A Dissertation entitled
Mathematical Models of the Activated Immune System during HIV Infection
by Megan Powell
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics
Dr. H. Westcott Vayo, Committee Chair
Dr. Joana Chakraborty, Committee Member
Dr. Marianty Ionel, Committee Member
Dr. Denis White, Committee Member
Dr. Patricia R. Komuniecki, Dean College of Graduate Studies
The University of Toledo May 2011 Copyright 2011, Megan Powell
This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Mathematical Models of the Activated Immune System during HIV Infection by Megan Powell
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics The University of Toledo May 2011
HIV is a virus currently affecting approximately 33.3 million people worldwide.
Since its discovery in the early 1980s, researchers have strived to find treatment that helps the immune system eradicate the virus from the human body. A great deal of advances have been made in helping HIV infected individuals from advancing to
AIDS, but no cure has yet been found. Researchers have found that the immune system is in a chronic state of activation during HIV infection and believe this could be a major contributor to the decline of immune system cell populations. Using analysis of systems of Ordinary Differential Equations, this paper serves to better understand the dynamics of the activated immune system during HIV infection. Both current and possible future therapies are considered.
iii This thesis is dedicated to the memory of Umaer Basha. Acknowledgments
Thank you to my parents, Mary Lou and Charles Powell for their unrelenting quest for the best education for me and my sister and always supporting my pursuit of higher education. To my sister, Jill Powell, for always believing that I would finish this dissertation and program. To my advisor, H. Westcott Vayo whose support and guidance throughout this program have been invaluable through five difficult years and without whom, this paper would not have been possible. To my committee members, Dr. Marianty Ionel, Dr. Denis White, and Dr. Joana Chakraborty for both inspiring me and helping this paper form into its final version. To Dr. Henry
Wente for being an inspiring instructor and nurturing my love of mathematics. To my fellow graduate student, Abdel Yousef, for his unselfishness in helping me through some very difficult problems. Finally, to Bounce for her constant companionship and my only guaranteed stability in a world of chaos.
v Table of Contents
Abstract iii
Acknowledgments v
Table of Contents vi
List of Figures ix
List of Abbreviations x
1 Introduction 1
1.1 The Immune System ...... 1
1.2 HIV Background ...... 2
1.3 Basic Model: Wodarz and Nowak ...... 4
1.4 Basic Model with Immune Response: Nowak and Bangham ...... 6
1.5 CD4 and CD8 T cell dynamics model: Vayo and Huang ...... 9
1.6 Models with Therapies ...... 10
1.6.1 Reverse Transcriptase Inhibitors ...... 10
1.6.2 Protease Inhibitors ...... 12
1.7 Models of Mutating Virus ...... 14
2 Dynamics of Healthy, Infected, Activated T Cells with Virus Effect 21
2.1 Definition of parameters ...... 21
vi 2.2 Numerical Values ...... 23
2.3 Model and Equilibrium Points without Treatment ...... 24
2.4 Equilibrium Points with Treatment ...... 30
2.4.1 Theoretical Therapy: Prevent activated CD8 T cells from killing
healthy CD4 T cells ...... 30
2.4.2 Existing Therapies: Protease Inhibitors and Reverse Transcrip-
tase Inhibitors ...... 34
2.5 Solutions and Graphs ...... 40
2.6 Discussion ...... 42
3 Dynamics of Naive, Effector, and Memory T cells without Virus
Effect 43
3.1 Definition of Variables and Parameters ...... 44
3.2 Model without virus and both naive and memory cells activating . . . 45
3.2.1 Solutions and Graphs ...... 48
3.2.2 Discussion ...... 50
3.3 Model without virus, only memory cells activating ...... 51
3.3.1 Solutions and Graphs ...... 55
3.3.2 Discussion ...... 57
4 Dynamics of Naive, Effector, and Memory T cells with Virus Effect 58
4.1 Definition of Parameters ...... 59
4.2 Before Treatment ...... 59
4.3 With Treatment ...... 61
4.4 Solutions and Graphs ...... 69
4.5 Discussion ...... 73
5 Conclusion and Future Research 74
5.1 Conclusion ...... 74
vii 5.2 Future Research ...... 75
References 77
A Sample Numerical Values from the Literature for Chapter 2 83
B Sample Numerical Values from Literature for Chapter 3 84
C Sample Numerical Values from Literature for Chapter 4 86
viii List of Figures
2-1 Infected cell population with HAART ...... 40
2-2 Activated T cell population with HAART ...... 41
2-3 Virus population with HAART ...... 41
3-1 Naive T cell population during chronic activation ...... 49
3-2 Effector T cell population during chronic activation ...... 49
3-3 Memory T cell population during chronic activation ...... 50
3-4 Recovering naive T cells with no naive cells activating ...... 55
3-5 Increasing effector T cells with no naive cells activating ...... 56
3-6 Memory T cells with no naive cells activating ...... 56
4-1 Naive T cells after HAART ...... 70
4-2 Memory T cells after HAART ...... 70
4-3 Effector T cells after HAART ...... 71
4-4 Infected T cells after HAART ...... 71
4-5 Infectious virus particles after HAART ...... 72
4-6 Non-infectious particles after HAART ...... 72
ix List of Abbreviations
AIDS ...... Acquired Immunodeficiency Syndrome CTL ...... Cytotoxic T Lymphocyte CD4 ...... Cluster of Differentiation 4 CD8 ...... Cluster of Differentiation 8 DNA ...... Deoxyribonucleic acid HAART ...... Highly Active Anti-retroviral Therapy HIV ...... Human Immunodeficiency Virus RNA ...... Ribonucleic acid
x Chapter 1
Introduction
1.1 The Immune System
The purpose of the immune system is to prevent infections and remove any ex- isting infections. During an immune response, the body defends itself by either destroying or rendering harmless any matter perceived as foreign. The body’s de- fense mechanisms consist of innate immunity, which help protect the body without needing to recognize the specific type of foreign matter, while adaptive immunity requires recognition of the foreign matter by lymphocytes, a type of white blood cell.
Bacteria, viruses, fungi, and parasites, collectively known as microbes, all stimulate adaptive immune responses but adaptive immune responses are also the major barrier to successful organ transplantation and blood transfusions. Humoral immunity, one type of adaptive immunity, is mediated by B lymphocytes which produce antibodies which help eradicate microbes before they are able to infect host cells. Cell-mediated immunity, another type of adaptive immunity, is mediated by T lymphocytes which help eliminate microbes that live inside infected cells. During their maturation in the thymus, T cells develop receptors specific to only one type of antigen, where an antigen is any molecule (often a protein) that can induce a specific immune response.
1 In order for a T cell receptor to combine with an antigen, the antigen must first be
processed and displayed by certain type of protein molecule (major histocompatibility
complex protein) found on antigen-presenting cells. Once the receptor and antigen
are bound the T cells become activated and multiply rapidly during a process called
clonal expansion. A fraction of the daughter cells differentiate into effector cells,
which launch an attack against the microbe expressing the antigen, and memory cells
which remain inactive until they encounter the antigen again at a later time. Surface
proteins expression define a particular cell where the standard notation is CD (cluster
of differentiation) and the number that designates that surface protein. Helper T
cells express the protein CD4 and once activated produce proteins called cytokines
that activate cytotoxic T lymphocytes (CTLs) that have the ability to kill infected
host cells. Cytotoxic T lymphocytes (or CD8+T cells) cannot function properly without stimulation by these cytokines. Therefore a dysfunction of the helper T cells will result in malfunctioning killer T cells as well [1],[37].
1.2 HIV Background
Human immunodeficiency virus (HIV) was first diagnosed in 1981. There are an estimated 1.1 million HIV positive people living in the United States today with more than 21% of them unaware of their infection [7]. In the world, there are an estimated 33.3 million HIV positive individuals and over 16 million orphans due to
AIDS. The virus is the most devastating in sub-Saharan Africa where in 2009, 72% of the 1.8 million HIV-related deaths occurred [35]
HIV is a retrovirus which has ribonucleic acid (RNA) as its nucleic core. One of HIV’s surface proteins, gp120, binds to the CD4 protein and a certain chemokine receptor. Therefore HIV preferentially (but not exclusively) infects helper T cells which express CD4. Once inside a helper T cell, the enzyme protease helps release
2 the virus’s RNA and the enzyme reverse transcriptase helps transcribe the RNA into
deoxyribonucleic acid (DNA) which is then integrated into the host cell’s DNA. Once
the infected cell is stimulated by an extrinsic source, it starts transcribing it’s own
DNA which inadvertently replicates the virus as well. The replication of the virus
inside the cell as well as immune system responses cause the death of the infected cell.
