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The New Face of AIDS

Permanent Link: http://ufdc.ufl.edu/UFE0021184/00001

Material Information

Title: The New Face of AIDS A Mathematical Modeling Approach to New Trends in HIV Treatment and Prevention
Physical Description: 1 online resource (56 p.)
Language: english
Creator: Melo, Fernanda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: aids, epidemiology, haart, heterogeneous, hiv, modeling, r0, treatment
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My study develops mathematical 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 HIV-related medical attention. My study focuses on exploring the effects of the most recent trend having a major 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 HIV-related 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 programmes at the entire population,provided that availability of testing and HIV-related medical attention is adequate.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Fernanda Melo.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Martcheva-Drashanska, Maia.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021184:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021184/00001

Material Information

Title: The New Face of AIDS A Mathematical Modeling Approach to New Trends in HIV Treatment and Prevention
Physical Description: 1 online resource (56 p.)
Language: english
Creator: Melo, Fernanda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: aids, epidemiology, haart, heterogeneous, hiv, modeling, r0, treatment
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My study develops mathematical 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 HIV-related medical attention. My study focuses on exploring the effects of the most recent trend having a major 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 HIV-related 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 programmes at the entire population,provided that availability of testing and HIV-related medical attention is adequate.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Fernanda Melo.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Martcheva-Drashanska, Maia.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021184:00001


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THE NEW FACE OF AIDS: A MATHEMATICAL MODELING APPROACH TO NEW
TRENDS IN HIV TREATMENT AND PREVENTION


















By
FERNANDA OLIVEIRA MELO


A 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

UNIVERSITY OF 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 ahr-l- .- 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~! ahr-l-ns 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

2-1 List of variables and parameters and their meaning ... .. .. 23

4-1 Simulation parameters and their value ranges .... ... .. 40

4-2 Specific parameter values used in simulations .... .. . 42

4-3 Reduced infectivity in individuals receiving HAART treatment .. .. .. .. 47

4-4 Prevalence of HIV infection with increasing prevention ... .. .. 48

4-5 Incidence of HIV infections with increasing prevention ... .. .. 48










LIST OF FIGURES

Figure page

2-1 Diagram of the model with treatment . ..... .. 25

3-1 The model without treatment: diagram . ... 28

4-1 Prevention focused on decreasing number of partners only. .. .. .. .. 43

4-2 Prevention focus on increasing correct and consistent condom use only. .. .. 44

4-3 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 single-stranded RNA
into double-stranded 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 HIV-related 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

HIV-related 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 HIV-related 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 v-wsi~ 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 ahr-l- .- 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 counter-effects 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 co-receptor is also necessary for the virus

to invade a cell. The main co-receptor, 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), life-threatening infections [8-10] There is usually an .-i-mptomatic 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 spill-over 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 middle-income 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 resource-limited 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 [15-17]. 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 HIV-positive 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 female-to-male 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 li-e 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 self-reported sure vi- the Centers for Disease Control and

Prevention (CDC) found that infected individuals who know of their HIV-positive status

have a significant reduction in high-risk behavior from before they learn about their










status [5]. However, other studies [16, 17] have shown that HIV-positive patients are

getting infected with hacterial STDs at rates comparable to or higher than those of

HIV-negative 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 HIV-negfative was 8.t:: in promiscuous women and "'

in non-promiscuous women for the same cohort [16]. A study in Nigeria found that 1 1' .

of HIV-positive individuals, but only 2.(1' of HIV-negative 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, cost-effective

prevention programmes of education and support directly on the sub-population receiving

HIV-related 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 HIV-related

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 non-homogenous

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), HIV-infected individuals who have not developed AIDS and are not

receiving treatment or HIV-related medical support (U), HIV-infected individuals under

treatment and/or other HIV-related 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 2-1.

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.(2-1)
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,



= s-p, ( p~yp+ ( y:jy N (2-2)



d~dT
= (b ( + p)T+c~)U (2-4)
dTi

di = v6U, + pT, ( + d)Ai (2-5)

j= 1 j= 1















Table 2-1: 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 HIV-infected individuals, with i average number of new sexual
partners per unit of time, who have not developed AIDS and are
not under treatment or HIV-related medical support
Ti HIV-infected individuals, with i average number of new sexual
partners per unit of time, who have not developed AIDS but are
receiving treatment and/or HIV-related medical support
A HIV-infected 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 HIV-infected 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 HIV-infected individuals who know their status receive
treatment and/or HIV-related medical support
asi rate at which HIV-infected individuals with i average number of new
sexual partners per unit of time develop AIDS without treatment
Yi rate at which HIV-infected 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 ((2-2)-(2-5)) for two discrete subclasses of susceptible

individuals, infected individuals not on treatment or HIV-related medical support, and

infected individuals undergoing treatment and/or receiving HIV-related 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 ((2-2)-(2-5)) simplifies to the following:

dS, nS,
= s -pS, (n1 -ps~qI, s( p~al) "(2-6)
d t n N, +s N,

dS, sS,
= As- pS n(1 psal, s(1- p~qI) *(2-7)
d t n N, +s N,
dl, nS,
=(n( -ps~,I + (1- p~as) -(p + a~n)I, (2-8)
d t n N, + s N,
dl, sS,
= (n1 -ps~qI, s( p~al) (p+ a)Is(2-9)
dt nNV, +sNV,
dA
= 0,l, + asl, (p + d)A (2-10)










A, + As pNV dA


(2-11)


2.2 The Model With Treatment but Without Change in Behavior

I will also consider the above model ((2-2)-(2-5)) including medical intervention and

treatment. The model is described by the flow chart diagram in Figure 2-1.


