Mathematical models of the activated immune system during HIV infection

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2011 Mathematical models of the activated immune system during HIV infection Megan Powell The University of Toledo Follow this and additional works at: Recommended Citation Powell, Megan, "Mathematical models of the activated immune system during HIV infection" (2011). Theses and Dissertations This Dissertation is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 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

3 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.

4 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

5 This thesis is dedicated to the memory of Umaer Basha.

6 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

7 Table of Contents Abstract iii Acknowledgments v Table of Contents vi List of Figures ix List of Abbreviations x 1 Introduction The Immune System HIV Background Basic Model: Wodarz and Nowak Basic Model with Immune Response: Nowak and Bangham CD4 and CD8 T cell dynamics model: Vayo and Huang Models with Therapies Reverse Transcriptase Inhibitors Protease Inhibitors Models of Mutating Virus Dynamics of Healthy, Infected, Activated T Cells with Virus Effect Definition of parameters vi

8 2.2 Numerical Values Model and Equilibrium Points without Treatment Equilibrium Points with Treatment Theoretical Therapy: Prevent activated CD8 T cells from killing healthy CD4 T cells Existing Therapies: Protease Inhibitors and Reverse Transcriptase Inhibitors Solutions and Graphs Discussion Dynamics of Naive, Effector, and Memory T cells without Virus Effect Definition of Variables and Parameters Model without virus and both naive and memory cells activating Solutions and Graphs Discussion Model without virus, only memory cells activating Solutions and Graphs Discussion Dynamics of Naive, Effector, and Memory T cells with Virus Effect Definition of Parameters Before Treatment With Treatment Solutions and Graphs Discussion Conclusion and Future Research Conclusion vii

9 5.2 Future Research 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

10 List of Figures 2-1 Infected cell population with HAART Activated T cell population with HAART Virus population with HAART Naive T cell population during chronic activation Effector T cell population during chronic activation Memory T cell population during chronic activation Recovering naive T cells with no naive cells activating Increasing effector T cells with no naive cells activating Memory T cells with no naive cells activating Naive T cells after HAART Memory T cells after HAART Effector T cells after HAART Infected T cells after HAART Infectious virus particles after HAART Non-infectious particles after HAART ix

11 List of Abbreviations AIDS Acquired Immunodeficiency Syndrome CTL Cytotoxic T Lymphocyte CD Cluster of Differentiation 4 CD Cluster of Differentiation 8 DNA Deoxyribonucleic acid HAART Highly Active Anti-retroviral Therapy HIV Human Immunodeficiency Virus RNA Ribonucleic acid x

12 Chapter 1 Introduction 1.1 The Immune System The purpose of the immune system is to prevent infections and remove any existing infections. During an immune response, the body defends itself by either destroying or rendering harmless any matter perceived as foreign. The body s defense 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

13 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 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

14 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 analyzed 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

15 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 dt = λ dx βxv dy dt = βxv ay dv dt = ky uv (1.1) 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

16 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 dt = λ dx. (1.2) Assuming the body has reached equilibrium before viral infection, the initial conditions of system (1.1) are x 0 = λ d, y 0 = 0, v 0 = 0. From the parameters, we see that the average lifetime of the uninfected cells is x = 1, and similarly, the average dx d lifetime of the infected cells is 1 a and the average lifetime of virus particles is 1 u. We can also determine that the total virus produced from one infected cell is ky ay = k a. In order for an infection to take hold, the basic reproductive number R 0, 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), R 0 = (βx) ( ) ( ) 1 1 (k) = βxk a u au where βx is the rate at which the virus infects healthy cells, 1 a (1.3) is the average lifespan of an infected cell, k is the rate one infected cell produces virus particles, and 1 u is the lifespan of a virus particle. Again, assuming the body is healthy and in equilibrium when the virus is first introduced to the body, x = λ d initially, so R 0 = βλk adu. The 5

