ThesisPDF Available

Intra-Host stochastic models for HIV dynamics and management

Authors:
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 6001400mm3in 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 =sT(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 (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 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(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=(1γ)βλN (1η)
δµ+βλ(1γ)and found that if R01 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 edescribes the probability
of infected target cells surviving the period of intracellular delay from tτto 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 =λδx(t)βx(t)v(t),
dy
dt =βx(tτ)v(tτ)eαy(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
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+
4200
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+
4200
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+
4200
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+
4200
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 λtand dies with probability δt. 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) = n1 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) = n1]P[X(t) = n1]
+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] = λt+o(∆t),
iii). P[X(t+ ∆t) = n1|X(t) = n] = δnt+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,PXn1(t) be the probability that the
population of healthy CD+
4cells has the value n1 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 PXn1(t) (birth)
which gives:
PXn(t+ ∆t) = {1(λt+δnt)}PXn(t) + {λt}PXn1(t)
+ (n+ 1)δtPXn+1 (t) + o(∆t)for n 0.
60
Rearranging the equation we have
PXn(t+ ∆t)PXn(t) = −{λ+}tPXn(t) + λtPXn1(t)
+ (n+ 1)δtPXn+1 (t) + o(∆t)for n 0.
Dividing through by ∆tand taking the limit as ∆t0 we have the following differ-
ence differential equation called the Kolmogorov forward difference differential equa-
tion for PXn(t)
P0
Xn(t) = −{λ+}PXn(t) + λPXn1(t)+(n+ 1)δPXn+1 (t)
=λPXn(t)δnPXn(t) + λPXn1(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 PXn1(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 0,0Px1, 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)zn1.
Multiplying equation (3.2) by znand summing over nwe have
X
n=1
P0
Xn(t)zn=λ
X
n=1
PXn(t)znδ
X
n=1
nPXn(t)zn+λ
X
n=1
PXn1(t)zn
+δ
X
n=1
(n+ 1)PXn+1 (t)Zn,
=λ
X
n=1
PXn(t)znδ
X
n=1
nPXn(t)zn
+λ
X
n=1
PXn1(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)(z1) + δ(1 z)Gx
∂z ,
and can be re-arranged to
∂Gx
∂t λ(z1)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(z1)Gx
.(3.3)
We can integrate the first two equations in equation (3.6) to have
Zδdt =Zdz
1z
ln|1z|=δt +C1.
63
Taking exponential on both sides we have
(1 z) = eδt+C1
which implies
(1 z)eδt =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
Zλ(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
eλ
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)eδt},
64
which becomes
Gx(z, t) = eλ
δzF{(1 z)eδt}.
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{1z}=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(z1)(1eδt)}{zeδt + 1 eδ 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(z1)(1eδt)}with mean λ
δ(1 eδt) and binomial probability
generating function {zeδt + 1 eδt)}n.
as t→ ∞, Gx(z, t) = eλ
δ(z1).(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 x1 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, y1 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 v1 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.
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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 (x1, y, v) (x, y, v)λt
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)eρτ 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)γt
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 eρτ .
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(λt+δxt+βxvt
+µvt+κyt+γt) + o(∆t)}Px,y,v (t)
+{λt+o(∆t)}Px1,y,v (t)
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+{δ(x+ 1)∆t+o(∆t)}Px+1,y,v (t)
+{βeρτ (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)
+{γt+o(∆t)}Px,y,v1(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) = −{λt+δxt+βxvt
+µvt+κyt+γt+o(∆t)}Px,y,v (t)
+{λt+o(∆t)}Px1,y,v (t)
+{δ(x+ 1)∆t+o(∆t)}Px+1,y,v (t)
+{β(x+ 1)(v+ 1)eρτ t+o(∆t)}Px+1,y1,v+1 (t)
+{κN(y+ 1)∆t+o(∆t)}Px,y+1,v 1(t)
+{γt+o(∆t)}Px,y,v1(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}Px1,y,v (t)
+{δ(x+ 1) + o(∆t)
t}Px+1,y,v (t)
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+{βeρτ (x+ 1)(v+ 1) + o(∆t)
t}Px+1,y1,v+1 (t)
+{Nκ(y+ 1) + o(∆t)
t}Px,y+1,v1(t)
+{γ+o(∆t)
t}Px,y,v1(t)
+{µ(v+ 1) + o(∆t)
t}Px,y,v+1 (t).
Taking the limit as ∆t0, 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)
+λPx1,y,v (t)
+δ(x+ 1)Px+1,y,v (t)
+βeρτ (x+ 1)(v+ 1)Px+1,y 1,v+1(t)
+Nκ(y+ 1)Px,y+1,v 1(t)
+γPx,y ,v1(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) = limt0Px,y,v (t+∆t)Px,y,v (t)
t
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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)
∂z1z2z3
=
X
x=1
X
y=1
X
v=1
xyvPx,y,v (t)zx1
1zy1
2zv1
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
λPx1,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