Figure 2-1. 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)(2-12)
dt ` nNV, + slV,


dS, sS,
= A, s -, A(t)
dt nNV, + slV,


(2-13)










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


(2-14)

(2-15)

(2-16)

(2-17)

(2-18)

(2-19)


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


(2-20)

(2-21)

(2-22)

(2-23)

(2-24)

(2-25)


dU "
dt
dU "
dt


A(t) n
X)nN


,









dA
= anU, + CsUs + YnTn + YsTs (p- + d)A (2-26)
dtV
= An + As pNV dA (2-27)









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 disease-free equilibrium

(DFE).
Theorem 1. The system (2-6) (2-11) has a unique disease-free 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 (2-6) (2-11) equal to zero, we

get S, =" and S, = So there exists a unique disease-free equilibrium given by

Eo = ( 0, 0, 0).
The following results concern with the local stability of the disease-free equilibrium

(DFE).
I now linearize the system (2-6) (2-11) 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 3-1. 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) (3-1)
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 .I-i-angind' ;cally stable. If condition

3-1 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 "(1-Pn)lln nh,
(pL+a,) nh,+sh,
s(1-ps)s sA" This leads to the following result:

Theorem 2. The DFE for the system (2-6) (2-11) 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 long-ternt 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 (1-s"!"' 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 (2-6) (2-11) 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 (2-6) (2-11) 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, (3-4)
n(s + s~ ~ l s5


0 = c0,l, + adsl (p + d) A (3-6)


0 = A, +A,-pNV- dA


(3-7)


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 (2-6)-(2-11) can only have two kinds of equilibria: the disease-free

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 (3-4) and (3-5) I get the following

n S,
(n( -ps~l,+ s1 psqs)= (p + a~n)I, (3-8)

sS,
(n( -psq,, s( -psal) =(p + cas)Is (3-9)
nlV, + s Ns
Let A = n(1 p,)rl,, + s(1 p,)rl,Is. Multiplying both sides of equation (3-8) by

"Op'" ""77 and both? sides of eqluations (3-9) by "il"j")" anld th~en adding (3-8) and (3-9) I

get


n(1- p~q, nS, S(1 prs~q S
A + (3-10)
p-1 cs nNV,+sN, p-1+ s nNV,+N
Then I can write equation (3-10) as

nS s
R"A R"A (3-11)


Thus I get the following relation (assuming A / 0)


R~nS, + R~sS, = nNV, + slV, (3-12)


Now I rewrite equations (3-2) and (3-3) using (3-12):

R"ns,
O = R"As R~IpS, A (3-13)

R" sS
0 = R"As R~pS, A OUU (3-14)
RM S, + Es S,
Now adding equations (3-13) and (3-14) I get the following


0 =R"As R"As "pS "S (3-15)









which can be written in the form


p(R~"S, + R"S,) =R"As + R"As A (3-16)


Equation (3-14) can be rewritten as

0 = As pS, A (3-17)
RM S, + gs S,
N ow,, I multiply, equation (3-1 7)\ byT R nSn RSsS to get the following

\ = (As p~ )(RnS, + RS o) AsS (3-18)


Now rewriting (3-18) using (3-16) as follows

0 = (s ps)(R s+ n( O"As + gs -\ A R"S,)) AsS, (3-19)



0 A p(s ),+ n(RA + U"" A) AsS, (3-20)

Multiplying everything out in (3-20) I get the following quadratic equation in S,

O =( n)R, ca + (mnR", nRASh (8 M)R"Sh + (s n)A)Ss

n~h,(3-21'
*(A,R"+ AsR"- A)

Since s > a there is a unique positive solution to equation (3-21) 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 (3-4) and (3-5) where the denominator has been replaced

with the left hand side of equation (3-12).

nS,
X (p + a~n)I, (3-22)
nR gS, +sR gS,










X = (p + a~s)Is (3-23)
nn"S, sR:Ss


A nS,
In (3-24)
p + annR"S, +sRgS,
A s S,
Is (3-25)
p + a nR"S,+ sRgS,
Recalling (3-12) I have

R~a, +R"sS, = n(S, + I,) + s(S, + Is) (3-26)




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)n-von\ f ,(A)+(R 1)s fs(A) = os\

A 8 nzX 82 sasX (3-28)
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 (3-29)
A n2 nX 82 sfsx
G(A) = +(3-30)

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 (3-2) and (3-3).

n S,
A =ci As pS, (3-31)
R~nS, + Rss

sS,
A n A pS, (3-32)
RMnS, + RgsS,
So the quotient s" (An-pS"). ThuS f"(A) (An-pf"(A))
Ss n(As-pLS,) fs(A) n(An --pfs(A))









Now let A A *-
.fn(A) sA,
him (3-33)

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 (2-6)-(2-11).

There is at least one A with 0 < A < A* for which equation (3-21) 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 (3-16) I have pRgS, = R"As + Rgn, A**

At A** from (3-18) we have 0 = (As pS,)R~sS, A**sS, and thus 0 = AsRg

p-R"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 (2-6)-(2-11).

Parameters of the models were estimated from the literature and are summarized in

Table 4-1. Values for the simulations where chosen within these ranges. Below, I discuss

briefly these estimates.