17 equilibrium points of this system are given by x = au βk = λ dr 0 y = λ a du βk = (R 0 1)du βk v = λk au d β = (R 0 1)d. (1.4) β At equilibrium, the reproductive ratio R becomes βx k au = 1. Therefore, at equilibrium, each infected cell is on average giving rise to one additional infected cell. Assuming R 0 is much bigger than one, x should be much smaller than x 0, thus this model does not explain individuals infected with HIV who a relatively constant number 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 (CT L) 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

18 dx dt = λ dx βxv dy dt = βxv ay pyz dv dt = ky uv dz dt = cyz bz (1.5) 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

19 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 CT L proliferation in response to an antigen b is the rate of CT L decay in absence of antigen stimulation Notice that in order for the rate of the CT L response to increase (i.e. dz dt > 0), cy must be greater than b. If cy < b (where y is the equilibrium point defined in (1.4 ), then a CT L response may be slightly activated, but equilibrium point (1.4) can be reached without any CT L response. Assuming that cy > b, the equilibrium point of system (1.5) is x = λcu cdu + βbk ŷ = b c v = bk cu ẑ = ( ) ( ) 1 λβck p cdu + βbk a. (1.6) Theorem 1. The equilibrium number of infected cells will be reduced with a CT L response. 8

20 Proof. Without a CT L response, the equilibrium number of infected cells is y. Assuming there are y infected cells, an increasing CT L response will occur if dz dt = cy z bz > 0. In order for this to occur, we must have z > 0 and cy > b y > b c. But the equilibrium number of infected cells with a CT L response is ŷ = b c. Thus y > ŷ i.e., the equilibrium number of infected cells without a CT L response is higher than the equilibrium number of infected cells with a CT L 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 dt = [a 1 (x 0 C 4 ) + a 2 (y 0 C 8 )]C 4 δ 1 C 4 (1.7) dc 8 dt = a 3 (x 0 + y 0 C 4 C 8 )C 8 (1.8) where C 4 (t): The amount of CD4 + T cells at time t. C 8 (t): The amount of CD8 + T cells at time t. x 0 : The standard amount of CD4 + T cells in a healthy human body. y 0 : The standard amount of CD8 + T cells in a healthy human body. a 1, a 2, a 3 : Positive constants, a 1 < a 2. δ 1 : Depletion rate of CD4 + T cells. x 0 C 4 (t): The difference in the standard amount of CD4 + T cells and amount of CD4 + T cells at time t. 9

21 y 0 C 8 (t): The difference in the standard amount of CD8 + T cells and amount of CD8 + T cells at time t. (x 0 + y 0 ) (C 4 + C 8 ): 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 therapeutic 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 equilibrium 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 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 recall that β is the rate at which the virus infects healthy cells. Then with dx dt = 0, 10

22 system (1.1) becomes dy dt dv dt = ay = ky uv (1.9) 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 e ut, we get ve ut + uve ut = ky e at e ut d dt (veut ) = ky e at e ut ve ut = ky e (u a)t u a + c. Substituting in the initial condition v(0) = v and using the fact that y = v ( u k ), after some rearrangement, we get v(t) = v u a (ue at ae ut ). 11

23 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 dt dv dt dw dt = βxv ay = uv = ky uw (1.10) 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 dt = βx (v e ut ) ay Letting b = βx v and multiplying by an integrating factor of e at, we get 12

24 dy dt eat + aye at = be ut e at d dt (yeat ) = be (a u)t ye at = be(a u)t a u + c. Substituting in the initial condition y(0) = y, we get y(t) = b a u (e ut e at ) + y e at. Then dw dt = bk a u (e ut e at ) + ky e at uw. Multiplying by an integrating factor of e ut, we have dw dt eut + uwe ut = bkeut a u (e ut e at ) + ky e ut e at d dt (weut ) = kb a u + (ky bk a u )e(u a)t. So we ut = kbt ( a u + ky bk ) ( ) e (u a)t + c. a u (u a) Using the initial condition w(0) = 0, we have ( w(t) = kbte ut a u + ky bk ) ( ) e (u a)t e ut + ky e ut a u (u a) a u bke ut (a u) 2. Using y = v ( u k ) and substituting βx v back in for b, the equation for non-infectious virus simplifies to 13