HIV causes many uninfected helper T cells to die as well, yet the mechanism for this
depletion remains unknown. The body is initially able to replace the dying helper
T cells but the immune system responses are unable to control the replicating virus
and eventually the number of T cells decreases. An individual is considered to have
AIDS when the helper T cell count falls below 200 per cubic millimeter where the
normal amount is 1000 to 1500 helper T cells per cubic millimeter [1], [37] . At this
point the entire immune system is compromised and leaves the individual extremely
vulnerable to other infections and high levels of cytokines in the system cause weight
loss, lethargy, and fever. Without treatment, most people die within two years of
the onset of AIDS [27], [37].
A better understanding of how HIV ultimately defeats the immune system has
been the purpose of many mathematical models that have been developed and ana-
lyzed since the discovery of the virus. Many articles focus on the interplay between
healthy CD4+T cells, infected CD4+T cells, and virus particles. Some authors also include the dynamics of immune responses and discuss how a mutating virus may contribute to the virus’ eventual control of the immune system. In this section we discuss some of these different models. For the remainder of this paper, T cells will refer to CD4+T cells (helper T cells) unless otherwise noted. The original work of this paper focuses on how the activated immune system affects the body’s inability to fully fight off an infection by HIV.
3 1.3 Basic Model: Wodarz and Nowak
We first consider a model put forth by Wodarz and Nowak [39]. This model simplifies the immune system dynamics to healthy T cells, infected T cells, and virus particles.
dx = λ − dx − βxv dt
dy = βxv − ay dt
dv = ky − uv (1.1) dt where
• t is time, usually measured in days
• x is the number of healthy T cells
• y is the number of HIV infected T cells
• v is the number of virus particles
• λ is the rate at which T cells are produced
• d is the natural death rate of healthy T cells
• β is the rate at which virus infects healthy cells
4 • a is the natural death rate of infected cell
• k is the rate at which infected cells produce virus particles
• u is the rate at which virus particles are removed from the body.
According to Nowak and Bangham [26], the third equation in system (1.1) should also include a term taking into consideration the rate at which virus particles are absorbed by host cells, but this rate is negligible when the virus load is large. Before the virus infects the human body, system (1.1) would be reduced to the equation
dx = λ − dx. (1.2) dt
Assuming the body has reached equilibrium before viral infection, the initial condi-
λ tions of system (1.1) are x0 = d , y0 = 0, v0 = 0. From the parameters, we see x 1 that the average lifetime of the uninfected cells is dx = d , and similarly, the average 1 1 lifetime of the infected cells is a and the average lifetime of virus particles is u . We ky k can also determine that the total virus produced from one infected cell is ay = a . In
order for an infection to take hold, the basic reproductive number R0, defined as the average number of secondary infections produced when one infected cell is introduced
to a host population, when almost all cells are healthy, must be greater than one.
For system (1.1),
1 1 βxk R = (βx) (k) = (1.3) 0 a u au
1 where βx is the rate at which the virus infects healthy cells, a is the average lifespan 1 of an infected cell, k is the rate one infected cell produces virus particles, and u is the lifespan of a virus particle. Again, assuming the body is healthy and in equilibrium
λ βλk when the virus is first introduced to the body, x = d initially, so R0 = adu . The
5 equilibrium points of this system are given by
au λ x∗ = = βk dR0
λ du (R − 1)du y∗ = − = 0 a βk βk
λk d (R − 1)d v∗ = − = 0 . (1.4) au β β
βx∗k At equilibrium, the reproductive ratio R becomes au = 1. Therefore, at equi- librium, each infected cell is on average giving rise to one additional infected cell.
∗ Assuming R0 is much bigger than one, x should be much smaller than x0, thus this model does not explain individuals infected with HIV who a relatively constant num- ber of healthy CD4+T cells for many years, before declining. The variability of how long it takes an infected individual to have significant T cell decline is the focus of much of the research done on HIV infection.
1.4 Basic Model with Immune Response: Nowak
and Bangham
Nowak and Bangham [26] discuss a model based on the basic model (1.1) which also includes the cytotoxic T lymphocyte (CTL) immune response, the main immune factor that limits virus replication. Recall that cytotoxic T lymphocytes, or killer
T cells, are activated by cytokines released by helper T cells and kill helper T cells infected by the virus. The model is given by
6 dx = λ − dx − βxv dt
dy = βxv − ay − pyz dt
dv = ky − uv dt
dz = cyz − bz (1.5) dt
where
• t is time, usually measured in days
• x is the number of healthy T cells
• y is the number of HIV infected T cells
• v is the number of virus particles
• λ is the rate at which T cells are produced
• d is the natural death rate of healthy T cells
• β is the rate at which virus infects healthy cells
• a is the natural death rate of infected cell
7 • k is the rate at which infected cells produce virus particles
• u is the rate at which virus particles die
• z is the abundance of virus specific CT Ls
• p isthe rate at which CT Ls kill infected cells
• c is the rate of CTL proliferation in response to an antigen
• b is the rate of CTL decay in absence of antigen stimulation
dz Notice that in order for the rate of the CTL response to increase (i.e. dt > 0), cy must be greater than b. If cy∗ < b (where y∗ is the equilibrium point defined in
(1.4 ), then a CTL response may be slightly activated, but equilibrium point (1.4)
can be reached without any CTL response. Assuming that cy∗ > b, the equilibrium
point of system (1.5) is
λcu x = b cdu + βbk
b y = b c
bk v = b cu
1 λβck z = − a . (1.6) b p cdu + βbk
Theorem 1. The equilibrium number of infected cells will be reduced with a CTL response.
8 Proof. Without a CTL response, the equilibrium number of infected cells is y∗. As-
∗ dz suming there are y infected cells, an increasing CTL response will occur if dt = ∗ ∗ ∗ b cy z − bz > 0. In order for this to occur, we must have z > 0 and cy > b ⇒ y > c . But the equilibrium number of infected cells with a CTL response is y = b . Thus b c ∗ y > yb i.e., the equilibrium number of infected cells without a CTL response is higher than the equilibrium number of infected cells with a CTL response.
1.5 CD4 and CD8 T cell dynamics model: Vayo
and Huang
Vayo and Huang [17] offer models representing the dynamics between CD4+T cells
and CD8+T cells in the healthy body and during HIV infection. The model they
put forth for the dynamics during HIV infection is
dC 4 = [a (x − C ) + a (y − C )]C − δ C (1.7) dt 1 0 4 2 0 8 4 1 4
dC 8 = a (x + y − C − C )C (1.8) dt 3 0 0 4 8 8
where
+ C4(t): The amount of CD4 T cells at time t.
+ C8(t): The amount of CD8 T cells at time t.
+ x0: The standard amount of CD4 T cells in a healthy human body.
+ y0: The standard amount of CD8 T cells in a healthy human body.
a1, a2, a3: Positive constants, a1 < a2.
+ δ1: Depletion rate of CD4 T cells.
+ x0 − C4(t): The difference in the standard amount of CD4 T cells and amount of CD4+T cells at time t.
9 + y0 − C8(t): The difference in the standard amount of CD8 T cells and amount of CD8+T cells at time t.
(x0 + y0) − (C4 + C8): The difference in the standard total number of T cells and the amount at time t.
1.6 Models with Therapies
There currently are drug therapies that assist the immune system in decreasing the number of infected cells and help reduce viral load. Two main types of thera- peutic drugs used today are reverse transcriptase inhibitors, which prevent the virus from infecting new cells, and protease inhibitors, which prevent infected cells from producing infectious virus particles. A combination of medication including these two types of drugs are called HAART, highly active anti-retroviral therapy [37]. Wodarz and Nowak [39] discuss both of these therapies. They assume that each therapy is applied enough after the initial infection that x, y, and, v have reached their equilib- rium values x∗, y∗, v∗ found in (1.4). Since the lifespan of a healthy cell is generally significantly longer than the life span of an infected cell or virus particle, we assume in all the following cases that the x value remains constantly x∗ in the time frame under consideration.
1.6.1 Reverse Transcriptase Inhibitors
The first therapy discussed is reverse transcriptase inhibitors. For simplicity, we assume the therapy is completely effective, thus with treatment, β = 0 where we
dx recall that β is the rate at which the virus infects healthy cells. Then with dt = 0,
10 system (1.1) becomes
dy = −ay dt dv = ky − uv (1.9) dt
with initial conditions
y(0) = y∗
v(0) = v∗.
We can solve the system by first solving the first equation for y.
y(t) = y∗e−at.