Partner acquisition rates vary greatly within populations of non-monogamous 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 non-monogamous 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 anti-retroviral 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 4-1: 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 month-l Natural death rate and removal from [24]
sexually active class
d 0.125 month-l 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 0.0667month-1 Rate at which HIV-infected individuals [3, 7]
undergo testing and know their status
6 1.-11'(ma I') Proportion of HIV-infected individuals [3]
who know their status and receive
treatment and/or other medical support
asi 0.01 month-l Rate at which HIV-infected individuals [32]
develop AIDS without treatment or
HIV-related medical support
Yi 0.005 0.006 month-l Rate at which HIV-infected individuals [19]
develop AIDS with the treatment and/or
HIV-related 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 4-1 and suninarize

them in Table 4-2. 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 4-1 and 4-2. 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 4-2: 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 HIV-infected individuals undergo testing
and know their status
Rate at which of HIV-infected highly active individuals
undergo testing and know their status
Proportion of HIV-infected individuals who know their
status and receive treatment and/or other medical
support
Rate at which moderately sexually active HIV-infected
individuals develop AIDS without intervention of
treatment or HIV-related medical support
Rate at which highly sexually active HIV-infected
individuals develop AIDS without intervention of
treatment or HIV-related medical support
Rate at which moderately sexually active HIV-infected
individuals develop AIDS with the intervention of
treatment and/or HIV-related medical support
Rate at which highly sexually active HIV-infected
individuals develop AIDS with the intervention of
treatment and/or HIV-related 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 4-1.


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 4-2.


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 4-3. 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 4-3.

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. one-quarter 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

HIV-positive 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 4-3: 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 ahr-l- .- caries the risk of producing multiresistant strains.

4.3 The Model With Treatment and Change in Behavior: Incorporating
Prevention Programmes in HIV-Related Medical Intervention

Now I allow for a behavior change in individuals who come to know of their

HIV-positive status and receive treatment. Consider the model (2-20) (2-26). As a

result of expanded treatment access, millions of people living with HIV are periodically

visiting health-care 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 4-4 and Table 4-5.

Table 4-4: 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 4-5: 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 HIV-related medical

attention. This approach to prevention is cost-effective, 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 HIV-related 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, well-defined 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 ak-a-l-s 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.

Although the possibility that there exists variation in susceptibility of individuals was

not explored in these models, it is increasingly important to acknowledge that possibility

as supported by [24] and [37]. I intend to further explore this in future research.










Although my study sought to improve the accuracy of contact patterns in the

population by creating discrete subclasses of individuals with varying number of new

sexual partners, future research is desirable that further improves the accuracy with which

patterns of contact in STD dynamics models are formulated. I intend to explore a network

approach to formation of sexual partnerships to better describe contact patterns in STD

models.









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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.





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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 2-1Listofvariablesandparametersandtheirmeaning ................ 23 4-1Simulationparametersandtheirvalueranges ................... 40 4-2Specicparametervaluesusedinsimulations .................... 42 4-3ReducedinfectivityinindividualsreceivingHAARTtreatment .......... 47 4-4PrevalenceofHIVinfectionwithincreasingprevention .............. 48 4-5IncidenceofHIVinfectionswithincreasingprevention .............. 48 6

PAGE 7

Figure page 2-1Diagramofthemodelwithtreatment ........................ 25 3-1Themodelwithouttreatment:diagram ....................... 28 4-1Preventionfocusedondecreasingnumberofpartnersonly. ............ 43 4-2Preventionfocusonincreasingcorrectandconsistentcondomuseonly. ..... 44 4-3Preventionfocusedonchangingrecruitmentonly. ................. 45 7

PAGE 8

Neoplasm Anabnormalgrowthoftissue,atumor.Immunedeciency Inabilityoftheimmunesystemtofunctionproperly.Resultsingreatersusceptibilitytodisease.Immunosuppression Reducedimmunesystemresponsetopathogens,suchasvirus,bacteriaorfungi.Seropositive IndividualinfectedwithHIV.Retrovirus RNAvirusthatsynthesizesDNAthroughreversetranscriptase.Lentivirus Typeofretroviruscharacterizedbypresentingalongintervalbetweeninfectionandtheonsetofsymptomsinhosts.ReverseTranscriptase Enzymeresponsiblefortranscriptionofsingle-strandedRNAintodouble-strandedDNA.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/orreceivingHIV-relatedmedicalattention.MystudyfocusesonexploringtheeectsofthemostrecenttrendhavingamajorimpactintheHIV/AIDSepidemic:HAARTtreatment.Specically,throughsimulation,itdemonstratestheimpactofintegratingpreventioneortswiththedistributionoftreatmentandHIV-relatedmedicalsupport.ItshowsthateducatingHIVpatientsaboutrisksofHIVtransmission,reliabilityofcondoms,andsafersexualbehaviormayeectivelydecreasetheprevalenceofHIVinthepopulationaseectivelyasaimingthesepreventionprogrammesattheentirepopulation,providedthatavailabilityoftestingandHIV-relatedmedicalattentionisadequate. 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.Asecondaryco-receptorisalsonecessaryforthevirustoinvadeacell.Themainco-receptor,presentinallcelltypesthatcanbeinfectedby 13

PAGE 14

8 ].Astheinfectiondevelops,theviruscausesdepletionandincreasingdisruptionoftheimmunesystemopeningmanydoorsfortheonsetofopportunistic(Table 1 ),life-threateninginfections[ 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.Bytheendof20051.3millionpeopleinlow-andmiddle-incomecountrieshavereceivedaccesstoantiretroviraltherapy[ 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,increasesinthequalityoflifeandlifeexpectancyforHIV-positiveindividualsundertreatmenthasreshapedthefaceoftheepidemic.HAARThasbeenabletoincreaselifeexpectancyandsignicantlyreducevirusloadinHIVpositivepatients,sometimestolevelsbelowthosethatcanbedetectedbyHIVtests[ 7 ].Inwomen,lowbloodplasmavirusloadhasbeencorrelatedwithlowervaginalvirusload,indicatingthepossibilityofalowerriskofperinatalandfemale-to-maleheterosexualtransmissionfromwomenundertreatment[ 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. Basedonastudyusingself-reportedsurveys,theCentersforDiseaseControlandPrevention(CDC)foundthatinfectedindividualswhoknowoftheirHIV-positivestatushaveasignicantreductioninhigh-riskbehaviorfrombeforetheylearnabouttheir 17