25 w(t) = We know at equilibrium, dy dt rewrite the above equation as ( ) ( ) kβx v t uv e ut + a u a u + βx v k (e at e ut ). (a u) 2 = 0, which implies βx v = ay = auv, thus we can k ( ) ( ) autv auv w(t) = e ut + a u (a u) uv (e at e ut ). 2 a u Thus the total amount of virus is given by v(t) + w(t) = v [e ut + ( ) ( ) aut u e ut 2 (e + at e ut)]. a u (a u) 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 dt = λ dx xn i=1β i v i dy i dt = β ixv i ay i py i z i dv i dt = k iy i uv i 14

26 dz i dt = cy iz i bz i (1.11) 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 CT L proliferation in response to an antigen b is the rate of CT L decay in absence of antigen stimulation v i is the amount of the specific strain (mutation) i of the virus y i is the amount of virus (strain i) infected cell z i is the CT L response specific to strain i 15

27 β i is the rate at which virus strain i infects healthy cells k i 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 dt = c iv i b i x i x i n j=1 u j v j dv i dt = v i(r p i x i s i z) dz dt =n i=1 k j v j bz z n j=1u j v j (1.12) where here v i represents each specific virus mutant strain where i = 1...n, where v = v i represents the total amount of virus x i represents the immune response specific to virus strain v i z represents the immune responses directed at all virus strains r is the average rate of replication of all virus strains p i is the efficiency of each strain-specific immune response s i is the efficiency of the global immune system response against virus strain i 16

28 c i is the rate at which each strain-specific immune response is evoked b i represents the decay of strain-specific immune responses in the absence of stimulation u j represents the ability for the virus to impair strain-specific immune system response j. k j 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 (x i ) 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 p i with p s i with s c i with k b i with b u j with u k j with k. The simplified model then becomes 17

29 dx i dt dv i dt dz dt = kv i bx i uvx i = v i (r px i sz) (1.13) = k v bz uvz. (1.14) From the above, we can find the rate at which the total virus population changes dv dt = v i (r px i sz) = r v i p v i x i sz v i = rv p v i x i szv (1.15) but we assume that x i 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 kv i bx i uvx i = 0 x i = kv i b + uv and k v bz uvz = 0 z = k v b + uv. Therefore we have dv dt = rv pk v 2 b + uv i sk v 2 b + uv. 18

30 Multiplying the middle term by v 2 /v 2, we have dv dt pkv2 ( ) vi 2 = rv sk v 2 b + uv v 2 b + uv. (1.16) The quantity ( v 2 i v 2 ) 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 we have, the lower D is where D is always between 0 and 1 with D = 1 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 dt = rv pkv2 b + uv D sk v 2 b + uv. (1.17) So now we can compute the steady-state of the total virus population, rv pkv2 b + uv D sk v 2 b + uv v = = 0 rb pkd + sk ru. (1.18) We notice as D decreases (diversity increases), the equilibrium total virus population increases. We now consider three cases of virus diversity. 1. ru > sk + 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

31 (sk ) and strain-specific responses (pk). Thus the virus replicates to high levels immediately without an asymptomatic stage. 2. sk > ru In this case, the global responses (sk ) alone can control the virus (ru) and strain-specific responses are not needed. 3. sk + pk > ru > sk In this case, both global (sk ) and strain-specific (pk) immune system responses can control the virus, whereas global responses (sk ) cannot. This control of virus replication changes when D is such that ru = sk + pkd, thus what is called the diversity threshold occurs when D = ru sk. (1.19) pk After the diversity threshold is reached, the immune system can no longer control the virus replication. 20