Then substituting into the second equation and multiplying by an integrating factor of eut, we get
veut + uveut = ky∗e−ateut d (veut) = ky∗e−ateut dt ky∗e(u−a)t veut = + c. u − a
∗ ∗ ∗ u Substituting in the initial condition v(0) = v and using the fact that y = v k , after some rearrangement, we get
v∗ v(t) = (ue−at − ae−ut). u − a
11 1.6.2 Protease Inhibitors
The second therapy discussed is protease inhibitors which prevent infected cells from producing infectious virus particles. After treatment is started, the infected virus particles only produce non-infectious particles we label w. Therefore we have a modified system (1.1) with c = 0 and the additional w term which results in
dy = βxv − ay dt dv = −uv dt dw = ky − uw (1.10) dt with initial conditions
y(0) = y∗
v(0) = v∗
w(0) = 0.
Note that the v term still appears since the previously infected particles will still be in the body for a small period of time after the treatment is applied. We solve this system as follows:
v(t) = v∗e−ut dy = βx∗(v∗e−ut) − ay dt
Letting b = βx∗v∗ and multiplying by an integrating factor of eat, we get
12 dy eat + ayeat = be−uteat dt d (yeat) = be(a−u)t dt be(a−u)t yeat = + c. a − u
Substituting in the initial condition y(0) = y∗, we get
b y(t) = (e−ut − e−at) + y∗e−at. a − u
Then dw bk = (e−ut − e−at) + ky∗e−at − uw. dt a − u
Multiplying by an integrating factor of eut, we have
dw bkeut eut + uweut = (e−ut − e−at) + ky∗eute−at dt a − u d kb bk (weut) = + (ky∗ − )e(u−a)t. dt a − u a − u
So kbt bk e(u−a)t weut = + ky∗ − + c. a − u a − u (u − a)
Using the initial condition w(0) = 0, we have
kbte−ut bk e(u−a)te−ut ky∗e−ut bke−ut w(t) = + ky∗ − + − . a − u a − u (u − a) a − u (a − u)2
∗ ∗ u ∗ ∗ Using y = v k and substituting βx v back in for b, the equation for non-infectious virus simplifies to
13 kβx∗v∗t −uv∗ βx∗v∗k w(t) = e−ut + + (e−at − e−ut). a − u a − u (a − u)2
dy ∗ ∗ ∗ auv∗ We know at equilibrium, dt = 0, which implies βx v = ay = k , thus we can rewrite the above equation as
autv∗ auv∗ uv∗ w(t) = e−ut + − (e−at − e−ut). a − u (a − u)2 a − u
Thus the total amount of virus is given by
aut u2 v(t) + w(t) = v∗ e−ut + e−ut + e−at − e−ut . a − u (a − u)2
1.7 Models of Mutating Virus
The question of how HIV eventually defeats the immune system has been at
the heart of AIDS research for many years. Many researchers have put forth the
theory that virus mutation, and the inability of the immune system to keep up with
the mutations, is one of the major factors contributing to the failure of the immune
system to eradicate the virus from the body. Nowak and Bangham [26] put forth
the model
dx = λ − dx − xn β v dt i=1 i i
dy i = β xv − ay − py z dt i i i i i
dv i = k y − uv dt i i i
14 dz i = cy z − bz (1.11) dt i i i where
• t is time, usually measured in days
• x is the number of healthy T cells
• y is the number of HIV infected T cells
• v is the number of virus particles
• λ is the rate at which T cells are produced
• d is the natural death rate of healthy T cells
• β is the rate at which virus infects healthy cells
• a is the natural death rate of infected cell
• k is the rate at which infected cells produce virus particles
• u is the rate at which virus is removed from the body
• z is the abundance of virus specific CT Ls
• p isthe rate at which CT Ls kill infected cells
• c is the rate of CTL proliferation in response to an antigen
• b is the rate of CTL decay in absence of antigen stimulation
• vi is the amount of the specific strain (mutation) i of the virus
• yi is the amount of virus (strain i) infected cell
• zi is the CTL response specific to strain i
15 • βi is the rate at which virus strain i infects healthy cells
• ki is the rate virus strain i is produced by infected cells.
Bittner, Bonhoeffer, and Nowak [5] offer a slightly different model where they only consider the virus strains and immune responses but not the virus infected cells. The model they offer is
dx i = c v − b x − x n u v dt i i i i ij=1 j j
dv i = v (r − p x − s z) dt i i i i
dz =n k v − bz − zn u v (1.12) dt i=1 j j j=1 j j
where here
P • vi represents each specific virus mutant strain where i = 1...n, where v = vi represents the total amount of virus
• xi represents the immune response specific to virus strain vi
• z represents the immune responses directed at all virus strains
• r is the average rate of replication of all virus strains
• pi is the efficiency of each strain-specific immune response
• si is the efficiency of the global immune system response against virus strain i
16 • ci is the rate at which each strain-specific immune response is evoked
• bi represents the decay of strain-specific immune responses in the absence of stimulation
• uj represents the ability for the virus to impair strain-specific immune system response j.
• kj is the rate at which the global immune system responses are evoked against strain j.
Wodarz and Nowak [39] simplify this model for analysis. When multiple strains exist in a body, a strain-specific response (xi) may actually respond to more than one strain, or not all. In this model, we assume that each strain-specific response is
responding to only one strain of the mutated virus. Furthermore, we assume many
of the parameters are the same for all strains of the virus and replace
• pi with p
• si with s
• ci with k
• bi with b
• uj with u
0 • kj with k .
The simplified model then becomes
17 dx i = kv − bx − uvx dt i i i dv i = v (r − px − sz) (1.13) dt i i dz = k0v − bz − uvz. (1.14) dt
From the above, we can find the rate at which the total virus population changes
dv X = v (r − px − sz) dt i i X X X = r vi − p vixi − sz vi X = rv − p vixi − szv (1.15)
but we assume that xi and z have converged to their steady state levels in a time frame shorter than the rate at which the entire population of virus changes, so we can use
∗ ∗ kvi − bxi − uvxi = 0 kv x∗ = i i b + uv and
k0v − bz∗ − uvz∗ = 0 k0v z∗ = . b + uv
Therefore we have dv pk X sk0v2 = rv − v2 − . dt b + uv i b + uv
18 Multiplying the middle term by v2/v2, we have
dv pkv2 X v2 sk0v2 = rv − i − . (1.16) dt b + uv v2 b + uv
2 P vi The quantity v2 represents the Simpson index D, which is a measure of biological diversity. The Simpson index, first introduced by Edward H. Simpson in 1949, is the probability that if individuals (viruses) are chosen at random from the general population they belong to the same species (strain) [33]. So the more virus strains
1 we have, the lower D is where D is always between 0 and 1 with D = n representing exactly n strains all occurring in the exact same abundance, and D = 1 representing only one virus strain present. We can rewrite the virus population change equation as
dv pkv2 sk0v2 = rv − D − . (1.17) dt b + uv b + uv
So now we can compute the steady-state of the total virus population,
pkv2 sk0v2 rv − D − = 0 b + uv b + uv rb v∗ = . (1.18) pkD + sk0 − ru
We notice as D decreases (diversity increases), the equilibrium total virus population increases. We now consider three cases of virus diversity.
1. ru > sk0 + pk
In this case, the virus replication (r) and damaging of the immune system (u)
overrides the immune system’s ability to control the virus both with global
19 (sk0) and strain-specific responses (pk). Thus the virus replicates to high levels
immediately without an asymptomatic stage.
2. sk0 > ru
In this case, the global responses (sk0) alone can control the virus (ru) and
strain-specific responses are not needed.
3. sk0 + pk > ru > sk0
In this case, both global (sk0) and strain-specific (pk) immune system responses
can control the virus, whereas global responses (sk0) cannot.
This control of virus replication changes when D is such that ru = sk0 + pkD, thus what is called the diversity threshold occurs when
ru − sk0 D = . (1.19) pk
After the diversity threshold is reached, the immune system can no longer control the virus replication.
20 Chapter 2
Dynamics of Healthy, Infected,
Activated T Cells with Virus Effect
The main characterization of HIV infection is the depletion of T cells. The mechanism for T cell depletion is still not fully understood by researchers. Some researchers have been exploring the theory that it is not the solely the virus that is responsible for the reduction of T cells but that the chronic activation of the immune system plays a crucial role as well[23] [14] [9]. Many of the T cells that are dying during HIV infection are not actually infected by the virus. This leads us to explore mathematical models which take into consideration the role of activated immune system cells as well as the virus in the depletion of T cells.