PAGE 18

5 ].However,otherstudies[ 16 17 ]haveshownthatHIV-positivepatientsaregettinginfectedwithbacterialSTDsatratescomparabletoorhigherthanthoseofHIV-negativepatients.InastudyinBrazil,theincidenceofnewgonorrheainfections(anacutebacterialSTD)inwomenwithknownHIVinfectionwas12.9%,whiletheincidenceamongwomenwhoareHIV-negativewas8.33%inpromiscuouswomenand2%innon-promiscuouswomenforthesamecohort[ 16 ].AstudyinNigeriafoundthat14%ofHIV-positiveindividuals,butonly2.0%ofHIV-negativeindividuals,testedpositiveforsyphilis[ 17 ]. TheBrazilianandNigerianresultsraisethequestionoftheeectivenessofchangesinbehaviorduetotheknowledgethatoneisseropositive(seeTable 1 ).EvenifpersonsthatbecomeawareoftheirHIVinfectionbelievethattheyaresubstantiallyincreasingprecautionstopreventthespreadofHIV[ 5 ],thefactthattheyareacquiringbacterialSTDssuggestthatthepreventionmeasurestheyaretakinghavenotbeensucient. ItisclearthenthatassumingthattreatmentwillpreventthespreadofHIVthroughreducedinfectivityorthroughchangeinbehaviorwithoutcounselingisnotideal.HereIwillarguethat,ifenoughtestingandmedicalcounselingareavailable,itispossibletoachievecomparablepreventionresultsbyfocusingtraditional,cost-eectivepreventionprogrammesofeducationandsupportdirectlyonthesub-populationreceivingHIV-relatedmedicalcare.Theseprogrammespreventnewinfectionsduetoincreasedcompliancewithcorrectandconsistentcondomuseandreducednumberofsexualpartners.IalsoillustratethatincorporatingeducationprogrammesintoHIV-relatedmedicalattentioncanbesubstantiallymoreeectivethanrelyinginreducedinfectivityalone.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

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24 ],[ 25 ],[ 26 ],[ 27 ]),ornon-homogenousmixingofindividuals(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

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Iconsideracompartmentalmodelofapopulationthatconsistsofsusceptibleindividuals(S),HIV-infectedindividualswhohavenotdevelopedAIDSandarenotreceivingtreatmentorHIV-relatedmedicalsupport(U),HIV-infectedindividualsundertreatmentand/orotherHIV-relatedmedicalsupportwhohavenotdevelopedAIDS(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

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2-1 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

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Listofvariablesandparametersandtheirmeaning Variable Description Description 23

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2{2 )-( 2{5 ))fortwodiscretesubclassesofsusceptibleindividuals,infectedindividualsnotontreatmentorHIV-relatedmedicalsupport,andinfectedindividualsundergoingtreatmentand/orreceivingHIV-relatedmedicalsupport.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

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dt=n+sNdA(2{11) 2{2 )-( 2{5 ))includingmedicalinterventionandtreatment.ThemodelisdescribedbytheowchartdiagraminFigure 2-1 Figure2-1. Diagramofthemodelwithtreatment For(t)=(n(1n)nUn+s(1s)sUs+n(1n)nTn+s(1s)sTs),theresultingmodelisgivenbythefollowingsystemofnonlineardierentialequations: 25

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dt=nUn+sUs+nTn+sTs(+d)A(2{18) dt=n+sNdA(2{19) 26

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dt=nUn+sUs+nTn+sTs(+d)A(2{26) dt=n+sNdA(2{27) 27

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2{6 )-( 2{11 )hasauniquedisease-freeequilibriumwhichisgivenby"o=(n 2{6 )-( 2{11 )equaltozero,wegetSn=n Thefollowingresultsconcernwiththelocalstabilityofthedisease-freeequilibrium(DFE). Inowlinearizethesystem( 2{6 )-( 2{11 )aroundtheDFE.TheJacobianmatrixforthissystem,whichIwillcallJfromhereon,attheDFEisgivenasfollows: Figure3-1. Themodelwithouttreatment:diagram 28

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thenthedeterminantofJoispositive. Ifn(1n)nnn 30

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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

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29 31 ]. HereImakeadistinctionbetweenthebasicreproductivenumberofthediseaseinthepopulation,theusualRo,orpopulationRo,andtheintrinsicRoofanindividualinthepopulation.ThepopulationRoisthemeasurementofreproductivesuccessofthepathogeninthehostpopulation.TheintrinsicRoisthenumberofsecondarycasesofthediseasethatoneinfectiveindividualwillproduceoveritsinfectivelifetimeinanentirelysusceptiblestablepopulation.ThelatteriscommonlyusedintheconstructionofthepopulationRoinhomogeneouspopulations[ 31 ]. Iconsiderthenumberofsecondarycasesthatoneindividualinthemoderatelyactivesubclass(In)willproduceoveritsentireinfectivelifetime.TheaveragerateatwhichaninfectiveindividualinInleavestheinfectiveclassisn+,andthereforetheaveragetimethatheorsheremainsinfectiveisgivenby1 32

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ThusIhaveRno=n(1n)n OnecanseenowthattheoverallpopulationRo(themeasurementofreproductivesuccessofthepathogenintheentirepopulation)thatwasderivedbytheanalysisofthelocalstabilityoftheDFEisgivenbyaweightedsumoftheintrinsicreproductivenumberofhostsineachofthetwohostclasses.Morespecically,Ro=Rno(nSn 23 ]. Thiscanbeunderstoodthefollowingway.Iconsideratypicalinfectiveindividualtobeanindividualchosenatrandomfromtheoriginalsusceptiblepopulationandthenmadeinfective.Anindividualischosenthroughasexualcontact.TheprobabilityofarandomsexualcontactbeingwithanindividualoftypeSiisii 33