32 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

33 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

34 Parameter x y z v λ η β a k γ Definition Number of healthy, naive T cells Number of infected T cells Number of healthy, activated T cells Number of virus particles Production rate coefficient of healthy cells Death rate coefficient of healthy naive cells Death rate of activated cell Rate virus infects healthy cells Death rate of infected cells Removal rate coefficient of virus µ Rate activated cells kill infected cells c b α m Rate of virus production by infected cells Rate activated cells kill healthy cells Rate of activation by HIV 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

35 2.3 Model and Equilibrium Points without Treatment The following model considers the dynamics of healthy naive T cells, infected T cells, healthy activated T cells, and virus particles dx dt = λ ηx bx (mz) axv αx dy dt = axv µy (mz) ky dz dt = αx βz dv dt = cy γv. (2.1) 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

36 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 x = 1 β (A α η) 2bα m z = 1 (A α η) (2.2) 2bm where A = 1 β (α2 β + βη 2 + 2αβη + 4bαλm). We notice that to guarantee both x and z are positive values, we need A > α + η. If all parameters are positive, then Proof. Suppose 1 β (α2 β + βη 2 + 2αβη + 4bαλm) > α + η. Then 1 β (α2 β + βη 2 + 2αβη + 4bαλm) α + η. 1 ( α 2 β + βη 2 + 2αβη + 4bαλm ) α 2 + 2αη + η 2 β 25

37 which implies ( α 2 + η 2 + 2αη + 4bαλm ) α 2 + 2αη + η 2 β implying 4bαλm β 0. But this contradicts the fact that all parameters are positive. Evaluating this equilibrium point numerically we have x = z = 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

38 which has eigenvalues labeled r 1, r 2, r 3, r 4 r 1 = , r 2 = , r 3 = , r 4 = (2.4) Because r 1 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 x = kβγ acβ αγµm y = A 1 a 2 c 2 kβ 2 ackαβγµm z = kαγ acβ αγµm v = A 1 a 2 ckβ 2 γ akαβγ 2 µm (2.5) 27

39 where A 1 = (a 2 c 2 β 2 λ + α 2 λγ 2 µ 2 m 2 + kα 2 βγ 2 µm ackαβ 2 γ ackβ 2 γη bk 2 αβγ 2 m + kαβγ 2 µηm 2acαβλγµm). (2.6) We assume all appropriate inequalities to make the equilibrium point positive. Evaluating equilibrium point (2.5) numerically, we have x = y = z = v = (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

40 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 which has eigenvalues labeled r 1, r 2, r 3, r 4 r 1 = i, r 2 = i, r 3 = , r 4 = (2.8) Here, all the equilibrium points have negative real parts so it is a strictly stable (stable and attractive) equilibrium point [4]. 29

41 2.4 Equilibrium Points with Treatment Now we see how the equilibrium points change as we apply existing and theoretical therapies 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 dt = λ ηx axv αx dy dt = axv µy (mz) ky dz dt = αx βz dv dt = cy γv. (2.9) 30

42 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 x = kβγ acβ mαγµ y = 1 (acβλ kαβγ kβγη mαλγµ) ackβ z = kαγ acβ mαγµ v = 1 (acβλ kαβγ kβγη mαλγµ). (2.10) akβγ Again, we assume all appropriate inequalities to assure a positive equilibrium point. Theorem 2. Let x 1, y 1, z 1, v 1 represent the values of equilibrium point (2.5) and x 2,y 2, z 2,v 2 represent the values of equilibrium point (2.10). Then x 1 = x 2, y 1 < y 2, z 1 = z 2, and v 1 < v 2. Furthermore, if bk is sufficiently small, then y 1 y 2 and v 1 v 2. 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. Proof. Upon inspection, we see that x 1 = x 2, z 1 = z 2. Now we consider y 1 y 2. Dividing, we have y 1 y 2 = A 1 (mαγµ acβ) (kαβγ acβλ + kβγη + mαλγµ) which expands to y 1 = A 2 + kαβγ 2 µηm bk 2 αβγ 2 m y 2 A 2 + kαβγ 2 µηm where A 2 = (acβ) 2 λ 2acαβλγµm ackαβ 2 γ ackβ 2 γη + α 2 λγ 2 µ 2 m 2 + kα 2 βγ 2 µm. 31