2.1 Definition of parameters
All lymphocytes arise from stem cells in the bone marrow but T cells mature in the thymus, an immune system organ located in the upper chest [1]. When T cells are first produced by the thymus they are called naive. Naive T cells have not previously responded to an antigen and circulate in the body waiting to find and respond to an antigen. If naive cells do not find an antigen to respond to,
21 they die within weeks or months and are replaced by new naive cells. This cycle of death and replacement leads to homeostasis, a fixed stable number of T cells in the body. We label the production rate of naive T cells by the thymus λ and the natural death rate of naive cells η. Once T cells are presented with an antigen by an antigen-presenting cell, they are activated and differentiate into effector cells and memory cells. Effector cells actively fight the infection but are short lived while memory cells remain inactive for many years but are quickly re-activated when the same antigen is encountered again [23]. We use α to designate the rate at which
HIV initiates naive T cells to activate and β for the death rate of an activated T cell.
HIV infects T cells and then uses the T cells to help reproduce and distribute more virus particles. In reproducing, HIV can disturb the function of the T cell enough that it kills the T cell. We label the rate cells are infected with HIV a, the rate the virus is produced by an infected cell c, and the rate infected cells are killed by HIV k. In fighting HIV, activated T cells generate cytokines which help activate CD8 T cells which then recognize and kill T cells infected with HIV [1]. We call the rate at which activated CD8 T cells kill infected T cells µ and theorize that CD8 T cells are also killing uninfected T cells at a rate b. The immune system not only kills infected cells, but tries to eliminate free virus particles from the body at a rate γ. In a healthy individual, the ratio of CD4 cells to CD8 cells is relatively constant [19].
We assume that the ratio of the two types of cells is a constant rate m. A summary of the parameters and variables used in this section is given in the chart below.
22 Parameter Definition
x Number of healthy, naive T cells
y Number of infected T cells
z Number of healthy, activated T cells
v Number of virus particles
λ Production rate coefficient of healthy cells
η Death rate coefficient of healthy naive cells
β Death rate of activated cell
a Rate virus infects healthy cells
k Death rate of infected cells
γ Removal rate coefficient of virus
µ Rate activated cells kill infected cells
c Rate of virus production by infected cells
b Rate activated cells kill healthy cells
α Rate of activation by HIV
m CD8/CD4 Ratio
2.2 Numerical Values
We consider various models to help describe the dynamics of the immune system during HIV infection. We will analyze with the parameters unknown but will also consider sample numerical values taken from the literature. A summary of numerical values used throughout this section is given in Appendix A.
23 2.3 Model and Equilibrium Points without Treat-
ment
The following model considers the dynamics of healthy naive T cells, infected T cells, healthy activated T cells, and virus particles
dx = λ − ηx − bx (mz) − axv − αx dt
dy = axv − µy (mz) − ky dt
dz = αx − βz dt
dv = cy − γv. (2.1) dt
Instead of introducing an additional variable of activated CD8 cells, we assume that the ratio of CD8 to CD4 T cells remains constant (m) and represent the number of activated CD8 cells by mz. We note that in this section, we are assuming that only naive cells are being infected by the virus. There is conflicting literature on how HIV infects T cells. Gowda et. al. [12] argue that T cells must be activated in order to be infected by HIV, Douek et. al. [10] suggest that HIV specific memory cells are the primary target of HIV, while Groot et. al. [13] show that the type of cell infected
24 depends on the strain of HIV. We will address HIV infecting different types of T cells at different rates later in the paper.
We look at investigating the equilibrium points of the system and compare how they change as we apply different therapies. We only consider equilibrium points where all variables and parameters are positive. If no virus is present (v = 0) there are no infected cells (y = 0) and we have the equilibrium point
1 β x = (A − α − η) 2bα m
1 z = (A − α − η) (2.2) 2bm where r 1 A = (α2β + βη2 + 2αβη + 4bαλm). β
We notice that to guarantee both x and z are positive values, we need A > α + η.
If all parameters are positive, then
r 1 (α2β + βη2 + 2αβη + 4bαλm) > α + η. β
Proof. Suppose
r 1 (α2β + βη2 + 2αβη + 4bαλm) ≤ α + η. β
Then
1 α2β + βη2 + 2αβη + 4bαλm ≤ α2 + 2αη + η2 β
25 which implies
4bαλm α2 + η2 + 2αη + ≤ α2 + 2αη + η2 β
implying
4bαλm ≤ 0. β
But this contradicts the fact that all parameters are positive.
Evaluating this equilibrium point numerically we have
x = 552.09
z = 0.496 87
v = y = 0. (2.3)
In order to determine the stability of this equilibrium point, we use the eigenvalues of the Jacobian matrix evaluated at the equilibrium point. The Jacobian is
−α − η − av − bmz 0 −bxm −ax av −µmz − k −µym ax . α 0 −β 0 0 c 0 −γ
With the given numerical values for the parameters and given equilibrium point, we have
26 −.001 177 4 0 −0.252 72 −0.149 06 0 −0.387 73 0 0.149 06 0.000 3 0 −0.333 3 0 0 100 0 −2
which has eigenvalues labeled r1, r2, r3, r4
r1 = 2. 750 2,
r2 = −.001405 8,
r3 = −0.333 07,
r4 = −5. 138 0. (2.4)
Because r1 does not have negative real parts, the equilibrium point is unstable [4]. If the virus has been able to establish an infection, we have the equilibrium point
kβγ x = acβ − αγµm
A y = 1 a2c2kβ2 − ackαβγµm
kαγ z = acβ − αγµm
A v = 1 (2.5) a2ckβ2γ − akαβγ2µm
27 where
2 2 2 2 2 2 2 2 2 2 A1 = (a c β λ + α λγ µ m + kα βγ µm − ackαβ γ
−ackβ2γη − bk2αβγ2m + kαβγ2µηm − 2acαβλγµm). (2.6)
We assume all appropriate inequalities to make the equilibrium point positive.
Evaluating equilibrium point (2.5) numerically, we have
x = 24. 635
y = 1.8833
z = .02 217 4
v = 94. 164. (2.7)
We can see that the number of healthy naive T cells has diminished significantly with infection. Therefore there are few T cells available to respond to a microbe other than HIV. This allows infections that are relatively benign to a healthy individual to become life threatening to an HIV infected individual. Again to determine the stability of this equilibrium point, we use the eigenvalues of the Jacobian matrix evaluated at the equilibrium point. The result is
28 −α − η − av − bmz 0 −bxm −ax av −µmz − k −µym ax . α 0 −β 0 0 c 0 −γ
With the given numerical values for the parameters and given equilibrium point, we have
−2. 638 4 × 10−2 0 −1. 127 7 × 10−2 −6. 651 5 × 10−3 2. 542 4 × 10−2 −0.332 58 −0.218 83 6. 651 5 × 10−3 0.000 3 0 −0.333 3 0 0 100 0 −2
which has eigenvalues labeled r1, r2, r3, r4
r1 = −.01172 8 + 0.083 97i,
r2 = −.01172 8 − 0.083 97i,
r3 = −0.33309,
r4 = −2.3357. (2.8)
Here, all the equilibrium points have negative real parts so it is a strictly stable
(stable and attractive) equilibrium point [4].
29 2.4 Equilibrium Points with Treatment
Now we see how the equilibrium points change as we apply existing and theoretical therapies.
2.4.1 Theoretical Therapy: Prevent activated CD8 T cells
from killing healthy CD4 T cells
Suppose a mechanism to prevent, or at least diminish, activated T cells from killing healthy naive T cells is discovered. If this is a perfectly performing mechanism, then we have b = 0 where b is rate activated cells kill healthy cells. Then system (2.1) is modified and becomes
dx = λ − ηx − axv − αx dt
dy = axv − µy (mz) − ky dt
dz = αx − βz dt
dv = cy − γv. (2.9) dt
30 We assume an infection has been established so we only consider the equilibrium
point where all variables and parameters are positive. This equilibrium point is
kβγ x = acβ − mαγµ
1 y = (acβλ − kαβγ − kβγη − mαλγµ) ackβ
kαγ z = acβ − mαγµ
1 v = (acβλ − kαβγ − kβγη − mαλγµ) . (2.10) akβγ
Again, we assume all appropriate inequalities to assure a positive equilibrium point.
