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Intra-Host stochastic models for HIV dynamics

and management

by

Rachel Waema Mbogo

A Research Thesis submitted for

the degree of

DOCTOR OF PHILOSOPHY IN BIOMATHEMATICS

to

Strathmore University

Nairobi, Kenya

November, 2013

Copyright c

November, 2013

Rachel Waema Mbogo

All Rights Reserved

Declaration

I, the undersigned, hereby declare that the work in this thesis is the result of my own

investigation, and that any work done by others or by myself previously has been

acknowledged and referenced accordingly.

Rachel Waema Mbogo

Strathmore Univesity

Center for Applied Research in Mathematical Sciences (CARMS)

This thesis has been approved by the supervisors as signed below.

Prof. Luboobi, L.S (Makerere University)

Principal Supervisor

Prof. Odhiambo J.W (Strathmore University)

Supervisor

Strathmore University

November, 2013

ii

Abstract

Mathematical models can facilitate the understanding of complex dynamics of many

biomedical systems such as epidemiology, ecology and virology. The main objective

of this study was to make use of mathematical models, this sought to develop and

use stochastic modeling to model HIV dynamics and its management. Mathematical

models can help to improve our understanding of dynamics of diseases such as in

HIV/AIDS by providing an alternative way to study the eﬀects of diﬀerent drugs, a

procedure which is otherwise risky or unethical when carried out on patients. Un-

tangling the dynamics between HIV and CD+

4cellular populations and molecular

interactions can be used to investigate the eﬀective points of interventions in the HIV

life cycle. Many existing models on within-Host HIV dynamics make simplifying as-

sumptions concerning the HIV targeted immune cells by assuming that the virus only

aﬀects CD+

4cells. However, other immune cells play also important roles in HIV pro-

gression. Also the deterministic models lack the eﬀect of randomness,hence stochastic

models are developed which are used to capture the dynamics of HIV infection and

examine various alternatives for the control and treatment of the virus.

The HIV and AIDS epidemic is extremely dynamic; this dynamism orthogonally com-

plicates interventions embraced for the management of the epidemic. This study was

ﬁrstly motivated by the fact that eradication of the HIV virus is not attainable with

the current available drugs and the focus has now shifted from eradication of the virus

to the management and control of the virus’ progression. Secondly, the treatment of

HIV is expensive and often times erratic. It is therefore necessary to develop a mathe-

matical model which when applied to HIV data accurately details the cost of treating

a HIV patient. Stochastic models are formulated which describe the interaction of

HIV, langerhans cells (LCs) and the CD+

4cells. Diﬀerence diﬀerential equations are

obtained and solved using probability generating function to obtain moments of the

numbers of uninfected immune cells, the HIV infected CD+

4T cells, and the free HIV

iii

particles at any time t. The model analysis showed that, eradication of HIV is not

possible without clearance of latently infected langerhans cells and the predicted rate

of decline in plasma HIV virus concentration depends on three factors: the death rate

of the virons, the eﬃcacy of therapy and the length of the intracellular delay. The

model produces interesting feature that successfully treated HIV patients will have

low, undetectable viral load.

Stochastic models based on Semi-Markov process were formulated describing the pro-

gression of HIV in an infected person. Firstly, Internal transition probabilities were

computed, which capture the probability of survival for the HIV patient under treat-

ment. It will also show the eﬀects of covariates:- age, follow up time and initial state

on the disease progression. Secondly, a model for the cost of treating a HIV patient

was formulated. This model captures the cost of managing a HIV patient given the

starting state and the treatment combination which is most eﬀective at each disease

state. Lastly, a Treatment Reward Model (TRM) was formulated, which gave insights

on the state, which the patient should be maintained so that they remain healthy,

productive, and non-infectious, and also the optimal or eﬀective time to initiate HIV

therapy.

iv

Contents

Declaration ................................... ii

Abstract..................................... iv

ListofFigures.................................. ix

ListofTables .................................. x

Acknowledgements ............................... xi

Dedication.................................... xii

1 Introduction 1

1.1 Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Process of mathematical modeling . . . . . . . . . . . . . . . . 2

1.1.2 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Markov Chain and processes . . . . . . . . . . . . . . . . . . . 5

1.1.4 Importance of Stochastic processes in HIV/AIDS Modeling . . 6

1.2 The HIV/AIDS pandemic . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Mechanism of HIV infection . . . . . . . . . . . . . . . . . . . 11

1.2.2 HIV replication Cycle . . . . . . . . . . . . . . . . . . . . . . 12

1.2.3 Role of Langerhans cells in HIV infection . . . . . . . . . . . . 16

1.2.4 HIV and treatment . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Problemstatement ............................ 21

1.4 Objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Signiﬁcance of the study . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Scopeofthestudy ............................ 23

1.7 Conceptual framework . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.8 Outlineofthestudy ........................... 26

2 Literature Review 27

2.1 Introduction................................ 27

2.2 Review of Intra-Host HIV models . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Kinetics of HIV infection . . . . . . . . . . . . . . . . . . . . . 28

v

2.2.2 Other factors associated with HIV Progression . . . . . . . . . 34

2.2.3 CD+

4T cells and viral load dynamics . . . . . . . . . . . . . . 35

2.3 Mathematical modeling of Intra-Host HIV dynamics . . . . . . . . . . 37

2.4 Stochastic Multi-state models for HIV progression . . . . . . . . . . 49

3In Vivo HIV dynamics 58

3.1 Introduction................................ 58

3.2 Healthy CD+

4cell population dynamics . . . . . . . . . . . . . . . . 58

3.2.1 Solving the Diﬀerence Diﬀerential Equation (DDE) . . . . . . 61

3.3 Intra-Host stochastic HIV interaction model . . . . . . . . . . . . . . 66

3.3.1 Variables and parameters for the model . . . . . . . . . . . . . 69

3.3.2 Probability Generating Function (PGF) of state variables . . . 75

3.3.3 Marginal Generating Functions(MGF) . . . . . . . . . . . . . 78

3.3.4 Numbers of immune cells and the virions . . . . . . . . . . . . 80

3.3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4 Intra-Host HIV dynamics under therapeutic intervention . . . . . . . 87

3.4.1 Variables and parameters for the model . . . . . . . . . . . . . 87

3.4.2 Marginal generating functions with therapeutic intervention . 90

3.4.3 Numbers of CD+

4T-cells and Virions under therapeutic inter-

vention............................... 91

3.4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 94

4 HIV, Langerhans and C D+

4cells dynamics 98

4.1 Assumptions................................ 100

4.2 Marginal generating functions . . . . . . . . . . . . . . . . . . . . . . 110

4.3 Numbers of Cells and the Virions . . . . . . . . . . . . . . . . . . . . 112

4.4 In Vivo HIV dynamics under therapeutic intervention . . . . . . . . . 114

4.5 Simulationresults............................. 116

5 Probability of HIV clearance 123

5.0.1 Solving the Diﬀerence Diﬀerential Equation (DDE) . . . . . . 127

vi

6 Semi-Markov process in HIV evolution 133

6.1 Introduction................................ 133

6.2 Semi-Markovprocess........................... 133

6.2.1 Attributes of semi-Markov process . . . . . . . . . . . . . . . . 134

6.2.2 HIV progression dynamics . . . . . . . . . . . . . . . . . . . . 138

6.3 HIV discrete time non-homogeneous Semi-Markov model . . . . . . . 143

6.4 HIV age-dependence semi-Markov model . . . . . . . . . . . . . . . . 150

6.5 Cost of HIV treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.5.1 Long run treatment cost . . . . . . . . . . . . . . . . . . . . . 156

6.5.2 HIV Treatment Reward Model (HTRM) . . . . . . . . . . . . 157

7 Discussion and recommendations 159

References 163

A Publications arising from this Thesis 181

vii

List of Figures

1.1 Schematic ﬂow of modeling process . . . . . . . . . . . . . . . . . . . 4

1.2 Global HIV/AIDS Statistics . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 HIVlifecycle ............................... 14

1.4 Conceptual Framework for the study . . . . . . . . . . . . . . . . . . 25

2.1 Natural Progression of HIV . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Literature summary on deterministic modeling . . . . . . . . . . . . . 48

2.3 HIV disease progression . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4 The model of the immunological stages a HIV/AIDS infected patient

cangointo. ................................ 52

2.5 Literature summary on stochastic modeling . . . . . . . . . . . . . . 56

3.1 Schematic diagram of Intra-Host HIV dynamics before treatment . . 68

3.2 Cell population dynamics before treatment . . . . . . . . . . . . . . 85

3.3 Cell population dynamics before treatment . . . . . . . . . . . . . . 86

3.4 Intra-Host HIV interaction with treatment . . . . . . . . . . . . . . . 89

3.5 Eﬀects of drug eﬃcacy on virus dynamics . . . . . . . . . . . . . . . 95

3.6 Eﬀects of drug eﬃcacy on virus dynamics . . . . . . . . . . . . . . . 96

4.1 The interaction of HIV virus and the immune system . . . . . . . . . 99

4.2 HIV dynamics when τ=2........................ 119

4.3 HIV dynamics when τ=0........................ 119

4.4 HIV dynamics when τ= 0 and drug eﬃcacy of 60% . . . . . . . . . . 120

4.5 HIV dynamics when τ= 2 and drug eﬃcacy of 60% . . . . . . . . . . 120

4.6 HIV dynamics with drug eﬃcacy of 85% . . . . . . . . . . . . . . . . 121

4.7 HIV dynamics when τ= 1 and drug eﬃcacy of 85% . . . . . . . . . . 121

6.1 HIV Multi-state Model with 5 immunological states and 20 transitions 140

6.2 Time dependent semi-Markov transition process . . . . . . . . . . . . 144

viii

6.3 Age dependent trajectory of a semi-Markov process . . . . . . . . . . 151

ix

List of Tables

3.1 Variables for the Intra-Host Stochastic model . . . . . . . . . . . . . 69

3.2 Parameters for the Intra-Host stochastic model . . . . . . . . . . . . 70

3.3 Intra-Host HIV transitions . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4 Parameter values for the Intra-Host HIV model . . . . . . . . . . . . 83

3.5 Variable values for the Intra-Host HIV model . . . . . . . . . . . . . 84

3.6 Intra-Host Model Variables . . . . . . . . . . . . . . . . . . . . . . . 88

3.7 Intra-Host model Parameters with treatment . . . . . . . . . . . . . . 88

4.1 Parameters for the Langerhans model . . . . . . . . . . . . . . . . . . 102

4.2 Variables for the Langerhans model . . . . . . . . . . . . . . . . . . . 103

4.3 Intra-Host HIV interaction . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4 Parameter Values for the Langerhans model . . . . . . . . . . . . . . 117

4.5 Variable values for the Langerhans model . . . . . . . . . . . . . . . . 118

6.1 HIV staging classiﬁcation according to CDC/WHO criteria . . . . . . 141

x

Acknowledgements

First and foremost, praises and thanks to the God, the Almighty, for His showers of

blessings throughout my research work to complete the research successfully.

I would like to express my deep and sincere gratitude to my research supervisors, Prof.

Livingstone Luboobi and Prof. John Odhiambo, for being so generous with their time,

and for their inspiring motivation which guided my research. I am especially grateful

for Prof. Luboobi’s eﬀort in training me to become a capable researcher. Also I

express my thanks to my director, Prof. Vitalis Onyango-Otieno for his support and

valuable prayers.

I am extremely grateful to my husband, Dr. Salesio Mbogo for his unfailing love,

patience and continuing support physically, mentally and spiritually. Thanks also

to Kanesa and Kyra for their constant love, patience and understanding. I owe my

brother, Prof. Timothy Waema a lot of respect and love for he has been my role

model and the reason why I am here.

My special thanks to the German Academic Exchange Service (DAAD), not only for

providing the funding which allowed me to undertake this research, but also for the

ﬂexibility the funding provided, giving me the opportunity to visit Germany and meet

other researchers. Finally, my thanks go to all the people who have supported me to

complete the research work directly or indirectly.

xi

Dedication

To my Family: Mbogo, Kanesa and Kyra

whose love and support sustained me throughout

xii

List of Abbreviations

Notation Full Name

AIDS Acquired Immune Deﬁciency Syndrome

APCs Antigen Presenting Cells

ART Antiretroviral Therapy

CBA Cost Beneﬁt Analysis

CBAM Cost Beneﬁt Analysis Model

DNA Deoxyribonucleic Acid

FDC Follicular Dendritic cell

FI Fusion Inhibitors

HAART Highly Active Antiretroviral Therapies

HIV Human Immunodeﬁciency Virus

LCs Langerhans cells

MCM Management Cost model

mRNA messenger Ribonucleic Acid

MTCT Mother-To-Child Transmission

NHSMRM Non-Homogeneous Semi-Markov Reward model

NHSMSM Non-Homogeneous Semi-Markov Stochastic model

ODEs Ordinary Diﬀerential Equations

PI Protease Inhibitors

PMTCT Prevention of Mother-To-Child Transmission

RNA Ribonucleic Acid

R0Basic Reproduction number

RTI Reverse Transcriptase Inhibitors

TRM Treatment Reward Model

UNAIDS United Nations Program on HIV/AIDS

vDNA viral Deoxyribonucleic Acid

xiii

Publications Arising From This Thesis

1. Mbogo, W. R., Luboobi, L. S., and Odhiambo, J. W. (2013a). Stochastic

model for in-host HIV dynamics with therapeutic intervention. Hindawi Pub-

lishing Corporation-ISRN Biomathematics, 2013 (1), 1 - 11.

2. Mbogo, W. R., Luboobi, L. S., and Odhiambo, J. W. (2013b). Mathematical

model for HIV and CD+

4cells dynamics in vivo. international journal of pure

and applied mathematics , 6 (2), 83 - 103.

3. Mbogo, W. R., Luboobi, L. S., and Odhiambo, J. W. (2014). Stochas-

tical model for Langerhans and HIV dynamics in vivo. Hindawi Publishing

Corporation-ISRN Applied Mathematics, 2014 (1), 1 - 10.

4. Mbogo, W. R., Chirove F., Wattanga, I.S.,Kipchumba B and Wangari I.M.

(2014). Mathematical model for Langerhans cells and HIV-1 evolution. Applied

Mathematics and computation, accepted.

5. Mbogo, W. R., Luboobi, L. S., and Odhiambo, J. W. (2014). Semi-Markov

model for evaluating the HIV patient treatment cost. Hindawi Publishing

Corporation- Journal of Applied Mathematics, under review.

6. Mbogo, W. R., Luboobi, L. S., and Odhiambo, J. W. (2014). Age dependent

Semi-Markov model for HIV evolution and treatment. Aplied Mathematics and

computation, under review.

xiv

Conference presentations arising from This Thesis

1. “Mathematical Model for Langerhans cells and HIV evolution”, at The 2nd

Strathmore International Mathematics Conference, August 12th 16th, 2013,

Strathmore, Kenya.