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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

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2{6 )-( 2{11 )canonlyhavetwokindsofequilibria:thedisease-freeequilibriumgivenby"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

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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
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+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

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lim!fn() Thatis,thelimitlim!fn() Inaddition,lim!F()=0.Consequently,G()>F(). TheIntermediateValueTheoremimpliesthatthereexistsatleastoneendemicequilibriumforthesystem( 2{6 )-( 2{11 ). Thereisatleastonewith00for00intheinterval0<. Atfromequation( 3{16 )IhaveRsoSs=Rnon+Rsos. Atfrom( 3{18 )wehave0=(sSs)RsosSssSsandthus0=sRsoRsoSs.SoIhaveRsoSs=sRso. SoIgetacontradiction:Rnon+sRso=sRsosinceRnon6=0. SoSn=fn()6=0forallwith0<<,andtheproofiscompleted. 38

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2{6 )-( 2{11 ). ParametersofthemodelswereestimatedfromtheliteratureandaresummarizedinTable 4-1 .Valuesforthesimulationswherechosenwithintheseranges.Below,Idiscussbrieytheseestimates. Partneracquisitionratesvarygreatlywithinpopulationsofnon-monogamoussexuallyactiveindividuals,rangingfrom1peryearto100partnersperyear[ 24 ].Theremovalrate,thattracksnaturaldeathrateandremovalfromthesexuallyactivepopulationbychangesinsexualbehaviorwastakenfrom[ 24 ].Theindividualsinquestionareassumedtobesexuallyactiveyoungadultswhoareexpectedtoliveanaverageof50yearsandengageinnon-monogamoussexualrelationsfor20years[ 24 ].Theparametersnands,thattrackthedurationofinfectivity,orthetimeittakesforaninfectedindividualtodeveloptheterminalsymptomsofAIDSandbethusremovedtotheclassofAIDSpatients(whoarenotsexuallyactive),aretakenclosetovaluesfoundin[ 32 ],withvariationsintroducedherecorrelatingwithlevelofsexualactivity.Thestudyin[ 32 ]waschosenfortheseparametersbecausethisstudywasperformedbeforetheintroductionofHAARTtherapyandmanyotheranti-retroviraltreatmentsandthereforedoesnotincorporatetheimpactoftreatmentsintheaveragetimetothedevelopmentofAIDS.Thevaluesforn,n,s,s,theprobabilityoftransmissionperpartner,dependonthenumberofcontactsperpartnerandtheprobabilityoftransmissionpercontact[ 24 ],aswellasinfectivityofseropositiveindividuals.Estimatesonthetransmissionprobabilitypersexualcontactrangefrom0.0003,fortransmissionfromfemaletomale,to0.08,fortransmissionfrommaletomale[ 24 ].Thenumberofcontactsperpartneristakenfrom[ 24 ]tobe2 39

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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%) ProportionofHIV-infectedindividualswhoknowtheirstatusandreceivetreatmentand/orothermedicalsupport[ 3 ]i0.01month1 32 ]i0.005-0.006month1 19 ] *T:Totalrecruitmentintosexuallyactivepopulation 40

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IchoseparametersforthesimulationswithintherangesinTable 4-1 andsummarizetheminTable 4-2 .Iinvestigatetwomajorpreventionfocalpoints:reducingnumberofpartnersandcondomuse.Ifurtherinvestigatepossiblecountereectsofthesefocusedpreventioneorts.Forthecaseofaneorttoreducethenumberofnewsexualpartnersperunitoftime,IexplorethechangesinthevalueofRowhenthereisnootherchangeinbehavior,andwhenindividualshaveareducedcompliancewiththecorrectandconsistentuseofcondomsbecausetheyfeelmoresecurewithfewerpartners.Similarly,Iconsiderthestrategyofpreventionthatfocusesonincreasingthecorrectandconsistentuseofcondoms.Isimulatethescenariowherethereisnootherchangeinbehaviorandthepossibleeectsofincreasedriskbehaviorinincreasednumberofnewpartnersperunitoftimethatcanresultfromovercondenceincondomprotection. ResultsofthesimulationaregiveninFigures 4-1 and 4-2 .Itcanbenotedthatgivenenoughcompliancewitheitherofthefocusedpreventionstrategies,Rodecreasesevengivensomecountereecttotheprogramme.However,itisalsoclearthatwhentherearesecondarychangesinbehavior,forexampleanincreaseinnumberofsexualpartnersduetoovercondenceincondomprotection,thepreventionstrategybecomeslessecient.Theresultsseemtosuggestthatthereshouldbesomeconcernwithpreventingsecondarycountereectivechangesinbehaviorwhenimplementinganypreventionprogramthatfocusesmostlyinoneareaofprevention,butalsothataslongassuchcountereectivechangesarekeptatmanageablelevels,thefocusedpreventionprogramisstillbetterthannopreventionatall. Theexplicitdivisionofthepopulationintotwodiscreteclassesbasedonthenumberofsexualpartnersthatindividualsacquire,allowsfortheinvestigationoftheeectofreducingtheproportionofthepopulationthatbecomeshighlysexuallyactive.Eorts 41

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Specicparametervaluesusedinsimulations ParameterValue Description Probabilityoftransmissionfrommoderatelysexuallyactiveinfectiveindividualpersusceptiblepartnerwithoutpreventivestrategiess0.02 Probabilityoftransmissionfromhighlysexuallyactiveinfectiveindividualpersusceptiblepartnerwithoutpreventivestrategies90% Condomecacy(intrinsic)un40% Compliancewiththeuseofcondomforindividualsinsteadyrelationshipsus60% Compliancewiththeuseofcondomforindividualsengagingincasualsexualrelationshipsn36% Condominducedprotection(i=ui)s54% Condominducedprotection(i=ui)n0:036month1 ProportionofHIV-infectedindividualswhoknowtheirstatusandreceivetreatmentand/orothermedicalsupportn0.09month1 42

