveinreducingthelikelihoodthatpeoplewithHIVwillengageinsexualactivitythatmightexposeotherstothevirus[ 36 ].Thus,itisveryusefultofocuspreventiononthisgroup. 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[37] Y.Huang,W.A.Paxton,S.M.Wolinsky,A.U.Neumann,L.Zhang,T.He,S.Kang,D.Ceradini,Z.Jin,K.Yazdanbakhsh,K.Kunstman,D.Erickson,E.Dragon,N.R.Landau,J.Phair,D.D.Ho,R.A.Koup.TheroleofamutantCCR5alleleinHIV1transmissionanddiseaseprogression.2(11)NatMed.(1996)1240. 55 PAGE 56 FernandaOliveiraMelowasbornonApril29,1981inBraslia,Brazil.Thedaughteroftwomedicaldoctors,shedevelopedapassionforthesciencesearlyon.Herearlyschoolyearswerespentmostlyinasmallbilingualschool,EscoladasNac~oes,inherhometown.InhighschoolsheenrolledintheAmericanSchoolofBraslia.In1998,shemovedtotheUnitedStateswithherparents.ShelivedinOrlando,Floridafor3years.ShecompletedhighschoolatDr.PhillipsHighSchoolandreceivedanassociateinartsdegreefromValenciaCommunityCollege.UponreceivingherA.A.shemovedtoGainesville,FloridatobeginherundergraduatestudiesinmathematicsattheUniversityofFlorida.InAugust2003,attheendofherjunioryear,shemarriedEdgarMelo.FernandawasawardedabachelorofsciencedegreeinmathematicswithaminorineducationinAugust2004.Shegraduatedwithmagnacumlaude.Shespecializedinappliedmathematicsandproduced,underthesponsorshipofDr.SergueiPilyugin,anundergraduatethesisentitledAnSIRmodelwithdiscreteimmunitysubclasses.InApril2005,herrstson,Lucas,wasborninKissimmee,Florida.ShestartedgraduateschoolattheMathematicsDepartmentintheUniversityofFloridain2005.Duringhertwoyearsasagraduatestudentthere,shehadtheopportunitytotakeindividualresearchcoursesunderbiomathematicianssuchasDr.MaiaMartcheva,Dr.SergueiPilyuginandDr.PatrickDeLenher.UpongraduatingwithherM.S.inappliedmathematics,FernandawillbemovingtotheZoologyDepartmentattheUniversityofFloridatocompleteadoctoratedegreeunderthementorshipoftheoreticalbiologistsDr.BenjaminM.BolkerandDr.RobertHolt.Thereshehopestofurtherherresearchintheareaofinfectiousdiseasedynamics,combiningtheoreticalbiologywithherknowledgeofbiomathematics. 56 