2. “Stochastic model for In-Host HIV virus dynamics with Therapeutic interven-

tion”, at The 2nd EAUMP Conference, 22nd 25th August 2012, Arusha, Tan-

zania.

3. “Stochastic modelling of HIV dynamics within an individual and its manage-

ment”, at the International Mathematics Research Meeting , July 23 -27, 2012,

Strathmore University, Kenya

xv

Chapter 1

Introduction

This chapter deals with the general introduction to mathematical modeling and the

impact of mathematical models in modeling disease progression and especially the

evolution of HIV/AIDS. Mathematical modeling especially stochastic modeling based

on generating functions, which is the main tool used in the study is introduced in

this chapter. Also Markov proceses as a modeling tool is introduced. Biological

background of HIV/AIDS and connection to mathematical modeling will be discussed.

Lastly, a framework summarizing the study is provided.

1.1 Mathematical modeling

A mathematical model is a description of a system or process using mathematical

concepts and language. The process of developing a mathematical model is known as

mathematical modelling, that is, mathematical modeling is the use of mathematics

to describe real-world phenomena. Mathematical models are used not only in the

natural sciences (such as physics, biology, earth science, meteorology) and engineering

disciplines (e.g. computer science, artiﬁcial intelligence), but also in the social sciences

(such as economics, psychology, sociology and political science); physicists, engineers,

statisticians, operations research analysts and economists use mathematical models

most extensively. A model may help to explain a system and to study the eﬀects of

diﬀerent components, and to make predictions about behaviour.

Mathematical models are characterized by assumptions about variables (the quanti-

ties that change), parameters (the quantities that do not change) and functional forms

1

(the relationship between the variables and parameters). These models are used to

guess how a system would work or how the system would react to certain variables

and parameters. Mathematical modeling can be a powerful tool for understanding

biologically observed phenomena which cannot be understood by verbal reasoning

alone. The models are increasingly being recognized within the biomedical sciences

as important tools that can aid the understanding of biological systems.

Mathematical models can take many forms, including but not limited to dynamical

systems, statistical models, diﬀerential equations, or game theoretic models. These

and other types of models can overlap, with a given model involving a variety of ab-

stract structures. In general, mathematical models can be categorized as deterministic

or stochastic models. Deterministic models ignore random variation, in other words,

they have no components that are inherently uncertain, that is, no parameters in the

model are characterized by probability distributions, as opposed to stochastic models.

For ﬁxed starting values, a deterministic model will always produce the same result.

A stochastic models are statistical in nature and so may predict the distribution of

possible outcomes. They produce many diﬀerent results depending on the actual

values that the random variables take in each realization.

1.1.1 Process of mathematical modeling

Mathematical modeling is the art of translating problems from an application area

into tractable mathematical formulations whose theoretical and numerical analysis

provide insights, answers, and guidance useful for the originating application. How

well any particular objective is achieved depends on both the state of knowledge about

a system or process and how well the modelling is done. It involves taking whatever

knowledge you may have of mathematics and of the system of interest and using that

knowledge to create something. Mathematical modeling is a series of steps taken to

convert an idea ﬁrst into a conceptual model and then into a quantitative model.

Conceptual model represents our ideas about how the system/process works. It is

2

expressed visually in a model diagram, typically involving boxes (state variables)

and arrows (material ﬂows or causal eﬀects).

Equations are developed for the rates of each process and are combined to form a

quantitative model consisting of dynamic (that is, varying with time) equations

for each state variable.

The dynamic equations can then be studied mathematically and solved analytically

or translated into computer code to obtain numerical solutions for state variable

trajectories.

One of the most useful ways to view mathematical modeling is as a process, as

illustrated in Figure 1.1

3

Figure 1.1: Schematic ﬂow of modeling process

Source: Author

4

1.1.2 Stochastic processes

A Stochastic process is a statistical process involving a number of random variables

depending on a variable parameter (which is usually time). In probability theory,

a stochastic process, or sometimes a random process, is the counterpart to a deter-

ministic process (or deterministic system). Instead of dealing with only one possible

“reality” of how the process might evolve with time (as is the case, for example, for

solutions of an ordinary diﬀerential equation), in a stochastic or random process there

is some indeterminacy in its future evolution described by probability distributions.

This means that even if the initial condition (or starting point) is known, there are

many possibilities the process might go to, but some paths are more probable and

others less.

1.1.3 Markov Chain and processes

A Markov chain is a type of stochastic process where probabilities of an event evolve

over time (that is, the probability of the next state given the current state and the

entire past depends only on the current state). Accordingly, the nature of this process

is probabilistic (stochastic) and dynamic. Markov chains (or processes) are used to

describe and forecast the behavior of an extremely wide variety of entities using the

probability distributions of their states. A Markov model uses either a discrete-time

stochastic process called a Markov chain or a continuous-time stochastic process called

a Markov process. In a continuous-time stochastic process, the states of a system can

be viewed at any time, not just at discrete instants in time. The Markovian property

is a constraint on the memory of the process: knowing the immediate past means

the earlier outcomes are no longer relevant. A semi-Markov process is a process that

changes states according to a discrete time Markov Chain (Xn) but stays at each state

for a random amount of time with distribution Hn. A Continuous Time Markov Chain

(CTMC) is a special case of a semi-Markov process in which the transition times are

exponentially distributed. The semi-Markov model is a useful tool to predict the

5

clinical progression of a disease, (see Di Biase et al. (2007)). It consists mainly of

computing the probability of a patient being into one of the possible stages of the

disease for a certain time and the probability that the subject might survive for a

time t. The most important property of semi-Markov processes is that they enable

you to consider not only the randomness in the diﬀerent states in which the infection

can evolve into but also the randomness of the time spent in each state. For more

literature (see Janssen and Manca (2006)). When the transitions in a Markov process

do in fact depend upon the amount of time that has elapsed, the process is said to be

non-homogeneous, otherwise if the probabilities of transition between states do not

depend on time, then the process is a homogeneous or stationary Markov chain.

1.1.4 Importance of Stochastic processes in HIV/AIDS Mod-

eling

There are real beneﬁts to be gained in using stochastic rather than deterministic

models. Our research will be grounded in stochastic modeling under the following

considerations:

a). Real life is stochastic rather than deterministic, particularly when modelling bio-

logical phenomena such as internal HIV viral dynamics. This is because diﬀerent

cells and infective virus particles reacting in the same environment multiply or die

at diﬀerent depending on one’s genes (Dalal, Greenhalgh, & Mao, 2009). Semi-

Markov models are more ﬂexible and model situations realistically since they relax

the Markov property.

b). The randomness both in the diﬀerent states of the infection and in the time spent

in each state. Markov models assume constant transition probabilities, where

as semi-Markov models overcome this diﬃculty and allow inclusion of diﬀerent

distributions of time spent in each state.

c). The randomness in the evolution of the infection taking into account the diﬀerent

6

ages of the patients. Transition probabilities in a semi-Markov process depend

not only on the current state but also on the time spent in that state.

d). Markov chains are used in studying ageing systems, the AIDS virus internal pro-

gression can be seen as an ageing system hence modeled using Semi-Markov pro-

cess.

e). In the absence of treatment, the future of the patient depends only on the present

state and not on all his previous history. This property is peculiar to homogeneous

semi - Makov process.

f). The time spent at each progression state of the virus is random, which is a prop-

erty for Markov processes.

g). Stochastic models provide more useful information than deterministic models as

by running a stochastic model many times we can build up a distribution of the

predicted outcomes, for example the number of infected cells at time t, whilst a

deterministic model will just give a single predicted value. Having a distribution

for the predicted outcomes is more versatile as it helps us examine practically

important essentially stochastic quantities, for example the variance of the number

of infective virus particles at a given time and the probability that the infective

virus particles have died out at a given time, which cannot be examined using

deterministic models. Even quantities such as the expected values of the number

of cells can be more accurately modelled using stochastic models because they

include the eﬀect of random variation on these quantities which deterministic

models cannot (Dalal, Greenhalgh, & Mao, 2008).

1.2 The HIV/AIDS pandemic

HIV is the acronym for Human Immunodeﬁciency Virus, which is a retrovirus of

the lentivirus family that was unknown until the early 1980’s, and that causes Ac-

7

quired Immune Deﬁciency Syndrome (AIDS). “Acquired” implies that it is an ac-

quired condition or infection, not something transmitted or inherited through the

genes. “Immune” implies that the virus aﬀects the body’s immune system, which is

responsible for protecting the body from germs such as bacteria, fungi and viruses.

“Deﬁciency” implies that the virus makes the immune system deﬁcient (fail to work

properly). “Syndrome” implies that someone with AIDS may experience a wide range

of diﬀerent diseases and opportunistic infections from time to time.

First reported in 1981 in the United States, HIV/AIDS has spread around the world

to infect millions of persons and it has become a major worldwide epidemic. The

result of HIV infection is relentless destruction of the immune system. By killing or

damaging cells of the immune system, HIV progressively destroys the body’s ability

to ﬁght infections, (see (Dalal et al., 2009)). All HIV infected persons are at risk

of illness and death from opportunistic infectiouns and neoplastic complications as a

result of the inevitable manifestations of AIDS. Retroviruses are unable to replicate

outside of living host cells because they contain only Ribonucleic Acid (RNA) and

and do not contain Deoxyribonucleic Acid (DNA). The variant of HIV that is the

cause for almost all infections is known as HIV-1.

According to 2012 estimates from UNAIDS, globally around 30.6 million adults and

3.4 million children [31.4 –35.9 million] people were living with HIV at the end of

2011. Worldwide, the number of people newly infected continues to fall: the number

of people (adults and children) acquiring HIV infection in 2011 (2.5 million [2.2 –2.8

million]) was 20% lower than in 2001, (see (Global Report: UNAIDS Report on the

Global AIDS epidemic: 2012 , n.d.)).

Sub-Saharan Africa is by far the region most-aﬀected by the AIDS epidemic. The

region has just over 12% of the world’s population, but is home to 69% ( accounts

for two-thirds ) of all people living with HIV worldwide. An estimated 1.9 million

adults and children became infected with HIV during 2011, contributing to a total

of 22.9 million people living with HIV in the region. Adult HIV prevalence varies

considerably across sub-Saharan Africa - from 0.2% in Madagascar to almost 26% in

8

Swaziland. An estimated 1.2 million people died from AIDS-related illnesses in 2011.

Antiretroviral therapy has had a signiﬁcant impact on the number of deaths from

AIDS; the scale-up of treatment contributed to a 29% decline in AIDS-related deaths

between 2005 and 2010. The scale-up of the prevention of mother-to-child transmis-

sion (PMTCT) programmes has also contributed to a decline in the number of new

HIV infections and AIDS-related deaths among children. Women are particularly af-

fected by HIV in sub-Saharan Africa; an estimated 59% of people living with HIV in

the region are women (Global Report: UNAIDS Report on the Global AIDS epidemic:

2011 , n.d.).

The statistics of the number of people living with HIV/AIDS worldwide is summarized

in Figure 1.2.

Figure 1.2: Global HIV/AIDS Statistics

Source: (Figure adapted from UNAIDS, 2012)

9

HIV ranks as one of the most destructive microbial scourges in human history and

poses a formidable challenge to the biomedical research and public health communities

in the world, (see (Grover & Das, 2011)). It has reached pandemic proportions, as

no country in the world is free from HIV/AIDS, (Fauci, 1999). One of the most

urgent world wide public health problems is the management of the HIV/AIDS. The

retrovirus reproduces inside the CD+

4cell and releases copies of itself into the blood.

It can be challenging to treat as the virus can rapidly mutate (alter) into new strains

of virus. The virus attacks the body’s immune system,through the CD+

4cells, that

are found in the blood. CD+

4cells are responsible for alerting the CD8+ T cells that

are responsible for ﬁghting foreign particles in the body. After they become infected,

the CD+

4cells are destroyed by HIV. Even though the body will attempt to produce

more CD+

4cells, their numbers will eventually decline and the immune system will

stop working.

HIV is spread through the exchange of bodily ﬂuids. There are deﬁnable risks for

HIV infection based upon the major known modes of transmission which are: sexual

contact (homosexual or heterosexual) with an infected person also called horizontal

transmission, direct contact with HIV infected blood or ﬂuid- in places where blood

products are not screened, there is a risk to recipients, and lastly, transmission from

an infected mother to her child, called mother to child transmission (MTCT) or ver-

tical transmission or perinatal infection. However, there are three major mechanism

of vertical transmission of HIV, infection through the placenta, known as in utero in-

fection; infection during birth known as intra-partum infection and infection through

breastfeeding, known as post-partum, (see (Waema & Olowofeso, 2005)).

Sexual contact with persons whose HIV viral load is greater, either with early infection

or in the late stage of clinical AIDS, increases the transmission risk. The presence

of cervical ectopia, oral contraceptive use, or pregnancy in women, intact foreskin

in men, and genital ulcer disease in either sex increases the risk for HIV infection.

Genital ulcers provide a more direct route to lymphatics draining to lymph nodes

containing many CD+

4lymphocytes and follicular dendritic cells. Tissue trauma

10

during intercourse does not appear to play a role in HIV transmission.

Although there are many complicating factors behind the spread of HIV, we still

believe that relevant mathematical models can provide a good insight of the dynamics

of the spread and management of the virus . If we can provide a satisfactory proﬁle

of this dynamics it will certainly help scientists to make timely remedial actions

(Tapadar & Ghosh, 2008).

1.2.1 Mechanism of HIV infection

Once the HIV virus enters the body it attacks CD+

4T cells of the human immune

system. These CD+

4T cells which are important constituent of the immune system

help in recognizing the foreign invasion, (Singh, Mishra, Dwivedi1, & Dwivedi, 2009).

The process of infection with HIV and CD+

4T cells completes in following steps: ﬁrst

of all the virus is attracted to a cell with the appropriate CD+

4receptor molecules

where it attaches itself to the susceptible cell membrane or by endocytosis and then

enters the cell. This process is facilitated by the proteins (gp120 and gp41) present

on the cell membrane. Then it transfuses the viral RNA and other necessary proteins

into the host cell. This process is called fusion. Thereafter, the viral RNA is reverse-

transcribed into DNA (this process is known as Reverse Transcription), which then

integrates with the host cell DNA. At this stage virus commandeers the mechanism

of the cell and starts producing its copies when activated. These copies are in form of

poly-protein which are cut into active proteins using protease. Finally the viral RNA

and these protein assemble near cell membrane and virus particles bud out of the

cell. These virus particles then mature and infect other cells, (Srivastava, Banerjee,

& chandra, 2012).