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Preventionfocusedondecreasingnumberofpartnersonly.Letpbethepercentagedecreaseinaveragenumberofnewsexualpartnersperunitoftime.Thus,averagenumberofnewsexualpartnersperunitoftimeofmoderatelysexuallyactiveclasshereisgivenbyn(1p)andaveragenumberofnewsexualpartnersperunitoftimeofhighlysexuallyactiveclassisgivenbys(1p).GraphshowsRoversusp.Letqbethepercentagedecreaseincorrectandconsistentcondomuse.Thencondomuseprotectionisgivenby1(un(1q))formoderatelyactivegroupand1(us(1q))forhighlyactivegroup.Ishowcurvesforq=p,q=0:5pandq=2p.ThelineS1isthebaselinecasewherethereisareductioninnumberofpartnersandnootherchangeinbehavior.CurvesS2-S4illustrateareductioninnumberofpartnerscoupledwithanincreasingreductioninthecorrectandconsistentuseofcondoms. 43

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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.CurvesS2-S4illustrateanincreaseincondomusecoupledwithanincreaseinnumberofnewsexualpartnersperunitoftimeasaresultofpossibleovercondencethattheuseofcondomswillprotectagainstthetransmissionofHIV. 44

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Preventionfocusedonchangingrecruitmentonly.Changesinrecruitmentproportionscanbeusedasapreventionstrategywithlowpossiblecountereects. toeducateyoungpeoplewhoareabouttobecomesexuallyactiveaboutsafersexualpracticesinvolvingalownumberofdierentsexualpartnershasaclearimpactindiseasepreventionasillustratedinFigure 4-3 Theparametersnandsrefertotherateatwhichsexuallyactiveindividualsgettested.AccordingtotheU.S.DepartmentofHealthandHumanServices[ 7 ]andtheUNAIDSReportontheGlobalAIDSEpidemic[ 4 ],intheU.S.one-quarterofthose 45

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3 ].IfocusonthecurrentscenarioinU.S.andlettreatmentavailabilitybeat100%.AccordingtotheCDC,HAARTtreatmentcanjustaboutdoublelifeexpectancyofseropositiveindividuals[ 19 ]. First,Iexploretheeectsoftreatmentintheabsenceofanyfurtherpreventionstrategy.Someliteraturesuggeststhateventhoughtreatmentlowerstheplasmavirusloadofseropositivepatients,thisreductiondoesnotcorrelatewithareductioninvirusloadinotherbodyuids,suchassemen[ 13 ].However,somestudiesofcoupleswithdiscordantserostatus,individualswithlowervirusloadhavealowerprobabilityoftransmittingHIV[ 5 ],whileitisnotclearbyhowmuch.Ishowtheeectsofreducedinfectivityofindividualsundergoingtreatment(Table 4-3 ).Theincidencewithouttreatmentwasof146individualsin100,000.WithoutreductionofinfectivityorchangeinriskbehaviortheintroductionoftreatmentincreasesincidenceofHIVinfections.Withadecreaseininfectivity,introducingtreatmentreducestheincidenceofHIVinfection.ItisimportanttorecallthatnotallindividualswhoreceiveHIVrelatedmedicalattentionareactuallytakingHAARTdrugs.SomehealthprofessionalsdelaythestartofHAARTbutstillprovidemedicalattentionintheformofviralloadscreening,bacterialSTDstreatment,andtreatmentofopportunisticinfections.Forthissimulation,however,IassumeindividualsreceivingmedicalinterventionareactuallyreceivingHAARTdrugs. 46

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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,millionsofpeoplelivingwithHIVareperiodicallyvisitinghealth-caredeliverysitestomonitortheirdiseaseandtreatmentprogress[ 3 ].Ibelievethisprovidesanimportantpaththroughwhichtoincorporatepreventionprogrammesthatfocusonbehaviorchange.IncountriesliketheUnitedStates,whereonlyabout25%ofHIVinfectionsareunknown,manyindividualswhoknowoftheirseropositivestatushavetheopportunitytoreceivemedicalsupportandtreatmenttoimprovetheirlifestyleandincreasetheirlongevity.IexploretheoutcomeofintegratingchangeofbehaviorpreventionprogrammesfocusedontheseindividualswiththedeliveryoftreatmentandHIVmedicalsupport. Assumingeectivepreventioneortsimplementedthroughintegrationofpreventionandmedicalsupport,ingeneral,assumethatnonandso>s.Infectivityisassumeddobereducedby50%fortreatedindividuals. 47

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4-4 andTable 4-5 Table4-4: 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% Table4-5: 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

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HereIreachmymainresult:giventhattestingandtreatmentavailabilityarecomparabletotheUnitedStates,itisnotnecessarytoeectivelyreachtheentirepopulationwithincreasedpreventionprogrammestoreducetheprevalenceandincidenceofHIV.ItissucienttoeectivelychangethebehaviorofthesubgroupofthepopulationcomposedofindividualsofknownseropositivestatuswhoarereceivingHIVrelatedmedicalsupportandcounseling.Butmorethanthat,theseresultsshowthatcouplingpreventioneortswithtreatmentdeliverycouldbeasaseectiveastargetingpreventiontotheentirepopulation. ThisconclusiondemonstrateshowlargeanimpactindividualsofknownseropositivestatuswhoreceivemedicalinterventioncanhaveinthepreventionofHIV.Byactivelyincorporatingandimprovingpreventionstrategies,individualswhoareawareoftheirseropositivestatuscaneectivelyhelpreducetheprevalenceofHIVinthepopulationandslowtheepidemic.ItisveryimportantthennottoneglectHIVpreventionrelatededucationfortheseindividuals.ItisclearfromreportsoftheCDC[ 5 ]thatindividualswhobecomeawareoftheirseropositivestatusdoactivelytrytopreventfurthertransmissionofthedisease.Therefore,itisnecessaryforsocietytoprovideappropriatetoolsfortheseindividualstoeectivelyhelpintheghtagainstHIV/AIDS.Itisnecessarytoprovideclearinformation,resources,andsupporttothoseindividualswhoareinfectiveandwhoarereceivingHIVrelatedmedicalattention. 49