Theorem 2. Let x1, y1,z1, v1 represent the values of equilibrium point (2.5) and x2,y2,
z2,v2 represent the values of equilibrium point (2.10). Then x1 = x2, y1 < y2, z1 = z2, and v1 < v2. Furthermore, if bk is sufficiently small, then y1 ≈ y2 and v1 ≈ v2. In other words, preventing activated CD8 T cells from killing healthy CD4 T cells will not change the equilibrium number of healthy naive and activated T cells and will increase the equilibrium number of infected cells and virus particles, although this increase may not be significant.
y1 Proof. Upon inspection, we see that x1 = x2, z1 = z2. Now we consider . Dividing, y2 we have y A 1 = 1 y2 (mαγµ − acβ)(kαβγ − acβλ + kβγη + mαλγµ)
which expands to 2 2 2 y1 A2 + kαβγ µηm − bk αβγ m = 2 y2 A2 + kαβγ µηm
2 2 2 2 2 2 2 2 2 where A2 = (acβ) λ − 2acαβλγµm − ackαβ γ − ackβ γη + α λγ µ m + kα βγ µm.
31 Factoring the last two terms in the numerator and rewriting the last term in the
denominator, we have
2 y1 A2 + kmαβγ (µη − bk) = 2 . y2 A2 + kmαβγ (µη)
We notice that all terms but the last are identical in the numerator and denominator.
y1 Since all parameters are positive, µη − bk < µη ⇒ < 1 ⇒ y1 < y2. Furthermore, y2
y1 v1 if bk is sufficiently small, µη − bk ≈ µη ⇒ ≈ 1 ⇒ y1 ≈ y2.Next we consider . y2 v2 Dividing, we have
v A 1 = 1 v2 (mαγµ − acβ)(kαβγ − acβλ + kβγη + mαλγµ)
v1 y1 We notice = so we can also conclude that v1 < v2 and if bk is sufficiently small, v2 y2
then v1 ≈ v2.
Evaluating (2.10) numerically, we have
x = 24. 635
y = 1. 884 1
z = 0.0 2217 4
v = 94. 203. (2.11)
Finding stability, we find the Jacobian
32 −α − η − av 0 0 −ax av −µmz − k µym ax α 0 −β 0 0 c 0 −γ
which numerically is
−2. 638 5 × 10−2 0 0 −6. 651 5 × 10−3 2. 543 5 × 10−2 −0.332 53 0.218 93 6. 651 5 × 10−3 0.000 3 0 −0.333 3 0 0 100 0 −2
and has eigenvalues labeled r1, r2, r3, r4
r1 = −.01150 8 + .08462 6i,
r2 = −.01150 8 − .08462 6i,
r3 = −0.333 50,
r4 = −2. 335 7. (2.12)
All the eigenvalues have negative real part, thus this is a strictly stable (stable and
attractive) equilibrium point [4].
Comparing the numerical equilibrium points (2.7) and (2.11) we notice that the
equilibrium points are nearly identical (since bk = 0.0002145) and both are strictly
stable. We conclude from Theorem 2, with the support of the numerical values of
33 the equilibrium points that applying a therapy of preventing CD8 cells from killing healthy, naive T cells will not improve the overall health of an HIV infected individual.
2.4.2 Existing Therapies: Protease Inhibitors and Reverse
Transcriptase Inhibitors
Now we consider the existing drug therapies of protease inhibitors and reverse transcriptase inhibitors. If completely effective, protease inhibitors prevent infected cells from producing infectious virus particles, only non-infectious particles we label w. We assume the non-infectious particles are produced and removed at the same rates the infectious particles are (i.e. c and γ apply to w as they did for v). Reverse transcriptase inhibitors help prevent the virus from infecting new cells. Together these two medications are part of a treatment regimen called HAART, highly active antiretroviral therapy. We assume both therapies are completely effective, so we set a = c = 0 where a is the rate at which virus infects healthy cells and c is the rate of virus production by an infected cell. Then the modified system becomes:
dx = λ − ηx − bx (mz) − αx dt
dy = −µy (mz) − ky dt
dz = αx − βz dt
34 dv = −γv dt
dw = cy − γw. (2.13) dt
The lifespan of a healthy naive T cell is significantly longer than that of an infected
T cell, activated T cell, or virus particle. Therefore, as Wodarz and Nowak [39] did,
we assume x is constant in the time frame under consideration. We assume all values
are at the equilibrium point (2.5) at the time treatment is started. Therefore the initial conditions for this system are
kβγ x(0) = acβ − αγµm
A y(0) = 1 a2c2kβ2 − ackαβγµm
kαγ z(0) = acβ − αγµm
A v(0) = 1 (2.14) a2ckβ2γ − akαβγ2µm
∗ ∗ where A1 is as previously defined in (2.5). We label x(0) = x , y(0) = y , z(0) = z∗, v(0) = v∗. We solve the system as follows. First we solve for v(t)
35 dv = −γv dt
v(t) = v∗e−γt
and z(t)
dz = αx∗ − βz. dt
Rearranging and multiplying by an integrating factor of eβt, we have
dz eβt + βzeβt = ax∗eβt dt d zeβt = ax∗eβt dt ax∗ zeβt = eβt + c β 1 ax∗ z(t) = + c e−βt β 1 ax∗ z(0) = + c = z∗ β 1 βz∗ − ax∗ c = 1 β ax∗ βz∗ − ax∗ z(t) = + e−βt. (2.15) β β
dy In order to solve for y in dt = −µy (mz) − ky, we substitute in for z and obtain
dy ax∗ βz∗ − ax∗ = y −µm + e−βt − k dt β β
36 dy −µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt = + dt y β β
−µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt ln y = t + + c β β2 2
−µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt y(t) = c exp t + 2 β β2
(−µmβz∗ + µmax∗) y(0) = c exp = y∗ 2 β2
(µmβz∗ − µmax∗) c = y∗ exp 2 β2
(µmβz∗ − µmax∗) −µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt y(t) = y∗ exp exp t + β2 β β2
(µmβz∗ − µmax∗) −µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt y(t) = y∗ exp exp t + . β2 β β2 (2.16)
dw In order to solve for w in dt = cy − γw , we substitute in for y. For simplicity, we let
37 (µmβz∗ − µmax∗) A = y∗ exp β2
−µmax∗ − kβ B = β
(−µmβz∗ + µmax∗) C = . β2
Therefore, we have
y(t) = AeBt+Ce−βt . (2.17)
Solving for w, we have
dw −βt = cAeBt+Ce − γw dt
dw −βt eγt + γweγt = cAeBt+Ce eγt dt
d −βt weγt = Ace(B+γ)t+Ce dt
Z w(t) = e−γt Ace(B+γ)t+Ce−βt dt. (2.18)
Therefore our solution set is
(µmβz∗ − µmax∗) −µmax∗ − kβ (−µmβz∗ + µmax∗) e−βt y(t) = y∗ exp exp t + β2 β β2
ax∗ βz∗ − ax∗ z(t) = + e−βt β β
38 v(t) = v∗e−γt
Z w(t) = e−γt Ace(B+γ)t+Ce−βt dt. (2.19)
Although we are only considering a time frame in which the healthy, naive T cells remain relatively constant (x = x∗) we can consider what happens as time tends to infinity.
lim y(t) = 0 t→∞
lim v(t) = 0 t→∞
ax∗ lim z(t) = t→∞ β
lim w(t) = 0 (2.20) t→∞
.
We consider the solution set (2.19) numerically, using the initial conditions
x(0) = x∗ = 24.635
y(0) = y∗ = 1.8833
z(0) = z∗ = .02 217 4
39 v(0) = v∗ = 94. 164. (2.21)
from (2.7) and the parameter values listed in Appendix A.
2.5 Solutions and Graphs
We obtain the solutions with corresponding graphs (when possible) as shown below. All cells are per cubic millimeter of blood and the time frame is given in days.
Figure 2-1: Infected cell population with HAART
y(t) = 1.8848 exp −0.332 32t − 7.731 3 × 10−4e−0.333 3t
40 Figure 2-2: Activated T cell population with HAART
z(t) = .002217 7e−0.333 3t + .01 9956
Figure 2-3: Virus population with HAART v(t)=94. 164e−2t
41 Z w(t) = 3, 774, 000e−2t exp 1. 667 7t − .0007731 3e−0.333 3t dt
2.6 Discussion
We notice that the viral load (v) and number of infected cells (y) are decreasing rapidly following treatment with HAART, and the number of activated T cells (z) are decreasing less dramatically with time. This conclusion is consistent with researchers’
findings that following treatment by HAART, viral load can be decreased to below detectable limits (less than 50 copies per microliter) and that the number of activated
T cells decreases [14] [2]. The limits in (2.20) indicate that the number of activated T cells (z) will remain proportional to the number of healthy naive T cells (x) despite the amount of virus (v) and infected cells (y) present, implying that HIV is causing a chronic state of T cell activation. Although this model accurately reflects what is known about viral load and T cell changes following HAART, it does not explain why
HAART is unable to completely eradicate the virus from the body. Many researchers believe this is due to a latent reservoir that HIV is able establish during early infection and that full eradication of the virus will only be possible if these reservoirs can be eradicated [25], [32], [8]. Modeling the effects of latent infection is a possibility for future research. In the next chapter, we analyze further models which include naive, activated, and memory cell dynamics during infection.