Once the HIV proviral DNA is within the infected cell’s genome, it cannot be elim-

inated or destroyed except by destroying the cell itself. A viral genome integrated

into the DNA of the host cell is called provirus, (Giorgi, 2011). The HIV provirus is

then replicated by the host cell. The infected cell can then release virions by surface

11

budding, or infected cells can undergo lysis with release of new HIV virions which

can then infect additional cells. During the process of production of virus some CD+

4

T cells burst due to viral load, this causes depletion of CD+

4T cells, (Stengel, 2008).

The probability of infection is a function of both the number of infective HIV virions

in the body ﬂuid which contacts the host as well as the number of cells available at

the site of contact that have appropriate CD+

4receptors.

HIV has the additional ability to mutate easily, in large part due to the error rate of

the reverse transcriptase enzyme, which introduces a mutation approximately once

per 2000 incorporated nucleotides. This high mutation rate leads to the emergence

of HIV variants within the infected person’s cells that can resist immune attack, are

more cytotoxic, can generate syncytia more readily, or can resist drug therapy. Over

time, diﬀerent tissues of the body may harbor diﬀering HIV variants. HIV eﬃciently

replicates only in living cells; the virus has no independent existence without infecting

and replicating within human cells.

1.2.2 HIV replication Cycle

The HIV replication cycle is as shown in Figure 1.3 and described as follows.

Entry to the cell The virus binds itself to the target monocyte/macrophages and

CD+

4T-cells by absorbing its surface proteins (the envelop gp120 protein) to

two host-cell receptors (proteins): the CD+

4molecule receptor and CCR5 or

CXCR4 co-receptors. Once HIV has attached to the host cell, the HIV-RNA

and enzymes such as reverse transcriptase (RT), integrase (IN), protease (PR),

are able to enter into the host cell (cytoplasmic compartment) (Santoro, 2009).

Replication and transcription The RT converts the single-stranded RNA genome

of the virus into double-stranded DNA, which is used to make double-stranded

viral DNA intermediate (vDNA) in a process known as reverse transcription;

this is prone to errors (mutations). The new vDNA is transported into the

12

cell nucleus to be integrated into the host chromosome. At this stage, the

virus is known as a provirus. The integration process requires the IN enzyme.

Thereafter, the virus can enter a latent stage of HIV infection, because the

proviral DNA remains permanently within the target cell in either a productive

or latent state. The factors that may aﬀect this stage are the HIV variant, the

cell type, and the expression capacity of the host cell (Nowak & May, 2000).

Eventually, the transcription of the HIV genomic materials and viral proteins

forms the HIV messenger (mRNA) and proteins required for the assembly of the

virus. This production is exported from the cell nucleus into the cell cytoplasm.

This process remains poorly understood because it involves many viral proteins

(Santoro, 2009).

Assembly and release The new mRNA codes for the new viral proteins that will

contribute to the reconstruction of the HIV-RNA. The viral proteins help the

mRNA and the reconstruction proteins to transport into the cell membrane

side. The structural components of the virus accumulate at the membrane of

the infected cell to construct the HIV virion. Leftover proteins (cleaved by the

protease) associated with the inner surface of the host-cell membrane, along

with the HIV RNA, are released to form a bud from the host cell, and can

proceed to infect other healthy cells (Santoro, 2009).

In order to understand how to manage HIV, we need to know the in-host HIV dynam-

ics in an infected person. A schematic diagram of HIV virus life cycle is presented in

Figure 1.3.

13

Figure 1.3: HIV life cycle

Source: Figure adapted from NIAID-www.niaid.nih.gov

14

HIV uses RNA to encode its genetic material. A complex of two proteins, gp120

and gp41, present on the viral surface enables the virus to fuse with the target cell

membrane. Once the virion enters a cell, the enzyme called reverse transcriptase

transcribes the viral RNA into double-stranded DNA, which then enters the host

genome. The time from when the virus enters the target cell to when the ﬁrst virions

start budding out has been estimated to last about 24 hours, (Markowitz, Louie,

Hurley, Sun, & Di Mascio, 2003) , although in some cases the virus can lie dormant

inside the cell for years. The infected cell will on average survive another 24 hours,

during which it will produce a large number of new viral particles. HIV is then rapidly

cleared, roughly in 45 minutes, and on average, only between 6 and 10 virions will go

on to successfully infect new cells, (Ribeiro et al., 2010).

Humans reproduce their genes in the form of DNA, not RNA. The DNA is converted

into RNA to make proteins, but when the body wants to make a copy of our genes,

it needs DNA. Since humans dont have a way to copy RNA, the HIV virus needs to

change its RNA to DNA in order to hijack our replication system. It carries its own

personal copy machine, the enzyme reverse transcriptase, to do just that. Reverse

transcriptase is like a little tool that crawls along the RNA molecule and makes a

matching strand of DNA, one bit at a time. The bits are individual DNA nucleotides

that ﬂoat around in within human cells.

Once a CD+

4T-cell has been infected by a virus, its normal operations are impaired.

Ultimately the number of CD+

4T-cells in the body diminishes, and since these cells are

critical for immune system response, its easy for people with AIDS to catch diseases

that would normally be easily defeated by the natural immune response. Since the

HIV virus mutates readily, its diﬃcult to formulate an eﬀective vaccine.

Mathematical modeling can be used to gain insights of understanding the complex

dynamics of many biomedical systems such as in epidemiology, ecology and virol-

ogy. These models can help to improve our understanding of internal dynamics of

HIV/AIDS. With that in mind, this study aims to develop stochastic models that are

used to examine various alternatives for the control and treatment of HIV/AIDS.

15

1.2.3 Role of Langerhans cells in HIV infection

HIV is a devastating human pathogen that causes serious immunological diseases in

humans around the world. The virus is able to remain latent in an infected host for

many years, allowing for the long-term survival of the virus and inevitably prolonging

the infection process, (Coleman & Wu, 2009). The location and mechanisms of HIV

latency are under investigation and remain important topics in the study of viral

pathogenesis. Given that HIV is a blood-borne pathogen, a number of cell types

have been proposed to be the sites of latency, including resting memory CD+

4T

cells, peripheral blood monocytes, dendritic cells (including langerhans cells) and

macrophages in the lymph nodes, and haematopoietic stem cells in the bone marrow,

(Corine, Coleman, Wang, & Wu, 2012). This study updates the latest advances in

the study of HIV interactions with langerhans cells, and highlights the potential role

of these cells as viral reservoirs and the eﬀects of the HIV-host-cell interactions on

viral pathogenesis.

Despite advances in our understanding of HIV and the human immune response in

the last 25 years, much of this complex interaction remains elusive. CD+

4T-cells

are targets of HIV, and are also important for the establishment and maintenance of

an adaptive immune response, (Hogue et al., 2008). The skin and mucosa are the

ﬁrst line of defense of the organism against external agents, not only as a physical

barrier between the body and the environment but also as the site of initiation of

immune reactions. The immunocompetent cells which act as antigen-presenting cells

are Langerhans cells (LCs). Infection of LCs by HIV is relevant for several reasons.

Firstly, LCs of mucosal epithelia may be among the ﬁrst cells to be infected following

mucosal HIV exposure and Secondly, LCs may serve as a reservoir for continued

infection of CD+

4T cells, especially in lymph nodes where epidermal LCs migrate

following antigenic activation, (Ramazzotti et al., 1995).

Many indirect and/or direct experimental data have shown that LCs may be a priv-

ileged target, reservoir and vector of dissemination for the HIV from the inoculation

16

sites (mucosa) to lymph nodes where the emigrated infected LCs could infect T lym-

phocytes, (see (Zoeteweij & Blauvelt, 1998; Arrighi, 2004)). Originated from the

bone marrow, LCs migrate to the peripheral epithelia (skin, mucous membranes)

where they play a primordial role in the induction of an immune response and are

especially active in stimulating naive T lymphocytes in the primary response through

a speciﬁc cooperation with CD+

4-positive lymphocytes after migration to proximal

lymph nodes, (Schmitt & Dezutter-Dambuyant, 1994). Apart from many plasma

membrane determinants, LCs also express CD+

4molecules which make them suscep-

tible targets and reservoirs for HIV, (see (Dezutter-Dambuyant & Colette, 1995)).

Once infected, these cells due to their localization in areas at risk (skin, mucous

membranes), their capacity to migrate from the epidermal compartment to lymph

nodes and their ability to support viral replication without major cytopathic eﬀects,

could play a role of vector in the dissemination of virus from the site of inoculation

to the lymph nodes and thereby to contribute to the infection of T lymphocytes, for

more reading, (Dezutter-Dambuyant & Colette, 1995).

Langerhans cells (LCs) which are members of the dendritic cells family and are pro-

fessional antigen-presenting cells (APCs), reside in epithelial surfaces such as the skin

and act as one of the primary, initial targets for HIV infection, (see (Saksena et al.,

2010; Zhang, 2005)). They specialize in antigen presentation and belong to the skin

immune system (SIS) and play a major role in HIV pathogenesis. As part of the

normal immune response, LCs capture virions at the site of transmission in the mu-

cosa(peripheral tissues), and migrate to the lymphoid tissue where they present to

naive T cells and hence are responsible for large-scale infection of CD+

4T lymphocytes

(reviewed in (Sewell & Price, 2001; Wilﬂingseder, Banki, Dierich, & Stoiber, 2005;

Donaghy, Stebbing, & Patterson, 2004)). These cells play an important role in the

transmission of HIV to CD+

4cells, (Hlavacek, Stilianakis, & Perelson, 2000), thus, LC

-CD+

4+ cell interactions in lymphoid tissue, which are critical in the generation of

immune responses, are also a major catalyst for HIV replication and expansion. This

replication independent mode of HIV transmission, known as trans-infection, greatly

17

increases T cell infection in vitro and is thought to contribute to viral dissemination

in vivo, (McDonald, 2010).

The Langerhans cell is named after Paul Langerhans, a German physician and anatomist,

who discovered the cells at the age of 21 in 1868 while he was a medical student,

(Romani, Brunner, & Stingl, 2012). The uptake HIV by professional antigen pre-

senting cells (APCs) and subsequent transfer of virus to CD+

4T cells can result in

explosive levels of virus replication in the T cells. This could be a major pathogenic

process in HIV infection and development of AIDS. This process of trans (Latin; to

the other side) infection of virus going across from the APC to the T cell is in con-

trast to direct, cis (Latin; on this side) infection of T cells by HIV, (see (Rinaldo,

2013)). Langerhans cell results in a burst of virus replication in the T cells that is

much greater than that resulting from direct, cis infection of either APC or T cells, or

trans infection between T cells. This consequently shows that Langerhans cells may

be responsible for the quick spread of HIV infection.

The individual cells of the immune system are highly interactive, and the overall

function of the system is a product of this multitude of interactions. The interplay

between HIV and the immune system is particularly complicated, as HIV directly

interacts with many immune cells, altering their functions, ultimately subverting the

system at its core, (Hogue et al., 2008). Because of this complexity, the immune re-

sponse and its interaction with HIV are naturally suited to a mathematical modelling

approach. Elucidating the mechanisms of LC-HIV-CD+

4T cells interactions is cru-

cial in uncovering more details about host-HIV dynamics during HIV infection. To

explore the role of LCs in HIV infection, we ﬁrst develop a stochastic model of HIV

dynamics in vivo before therapy. Next, we introduce therapeutic intervention and

ﬁnally investigate which parameters and/or which interaction mechanisms strongly

aﬀect the infection dynamics.

Mathematical models can facilitate the understanding of complex biomedical systems

such as in HIV/AIDS. Untangling the dynamics between HIV and the immune cellular

populations and molecular interactions can be used to investigate the eﬀective points

18

of interventions in the HIV life cycle. With that in mind, we will develop state

transition systems dynamics and stochastic models that can be used to understand

the dynamics of HIV infection and therapy. We start by describing a model for the

interaction of HIV with CD+

4T cells that considers three populations: (uninfected T

cells, actively infected T cells, free virus). We explore how mathematical models can

be used to design speciﬁc treatment regimes that can boost anti-viral immunity and

induce long-term virus control.

1.2.4 HIV and treatment

Recently, highly active antiretroviral therapies (HAART) have become available for

HIV infected patients. The HAART treatments can rapidly reduce the amount of

virus in the plasma of infected individuals to a very low level. Without treatment, the

number of CD+

4cells per unit volume is expected to decrease with time since infection

by the virus. However on introduction of treatment interventions, the process is

expected to reverse with the counts increasing to return to the normal level.

An understanding of typical course of the infection within an individual and the

quantitative information about latent period, infectious period or in some cases in-

cubation period is essential in designing eﬀective treatment strategies. According to

the current international guidelines, initiation of the an anti-retroviral treatment is

based on both clinical-immunological parameters and HIV viremia (viral load), they

state that, antiretroviral therapy should be initiated for patients with CD+

4cell count

below 350 cells/µL. In this study, we will study the implication of initiating therapy

at an earlier state. Then, since an infected individual being in any of the immunolog-

ical states, usually begins an antiviral treatment, the patient cost evaluation problem

arises. The peculiarity of this kind of treatment is that, once it started, it will contin-

ues till the patient death, also if the medical tests show an improvement of the CD+

4

count, then the patient has a recovery from the stage where he is, (Di Biase et al.,

2007). Knowing the state in which the patient is helps a practitioner to decide on the

19

correct dosage for the patient and also on the time to initiate therapy.

In recent years, anti retroviral drugs have been developed which attack diﬀerent

phases of virus life cycle in CD+

4T cell so as to clear the virus or to keep it at low

level. These drugs include Fusion Inhibitors (FI), Reverse Transcriptase Inhibitors

(RTI), and Protease Inhibitors (PI). FI inhibits the fusion of viral particles in the

host cell and causes low infection rate. RTI inhibits the reverse transcription of

viral RNA into DNA, which decreases the number of productively infected cells.

PI inhibits the cleaving of polyprotein encompassing reverse transcriptase, protease,

integrase, and other matrix proteins into functional units. This causes the production

of noninfectious viral particles. Generally a combination of these drugs is used to get

better results and to keep virus to an undetectable level, (Srivastava & Chandra,

2010).

The knowledge of principal mechanisms of viral pathogenesis, namely the binding of

the retrovirus to the gp120 protein on the CD+

4cell, the entry of the viral RNA into

the target cell, the reverse transaction of viral RNA to viral DNA, the integration of

the viral DNA with that of the host, and the action of viral protease in cleaving viral

proteins into mature products have led to the design of drugs (chemotherapeutic

agents) to control the production of HIV. Various models have been proposed to

understand the eﬀect of drug therapy on viremia level, (De Leenheer & Smith, 2003;

Srivastava & Chandra, 2010). Here it may be pointed out that the eﬀect of RTI does

not decrease the infection rate, as has been considered in previous models, (Perelson

& Nelson, 1999; Ouifki & Witten, 2007). Rather due to presence of RTI the cells

which have got infected with virus will cease to become productively infected to

produce virus copies and they will either become susceptible for infection again or

will be cleared due to immune response. (Srivastava & Chandra, 2010) classiﬁed the

cells in infected class into two subclasses: Pre-RT class and Post-RT class (depending

whether in infected cell reverse transcription has been completed or not), and studied

the dynamics of the model.