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Themodelwithtreatmentanddierentiatedrateofpartneracquisition,whichtomyknowledgehasnotbeenpreviouslystudied,accountsforheterogeneitiesamongindividualsintheirbehaviortowardsacquisitionofnewpartnersanduseofcondomswhileexploringtheeectsoftreatmentandmedicalsupport,withandwithoutpreventioncounseling,ontheoveralldiseasedynamics.Themodelwithouttreatmentaccountsforheterogeneitiesamongindividualsintheirbehaviortowardsacquisitionofnewpartnersanduseofcondoms,butemphasizesthesemechanismsexplicitly,especiallyintheexplicitrenderingofRo,andgivesfurtherinsightonpreventionstrategies. ForthemodelwithouttreatmentIderivedtheexplicitformulaforthebasicreproductionnumberandprovedexistenceofatleastoneendemicequilibriumwhenRo>1.IdiscussthevalueofthebasicreproductionnumberinviewofothermoretraditionalepidemiologicalpapersandthemeaningofthisvalueofRo. Usingsimulations,IexaminedthetransmissiondynamicsofthediseaseinapopulationwhereHAARTisintroduced,withandwithoutfocalpreventioneorts.ThemainresultIreachisthatpreventionprogrammesdonotneedtoreachtheentirepopulation,butonlyneedtofocusonthegroupreceivingHIV-relatedmedicalattention.Thisapproachtopreventioniscost-eective,sincethetargetpopulationissignicantlysmallerthanthetotalpopulation,sincethereisaclearchannelforthedeliveryoftreatment,andsincethischanneldoesnotrelyonthecostlyantiretroviraldrugsforreducedinfectivity.Couplingpreventionprogrammes,suchascounselingandeducationprogrammes,withthedeliveryofHIV-relatedmedicalattentioncanliberateHAARTfrombeingusedasapreventionstrategy,sothatthestartofHAARTdrugscanbedelayedifnecessaryandtheriskofproducingmultiresistantHIVstrainscanbereduced.Thisapproachpreventsseropositivepatientsfromincreasingriskybehaviorduetoovercondenceintreatmentandreducedinfectivity.Therealsoexistsaclear 50

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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. Althoughthepossibilitythatthereexistsvariationinsusceptibilityofindividualswasnotexploredinthesemodels,itisincreasinglyimportanttoacknowledgethatpossibilityassupportedby[ 24 ]and[ 37 ].Iintendtofurtherexplorethisinfutureresearch. 51

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[1] CentersforDiseaseControlandPrevention(CDC).PneumocystisPneumonia-LosAngeles.MorbidityandMortalityWeeklyReports(MMWR).30(1981)250. [2] CentersforDiseaseControlandPrevention(CDC).Follow-UponKaposi'sSarcomaandPneumocycstisPneumonia.MorbidityandMortalityWeeklyReports(MMWR).30(1981)409. [3] JointUnitedNationsProgrammeonHIV/AIDS(UNAIDS).ReportontheGlobalAIDSepidemic.(2006) [4] JointUnitedNationsProgrammeonHIV/AIDS(UNAIDS).AIDSEpidemicUpdate:December2006.(2006). [5] CentersforDiseaseControlandPrevention(CDC).RevisedRecommendationsforHIVTestingofAdults,Adolescents,andPregnantWomeninHealth-CareSettings.MorbidityandMortalityWeeklyReports(MMWR).55(2006)1. [6] JointUnitedNationsProgrammeonHIV/AIDS(UNAIDS).25YearsofAIDS:FactSheet.(2006) [7] NationalInstituteofAllergyandInfectiousDiseases(NIAID).NationalInstitutesofHealth(NIH).U.S.DepartmentofHealthandHumanServices(HHS).HIVinfectionandAIDS:Anoverview.Exploring.(2005). [8] N.A.Campbell,J.B.Reece.Biology.7thed,BenjaminCummings,SanFrancisco,CA.(2005). [9] NationalInstituteofAllergyandInfectiousDiseases(NIAID).NationalInstitutesofHealth(NIH).U.S.DepartmentofHealthandHumanServices(HHS).HowHIVcausesAIDS.Exploring.(2004). [10] J.X.Velasco-Hernandez,J.A.Garcia,D.E.Kirschner.RemarksonModelingHost-ViralDynamicsandTreatmentin:Volume125:MathematicalApproachesforEmergingandReemergingInfectiousDiseasesPartI:AnIntroductiontoModels,Methods,andTheory.Springer-Verlag,NewYork,(2002)287. [11] amfAR,TheFoundationforAIDSResearch.2005AnnualReport.amfARPublicInformationDepartment.NewYork,NewYork.(2005) [12] amfAR,TheFoundationforAIDSResearch.FactsforLife:WhatyouandthepeopleyoucareaboutneedtoknowaboutHIV/AIDS.amfARPublicInformationDepartment.NewYork,NewYork.(2005) [13] H.Zhang,G.Dornadula,M.Beumont,L.Livornese,B.VanUitert,K.Henning,R.J.Pomerantz.HumanImmunodeciencyVirusType1intheSemenofMenReceivingHighlyActiveAntiretroviralTherapy.TheNewEnglandJournalofMedicine(NEnglJMed).339(1998)1803. 53