Conclusion 3. HAART dramatically decreases the number of infected cells, activated cells, and virus particles.
42 Chapter 3
Dynamics of Naive, Effector, and
Memory T cells without Virus
Effect
In this section, we analyze the dynamics of an immune response to HIV focusing on naive, effector, and memory cells but neglecting the virus. Immune responses to any antigen traditionally consist of multiple stages. First, the naive T cells locate and recognize the antigen of the microbe. Following the appropriate signals, the naive cells become activated and multiply quickly. This phase is referred to as clonal expansion. Many of these daughter cells differentiate into activated effector cells which launch an attack against all the antigens that are recognized infected cells infected with the recognized antigen. Others differentiate into memory cells which remain in the body, ready to recognize the antigen again quickly if it returns.
Once the infection is cleared from the body, most effector cells which participated in the attack against the antigen die by a process called apoptosis in order to prevent the immune system from attacking excessively [1], [37]. For reasons that are still not completely understood, the immune system is unable to completely eliminate
43 HIV from the body, therefore the immune response is always activated [40]. As a result of this chronic activation, naive cells may be constantly differentiated into effector cells to help fight HIV, leaving fewer naive cells to respond to other infections.
Furthermore, the increased pool of effector cells may lead to a decreased pool of memory cells which are the quickest to respond to antigen presence. In this section we explore a model to help understand the effects of the chronic activation of the immune system. A summary of the parameters and variables used in this section is given in the chart below.
3.1 Definition of Variables and Parameters
Parameter Definition
x Naive cells
v Virus particles
z Effector cells (activated cells)
r Memory cells
λ Production rate of naive cells
η Death rate of naive cells
β Death rate of effector cells
βm Death rate of memory cells
en Effector cells produced in the activation of a naive cell
em Effector cells produced in the activation of a memory cell
αn Rate of activation of naive cells by HIV
αm Rate of activation of memory cells by HIV q Rate of conversion of effector cell to memory cell
44 3.2 Model without virus and both naive and mem-
ory cells activating
We consider a system that includes naive, effector, and memory cells but neglects
effects of the virus infecting T cells in order to gain a better understanding of the role
of the immune response in the depletion of T cells. We continue to assume that naive
T cells (x) are being produced at a constant rate λ and are removed from the naive T
cell population either by natural death (at rate ηx) or by becoming activated (at rate
αnx). Similarly, we assume that memory cells (r) are removed from the population by either natural death (at rate βmr) or by activation (at rate αmr). Both naive cells and memory cells become activated by an antigen and multiply as represented
by the terms en(αnx) and em(αmr). We represent effector cells dying by βz and differentiated into memory cells by qz. Our model is
dx = λ − ηx − α x dt n
dz = e (α x) + e (α r) − βz − qz dt n n m m
dr = qz − α r − β r. (3.1) dt m m
Because this is a non-homogeneous linear system, we can solve by analytic methods.
The matrix associated with the complementary system is −η − αn 0 0 e α −β − q e α , n n m m 0 q −αm − βm which has the following eigenvectors (ξ1, ξ2, ξ3) with associated eigenvalues (k1, k2, k2):
45 − B1 qαnen 1 ξ1 = − (η − α + α − β ) , q m n m 1
k1 = −η − αn,
0 1 ξ2 = − (q + β − α − β + B ) , 2q m m 2 1
−1 k = (q + β + α + β + B ) , 2 2 m m 2
0 1 ξ3 = (α − β − q + β + B ) , 2q m m 2 1
1 k = B , 3 2 2 where
2 2 B1 = qη − qαm + qαn − qβm − η − αn + βη − βαm + βαn − ββm
+ηαm − 2ηαn + ηβm + αmαn + αnβm + qαmem
46 and
2 2 2 B2 = [2qβ − 2qαm − 2qβm + β + αm + βm.
2 1/2 −2βαm − 2ββm + 2αmβm + q + 4qαmem] .
We find a constant particular solution of the non-homogeneous equation of the form A B where C
λ A = , η + αn
λα e α + β B = n n m m , η + αn qαm + qβm + βαm + ββm − qαmem
qλα e C = n n . (η + αn)(qαm + qβm + βαm + ββm − qαmem)
Then with constants c1,c2,c3 we can write the solution to our system as
c B x(t) = − 1 1 e(−η−αn)t + A, qαnen
47 c z(t) = − 1 (η − α + α − β ) e(−η−αn)t q m n m c2 1 (−q−β−α −β −B )t − (q + β − α − β + B ) e 2 m m 2 2q m m 2
c3 1 tB + (α − β − q + β + B ) e 2 2 + B, 2q m m 2
1 1 (−η−αn)t t(−q−β−αm−βm−B2) tB2 r(t) = c1e + c2e 2 + c3e 2 + C.
3.2.1 Solutions and Graphs
A healthy individual has 500 to 1500 T cells per mm3 of blood so we assume that
1 mm3 of blood has 1000 T cells. Naive T cells make up approximately 70% of T cells, memory cells 1%, leaving 29% as effector cells [3]. Therefore we use the initial values x(0) = 700, z(0) = 290, r(0) = 10. Using these initial conditions and the numerical values listed in the appendix, we find the following solutions with their corresponding graphs.
48 Figure 3-1: Naive T cell population during chronic activation
x (t) = 698. 85e−1. 201 4t + 1. 141 8
Figure 3-2: Effector T cell population during chronic activation
z(t) = 1462.7e2.5514t − 64.142e−4.9186t − 1094.1e−1.2014t − 14.522
49 Figure 3-3: Memory T cell population during chronic activation
r(t) = 27. 963e−4. 918 6t − 156. 31e−1. 201 4t + 139. 8e2. 551 4t − 1. 451 8
3.2.2 Discussion
We notice that the number of naive T cells falls quickly while the number of effector and memory T cells continue to increase. While during the typical course of
HIV infection, the number of naive T cells does not decrease to zero as quickly as the graph suggests. A critically low number of naive T cells will significantly compromise an individual’s ability to launch an immune response against any infection. In the following section we explore what happens if it was possible to inhibit naive T cells from being constantly activated by HIV.
Conclusion 4. During untreated HIV infection, naive T cell populations decline while effector and memory cell populations rise.
50 3.3 Model without virus, only memory cells acti-
vating
During a standard infection, T cells are activated, fight the infection, then either
die or remain as memory cells. The last part of this sequence happens in the days
following the clearance of the virus from the body. HIV is not completely cleared
from the body so the immune system stays chronically activated. The following
model suggests what would happen if it was possible to keep naive cells from contin-
ually activating, and leaving memory cells to activate and fight the infection. Our
assumption for model (3.1) was 1000 T cells per cubic mm of blood. We assume
that the body has been infected for some time and the total number of T cells has
decreased to 400. We also assume that the percentage of each type of T cells has
changed. We assume that the percent of naive T cells has dropped from 70% to
40%, the percent of effector cells has increased from 29% to 50% and that the percent
of memory cells has increased from 1% to 10%. The parameter αn represented the activation rate of naive cells, so we assume this value is now zero. Our modified
system becomes
dx = λ − ηx dt
dz = e (α r) − βz − qz dt m m
dr = qz − α r − β r. (3.2) dt m m
To solve this system, we first solve for x.
51 dx = λ − ηx dt λ x(t) = + ce−ηt η λ x = + c 0 η λ c = x − 0 η λ λ x(t) = + x − e−ηt. η 0 η
We then consider the system
dz = e (α r) − βz − qz dt m m
dr = qz − α r − β r dt m m
which has corresponding matrix
−β − q e α m m . q −αm − βm
The eigenvectors and eigenvalues of this matrix are
− 1 (q + β − α − β + B ) 2q m m 3 ξ1 = 1
1 1 1 1 1 k = − q − β − α − β − B 1 2 2 2 m 2 m 2 3
and
52 1 (α − β − q + β + B ) 2q m m 3 ξ2 = 1
1 1 1 1 1 k = B − β − α − β − q 2 2 3 2 2 m 2 m 2 where
2 2 2 B3 = [2qβ − 2qαm − 2qβm + β + αm + βm
2 1/2 −2βαm − 2ββm + 2αmβm + q + 4qαmem] .