Mathematical models provide an alternative way to study the eﬀects of diﬀerent drugs,

20

a procedure which is otherwise risky or unethical when carried out on patients. Apart

from providing an alternate route that does not violate any rights, mathematical

models also provide clinicians with almost instant results on studies that would have

required several months or even years when conducted in animals or human beings.

Such models have helped clinicians in making the complex choices involved in treating

HIV-infected patients.

1.3 Problem statement

This study was ﬁrstly motivated by the fact that eradication of the HIV virus is

not attainable with the current available drugs and the focus has now shifted from

eradication of the virus to the management and control of the virus’ progression.

The question is how to maximally suppress the viral load with minimum use of the

treatment therapies. The other question is how to allocate the limited budget to

minimize the number of new HIV cases and also to improve the lives of those infected

by the disease. Secondly, the treatment of HIV is expensive and often times erratic.

It is therefore necessary to develop a mathematical model which when applied to HIV

data accurately details the cost of treating a HIV patient. Mathematical modeling

can be used to gain insight on the dynamics of the spread and management of the

virus whose analysis can be used to answer these questions. Existing deterministic

models lack the eﬀect of randomness, and hence we developed stochastic models that

are used to capture the dynamics of HIV infection and examine various alternatives

for the control and treatment of HIV/AIDS.

1.4 Objectives of the study

1. The Main objective of this study is to make use of mathematical models and

in particular develop stochastic models for the In vivo HIV dynamics and its

management. Mathematical models can help to improve our understanding of

21

dynamics of diseases such as in HIV/AIDS by providing an alternative way

to study the eﬀects of diﬀerent drugs, a procedure which is otherwise risky or

unethical when carried out on patients. These models will relate the various

dynamics of the epidemic to the cost implications over time and regionally.

2. The study was divided into the following speciﬁc objectives:-

a). To formulate a stochastic Virus-Host interaction model to describe the distribu-

tion of the virons and the immune system cells after a certain period of time given

the initial distributions and with treatment therapies.

b). To formulate Non-Homogeneous Semi-Markov Stochastic models (NHSMSM) to

describe the progression of the disease as deﬁned by the CD+

4cell count and viral

load for those infected and how this can be controlled.

c). To formulate a Management cost model (MCM). This model when ﬁtted with real

data will give insights on which treatment combinations incurs minimum cost and

which treatment combination is most eﬀective at each disease state.

d). To formulate a cost Beneﬁt Analysis Model (CBAM), which we deﬁne as Treat-

ment Reward Model (TRM). This model when ﬁtted to data will be able to show

the optimal or eﬀective time to initiate therapy.

1.5 Signiﬁcance of the study

The results of this study will provide insight on the timing of interventions geared

towards the management of the HIV epidemic. The stochastic reward model for-

mulated will show at what stage, should the infected persons start the treatment

therapy for their incubation period to be prolonged and also what treatment therapy

combinations are necessary. The cost model can help the government and donors to

make informed decisions about resource allocation in planning and evaluating control

22

strategies for the disease (budget plan for the acquisition of the ARTs, VCT centers,

nutritional support, etc).

1.6 Scope of the study

The study covers mathematical modeling of HIV evolution and in particular, to study

the internal dynamics of HIV virus, stochastic models were employed. As most real

world problems are not deterministic including stochastic eﬀects into the model gave

us a more realistic way of modelling viral dynamics, particularly when modelling

biological phenomena such as internal HIV viral dynamics. Probability Generating

function was used as a tool to solve nonlinear diﬀerential equations which were derived

in the formulation process. The study also employed Semi-Markov process in deriving

the cost and reward models for HIV management.

One of the limitations faced in this study was availability of real data. The data

collected was not good enough to produce desirable results, therefore the study has

relied on data from literature and simulations.

1.7 Conceptual framework

This study develops stochastic models for the study of Intra-Host HIV dynamics. For

the HIV-Host interaction, stochastic models are formulated describing the interaction

of HIV and the CD+

4cells. Diﬀerence equations are obtained and solved using proba-

bility generating function to obtain moments of the numbers of uninfected CD+

4cells,

the HIV infected CD+

4T cells, and the free HIV particles at any time t.

For the disease progression, stochastic models based on Semi-Markov process are

formulated describing the progression of HIV in an infected person. Semi- Markov

process is used in the study to derive a conditional probability of a patient transition-

ing from one disease state to another. Also the Semi-Markov process is used to derive

23

the disease management cost model and the Treatment Reward Model (TRM)for an

HIV patient on treatment.

The conceptual framework for the study is summarized in Figure 1.4. The green

arrows show the route used to model the HIV-Host interactions and the red arrows

show the route for modeling HIV disease progression in an infected person.

24

Figure 1.4: Conceptual Framework for the study

Source: Author

25

1.8 Outline of the study

In this study, stochastic models for In-Host HIV evolution are developed. Reviews

on existing models in literature on HIV evolution are discussed in Chapter 2. In

Chapter 3, stochastic models are formulated describing the interaction of HIV and the

CD+

4cells. Partial diﬀerential equations are obtained for the probability generating

function of the numbers of uninfected CD+

4cells, the HIV infected CD+

4T cells,

and the free HIV particles at any time t. In the same Chapter, moments for the

variable’s populations are derived in the absence of therapeutic intervention and also

in a therapeutic environment.

The interaction of HIV, LCs and CD+

4cells is discussed in Chapter 4. Partial diﬀer-

ential equations are obtained for the probability generating function of the numbers

of immune cells, and the free HIV particles at any time t. In the same chapter, mo-

ments for the variables are derived in the absence of therapeutic intervention and also

in a therapeutic environment, simulations of the models provide a numerical illustra-

tion to demonstrate the impact of intercellular delay and therapeutic intervention in

controlling the progression of HIV . Chapter 5 presents a model for the probability

of extinction of HIV virus.

Chapter 6 brieﬂy introduces Semi-Markov processes, then derivation of conditional

probability of semi-Markov process is presented. The model is used to determine the

probability of an HIV patient moving from one disease state to another. The disease

management cost model is formulated in this chapter and cost beneﬁt for the patient

on treatment is also presented. Some concluding remarks follow in Chapter 7.

26

Chapter 2

Literature Review

2.1 Introduction

This chapter reviews existing models in literature on HIV/AIDS internal virus dy-

namics. Review on deterministic and stochastic models employed in literature on

HIV/AIDS modeling will be considered. Summary of the models so far applied on

HIV internal dynamics will be presented.

2.2 Review of Intra-Host HIV models

The primary target of HIV is the immune system itself, which is gradually destroyed.

Viral replication actively continues following initial HIV infection, and the rate of

CD+

4lymphocyte destruction is progressive Klatt (2002). Clinically, HIV infection

may appear “latent” for years during this period of ongoing immune system destruc-

tion. During this time, enough of the immune system remains intact to provide im-

mune surveillance and prevent most infections. Eventually, when a signiﬁcant number

of CD+

4lymphocytes have been destroyed and when production of new CD+

4cells

cannot match destruction, then failure of the immune system leads to the appearance

of clinical AIDS, (Klatt, 2002).

Infection with HIV is sustained through continuous viral replication with reinfection

of additional host cells. Both HIV in host plasma and HIV-infected host cells appear

to have a short lifespan; and late in the course of AIDS the half-life of plasma HIV

27

is only about 2 days, see (Klatt, 2002). Thus, persistent viremia requires continuous

reinfection of new CD+

4lymphocytes followed by viral replication and infected host

cell turnover. This rapid turnover of HIV and CD+

4lymphocytes promotes origin of

new strains of HIV within the host from mutation of HIV.

HIV infects cells in the immune system and the central nervous system. One of the

main type of cells that HIV infects is the T helper lymphocyte. These cells play

a crucial role in the immune system, by coordinating the actions of other immune

system cells (“Pathophysiology of the human immunodeﬁciency virus”, n.d.). A large

reduction in the number of T helper cells seriously weakens the immune system. HIV

infects the T helper cell because it has the protein C D+

4on its surface, which HIV

uses to attach itself to the cell before gaining entry. This is why the T helper cell

is sometimes referred to as a CD+

4lymphocyte (Koppensteiner, Brack-Werner, &

Schindler, 2012). Once it has found its way into a cell, HIV produces new copies of

itself, which can then go on to infect other cells. Over time, HIV infection leads to a

severe reduction in the number of T helper cells available to help ﬁght disease. The

number of T helper cells is measured by having a CD+

4test and is referred to as the

CD+

4count. It can take several years before the CD+

4count declines to the point

that an individual is said to have progressed to AIDS.

2.2.1 Kinetics of HIV infection

The clinical course of untreated HIV infection proceeds through four stages. The

diﬀerent clinical stages of HIV infection reﬂect patient immunological and virological

status. The immunological status is determined by the absolute (cells/µL) or relative

( lymphocytes) number of CD+

4T-cell count (Flandre et al., 2007). The virologi-

cal status is guided by the number of copies of HIV RNA per milliliter of plasma

expressed as copies/µL. Although HIV ultimately resides within cells, the plasma

measurement is an accurate reﬂection of the burden of infection and the magnitude

of viral replication. It is used to assess the risk of disease progression and can help

28

guide initiation of therapy. It is critical in monitoring virologic response to ART.

These two surrogate markers are currently widely used in HIV medicine to monitor

progression of the infection.

The four distinct HIV progression stages are: primary infection, clinically asymp-

tomatic stage, symptomatic HIV infection, and progression from HIV to AIDS.

STAGE 1: Primary HIV Infection (PHI)

This is the early phase (acute infection) of infection with CD+

4count above 500

cells/ µL, this is during the ﬁrst 2 - 6 weeks after HIV enters the host. The period

is characterized by massive uncontrolled HIV replication in blood, resulting in high

levels of free HIV virions circulating in the plasma (millions copies/µL). Existing

scientiﬁc knowledge show that, during this stage there is a large amount of HIV in the

peripheral blood and the immune system begins to respond to the virus by producing

HIV antibodies and cytotoxic lymphocytes. This process is known as seroconversion.

A normal CD+

4count is 1,000 per µL, with a range of 600-1400 per µL. The CD+

4

count falls during primary infection, then returns to near or lower than normal levels.

Eventually, it slowly decreases, taking many years to reach the level of 200 per µL

that characterizes AIDS, (Callaway & Perelson, 2002). After initial infection with

HIV, the viral Load (the number of virions in the peripheral blood) quickly picks to

very high levels usually greater than 100,000 copies/µL followed by a fall of the helper

T lymphocytes (CD+

4T cells) (Perelson, Neumann, & Markowitz, 1996). Patients

usually show symptoms similar to ﬂu in this phase. Then during the 9th to the

12th week, the viral load experiences a sharp decline and the number of CD+

4T cells

returns almost to their normal level. With the viral load falling to a very low level, the

clinical symptoms disappear. However, the viruses cannot be completely eliminated

after primary infection unlike most other viral infections.

Existing scientiﬁc knowledge show that, primary HIV infection may go unnoticed in

at least half of cases or produce a mild disease which quickly subsides, followed by

29

a long clinical “latent” period lasting years. Prospective studies of acute HIV in-

fections show that fever, lymphadenopathy, pharyngitis, diﬀuse erythematous rash,

arthralgia/myalgia, diarrhea, and headache are the commonest symptoms seen with

acute HIV infection. These symptoms diminish over 1 to 2 months. The symptoms

of acute HIV infection resemble an infectious mononucleosis-like syndrome. Symp-

tomatic acute HIV infection is more likely to occur in persons who acquired HIV

infection through sexual transmission.

Generally, within 3 weeks to 3 months the immune response is accompanied by a

simultaneous decline in HIV viremia. Both humoral and cell mediated immune re-

sponses play a role. The CD+

4lymphocytes rebound in number, but not to pre-

infection levels. Seroconversion with detectable HIV antibody by laboratory testing

accompanies this immune response, sometimes in as little as a week, but more often

in two to four weeks. Prolonged HIV infection without evidence for seroconversion,

however, is an extremely rare event as is indicated in existing scientiﬁc knowledge.

STAGE 2: Clinically Asymptomatic Stage

The intermediate phase (clinical latency) when CD+

4count falls between 350 and

500 cells/ µL, which is a long period of the asymptomatic stage lasting on average

between 8 and 10 years or more, as its name suggests, is free from major symptoms,

although there may be swollen glands. Existing scientiﬁc knowledge indicate that,

the level of HIV viral load in the peripheral blood drops gradually in this stage to

the range of 20,000 - 60,000 copies/µL and remains more or less steady, but people

remain infectious and HIV antibodies are detectable in the blood, so antibody tests

will show a positive result. During this phase, viral load remains very low, while the

CD+

4+ T cell population continues to decline slowly until it becomes lower than a

critical value. Research has shown that HIV is not dormant during this stage, but is

very active in the lymph nodes. Patients feel relatively well during this period.

30

STAGE 3: Symptomatic HIV Infection

This stage occurs when the host’s immune system is immunocompromised to such a

degree that without an initiation of Highly Active Anti-Retroviral Therapy (HAART)

the infection is going to progress to Acquired Immunodeﬁciency Syndrome (AIDS) at

rapid pace. This is the stage when CD+

4count falls between 200 and 350 cells/µL.

Over time the immune system becomes severely damaged by HIV. This is thought to

happen for three main reasons:

i). The lymph nodes and tissues become damaged or ’burnt out’ because of the

years of activity.

ii). HIV mutates and becomes more pathogenic, in other words stronger and more

varied, leading to more T helper cell destruction.

iii). The body fails to keep up with replacing the T helper cells that are lost.

As the immune system fails, so symptoms develop. Initially many of the symptoms

are mild, but as the immune system deteriorates the symptoms worsen. Scientiﬁc

knowledge show that, symptomatic HIV infection is mainly caused by the emergence

of certain opportunistic infections that the immune system would normally prevent.

This stage of HIV infection is often characterised by multi-system disease and in-

fections that occur in almost all body systems. Treatment for the speciﬁc infection

is often carried out, but the underlying cause is the action of HIV as it erodes the

immune system. Unless HIV itself can be slowed down the symptoms of immune

suppression will continue to worsen.

STAGE 4: Progression from HIV to AIDS

Finally the advanced phase (AIDS) with CD+

4less than 200 cells/µL, at this stage,

the viral load climbs up again, resulting in an onset of AIDS. In such a ﬁnal phase,

the impaired immune system can no longer ﬁght oﬀ infections, and patients usually

die from a variety of opportunistic infections that would normally be cleared.