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LisaR.Metsch,MargaretPereyra,CarlosdelRio,LyttGardner,WayneA.Duus,GordonDickinson,PeterKerndt,PamelaAnderson-Mahoney,SteanieA.Strathdee,AlanE.Greenberg,fortheAntiretroviralTreatmentandAccessStudyGroup(ARTAS).DeliveryofHIVPreventionCounselingbyPhysiciansatHIVMedicalCareSettingsin4USCities.AmJPublicHealth.94(7)(2004)1186 [15] M.C.Boily,F.I.Bastos,K.Desai,B.Masse.ChangesinthetransmissiondynamicsoftheHIVepidemicafterthewide-scaleuseofantiretroviraltherapycouldexplainincreasesinsexuallytransmittedinfections:resultsfrommathematicalmodels.SexTransmDis.31(2004)100. [16] Brazil,MinistriodaSade-SecretariadeVigilnciaemSade/ProgramaNacionaldeDSTeAIDS.Tabelascontendoosdadosdevaginosebacteriana,candidase,slis,HPV,HIV,HBVeHCV.(2004). [17] UnekeJesse,ChigozieandOgbu,OgbonnayaandAlo,MosesandAriom,Thaddeus(2006)SyphilisserologyinHIV-positiveandHIV-negativeNigerians:Thepublichealthsignicance.OnlineJournalOfHealthAndAlliedSciences5(2). [18] C.E.Hart,J.L.Lennox,M.Pratt-Palmore,T.C.Wright,R.F.Schinazi,T.Evans-Strickfaden,T.J.Bush,C.Schnell,L.J.Conley,K.A.Clancy,andT.V.Ellerbrock.CorrelationofHumanImmunodeciencyVirusType1RNALevelsinBloodandtheFemaleGenitalTract.JInfectDis.179(1999)871. [19] PanelonClinicalPracticesforTreatmentofHIVInfection(October6,2005).GuidelinesfortheUseofAntiretroviralAgentsinHIV-1-InfectedAdultsandAdolescents.DepartmentofHealthandHumanServices.(2005). [20] CentersforDiseaseControlandPrevention(CDC).HIVPrevention:HIVanditstransmission.(1999). [21] CentersforDiseaseControlandPrevention(CDC).HIVanditstransmission.(1999). [22] T.C.Quinn,M.J.Wawer,N.Sewankambo.Viralloadandheterosexualtransmissionofhumanimmunodeciencyvirustype1.RakaiProjectStudyGroup.NEnglJMed342(2000)921. [23] C.Castillo-Chavez,Z.Feng,W.Huang,OnthecomputationofRoanditsroleonglobalstability,in:C.Castillo-Chvez,withS.Blower,P.vandenDriessche,D.Kirschner,A.-A.Yakubu(eds.),MathematicalApproachesforEmergingandReemergingInfectiousDiseases:AnIntroduction,Springer-Verlag,(2002)229. [24] J.M.Hyman,J.Li,E.A.Stanley,ThedierentialinfectivityandstagedprogressionmodelsforthetransmissionofHIV,Math.Biosci.155(1999)77. 54

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J.M.Hyman,JLi.ThereproductivenumberforanHIVmodelwithdierentialinfectivityandstagedprogression.LinearAlgebraanditsapplications.398(2005)101. [26] J.M.Hyman,J.Li.Dierentialsusceptibilityepidemicmodels.J.Math.Biol.50(2005)626. [27] J.M.Hyman,J.Li.Dierentialsusceptibilityandinfectivityepidemicmodels.Math.Biosci.andEng.3(2006)89. [28] J.MHyman,J.Li.BehaviorchangesinSISSTDmodelswithselectivemixing.SIAMJ.Appl.Math.57(1997)1082. [29] O.Diekmann,J.A.P.Heesterbeek,MathematicalEpidemiologyofInfectiousDisease;ModelBuilding,AnalysisandInterpretation,Wiley,NewYork,(1999). [30] L.J.S.Allen,AnIntroductiontoMathematicalBiology,PearsonPrenticeHall,UpperSaddleRiver,NJ,(2007). [31] P.vandenDriessche,J.Watmough,Reproductionnumbersandsub-thresholdendemicequilibriaforcompartmentalmodelsofdiseasetransmission,Math.Biosci.180(2002)29. [32] J.M.Logini,W.S.Clark,M.Haber,R.Horsburgh.ThestagesofHIVinfection:Waitingtimesandinfectiontransmissionprobabilities,in:Castillo-Chaves,Levin,Shoemaker(Eds.),MathematicalApproachestoAIDSEpidemiologyLectureNotesinBiomathematics,vol.83.Springer.(1989)111. [33] R.M.Anderson,R.M.May.Infectiousdiseasesofhumans:Dynamicsandcontrol.Oxford,NewYork.(1991). [34] M.D.Guimaraes,A.Munoz,C.Boschi-Pinto,E.A.Castilho.HIVinfectionamongfemalepartnersofseropositivemeninBrazil.RiodeJaneiroHeterosexualStudyGroup.AmJEpidemiol1995;142(5):538-47. [35] J.E.Anderson,R.Wilson,T.S.Jones,P.Barker.CondomuseandHIVriskbehaviorsamongUSadult:datafromanationalsurvey.Fam.Plan.Perspect.31(1999)24-28. [36] G.Marks,N.Crepaz,J.W.Sentertt,R.S.Janssen.Meta-analysisofhigh-risksexualbehaviorinpersonsawareandunawaretheyareinfectedwithHIVintheUnitedStates:ImplicationsforHIVpreventionprograms.JAcquirImmuneDecSyndr.39(4)(2005)446. [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.TheroleofamutantCCR5alleleinHIV-1transmissionanddiseaseprogression.2(11)NatMed.(1996)1240. 55

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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