Therefore, with constants c4 and c5, we have
c 1 z(t) = − 4 (q + β − α − β + B ) exp − (q + β + α + β + B ) t 2q m m 3 2 m m 3 c 1 + 5 (α − β − q + β + B ) exp − (q + β + α + β − B ) t 2q m m 3 2 m m 3
1 r(t) = c exp − (q + β + α + β + B ) t 4 2 m m 3 1 +c exp − (q + β + α + β − B ) t 5 2 m m 3
Using the initial conditions to solve for c1 and c2 we have
53 c c z = − 4 (q + β − α − β + B ) + 5 (α − β − q + β + B ) 0 2q m m 3 2q m m 3
r0 = c4 + c5
which yields
1 c4 = − (qr0 + 2qz0 + βr0 − αmr0 − βmr0 − B3r0) 2B3 and 1 c5 = (qr0 + 2qz0 + βr0 − αmr0 − βmr0 + B3r0) . 2B3
So our solution set is
λ λ x(t) = + x − e−ηt η 0 η
c 1 1 1 1 1 z(t) = − 4 (q + β − α − β + B ) exp − q − β − α − β − B t 2q m m 3 2 2 2 m 2 m 2 3 c 1 1 1 1 1 + 5 (α − β − q + β + B ) exp B − β − α − β − q t 2q m m 3 2 3 2 2 m 2 m 2
1 1 1 1 1 r(t) = c exp − q − β − α − β − B t (3.3) 4 2 2 2 m 2 m 2 3 1 1 1 1 1 +c exp B − β − α − β − q t (3.4) 5 2 3 2 2 m 2 m 2
with c4, c5, andB3 as defined above.
54 3.3.1 Solutions and Graphs
Using the parameter values listed in the chart in Appendix B and the initial
conditions x0 = 400(.40) = 160, z0 = 400(.50) = 200, r0 = 400(.10) = 40 we obtain the following solutions with corresponding graphs.
Figure 3-4: Recovering naive T cells with no naive cells activating
x(t) = 1000.0 − 840.0e−.0013717t
55 Figure 3-5: Increasing effector T cells with no naive cells activating
z(t) = 239.29e−.0013717t − 39.292e−4.9186t
Figure 3-6: Memory T cells with no naive cells activating
r(t) = 17. 13e−4.918 6t + 22. 87e0.184 2t
56 3.3.2 Discussion
We notice that because they are not being constantly activated, the number of naive cells is able to recover back to a normal value of 1000. The number of effector cells continues to increase because the body is continuously trying to fight the HIV infection. Memory cells initially decrease because they are reactivated into effector cells, but we are assuming effector cells can multiply quickly and return to the resting memory cell population. We conclude that chronic activation of T cells is indeed playing a crucial role in the depletion of T cells during HIV infection. In this section, we neglected the effects of the virus infecting which we will explore in the next chapter.
Conclusion 5. Chronic activation of the immune system is a significant factor in the decrease in the overall T cell population, specifically the naive T cell population.
57 Chapter 4
Dynamics of Naive, Effector, and
Memory T cells with Virus Effect
In the previous chapters, we considered models taking into consideration the dy- namics of healthy naive T cells, infected T cells, healthy effector T cells, and virus particles both before and after treatment with HAART. In the models in Chapter 2, we assumed only naive T cells were becoming infected with HIV and did not consider the role of memory cells in immune system dynamics. In Chapter 3, we considered naive, effector, and memory cells alone without the virus. The following model is a combination of the models put forth in Chapters 2 and 3 with some modifications.
Different from Chapter 2, we assume naive, effector, and memory cells all susceptible to infection by HIV, albeit possibly at different rates. We also have a modified naive
T cell production term. There is no clear consensus from immunologists at what rate
T cells are produced. It has been theorized that it is constant, periodic, diminished with time and others. In this section, we use a T cell production rate that is limited by a maximum value of T cells. In addition, we have changed activation rates of
T cells to be proportional to the amount of virus in the system as opposed to the number of specific type of T cells. A summary of the parameters and variables used
58 in this section is given in the chart below.
4.1 Definition of Parameters
Parameter Definition
x Naive cells
v Virus particles
z Effector cells (activated cells)
r Memory cells
λ Production rate of naive cells
η Death rate of naive cells
β Death rate of effector cells
βm Death rate of memory cells
an Rate virus infects naive cells
ae Rate virus infects effector cells
am Rate virus infects memory cells
en Effector cells produced in the activation of a naive cell
em Effector cells produced in the activation of a memory cell
αn Rate of activation of naive cells by HIV
αm Rate of activation of memory cells by HIV q Rate of conversion of effector cell to memory cell
4.2 Before Treatment
For an infected individual who has not started treatment with HAART we having the following system.
59 dx x = λ 1 − − ηx − a xv − α v dt M n n
dr = qz − α v − β r − a rv dt m m r
dz = e (α v) + e (α v) − βz − a zv − qz dt n n m m e
dy = a xv + a zv + a rv − ky dt n e r
dv = cy − γv. dt
In this model, healthy naive T cells production is limited by a maximum value
M and the number of healthy naive T cells (x) is diminished by natural death (ηx), infection by HIV (anxv), and activation (αnv). Effector cells (z) are a result of naive (x) and memory cells (r) becoming activated and then multiplying (en (αnv)
and em (αmv)). Natural death (βz), infection (aezv), and differentiation into memory cells (qz) all decrease the amount of effector cells fighting the infection. Memory cells
(r) differentiate from effector cells (qz) and either die naturally (βmr), are infected
(arrv), or become activated (αm). The population of infected cells (y) result from
infection of naive, effector, and memory cells (anxv, aezv, arrv respectively) and die
60 by apoptosis at a rate ky. Virus particles are produced by infected cells (cy) and are
removed from the body at a rate γv.
4.3 With Treatment
We apply HAART and consider the following modified system. Recall that
HAART includes protease inhibitors and reverse transcriptase inhibitors. Protease
inhibitors help prevent infected cells from producing infectious virus particles, only
non-infectious particles we label w. We assume the non-infectious particles are pro-
duced and removed at the same rates the infectious particles are (i.e. c and γ apply
to w as they did for v). Reverse transcriptase inhibitors help prevent the virus from
infecting new cells. We again assume that that both therapies are working perfectly.
Therefore, because of the reverse transcriptase inhibitors, we set an, ae, am equal to
zero. We use initial conditions x(0) = x0, r(0) = r0, z (0) = z0, y(0) = y0, v(0) = v0, w(0) = 0. The modified system is
dx x = λ 1 − − ηx − α v dt M n
dr = qz − α v − β r dt m m
dz = e (α v) + e (α v) − βz − qz dt n n m m
61 dy = −ky dt
dv = −γv dt
dw = cy − γw. (4.1) dt
We first solve the equation
dv = −γv dt
−γt v(t) = v0e
We then substitute for v in the equation
dx −λ = − η x − α v + λ dt M n
dx −λ = − η x − α v e−γt + λ dt M n 0
dx λ + + η x = −α v e−γt + λ dt M n 0
62 λ Let A = M + η.
dx eAt + AeAtx = −α v e−γteAt + λeAt dt n 0
d xeAt = −α v e−γteAt + λeAt dt n 0
−α v λ xeAt = n 0 e(A−γ)t + eAt + C A − γ A
−α v λ x = n 0 e−γt + + Ce−At A − γ A
−α v λ x(0) = n 0 + + C = x A − γ A 0
λ α v C = x − + n 0 0 A A − γ
−α v λ λ α v x(t) = n 0 e−γt + + x − + n 0 e−At. A − γ A 0 A A − γ
63 We then solve for the remaining variables by considering the matrix corresponding to the system
dr = qz − α v − β r dt m m
dz = e (α v) + e (α v) − βz − qz dt n n m m
dy = −ky dt
dv = −γv dt
dw = cy − γw. dt
The matrix is
−β q 0 −α 0 m m 0 −β − q 0 (e α + e α ) 0 n n m m 0 0 −k 0 0 . 0 0 0 −γ 0 0 0 c 0 −γ
64 with eigenvectors and corresponding eigenvalues as below,
− q q+β−βm 1 ξ1 = 0 , 0 0
k1 = −q − β,
0 0 ξ2 = 0 , 0 1
k2 = −γ,
65 γαm−βαm−qαm+qαmem+qαnen − 2 qγ−qβm−γ +βγ−ββm+γβm αmem+αnen q+β−γ ξ3 = 0 , 1 0
k3 = −γ,
1 0 ξ4 = 0 , 0 0
k4 = −βm,
66 0 0 1 ξ5 = , − c (k − γ) 0 1
k5 = −k.