31

In summary, during acute infection an initial peak of HIV RNA (viral load) and a

massive depletion in CD+

4cell count is observed and after 3-6 months of infection the

viral load and CD+

4cell count reach a more stable level (steady state). Without an-

tiretroviral treatment the asymptotic chronic state lasts for around 10 years (Wolbers

et al., 2010) before an increase in viral load and an accelerated decrease in CD+

4cell

count is observed. Ultimately, the progressive destruction of the immune system leads

to the so-called Acquired Immunodeﬁciency Syndrome (AIDS) and death as shown

in Figure 2.1.

Figure 2.1: Natural Progression of HIV

The viral load is shown in red, and the CD+

4cell counts in blue. (Figure adapted

from Giorgi, 2011)

32

Unlike most infections in past epidemics, AIDS is distinguished by a very long latent

period before the development of any visible signs of infection, (Klatt, 1998). Dur-

ing this phase, there is little or no viral replication detectable in peripheral blood

mononuclear cells and little or no culturable virus in peripheral blood. The CD+

4

lymphocyte count remains moderately decreased. However, the immune response

to HIV is insuﬃcient to prevent continued viral replication within lymphoid tissues.

Tests for HIV antibody will remain positive during this time but p24 antigen tests

are usually negative. There is no evidence to suggest that seroconversion, or loss of

antibody, occurs in HIV infected persons (Giorgi, 2011).

The average HIV-infected person may have an initial acute self-limited illness, may

take up to several weeks to become seropositive, and then may live up to 8 or 10

years, on average, before development of the clinical signs and symptoms of AIDS,

(Klatt, 1998). Persons infected with HIV cannot be recognized by appearance alone,

are not prompted to seek medical attention, and are often unaware that they may

be spreading the infection. There has been no study to date that shows a failure of

HIV-infected persons to evolve to clinical AIDS over time, though the speed at which

this evolution occurs may vary.

At least 10% of persons infected with HIV are “long survivors” who have not had

signiﬁcant progressive decline in immune function. Findings include: a stable CD+

4

lymphocyte count, negative plasma cultures for HIV, a strong HIV neutralizing anti-

body response, and a strong virus-inhibitory CD8 lymphocyte response (Klatt, 1998).

In addition,existing scientiﬁc research show that, the lymph node architecture is main-

tained without either the hyperplasia or lymphocyte depletion common to progression

to AIDS. Though peripheral blood mononuclear cells contain detectable HIV and viral

replication continues in long survivors, though their viral burden is low.

The development of signs and symptoms of AIDS typically parallels laboratory testing

for CD+

4lymphocytes (Akinbami et al., 2013). A decrease in the total CD+

4count

below 500/microliter presages the development of clinical AIDS, and a drop below

200/microliter not only deﬁnes AIDS, but also indicates a high probability for the

33

development of AIDS-related opportunistic infections and/or neoplasms. Plasma HIV

RNA increases as plasma viremia becomes more marked. The risk for death from HIV

infection above the 200/microliter CD+

4level is low.

2.2.2 Other factors associated with HIV Progression

Although the CD+

4count and HIV viral load are the most important predictors of HIV

progression, it is increasingly recognized that a number of other factors, and likely

others that remain unknown, contribute to disease progression in HIV infection,.

These could include, genetic factors, age, observational times, other immune cells etc.

Genetic factors

Host genetic factors have been shown to alter the rate of HIV progression. Existing

scientiﬁc research show that, various human leukocyte antigen (HLA) alleles have

been associated with faster or slower progression rates. Genetic polymorphisms also

play a role. For example, CCR5 is a chemokine receptor that can serve as a coreceptor

for HIV entry into the CD+

4cell. A naturally occurring variant allele for CCR5 has

a 32 base pair deletion. Several existing research show that individuals who are

heterozygous for this allele have slower progression of HIV disease.

Age

Several studies have shown a higher risk of morbidity and mortality in older patients.

When followed from seroconversion, older patients demonstrate faster disease pro-

gression compared with younger patients. Older patients also are found to have a less

robust increase in the CD+

4count in response to ART. These observations have led

to the recommendation to consider age as a factor in determining when to initiate

therapy.

34

2.2.3 CD+

4T cells and viral load dynamics

The CD+

4T-cells are a type of T-lymphocyte (white blood cells) that helps coordinate

the immune system’s response to infection and disease . These cells express a molecule

called CD+

4on their surfaces, which allow them to detect foreign substances, including

viruses that enter the body. Unfortunately CD+

4cells (also called T cells or T-helper

cells) are the primary targets of the HIV virus. The CD+

4count is the number of

CD+

4cells per microliter (L) of blood and is an excellent indicator of how healthy the

immune system is and whether a person is at risk of getting certain infections. It is

indicated in cells per mm3, and it is measured by taking a blood sample. Although

the normal number of CD+

4cells varies from individual to individual, it is usually

between 800 and 1 500 cells per µL. HIV patients who are otherwise healthy and

symptom-free should have their CD+

4cell count and viral load tested about two to

four times a year. However, symptomatic patients should be tested more frequently

to evaluate both the risk of opportunistic infections and the response to HIV drug

treatments.

The viral load refers to the actual number of HIV viruses in the blood. The viral

load can be “counted” by doing blood tests. The virus count is indicated in copies

per ml. The HIV viral load is the best indicator of how active HIV is in the patient’s

body (prognosis), and it is also used to measure a person’s response to antiretroviral

treatment. There is a very special relationship between the viral load and the CD+

4

cell count, and if considered together, they can predict whether a person’s journey

towards AIDS (the ﬁnal stage of disease) will be rapid or slow (Van, 2008). The CD+

4

count and the viral load measurement are both fundamental markers of the state of an

HIV infected patient. The potential of these immunological and virological reservoirs

determines the way the patients are handled, see (Mathieu, Foucher, Dellamonica, &

Daures, 2007).

Viral load and CD+

4cells vary together, which means that a higher viral load will

lead to a lower CD+

4count (because the virus destroys the CD+

4cells), while a lower

35

viral load will go hand in hand with a higher CD+

4cell count (because less viruses in

the blood give the immune system a chance to built up its resources again). Disease

progression (the extent to which an HIV infected person gets sick with opportunistic

diseases and infections) will depend on the viral load as well as the CD+

4cell count

in the blood. The higher the viral load, and the lower the CD+

4cell count, the easier

it will be for all kinds of infections to attack the body. The progression to the ﬁnal

phase of Aids (and death) will therefore be much faster with a high viral load. On the

other hand: An HIV-infected person with a low viral load and a high CD+

4count can

stay healthy for many years, because the immune system is strong enough to ﬁght oﬀ

infections.

Antiretroviral therapy (ART) directly aﬀects the activity of HIV in the body and will

lower the viral load. With less HIV present, the body is able to produce more CD+

4

cells and improve the immune system. An undetectable viral load does not mean HIV

is cured or that the patient is not infectious to others. It means that virus cannot be

detected in the blood, although it exists in other parts of the body (Price & Spudich,

2008). If the patient’s CD+

4count increases with successful ART (Antiretroviral

therapy), he or she may be protected from infections and other illnesses related to

HIV. CD+

4and viral load values show disease staging and also guides decision to

start or defer ART. Most viruses, after infecting humans, are rapidly cleared during

the initial acute stage of infection and after some time (about six weeks) establish a

long-term asymptomatic chronic infection due to a mutual compromise of the virus

and the immune system.

CD+

4T-Cell count test

The CD+

4T-cell count serves as a surrogate for T-cell mediated immune response

assays in monitoring the progression of HIV’s response to therapy. The test deter-

mines the counts of CD+

4T-cell per cubic millimetre of a blood sample. An average

normal CD+

4T- cell count is 1000mm3, with a range of 600−1400mm3in the RPAH

laboratory. This count falls during primary infection and then usually returns to near

36

normal levels (Callaway & Perelson, 2002). Then, untreated, the CD+

4T-cell count

falls gradually to about 200mm3, or even less, and at this level the incidence of OIs

arises; this phase is known as the AIDS incident stage. This reduction is associated

with the hyper-activation of CD8 T-cells, which may kill HIV- infected cells (Santoro,

2009). The CD8 T-cell response is thought to be important in controlling the infec-

tion; however, the CD8 T-cell counts decline similarly to those of CD+

4T-cells over

prolonged periods.

Viral load test

The nucleic acid-based HIV viral load (VL) test is used to decide when to start an

HIV therapy regimen. The test determines the number of HIV copies in a blood

sample. VL is a strong predictor of the likelihood of disease progression and provides

strong prognostic value, when paired with CD+

4T-cell counts. Therefore, the current

guidelines of US NIH and WHO for monitoring the HIV infection in developed coun-

tries advocate the use of VL assays for determining initiation of treatment regimens,

monitoring the responses to these therapies, and switching drug regimens (Global Re-

port: UNAIDS Report on the Global AIDS epidemic: 2012 , n.d.); (Santoro, 2009).

Many assays methods have been developed and shown to be robust and their use

attests to the diagnostic power of VL.

2.3 Mathematical modeling of Intra-Host HIV dy-

namics

Since HIV pandemic ﬁrst became visible, many mathematical models have been de-

veloped to describe the immunological response to infection with human immunodeﬁ-

ciency virus (HIV). Mathematical modeling has proven to be valuable in understand-

ing the dynamics of infectious diseases with respect to host-pathogen interactions.

HIV which was ﬁrst discovered in 1980, has spread relentlessly throughout the world

37

and now is a major epidemic worldwide as is shown in several exixting scientiﬁc re-

search. HIV spreads by attacking the immune system, in particular by depleting the

CD+

4T cells. The pathogenesis of HIV infection is a function of the virus life cycle,

the host cellular environment, and quantity of virus in the infected individual (Dalal

et al., 2008). Factors such as age or genetic diﬀerences among individuals, the level

of virulence of an individual strain of virus, and co-infection with other microbes may

inﬂuence the rate and severity of disease progression.

Cells with CD+

4receptors at the site of HIV entry become infected and viral replica-

tion begins within them. The infected cells can then release virions or infected cells

can undergo lysis to release new virions, which can then infect additional cells. CD+

4

cells, the primary targets of HIV, become infected as they encounter HIV. Active

replication of HIV occurs at all stages of the infection. This interaction between the

virus and the immune system is called HIV internal viral dynamics. Modeling the

interaction between HIV virus and the immune system has been a major area of re-

search for many years. Most previously studied models of internal HIV dynamics in

the literature have used deterministic diﬀerential equation models, ignoring stochastic

eﬀects.

Models used to study HIV infection have involved the concentrations of uninfected

target cells, T, infected cells that are producing virus, T∗, and virus, V. After pro-

tease inhibitors are given, virus is classied as either infectious, VI, i.e., not inﬂuenced

by the protease inhibitor, or as non-infectious, VNI , due to the action of the protease

inhibitor which prevents virion maturation into infectious particles. The general class

of models that have been studied by (Perelson et al., 1996; Nowak & Bangham, 1996;

Nowak, Bonhoeﬀer, Shaw, & May, 1997; Kirschner, 1996) have a form similar to:

dT

dt =s−T(dT+κV ),

dT ∗

dt =κT V −δT ∗,(2.1)

dV

dt =N δT ∗−cV.

38

where sis the rate at which new target cells are generated, dTis their speciﬁc death

rate and κis the constant rate characterizing their infection. Once cells are infected,

they assume that they die at rate δeither due to the action of the virus or the immune

system, and produce Nnew virus particles during their life, which on average has

length 1

δ. Lastly, they assumed that virus particles are cleared from the system at

rate c per virion.

Current HIV/AIDS deterministic models are derived from the ﬁrst model of viral pop-

ulation dynamics in host cells, see (Nowak & May, 2000). Nowak and May developed

the following three dimensional model for the HIV virus dynamics, encompassing

three variables: the population size of uninfected cells, infected cells, and free virions,

termed as x,yand v, respectively.

dx

dt = Λ −x(µx+αv),

dy

dt =αxv −µyy, (2.2)

dv

dt =κy −µvv.

System (2.2) has been derived to model in vivo dynamics of HIV, HBV, and other

virus (Perelson, Kirschner, & de Boer, 1993; Perelson et al., 1996; Bonhoeﬀer, May,

Shaw, & Nowak, 1997; Perelson & Nelson, 1999; Nowak & May, 2000; Tuckwell,

2004; Nowak & Bangham, 1996). It can also be considered as a model for the HTLV-

I infection if x(t), y(t), and v(t) are regarded as healthy, latently infected, and actively

infected CD+

4T cell pools (Nowak & Bangham, 1996; Wang, Li, & Kirschner, 2002).

According to this model, the free virus population vinfects the uninfected cells xat

a rate proportional to the product of their loads αxv , where αis the rate of infection

(changing healthy cells to infected ones). The infected cells ystart producing free

virions at a rate proportional to their load κy , where κis the production rate of

virions by the infected cells, and die at a rate µyy, while the free virions vare

removed from the system at rate µvv.

From the ODEs, they found that the productive number of system (2.2) is R0=

39

Λκα

µxµyµv, which describes the average number of newly infected cells generated from

one infected cell at the beginning of the infection process. In the analysis of their

dynamics they found out that if the basic reproduction number R0<1 , the infection-

free equilibrium E0= ( Λ

µx,0,0) is locally asymptotically stable and the disease dies

out; if R0>1 , then a unique chronic-infection equilibrium exists and is locally

asymptotically stable and the virus persists in the host. R0, the basic reproduction

number, is deﬁned as the expected number of secondary infected cells caused by

a single infected cell entering the disease- free population at equilibrium. Here a

secondary infected cell is a cell which is infected by an infective virus particle which

is produced by the initial infected cell.

The virus establish themselves among the target cells and infection becomes chronic.

In (Wang & Li, 2006) and (De Leenheer & Smith, 2003), global dynamics of system

(2.2) are established: the system is infection free if R0<1, and if R0>1, then the

chronic-infection equilibrium is globally asymptotically stable. In the basic model

(2.2), no distinction is made between target cells infected by the virus and virus

producing target cells; they are both labeled y(t).

To account for the time between viral entry into a target cell and the production

of new virus particles, which typically lasts for around 1 day for the HIV infection,

divisions of activated target cells have been incorporated into the basic model (2.2)

(Perelson et al., 1993; Perelson & Nelson, 1999; De Leenheer & Smith, 2003; Wang &

Li, 2006; Dixit, Markowitz, Ho, & Perelson, 2004). Models that include time delays

have been developed and investigated (Herz, Bonhoeﬀer, Anderson, May, & Nowak,

1996; Nelson & Perelson, 2002; Nelson, Murray, & Perelson, 2000; Culshaw & Ruan,

2000; Dixit et al., 2004; Wang, Zhou, Wu, & Heﬀernan, 2009). One distinct feature

of delay diﬀerential equation models is that delays typically destabilize an otherwise

stable equilibrium and cause sustained oscillation through Hopf bifurcations. In recent

studies of in-host viral model with intracellular delay and cell divisions (Culshaw &

Ruan, 2000);(Wang et al., 2009), it is shown that sustained oscillation can occur for

realistic parameter values. The paper by (Wang et al., 2009) also contains updated

40

review of literature of in-host viral modeling.

