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A Dissertation Entitled Mathematical Models of the Activated Immune 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] .
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