Our solutions then are
−α v λ λ α v x(t) = n 0 e−γt + + x − + n 0 e−At A − γ A 0 A A − γ
C1q −(q+β)t γαm − βαm − qαm + qαmem + qαnen −γt −βmt r(t) = − e +C3 − 2 e +C4e q + β − βm qγ − qβm − γ + βγ − ββm + γβm
α e + α e z(t) = C e−(q+β)t + C m m n n e−γt 1 3 q + β − γ
1 y(t) = C − (k − γ) e−kt 5 c
67 −γt v(t) = C3e
−γt −kt w (t) = C2e + C5e
Using the initial conditions, we have
C1q γαm − βαm − qαm + qαmem + qαnen r0 = − + C3 − 2 + C4 q + β − βm qγ − qβm − γ + βγ − ββm + γβm
α e + α e z = C + C m m n n 0 1 3 q + β − γ
1 y = C − (k − γ) 0 5 c
v0 = C3
w0 = C2 + C5.
Solving for the constants, we have
68 α e + α e C = z − v m m n n 1 0 0 q + β − γ
cy C = w + 0 2 0 k − γ
C3 = v0
2 βmr0 + qγr0 + qγz0 − qβmr0 − qαmv0 − qβmz0 C4 = + (γ − βm)(q + β − βm)
βγr0 − ββmr0 − γβmr0 − βαmv0 + αmβmv0 + qαmv0em + qαnv0en
(γ − βm)(q + β − βm)
−cy C = 0 5 k − γ
We assume as in the previous section that the body has been infected for some time and the total number of T cells has decreased to 400 and that the percent of naive T cells has dropped is 40%, the percent of effector cells is 50% and that the percent of memory cells is10%.
4.4 Solutions and Graphs
Using sample initial conditions and parameter values listed in Appendix C, we have the solutions
69 Figure 4-1: Naive T cells after HAART
x(t) = 0.687 38e−2.0t − 440. 68 exp −2. 286 2 × 10−3t + 599. 99
Figure 4-2: Memory T cells after HAART
r(t) = 1. 728 3e−2.0t + 150. 38e−0.001 1t − 112. 1e−0.733 3t (4.3)
70 Figure 4-3: Effector T cells after HAART
z(t) = 205. 2e−0.733 3t − 5. 203 6e−2.0t
Figure 4-4: Infected T cells after HAART y(t)=135. 23e−0.333 3t
71 Figure 4-5: Infectious virus particles after HAART v(t)=4. 12e−2.0t
Figure 4-6: Non-infectious particles with HAART
w(t) = 4056. 8e−0.333 3t − 4056. 8e−2.0t
72 4.5 Discussion
We notice with treatment that the naive and memory cells are able to recover back to a sustainable level. Effector cells decrease, presumably because they help the body fight the infection and once the number of virus particles is diminished, effector cells will die by apoptosis. Treatment also helps decrease both the infectious virus particles and over time the non-infectious virus particles.
Conclusion 6. HAART helps the naive and memory T cell populations return to normal levels.
73 Chapter 5
Conclusion and Future Research
5.1 Conclusion
AIDS is a devastating disease that is still not fully understood by researchers.
Mathematical modeling can help us better understand the dynamics of immune re- sponses to HIV. Nowak and Bangham’s model shows that cytotoxic T lymphocytes help decrease the equilibrium number of infected T cells but immune system responses are eventually defeated by the virus. Throughout the paper, we have shown that treatment by HAART significantly helps the immune system decrease the amount of virus in they body which is supported by the immunology literature.
Constant activation of the immune system contributes to the diminishment of the number of CD4+T cells (helper T cells) but it not clear whether it is healthy cells being killed by activated CD8+T cells (killer T cells) or exhaustion from constant turnover of T cells. We showed that if killer T cells were prevented from killing healthy helper T cells, it would not change the equilibrium number of naive and activated T cells and would actually increase the equilibrium number of infected cells and virus particles. Changes in helper T cell populations was the focus of the models presented here. We were able to show that the chronic activation of naive CD4+T
74 cells is playing a crucial role in the depletion of CD4+T cells.
Conclusion 7. HAART helps significantly diminish the virus and infected T cell
populations
Conclusion 8. Chronic activation of the immune system plays a significant role in
the depletion of the T cell population during untreated HIV infection.
Conclusion 9. Preventing killer T cells from killing helper T cells will not help the
overall T cell/virus dynamics.
5.2 Future Research
While HAART appears to be able to eradicate the virus from the body, when
it is stopped, the virus is able re-establish an infection due to a latent reservoir of
resting T cells. The latent reservoir is small (˜1 per 106 cells) but is very stable and
is theorized to be able to persist for up to 60 years despite treatment by HAART [6].
persist for up to 60 years despite treatment by HAART [6]. T cells. A variety of
T cell stimuli can later activate the latent HIV [38]. There are two main theories
why the latent reservoir persists for so long. The first is the extremely long half
life of the latently infected cells (44 months). The second theory is that low-level
viral replication continues in patients on HAART [25]. The two major obstacles the
latent reservoir provides are that HAART cannot eradicate the virus from the body
and drug-resistant strains of HIV remain in the reservoir, limiting treatment options
[32]. Understanding how significant a role latent infection plays in the persistence of
the virus is one possible avenue for further research.
Another avenue for future research is exploring how the CD8+T cell population changes and how the dynamics between CD8+T cells and CD4+T cells change during
HIV infection. While HIV primarily infects CD4+T cells, CD8+T cells have an
75 increased turnover and death rate during infection [15] and the virus may inhibit
naive CD8+T from fully differentiating into effector CD8+T cells, therefore limiting
their ability to kill the virus and infected cells [11],[21]. The inability of CD8+T cells to effectively kill infectious particles may be a contributor to HIV’s ability to evade eradication by immune system. Virus mutation may also inhibit CD8+T cells ability to recognize the virus and launch a response against it [21] Exploring the role of CD8+T cells and the virus’s ability to inhibit their full functioning is another possibility for future research.
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82 Appendix A
Sample Numerical Values from the
Literature for Chapter 2
Per cubic millimeter of blood (about 1000 CD4 cells) Values are upon infection. Parameter Value (per day) Source/Comments
λ 0.65 cells λ = 1000η
η 0.00065 cells [24]
b 0.00065 cells Assume b = η
a 0.00027 cells/virion [28]
k µ .165 Assume µ= 2 k 0.33 cells [28]
c 100 virions [28]
γ 2 [28]
α .0003 [24]
β 0.3333 [24] CD4 m .7042253521 [19] = 1.42 CD8
83 Appendix B
Sample Numerical Values from
Literature for Chapter 3
Parameter Value (per day) Source/Comments
λ 1000/729 = 1.3717 λ = 1000η
η 1/729 = 1.3717 × 10−3 [22], lifespan 729 days
αn 24/20 = 1.2 [18]Iezzi,20hr exposure
αm 24/6 = 4.0 Assume activates 3x faster than naive
1 β 3 = 0.33333 [24], Lifespan 3 days
βm 1/913 = .0011 [34], half life 2-3 years
en 2.4 [20], ˜10hrs per cell division
em 2.4 [22]Stock, naive and memory same q 1/2.5 = 0.4 [16]Hu, 2-3 days
84 Initial Value Value Source/Comment
x0 160 400(.40) = 160
r0 40 400(.50) = 200
z0 200 400(.10) = 40
y0 135.23 [31]
v0 log 4.12 [29]
w0 0
85 86 Appendix C
Sample Numerical Values from
Literature for Chapter 4
Parameter Value Source/Comments
λ 1000/729 = 1.3717 [22]
η 1/729 = 1.3717 × 10−3 λ = 1000η
αn 0.3333 [36], 3 days before proliferation
αm 0.3333 [36], 3 days before proliferation
1 β 3 = 0.33333 [24], lifespan 3 days
βm 1/913 = .0011 [34], half life 2-3 years
en 2.4 [20], ˜10 hrs per cell division
em 2.4 Stock, naive and memory cells same q 1/2.5 = 0.4 [16], 2-3 days for conversion
M 1500 [1]
an .00027 cells/virion [28], chosen so that virus persists
ar 2(.00027) = 0.000 54 Assumed to be infected twice as fast as naive cells
ae 2(.00027) = 0.000 54 Assumed to be infected twice as fast as naive cells k 0.33 [28]
c 50 [30]
γ 287 [28], lifespan 1/2 day