The above researchers, in their studies they did not look at the eﬀects of treatment

therapy on the virus dynamics and also they ignored the random eﬀect on the disease

progression. (Nelson & Perelson, 2002) studied a simpliﬁed version of (Nowak & May,

2000) deterministic model, they also assumed that the number of uninfected cells was

a constant and modelled the dynamics of infected cells and infective virus particles.

They introduce protease inhibitor drug therapy. To analyse the eﬀects of giving an

antiretroviral drug, the model equations were modiﬁed. Reverse transcriptase (RT)

inhibitors block the ability of HIV to successfully infect a cell. Protease inhibitors

(PI) cause the production of non-infectious viral particles. So, in the presence of these

drugs, the model equations became:

dx

dt = Λ −xµ −(1 −γRT )βxv,

dy

dt = (1 −γRT )βxv −αy, (2.3)

dv

dt = (1 −γP I )cy −γv.

The researchers here only considered ART treatment as the only treatment therapy,

again they did not incorporate random eﬀects. (Nelson & Perelson, 2002) outlined

the basic deterministic model they had discussed earlier. They discussed modiﬁcation

of this basic model to summarize the eﬀects of drug therapy on virus concentration

and introduced a time delay into the model. The model was then ﬁtted to data. They

were of the opinion that models that include intracellular delays are more accurate

representations of the biology and change the estimated values of kinetic parameters

when compared to models without delays. They developed and analysed a set of

models that included intracellular delays, combination antiretroviral therapy, and

the dynamics of both infected and uninfected T cells. They showed that for less

than perfect drug eﬀect, the value of the death rate of productively infected cells is

increased when data is ﬁtted with delay models compared to the values estimated

with a non-delay model. They also provided some general results on the stability of

41

the system.

(Tuckwell, 2004) discussed and analysed deterministic model including the term βxv

in Nowak’s third equation. They argued that Since the growth of a viral population

depends on the ability of the virus to penetrate new host cells, the simplest growth

model has the following three components for a given volume of tissue: x(t) = number

of uninfected cells; y(t) = number of infected cells; and v(t) = number of free virus

particles. with the above assumptions, the came up with the following modiﬁed

deterministic equations:

dx

dt = Λ −xµ −βxv,

dy

dt =βxv −αy, (2.4)

dv

dt =cy −γv −βxv.

where uninfected host cells are supplied at rate Λ and have a per cell death rate of

µ, the parameter βdescribes the rate at which virus infects host cells, c is the rate

at which free virions are produced per infected cell, αand γare the ”per capita”

rates of attrition of infected cells and virions, respectively. They slightly modiﬁed the

third equation version of the standard model formulated by (Nowak & May, 2000)

by adding the term βxv . The additional term βxv was to allow for the fact that

whenever a cell is attacked, a free virus must disappear. They also analysed and

computed the basic reproduction number R0. Again they did not consider the eﬀect

of treatment on the disease progression and the stochastic eﬀect was ignored.

(Dalal et al., 2009) extended the results in (Tuckwell, 2004) to include the eﬀect of

HAART. They discussed an ordinary diﬀerential equation model which described the

internal HIV viral dynamics in the presence of treatment such as antiretroviral drugs.

They proposed the following three dimensional model to describe the viral dynamics

in the presence of HIV infection and HAART. HAART is generally a combination of

42

reverse transcriptase inhibitor (RTI) drugs and protease inhibitor (PI) drugs:

dx

dt =λ−xδ −(1 −α)βxv,

dy

dt = (1 −α)βxv −κy, (2.5)

dv

dt = (1 −η)N κy −µv −(1 −α)βxv.

where (1 −α) is the reverse transcriptase inhibitor drug eﬀect, (1 −η) is the protease

inhibitor drug eﬀect, λis the total rate of production of healthy cells per unit time, δ

is the per capita death rate of healthy cells, βis the transmission coeﬃcient between

uninfected cells and infective virus particles, κis the per capita death rate of infected

cells, Nis the average number of infective virus particles produced by an infected

cell in the absence of treatment during its entire infectious lifetime and µis the per

capita death rate of infective virus particles.

Note that when a single infective virus particle infects a single uninfected cell the

virus particle is absorbed into the uninfected cell and eﬀectively dies, hence the term

(1 −α)βxv appears in all the three equations. They derived an expression for the

basic reproductive number R0,R0=(1−γ)βλN (1−η)

δµ+βλ(1−γ)and found that if R0≤1 then the

equilibrium with only uninfected cells was the unique equilibrium and was globally

asymptotically stable. They showed the global stability of the DFE when R0<1,

which was not shown by (Tuckwell, 2004). Then also gave the analytical results and

examined the probability of extinction using a branching process model.

All of the above models are deterministic models and do not incorporate random

eﬀects. As most real world problems are not deterministic including stochastic eﬀects

into the model gives us a more realistic way of modelling viral dynamics.

When HIV enters the body, it targets all the cells with CD+

4receptors including the

CD+

4T-cells. The knowledge of principal mechanisms of viral pathogenesis, namely

the binding of the retrovirus to the gp120 protein on the CD+

4cell, the entry of

the viral RNA into the target cell, the reverse transaction of viral RNA to viral

43

DNA, the integration of the viral DNA with that of the host, and the action of viral

protease in cleaving viral proteins into mature products have led to the design of

drugs (chemotherapeutic agents) to control the production of HIV.

Chronic HIV infection causes gradual depletion of the CD+

4T-cell poll, and thus

progressively compromises the host’s immune response to opportunistic infections,

leading to Acquired Immunodeﬁciency Syndrome (AIDS), (Yan & Xiang, 2012). With

the spread of the HIV-AIDS pandemic and in the absence of an ”eﬀective” vaccine

or cure, therapeutic interventions are still heavily relied on. Several research studies

have been carried out in the recent past, both theoretically and experimentally, to

analyse the impact of therapy on the viral load in HIV infected persons in order to

ascertain the eﬀectiveness of the treatment (see, for example, (Perelson & Nelson,

1999; Kirschner, 1996; Perelson et al., 1996; Mellors et al., 1997; Nijhuis et al., 1998;

Tan & Xiang, 1999; Bangsburg et al., 2004)). Their utility lies in the ability to predict

an infected steady state and examining the eﬀects that changes in parameters have

on the outcome of the system over time, to determine which parameters are most

important in disease progression, and further determine critical threshold values for

these parameters.

In HIV infected individuals, the infection exhibits a long asymptomatic phase (after

the initial high infectious phase) of approximately 10 years on average before the onset

of AIDS. During this incubation period which some call the clinical latency period,

the individuals appear to be well and may contribute signiﬁcantly to the spread

of the epidemic in a community (Venkata, Morire, Swaminathan, & Yeko, 2009).

Some clinical markers such as the CD+

4cell count and the RNA viral load (viraemia)

provide information about the progression of the disease in infected individuals. Also,

the clinical latency period of the disease may provide a suﬃciently long period during

which to attempt an eﬀective suppressive therapeutic intervention in HIV infections.

Various biological reasons lead to the introduction of time delays in models of disease

transmission. Time delays are used to model the mechanisms in the disease dynamics

(see for instance (Van-den Driessche & Watmough, 2002; Arino, Hbid, & Ait Dads,

44

2006)). Intracellular delays and the target-cell dynamics such as mitosis are two key

factors that play an important role in the viral dynamics. Mitosis in healthy or in-

fected target-cell population are typically modelled by a logistic term (Perelson et

al., 1993; Perelson & Nelson, 1999; De Leenheer & Smith, 2003; Nowak & Bangham,

1996; Wang & Li, 2006; Wang & Ellermeyer, 2006). Intracellular delays have been

incorporated into the incidence term in ﬁnite or distributed form (Culshaw & Ruan,

2000; Herz et al., 1996; Li & Shu, 2010a, 2010b; Nelson et al., 2000; Nelson & Perel-

son, 2002; Perelson et al., 1996; Wang et al., 2009). In (Wang & Ellermeyer, 2006;

De Leenheer & Smith, 2003) intra-host viral models with a logistic growth term with-

out intracellular delays are investigated, and it is shown that sustained oscillations

can occur through Hopf bifurcation when the intrinsic growth rate increases.

It is shown in (Culshaw & Ruan, 2000; Wang et al., 2009), in in-host models with

both a logistic growth term and intracellular delay, that Hopf bifurcations can occur

when the intracellular delay increases. In (Li & Shu, 2010a), using in-host models

with a general form of target-cell dynamics and general distributions for intracellular

delays, it is shown that the occurrence of Hopf bifurcation in these models critically

depends on the form of target-cell dynamics. More speciﬁcally, it is proved in (Li &

Shu, 2010b) that, if the target-cell dynamics are such that no Hopf bifurcations occur

when delays are absent, introducing intracellular delays in the model will not lead to

Hopf bifurcations or periodic oscillations.

To incorporate the intracellular delay phase of the virus life-cycle, (Li & Shu, 2010) as-

sumed that virus production occurs after the virus entry by a constant delay τ. They

came up with a basic in-host compartmental model of the viral dynamics containing

three compartments:- x(t), y(t) and v(t) denoting the populations of uninfected target

cells, infected target cells that produce virus, and free virus particles, respectively.

They further assumed Parameters δ,αand µare turnover rates of the x,yand v

compartments, respectively. Uninfected target cells are assumed to be produced at

a constant rate λ. They assumed also that cells infected at time twill be activated

and produce viral materials at time t+τ. In their model, the constant sis the death

45

rate of infected but not yet virus-producing cells, and e−sτ describes the probability

of infected target cells surviving the period of intracellular delay from t−τto t.

Constant κdenotes the average number of virus particles each infected cell produces.

With the above assumptions they came up with the following system of diﬀerential

equations.

dx

dt =λ−δx(t)−βx(t)v(t),

dy

dt =βx(t−τ)v(t−τ)e−sτ −αy(t),(2.6)

dv

dt =κy(t)−µv(t).

System (2.6) can be used to model the infection dynamics of HIV, HBV and other

virus (De Leenheer & Smith, 2003; Nowak & Bangham, 1996; Nowak & May, 2000;

Perelson et al., 1996; Perelson & Nelson, 1999; Tuckwell & Wan, 2004). It can also

be considered as a model for the HTLV-I infection if x(t), y(t), and v(t) are regarded

as healthy, latently infected, and actively infected CD+

4T cells (Nowak & Bangham,

1996; Wang & Li, 2006). From the ODEs, they found that the productive number of

system (2.6) is R0=λβκe−sτ

δαµ , which describes the average number of newly infected

cells generated from one infected cell at the beginning of the infection process. For

detailed description and derivation of deterministic model, as well as the incorporation

of intracellular delays, we refer the reader to (Li & Shu, 2010; Perelson & Ribeiro,

2013).

Various biological reasons lead to the introduction of time delays in models of disease

transmission. Time delays are used to model the mechanisms in the disease dynamics,

see for instance (Van-den Driessche & Watmough, 2002; Arino et al., 2006)). (Tan &

Xiang, 1999) developed a state-space model of HIV pathogenesis in HIV infected in-

dividuals undergoing combination treatment (i.e. a treatment with a combination of

anti- viral drugs such as AZT and Ritonavir which can inhibit either the reverse tran-

scriptase or the protease of HIV). Their model included a mathematical description

of the production of infectious free HIV and non-infectious free HIV, by extending

46

the model of (Perelson et al., 1996) and developing procedures for estimating and pre-

dicting the number of uninfected CD+

4cells, infectious free HIV, non-infectious free

HIV and HIV infected CD+

4cells. They not only extended the model by (Perelson et

al., 1996) to a stochastic model, but also applied their model to data of patients con-

sidered by (Perelson et al., 1996). Their model was discrete in time and comprised

a system of stochastic diﬀerence equations which were derived from the biological

speciﬁcations of the HIV-replication cycle.

(Tan & Xiang, 1999) developed a state-space model, (Kamina, Makuch, & Zhao, 2001)

considered a stochastic model of the growth of the HIV population which carries over

the principle of the virology of HIV, the life-cycle of HIV and allows the production

of non-infectious (defective) free HIV to reduce the severity of HIV in a HIV-infected

individual undergoing a combination of therapeutic treatment. Their aim was to

use a stochastic model obtained by extending the model of (Perelson et al., 1996) to

determine the number of uninfected T4 cells, infected T4 cells and free HIV in an

infected individual by examining the combined antiviral treatment of HIV. This is

important because it helps in determining the eﬃcacy of methods used in the research

areas of pathogenesis, progression and combined treatment of HIV. Obtaining the

variance and co-variance structures of variables representing the number of virus

producing cells, and the levels of infectious free and non-infectious free HIV was their

main contribution to their research. These variance and co-variance structures may

sometimes be diﬃcult to obtain for stochastic models (unlike the case with the mean

structure), but being able to obtain these structures is expected to shed light on the

type of relationship between the variables.

From the literature, many researchers have employed deterministic models to study

HIV internal dynamics, ignoring the stochastic eﬀects. Summary of the HIV-Host

interaction dynamic review is shown in Figure 2.2:

47

Figure 2.2: Literature summary on deterministic modeling

Source: Author

48

2.4 Stochastic Multi-state models for HIV pro-

gression

Stochastic models have not only been applied exhaustively in studying HIV/AIDS

in particular but in infectious diseases in general. Markov process is a tool that is

useful to study problems linked to the ageing of a system and semi-Markov model is a

useful tool to predict the clinical progression of a disease, see (Di Biase et al., 2007).

Markov processes and in particular the Semi-Markov models have been employed in

the ﬁeld of biomedicine, for example, in applications to prevent, screen, and design

cancer prevention trials, see in (Zelen & Davidov, 2000). (Mathieu, Loup, Dellam-

onica, & Daures, 2005), argued that, the use of reversible disease states deﬁned on

both CD+

4and V L levels is quite innovative. They considered four states which are

characteristics of the disease stages, the states were deﬁned on crossed values of both

Viral load and CD+

4cell count as shown below.

1. State I: V L ≤400 and CD+

4≤200

2. State 2: V L ≤400 and CD+

4>200

3. State 3: V L > 400 and CD+

4>200

4. State 4: V L > 400 and CD+

4≤200

They formulated continuous time homogeneous Markov process (HM) for evaluating

the evolution of HIV virus in a given patient at a given time. Transition intensities and

transition probabilities according to time were estimated. Distributions for sojourn

times in each State were characterized and the distribution of patients in states at

diﬀerent times was also predicted. Their analysis showed that, the state V L ≤400

and CD+

4>200 represents a stable state in the evolution of seropositive patients.

In their analysis, they ignored the fact that the disease progression is dependent on

other factors like time and age. (Mathieu et al., 2007) extended his previous study

49

(Mathieu et al., 2005) by modeling the HIV evolution using Non - Homogeneous

Semi- Markov Model (NHSMM) in continuous time. They considered four states of

the disease evolution and assumed that patients move through these states according

to ten transitions as in Figure 2.3:

Figure 2.3: HIV disease progression

50

They used a parametric approach and computed the interval transition probabilities.