43 Factoring the last two terms in the numerator and rewriting the last term in the denominator, we have y 1 = A 2 + kmαβγ 2 (µη bk). y 2 A 2 + kmαβγ 2 (µη) We notice that all terms but the last are identical in the numerator and denominator. Since all parameters are positive, µη bk < µη y 1 y 2 < 1 y 1 < y 2. Furthermore, if bk is sufficiently small, µη bk µη y 1 y 2 1 y 1 y 2.Next we consider v 1 v 2. Dividing, we have v 1 v 2 = A 1 (mαγµ acβ) (kαβγ acβλ + kβγη + mαλγµ) We notice v 1 v 2 = y 1 y 2 then v 1 v 2. so we can also conclude that v 1 < v 2 and if bk is sufficiently small, Evaluating (2.10) numerically, we have x = y = z = v = (2.11) Finding stability, we find the Jacobian 32

44 α η av 0 0 ax av µmz k µym ax α 0 β 0 0 c 0 γ which numerically is and has eigenvalues labeled r 1, r 2, r 3, r 4 r 1 = i, r 2 = i, r 3 = , r 4 = (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 = ) and both are strictly stable. We conclude from Theorem 2, with the support of the numerical values of 33

45 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 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 dt = λ ηx bx (mz) αx dy dt = µy (mz) ky dz dt = αx βz 34

46 dv dt = γv dw dt = cy γw. (2.13) 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 x(0) = kβγ acβ αγµm y(0) = A 1 a 2 c 2 kβ 2 ackαβγµm z(0) = kαγ acβ αγµm v(0) = A 1 a 2 ckβ 2 γ akαβγ 2 µm (2.14) where A 1 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

47 dv dt = γv v(t) = v e γt and z(t) dz dt = αx βz. Rearranging and multiplying by an integrating factor of e βt, we have dz dt eβt + βze βt = ax e βt d ( ) ze βt dt = ax e βt ze βt = ax β eβt + c 1 z(t) = ax β z(0) = ax β + c 1e βt + c 1 = z c 1 = βz ax β [ ] z(t) = ax βz β + ax e βt. (2.15) β In order to solve for y in dy dt = µy (mz) ky, we substitute in for z and obtain ( ( dy ax dt = y µm β + [ ] ) βz ax e βt β ) k 36

48 [ dy µmax y = kβ β [ ( µmβz + µmax ) e βt + β ]] dt ( ) µmax kβ ln y = t + ( µmβz + µmax ) e βt + c β β 2 2 [( ) ] µmax kβ y(t) = c 2 exp t + ( µmβz + µmax ) e βt β β 2 [ ] ( µmβz + µmax ) y(0) = c 2 exp = y β 2 [ ] (µmβz c 2 = y µmax ) exp β 2 [ ] [( ) ] (µmβz y(t) = y µmax ) µmax kβ exp exp t + ( µmβz + µmax ) e βt β 2 β β 2 [ ] [( (µmβz y(t) = y µmax ) µmax kβ exp exp β 2 β ) t + ( µmβz + µmax ) e βt β 2 (2.16) ]. In order to solve for w in dw dt = cy γw, we substitute in for y. For simplicity, we let 37

49 [ ] (µmβz A = y µmax ) exp β 2 ( ) µmax kβ B = β C = ( µmβz + µmax ) β 2. Therefore, we have y(t) = Ae Bt+Ce βt. (2.17) Solving for w, we have dw dt = cae Bt+Ce βt γw dw dt eγt + γwe γt = cae Bt+Ce βt e γt d ( ) we γt = Ace (B+γ)t+Ce βt dt w(t) = e γt Ace (B+γ)t+Ce βt dt. (2.18) Therefore our solution set is [ ] [( ) ] (µmβz y(t) = y µmax ) µmax kβ exp exp t + ( µmβz + µmax ) e βt β 2 β β 2 z(t) = ax β + [ βz ax β ] e βt 38