In their transition diagram, they assumed that the patients live forever, that is, they

ignored the death state. (Di Biase et al., 2007) in their study, they presented an

Homogeneous Semi - Markov Process (HSMP) approach to the dynamic evolution

of the HIV Virus Infection, as deﬁned by CD+

4levels. They improved the work of

(Mathieu et al., 2007) by introducing an absorbing state (death) in their transition

state graph. In order to predict the HIV/AIDS evolution, they employed the following

immunological states related to CD+

4count plus an absorbing state (the death of the

patient):

1. State I : C D+

4count >500 cells/microliter.

2. State II : 350 < CD+

4count ≤500 cells/microliter.

3. State III : 200 < CD+

4count ≤350 cells/microliter.

4. State IV : C D+

4count ≤200 cells/microliter.

5. D : Death (Absorbing state).

They assumed, therefore, that the HIV/AIDS infection shifts between ﬁve diﬀerent

degrees of seriousness. They came up with the transition states diagram as in Figure

2.4

51

Figure 2.4: The model of the immunological stages a HIV/AIDS infected patient can

go into.

In their analysis,they computed the probabilities of an infected person’s survival.

They assumed that CD+

4+ Cell count are the only markers of the disease progression

and also ignored the fact that the disease progression is dependent on other factors

like time and age.

(D’Amico, Di Biase, Janssen, & Manca, 2011) improved their previous work by includ-

ing viral load as another marker for disease progression and introduced non- homo-

geneity in their analysis. They analyzed HIV/AIDS dynamics, deﬁned by CD+

4levels

and viral load, by means of three diﬀerent non-homogeneous semi-Markov stochastic

models. The ﬁrst model focused on the patient’s age as a relevant factor to forecast

the transitions among the diﬀerent levels of seriousness of the disease. The second one

considered the disease evolution based on the chronological time. The third model

which was the most powerful, considered the two previous features simultaneously.

The computed conditional probabilities showed the diﬀerent responses of the subjects

depending on their ages and the elapsing of time. In order to predict the HIV/AIDS

evolution, they employed the following immunological states related to CD+

4count

plus an absorbing state (the death of the patient):

52

1. State I: V L ≤400 and CD+

4≤200

2. State 2: V L ≤400 and CD+

4>200

3. State 3: V L > 400 and CD+

4>200

4. State 4: V L > 400 and CD+

4≤200

5. State 5: Death (absorbing state)

They assumed, therefore, that the HIV/AIDS infection shifts between ﬁve diﬀerent

degrees of seriousness. They used their previous transition states diagram Figure 2.4.

Few researchers have modeled HIV internal dynamics using stochastic processes, those

who have used stochastic models, have considered ART treatment as the only treat-

ment therapy and also none of them have looked at the cost of treating an HIV

patient and the revenue generated by such a treated person (in terms of advice to

other infected persons, care for the family, living example to the society etc). Also the

dynamic evolution of the infection has been analyzed in a homogeneous framework

using only immunological markers. This is the gap/problem this research will try to

address.

As extensions of Markov processes and renewal processes, semi-Markov processes are

widely applied and hence, an important methodology for modeling. Semi-Markov

models have extensively been studied and applied in ﬁnance, insurance, business ad-

ministration as well as manpower models. In biology and medicine, Semi-Markov

modeling has also been used in continuous time to study prognosis and the evolution

of diseases, (see (Moore & Pyke, 1968; Lagakos, Sommer, & Zelen, 1978; Dinse &

Lagakos, 1980; Mode & Pickens, 1988; Lawless & Fong, 1999; Joly & Commenges,

1999; Andersen, Esbjerg, & Sorensen, 2000; Escolano, Golmard, Korinek, & Mallet,

2000; Kang & Lagakos, 2004; Chen & Tien, 2004; Foucher, Mathieu, Philippe, Du-

rand, & Daur’es, 2005; Mathieu et al., 2007; Kang & Lagakos, 2007)). Typically these

methods assume the sample paths are continuously observed. However, it is often the

53

case where study individuals’ states are observed only at discrete time points with no

information about the occupied states in between observation times.

Recently, (Kang & Lagakos, 2007) developed methods for ﬁtting continuous-time

semi- Markov multi-state models to panel data. Their methods are illustrated with a

model of the natural history of oncogenic genital HPV infection in women using data

from the placebo arm of an HPV vaccine trial. Discrete-time semi-Markov models

have not received as much attention in the literature as continuous-time semi- Markov

models. In ﬁnance, for example, credit rating and reliability models are based upon

discrete time Semi-Markov theory like (Frydman, 1995a; Satten & Sternberg, 1999a;

Sternberg & Satten, 1999b; Satten, 1999; Barbu, Boussemart, & Limnios, 2004; Barbu

& Limnios, 2006; Di Biase et al., 2007). For studies with ﬁxed scheduled visits, such

as clinical trials, it is natural to model time as discrete.

Discrete-time models can have advantages over continuous-time models, such as

not requiring the speciﬁcation of guarantee times. (Frydman, 1995a; Satten, 1999;

Sternberg & Satten, 1999b) studied nonparametric estimators for discrete-time semi-

Markov unidirectional models with varying initial states in HIV data. They con-

sidered only unidirectional models, which may not be applied to complex disease

processes such as HPV and HIV where prior states may be revisited and states may

not be visited sequentially. Their methods extended the Markov models developed

by (DeGruttola & Lagakos, 1989; Frydman, 1992, 1995b) to a more generalizable

discrete-time semi-Markov framework by allowing the probability of transitioning

from the HIV positive state to AIDS to depend on the duration of HIV infection.

(Barbu et al., 2004; Barbu & Limnios, 2006) studied discrete-time multi-state bidi-

rectional semi-Markov models in ﬁnance. Their methods require parametric assump-

tions, only allow for incident infections, and do not address the possibility of missing

data.

The evolution of HIV is Multi-directional, that is recovery from one state to previous

state is reasonable. We propose to develop discrete-time non-homogeneous Semi-

54

Markov models that can be used to study HIV evolution.

A summary of the HIV evolution dynamics review is shown in Figure 2.5:

55

Figure 2.5: Literature summary on stochastic modeling

Source: Author

56

In our research, we considered a stochastic model for the interaction of HIV virus

and the immune system (In-Host HIV dynamic model) in an HIV-infected individual

undergoing a combination-therapeutic treatment. Our aim in this study was to use

a stochastic (since both the death rates of the cells and the virus are aﬀected by

many complicated biological phenomena we think that there is randomness involved

in these rates) model obtained by extending the deterministic diﬀerential equations

of (Dalal et al., 2009) by introducing stochastic eﬀects to their diﬀerential equations,

introduce intracellular delay in the model and also introduce other treatment therapies

to determine probability distributions, of the uninfected CD+

4cells, infected CD+

4

cells and the free HIV particles in an infected individual at any time t by examining

the combined antiviral treatment of HIV. Based on the model, we obtained joint

probability distributions and expectations of variables representing the numbers of

uninfected CD+

4cells,the HIV infected CD+

4T cells, and the free HIV particles at

any time t, and derived conclusions for the reduction or elimination of HIV in HIV-

infected individuals, which is one of the main contributions of this study.

For the disease progression, the study extended the work of (Di Biase, D’Amico,

Janssen, & Manca, 2009; D’Amico et al., 2011) by studying the non-homogeneous

semi-Markov models and then attaching a reward structure to the process, which

will help us analyze the cost of treating an HIV infected patient and conduct Cost

Beneﬁt Analysis. The use of reward stochastic processes allows us to associate a

certain amount of money (the cost of the treatment used for a HIV patient) to a state

occupancy of the system. Actually the reward associated to the treatment of an HIV

patient at any given state was considered also.

57

Chapter 3

In Vivo HIV dynamics

3.1 Introduction

This chapter discusses the methodology used to model the dynamics of HIV evolution.

The mathematical models for HIV - Immune cells interactions are formulated in this

chapter. The ﬁrst section starts with looking at the dynamics of the immune cells

and especially the CD+

4cells in the absence of the virus.

3.2 Healthy CD+

4cell population dynamics

Deﬁne X(t) as the number of healthy CD+

4cells at time t. In the modeling, changes

in the number of healthy cells is treated as an immigration and death process. In the

absence of the HIV, we will model the dynamics of the healthy CD+

4cells. For the

uninfected cell population, we suppose that healthy CD+

4cells are replenished from

thymus with probability λ∆tand dies with probability δ∆t. Let the incremental

change in the number of healthy cells during the small time period ∆tbe denoted by

∆X. Then this leads to the assumptions that :-

The probability that there are nhealthy CD+

4cells in an infected person during the

time interval (t, t + ∆t) is equal to the probability that one of the following events

occur:-

a). There were nhealthy CD+

4cells by time tand nothing happens during the time

interval (t, t + ∆t).

58

b). There were n-1 healthy CD+

4cells by time tand one healthy cell is produced

during the time interval (t, t + ∆t)

c). There were n+1 healthy CD+

4cells by time tand one healthy cell dies during the

time interval (t, t + ∆t)

The CD+

4cells dynamics is modelled as an Immigration and death process. Let

PXn(t) = P[X(t) = n], then a standard approach to solving PXn(t) is by using the

Kolmogorov diﬀerence diﬀerential equation. This approach makes use of assumptions

concerning the probabilities of various events occurring in a small interval of length

∆t.

Suppose we assume that the CD+

4cell population at time (t+ ∆t) is n, then X(t+

∆t) = n. Therefore, there are the following possibilities for the way this could occur

starting at time t:-

i). X(t) = nwith no change from tto t+ ∆t

ii). X(t) = n−1 with only a single production in ∆t

iii). X(t) = n+ 1 with only a single death in ∆t

iv). possibilities involving two or more independent changes in ∆t

These assumptions yield the expressions for the probability that X(t+ ∆t) = n

PXn(t+ ∆t) = P[X(t+ ∆t) = n],

=P[X(t+ ∆t) = n|X(t) = n]P[X(t) = n]

+P[X(t+ ∆t) = n|X(t) = n+ 1]P[X(t) = n+ 1]

+P[X(t+ ∆t) = n|X(t) = n−1]P[X(t) = n−1]

+P[X(t+ ∆t) = n|X(t)6=n, n −1, n + 1]P[X(t)6=n, n −1, n + 1].

59

The change in population size of the CD+

4cells is governed by the following conditional

probabilities

i). P[X(t+ ∆t) = n|X(t) = n] = 1 −(λ+δn)∆t+o(∆t),

ii). P[X(t+ ∆t) = n+ 1|X(t) = n] = λ∆t+o(∆t),

iii). P[X(t+ ∆t) = n−1|X(t) = n] = δn∆t+o(∆t),

where o(∆t) is the probability of more than one event.

It is assumed that ∆tis suﬃciently small to guarantee that the probability of more

than one event occurring in (t, t + ∆t) is negligible.

Let the probability distribution of the population size at time t be deﬁned by pXn(t) =

P r{X(t) = n/X(0) = i}, for i= 0,1, .....

From the assumptions above, the distribution of the cells is obtained by deriving

a system of stochastic diﬀerence diﬀerential equation (SDE)also called Kolmogorov

forward diﬀerence diﬀerential equation for the CD+

4cells dynamics, then summarize

the equation using a generating function and then directly solve the resulting lagrange

equation to get the probability distributions for the cell population.

Let PXn(t) = P rob[X(t) = n] be the probability that the random population of

healthy CD+

4cells has the value nat time t,PXn−1(t) be the probability that the

population of healthy CD+

4cells has the value n−1 at time t,and PXn+1(t) be the

probability that the population of healthy CD+

4cells has the value n+ 1 at time t.

Then from probability rules it follows that:

PXn(t+ ∆t)= PXn(t)(no birth or death) or PXn+1 (t) (death) or PXn−1(t) (birth)

which gives:

PXn(t+ ∆t) = {1−(λ∆t+δn∆t)}PXn(t) + {λ∆t}PXn−1(t)

+ (n+ 1)δ∆tPXn+1 (t) + o(∆t)for n ≥0.

60

Rearranging the equation we have

PXn(t+ ∆t)−PXn(t) = −{λ+nδ}∆tPXn(t) + λ∆tPXn−1(t)

+ (n+ 1)δ∆tPXn+1 (t) + o(∆t)for n ≥0.

Dividing through by ∆tand taking the limit as ∆t→0 we have the following diﬀer-

ence diﬀerential equation called the Kolmogorov forward diﬀerence diﬀerential equa-

tion for PXn(t)

P0

Xn(t) = −{λ+nδ}PXn(t) + λPXn−1(t)+(n+ 1)δPXn+1 (t)

=−λPXn(t)−δnPXn(t) + λPXn−1(t) + δ(n+ 1)PXn+1 (t).

where the range of nis all integers and the probability for negative nis assumed to

be zero (PXn(t) = 0 for n < 0 for non-negative initial population size Xn(0) >0).

For n= 0, then P0

X0(t) = −λPX0(t) + δPX1(t).

The prime indicates diﬀerentiation with respect to t. In equation (3.2), there are

three unknown probabilities, PXn(t), PXn+1(t) and PXn−1(t). Therefore this equation

cannot be solved directly, we resort to the method of probability generating function

(pgf).

3.2.1 Solving the Diﬀerence Diﬀerential Equation (DDE)

We now solve equation (3.2) using probability generating function (pgf). The proba-

bility generating function for a probability distribution PXn(t) is deﬁned as

61

Gx(z, t) =

∞

X

n=0

PXn(t)zn, for t ≥0,0≤Px≤1, z ≥1 (3.1)

Diﬀerentiating equation (3.3) with respect to tyields

∂

∂t Gx(z, t) =

∞

X

n=0

∂

∂t PXn(t)zn

=

∞

X

n=1

P0

Xn(t)zn.

Diﬀerentiating equation (3.3) with respect to zyields

∂Gx

∂z =

∞

X

n=0

nPXn(t)zn−1.

Multiplying equation (3.2) by znand summing over nwe have

∞

X

n=1

P0

Xn(t)zn=−λ

∞

X

n=1

PXn(t)zn−δ

∞

X

n=1

nPXn(t)zn+λ

∞

X

n=1

PXn−1(t)zn

+δ

∞

X

n=1

(n+ 1)PXn+1 (t)Zn,

=−λ

∞

X

n=1

PXn(t)zn−δ

∞

X

n=1

nPXn(t)zn

+λ

∞

X

n=1

PXn−1(t)zn+δ

∞

X

n=1

(n+ 1)PXn+1 (t)zn.

Using the partial derivatives of pgf, then we convert equation (3.4) to:

∂Gx

∂t −P0

X0(t) = −λ[Gx(z, t)−PX0(t)] −δ z ∂Gx

∂z

62

+λzGx(z, t) + δ[∂Gx

∂z −PX1(t)].