50 v(t) = v e γt 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 = y(0) = y = z(0) = z =

51 v(0) = v = (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) = exp ( t e 3t) 40

52 Figure 2-2: Activated T cell population with HAART z(t) = e t Figure 2-3: Virus population with HAART v(t)= e 2t 41

53 w(t) = 3, 774, 000e 2t exp ( t e t) 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

54 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

55 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 x v z r λ η β β m e n e m α n α m q Definition Naive cells Virus particles Effector cells (activated cells) Memory cells Production rate of naive cells Death rate of naive cells Death rate of effector cells Death rate of memory cells Effector cells produced in the activation of a naive cell Effector cells produced in the activation of a memory cell Rate of activation of naive cells by HIV Rate of activation of memory cells by HIV Rate of conversion of effector cell to memory cell 44

56 3.2 Model without virus and both naive and memory 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 α n x). Similarly, we assume that memory cells (r) are removed from the population by either natural death (at rate β m r) or by activation (at rate α m r). Both naive cells and memory cells become activated by an antigen and multiply as represented by the terms e n (α n x) and e m (α m r). We represent effector cells dying by βz and differentiated into memory cells by qz. Our model is dx dt = λ ηx α nx dz dt = e n (α n x) + e m (α m r) βz qz dr dt = qz α mr β m r. (3.1) Because this is a non-homogeneous linear system, we can solve by analytic methods. The matrix associated with the complementary system is η α n 0 0 e n α n β q e m α m, 0 q α m β m which has the following eigenvectors (ξ 1, ξ 2, ξ 3 ) with associated eigenvalues (k 1, k 2, k 2 ): 45

57 ξ 1 = B 1 qα ne n 1 q (η α m + α n β m ) 1, k 1 = η α n, ξ 2 = 0 1 2q (q + β α m β m + B 2 ) 1, k 2 = 1 2 (q + β + α m + β m + B 2 ), ξ 3 = 0 1 (α 2q m β q + β m + B 2 ) 1, k 3 = 1 2 B 2, where B 1 = qη qα m + qα n qβ m η 2 α 2 n + βη βα m + βα n ββ m +ηα m 2ηα n + ηβ m + α m α n + α n β m + qα m e m 46

58 and B 2 = [2qβ 2qα m 2qβ m + β 2 + α 2 m + β 2 m. 2βα m 2ββ m + 2α m β m + q 2 + 4qα m e m ] 1/2. We find a constant particular solution of the non-homogeneous equation of the form A B where C A = λ η + α n, ( ) λαn e n α m + β m B =, η + α n qα m + qβ m + βα m + ββ m qα m e m C = qλα n e n (η + α n ) (qα m + qβ m + βα m + ββ m qα m e m ). Then with constants c 1,c 2,c 3 we can write the solution to our system as x(t) = c 1B 1 qα n e n e ( η αn)t + A, 47

59 z(t) = c 1 q (η α m + α n β m ) e ( η αn)t c 2 2q (q + β α m β m + B 2 ) e 1 2 ( q β αm βm B 2)t + c 3 2q (α m β q + β m + B 2 ) e 1 2 tb 2 + B, r(t) = c 1 e ( η αn)t + c 2 e 1 2 t( q β αm βm B 2) + c 3 e 1 2 tb 2 + C Solutions and Graphs A healthy individual has 500 to 1500 T cells per mm 3 of blood so we assume that 1 mm 3 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

60 Figure 3-1: Naive T cell population during chronic activation x (t) = e t Figure 3-2: Effector T cell population during chronic activation z(t) = e t e t e t

61 Figure 3-3: Memory T cell population during chronic activation r(t) = e t e t e t 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