Since P0

X0(t) = −λPX0(t)+(δ+β)PX1(t), then on simpliﬁcation we have-

∂Gx

∂t =−λGx(z, t)−δz ∂Gx

∂z +λzGx(z, t) + δ∂Gx

∂z ,

which further becomes

∂Gx

∂t =λGx(z, t)(z−1) + δ(1 −z)∂Gx

∂z ,

and can be re-arranged to

∂Gx

∂t −λ(z−1)Gx(z, t)−δ(1 −z)∂Gx

∂z = 0.(3.2)

The equation (3.5) is a partial diﬀerential equation called lagrange partial diﬀerential

equation and the standard method of solution is to set up the auxiliary equations:-

First we think of the auxiliary equation of (3.5)

dt

1=dz

δ(1 −z)=dGx

λ(z−1)Gx

.(3.3)

We can integrate the ﬁrst two equations in equation (3.6) to have

Zδdt =Zdz

1−z

⇒ln|1−z|=δt +C1.

63

Taking exponential on both sides we have

(1 −z) = eδt+C1

which implies

(1 −z)e−δt =C1.

where C1is a constant of integration.

From the last two equations in equation (3.6), we have

Zdz

δ(1 −z)=ZdG

λ(1 −z)G

⇒Z−λ(1 −z)dz

δ(1 −z)=ZdG

G

⇒−λ

δZdz =ZdG

G

⇒−λ

δz= ln G+C2.

Now taking exponential on both sides we have

e−λ

δz+C2=Gx(z, t)

which implies that

C2=eλ

δzGx(z, t)

where C2is a constants of integration. Setting C2as a function of C1we arrive at

the most general solution of equation

eλ

δzGy(z, t) = F{(1 −z)e−δt},

64

which becomes

Gx(z, t) = e−λ

δzF{(1 −z)e−δt}.

where Fis an arbitrarily diﬀerentiable function. We had denoted that Xn(t) is the

size of the population at time tfor 0 ≤t≤ ∞, let the initial population at time t= 0

be Xn(0) = ithen

Gx(z, 0) = zi.

Therefore

F{1−z}=zi,for |z|<1.

For any θ,

θ= 1 −z

re-arranging the equation we have

z= 1 −θ

thus, it follows that

F(θ) = (1 −θ)i.

65

Now replacing θwith 1 −z, we have the following solution

Gx(z, t) = e{λ

δ(z−1)(1−e−δt)}{ze−δt + 1 −e−δ t)}n.(3.4)

The pgf in equation (3.7) above is a product of two components, a poisson probability

generating function e{λ

δ(z−1)(1−e−δt)}with mean λ

δ(1 −e−δt) and binomial probability

generating function {ze−δt + 1 −e−δt)}n.

as t→ ∞, Gx(z, t) = eλ

δ(z−1).(3.5)

Equation (3.8) is a pgf of a poisson distribution with mean λ

δand variance λ

δ. The

properties of the CD+

4cells population dynamics follows the properties of a Poisson

distribution.

3.3 Intra-Host stochastic HIV interaction model

The human immune system is complex and can mount diﬀerent types and intensities

of responses. In healthy individuals, the majority of human viral infections can be

fought by the body’s own immune system, but sometimes the response is so strong

that it can jeopardize the survival of the individual. In any case, it is important

to model the interaction between viruses and immune system because they provide

useful insights on the eﬀects of drug treatment and on optimizing medical procedures

which may beneﬁt patients. HIV is among the most studied viruses in biomedical

research and causes a tremendous global problem. A large number of parameters are

available from clinical trials which make HIV suitable for mathematical modeling and

testing.

66

In HIV infected individuals, the infection exhibits a long asymptomatic phase (after

the initial high infectious phase)of approximately 10 years on average before the

onset of AIDS Venkata et al. (2009). During this incubation period, which some

call the clinical latency period, the individuals appear to be well and may contribute

signiﬁcantly to the spread of the epidemic in a community. The RNA viral load and

the CD4 cells, which originate in the bone marrow and mature in the thymus gland

and which play a dominant role in the immune system of the human body, are the

two main clinical markers of disease progression in an HIV infected individual. It has

been observed in HIV infected individuals that as a consequence of HIV infection,

selective depletion of CD4 cells occur. When the CD+

4T cell count in such individuals

drop, these cells are unable to mount an eﬀective response of signaling the CD8cells

of the presence of the virions and consequently, the individual becomes susceptible to

opportunistic infections Venkata et al. (2009).

A typical interaction of HIV and immune response is shown in Figure 3.1.

67

Figure 3.1: Schematic diagram of Intra-Host HIV dynamics before treatment

Source: Author

68

Without exception, viruses cannot reproduce by themselves. Instead, upon successful

invasion of target cells (CD+

4lymphocytes), they use the machinery and metabolism

of a host cell to produce multiple copies of themselves and assemble within the cell.

Once free virions infect the immune system,they successfully enter a host cell and the

cell becomes infected. All cell resources are shared by the virus and the entire cell

becomes a virus factory. In other words, the viral genetic material, DNA or RNA, is

synthesized by the cell. The mature viral particles are then released into the blood

and a new virion’s life cycle begins.

To study the interaction of HIV virus and the immune system, we therefore need a

stochastic version of the deterministic model presented in the literature. A stochastic

process is deﬁned by the probabilities with which diﬀerent events happen in a small

time interval ∆t. In this study, there are two possible events in the formultion of the

model (production and death/removal) for each population (uninfected cells, infected

cells and the free virions). The corresponding rates in the deterministic model are

replaced in the stochastic version by the probabilities that any of these events occur

in a small time interval ∆t.

3.3.1 Variables and parameters for the model

The variables and parameters for the model are described as in Tables 3.1 and 3.2:

Table 3.1: Variables for the Intra-Host Stochastic model

State Variable Description

X(t) The concentration of uninfected (susceptible) cells at time t

Y(t) The concentration of infected cells at time t

V(t) The concentration of virus particles at time t

69

Table 3.2: Parameters for the Intra-Host stochastic model

Parameter Description

λThe total rate of production of healthy cells per unit time

δThe per capita death rate of healthy cells

βThe transmission coeﬃcient between uninfected cells and

infective virus particles

κper capita death rate of infected cells

γThe virus production rate due to risk behaviors

µThe per capita death rate of infective virus particles

NAverage number of virus particles produced by an infected cell

in the absence of treatment during its entire infectious lifetime

Let X(t) be the size of the healthy cells population at time t,Y(t) be the size of

infected cell at time tand V(t) be the size of the virions population at time t. In

the model to be formulated, it is now assumed that instead of rates of births and

deaths, there is a possibility of stochastic births or deaths of the heathy cells, infected

cells and the virus particles. Thus X(t), Y(t) and V(t) are time dependent random

variables. Let the probability of there being xhealthy cells, yinfected cells and v

virions in an infected person at time tbe denoted by the following joint probability

function:- Px,y,v (t) = P[X(t) = x, Y (t) = y, V (t) = v], for x, y, z = 0,1,2,3.....

The standard argument based on the forward Chapman-Kolmogorov diﬀerence diﬀer-

ential equation is used to obtain the joint probability function Px,y,v (t) , by considering

the joint probability Px,y,v(t, t + ∆t) . This joint probability is obtained as the sum

of the probabilities of the following mutually exclusive events:

a). There were xhealthy cells, yinfected cells and vvirions by time tand nothing

happens during the time interval (t, t + ∆t).

b). There were x−1 healthy cells, yinfected cells and vvirions by time tand one

healthy cell is produced from the thymus during the time interval (t, t + ∆t)

70

c). There were x+ 1 healthy cells, yinfected cells and vvirions by time tand one

healthy cell dies or is infected by HIV virus during the time interval (t, t + ∆t)

d). There were xhealthy cells, y−1 infected cells and vvirions by time tand one

healthy cell is infected by HIV virus during the time interval (t, t + ∆t)

e). There were xhealthy cells, y+ 1 infected cells and vvirions by time tand one

infected cell dies (HIV-infected cell bursts or undergoes a lysis)during the time

interval (t, t + ∆t)

f). There were xhealthy cells, yinfected cells and v−1 virions by time tand one

virion is produced(HIV-infected cell undergoes a lysis or the individual angages

in risky behaviours) during the time interval (t, t + ∆t)

g). There were xhealthy cells, yinfected cells and v+ 1 virions by time tand one

virion dies during the time interval (t, t + ∆t)

We now summarize the events that occur during the interval (t, t + ∆) together with

their transition probabilities in Table 3.3.

71

Table 3.3: Intra-Host HIV transitions

Possible transitions in host interaction of HIV and Immune system Cells

and corresponding probabilities

Event Population components Population components probability

(X,Y,V) at t (X,Y,V) at (t, t + ∆) of transition

Production of uninfected cell (x−1, y, v) (x, y, v)λ∆t

Death of uninfected cell (x+ 1, y, v) (x, y, v)δ(x+ 1)∆t

Infection of uninfected cell (x+ 1, y −1, v + 1) (x, y, v)β(x+ 1)(v+ 1)e−ρτ ∆t

Production of virions from (x, y + 1, v −1) (x, y, v)κN (y+ 1)∆t

the bursting infected cell

Introduction of Virions (x, y, v −1) (x, y, v)γ∆t

due to re-infection because

of risky behaviour

Death of virions (x, y, v + 1) (x, y, v)µ(v+ 1)∆t

If we let τto be the time lag between the time the virus contacts a target CD4 T cell

and the time the cell becomes productively infected (including the steps of successful

attachment of the virus to the cell, and penetration of virus into the cell). If we

also let ρto be the death rate of infected but not yet virus producing cell, then the

probability that the infected cell will survive to virus producing cell during the short

time interval τwill be given by e−ρτ .

The change in population size during the time interval ∆t, which is assumed to be

suﬃciently small to guarantee that only one such event can occur in (t, t + ∆t), is

governed by the following conditional probabilities;

Px,y,v (t+ ∆t) = {1−(λ∆t+δx∆t+βxv∆t

+µv∆t+κy∆t+γ∆t) + o(∆t)}Px,y,v (t)

+{λ∆t+o(∆t)}Px−1,y,v (t)

72

+{δ(x+ 1)∆t+o(∆t)}Px+1,y,v (t)

+{βe−ρτ (x+ 1)(v+ 1)∆t+o(∆t)}Px+1,y −1,v+1(t) (3.6)

+{κN(y+ 1)∆t+o(∆t)}Px,y+1,v −1(t)

+{γ∆t+o(∆t)}Px,y,v−1(t)

+{µ(v+ 1)∆t+o(∆t)}Px,y,v+1 (t).

Re-arranging equation (3.9), we have

Px,y,v (t+ ∆t)−Px,y,v (t) = −{λ∆t+δx∆t+βxv∆t

+µv∆t+κy∆t+γ∆t+o(∆t)}Px,y,v (t)

+{λ∆t+o(∆t)}Px−1,y,v (t)

+{δ(x+ 1)∆t+o(∆t)}Px+1,y,v (t)

+{β(x+ 1)(v+ 1)e−ρτ ∆t+o(∆t)}Px+1,y−1,v+1 (t)

+{κN(y+ 1)∆t+o(∆t)}Px,y+1,v −1(t)

+{γ∆t+o(∆t)}Px,y,v−1(t)

+{µ(v+ 1)∆t+o(∆t)}Px,y,v+1 (t).

Dividing through by ∆tyields

Px,y,v (t+ ∆t)−Px,y,v (t)

∆t=−{λ+δx +βxv +µv +κy +γ+o(∆t)

∆t}Px,y,v (t)

+{λ+o(∆t)

∆t}Px−1,y,v (t)

+{δ(x+ 1) + o(∆t)

∆t}Px+1,y,v (t)

73

+{βe−ρτ (x+ 1)(v+ 1) + o(∆t)

∆t}Px+1,y−1,v+1 (t)

+{Nκ(y+ 1) + o(∆t)

∆t}Px,y+1,v−1(t)

+{γ+o(∆t)

∆t}Px,y,v−1(t)

+{µ(v+ 1) + o(∆t)

∆t}Px,y,v+1 (t).

Taking the limit as ∆t→0, we have the following forward Kolmogorov partial

diﬀerence diﬀerential equation for Px,y,v(t).

P0

x,y,v (t) = −{λ+δx +βxv +µv +κy +γ}Px,y,v(t)

+λPx−1,y,v (t)

+δ(x+ 1)Px+1,y,v (t)

+βe−ρτ (x+ 1)(v+ 1)Px+1,y −1,v+1(t)

+Nκ(y+ 1)Px,y+1,v −1(t)

+γPx,y ,v−1(t) (3.7)

+µ(v+ 1)Px,y,v+1 (t).

Equation (3.10) is also called a master equation or the diﬀerence diﬀerential equation

with the initial condition

P0

0,0,0(t) = −(λ+γ)P0,0,0(t) + δP1,0,0(t) + µP0,0,1(t)

where P0

x,y,v (t) = lim∆t→0Px,y,v (t+∆t)−Px,y,v (t)

∆t

74

and the prime indicates diﬀerentiation with respect to t.

3.3.2 Probability Generating Function (PGF) of state vari-

ables

The Multivariate probability generating function for a joint probability distribution

Px,y,v (t) is deﬁned as

G(z1, z2, z3, t) =

∞

X

x=0

∞

X

y=0

∞

X

v=0

Px,y,v (t)zx

1zy

2zv

3.

Diﬀerentiating the pgf with respect to tyields

∂G(z1, z2, z3, t)

∂t =

∞

X

x=0

∞

X

y=0

∞

X

v=0

P0

x,y,v (t)zx

1zy

2zv

3.

Diﬀerentiating again the pgf with respect to z1, z2, z3yields

∂3G(z1, z2, z3, t)

∂z1∂z2∂z3

=

∞

X

x=1

∞

X

y=1

∞

X

v=1

xyvPx,y,v (t)zx−1

1zy−1

2zv−1

3

=

∞

X

x=0

∞

X

y=0

∞

X

v=0

(x+ 1)(y+ 1)(v+ 1)Px+1,y+1,v+1 (t)zx

1zy

2zv

3.

Multiplying equation (3.10) by zx

1zy

2zv

3and summing over x,y, and v, we have

∞

X

x=1

∞

X

y=1

∞

X

v=1

P0

x,y,v (t)zx

1zy

2zv

3=

∞

X

x=1

∞

X

y=1

∞

X

v=0

−{λ+δx +βxv +µv +κy +γ}Px,y,v (t)zx

1zy

2zv

3

75

+

∞

X

x=1

∞

X

y=1

∞

X

v=1

λPx−1,y,v (t)zx

1zy

2zv

3

+

∞

X

x=1

∞

X

y=1

∞

X

v=1

δ(x+ 1)Px+1,y,v (t)zx

1zy