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**HIV Structure**

**HIV Structure**

HIV has a dense cylindrical core. It is around 120nm in diameter (120 billionths of a meter; around 60 times smaller than a red blood cell) and 10kb in length and roughly spherical. It is composed of two copies of single-stranded RNA enclosed by a conical capsid comprising the viral protein p24 (figure 2.1). This conical capsid can be described in layman’s language as being bullet shaped. The RNA component is 9749 nucleotides long and it is surrounded by a plasma membrane of host-cell origin. The RNA is part of a protein-nucleic acid complex which is composed of the nucleo-protein p7 and the reverse transcriptase (RT) p66. The single-strand RNA is tightly bound to the nucleocapsid proteins p7 and enzymes such as reverse transcriptase (RT) i.e. p66, protease (PR) i.e. p11 and integrase (IT) i.e. p32 that are indispensable for the replication, proliferation and development of the viron.

The nucleocapsid (p7 and p6) associates with the genomic RNA (one molecule per hexamer) and protects the RNA from digestion by nucleases. The ends of each strand of HIV RNA has an RNA sequence called the long terminal repeat (LRT). The LRT has regions which act as switches to control production of new viruses. Surrounding this capsid is the matrix layer which is made up of the protein p17 and this ensures the integrity of the viron particle. Also enclosed within the viron particle are genes 18 such as Vpr, Nef, Vif, p7 and viral protease. These are genes that code the proteins used in controling the ability of the virus to infect a cell and produce new copies of virus and/or cause disease. The outer viral envelope which is formed when the capsid buds from the host cell, taking some of the host-cell membrane with it is a coat of lipoprotein membrane fat. Projecting from this viral envelop/membrane are 72 little spikes formed from the glycoproteins gp120 and gp 41 (HIV Wikipedia 2008, Hoffmann et al. 2007 and Smith 2008).

**HIV-1 Subgroup, Recombination and Epidemiological Structure**

Although once an individual becomes infected, eradication of the virus still remains impossible despite all the therapeutic advantages achieved during the last decade, knowledge of the epidemiological prevalence can still help to contain the disease to a certain degree (Hoffmann et al. 2007). There are three subtypes of HIV-1 namely: M group or the “major” group, O group or the “outlier” group and N group or the “new” group. Each group is divided into subtypes and their recombined subtype known as the circulating recombinant forms (CRFs). Given below are the HIV-1 subgroups and their epidemiological prevalence.

**A STOCHASTIC MODEL OF THE PERIOD OF THE GAY-LIFE**

**A STOCHASTIC MODEL OF THE PERIOD OF THE GAY-LIFE**

Consider a population of homosexuals consisting of susceptibles and infectives. Assume that at time t = 0, a new member who is tested HIV negative enters into the population and makes sexual contacts with members of the population. Assume further that his/her contacts occur at random time points which follow a Poisson process with parameter λ, λ > 0. Let the probability that the individual who has already had n contacts up to time t when he/she tested HIV positive for the first time in the interval (t, t+∆) be given by nµ∆ + ο(∆), µ > 0. Let the gay life period of the individual be represented by the random variable ST. in the next section, we obtain the probability density function (p.d.f) of ST.

**THE BACK-CALCULATION AND THE INFECTION RATE**

One of the methods used in estimating and projecting the infection rate from AIDS incidence data is the back-calculation method (Brookmeyer and Gail 1994). It is an important method of constructing rates of HIV infection and estimating current prevalence of HIV infection and future incidence of AIDS (Bacchetti et al. 1993). This method has been used by many mathematical scientists to obtain and predict the AIDS incidence of different populations. Amongst the work done are those of Verdecchia and Mariotto (1995) who modelled past HIV infections in Italy considering the interaction between age and calendar time. Anbupalam et 42 al. (2002) also used the Back calculation method to project future AIDS cases in Tamil Nadu by assuming that the incubation distribution was Weibul and Log-logistic. Ong and Soo (2006) estimated the HIV infection rates and projection in Malaysia while Lopman and Gregson (2008) used the Back-calculation method to reconstruct the historical trends in HIV incidence in Harare, Zimbabwe by using mortality data. They also attempted to determine the amount of peakness of HIV incidence and when the peakness occurred in Harare, Zimbabwe.

**The Life Cycle of HIV**

**The Life Cycle of HIV**

The HIV is a retrovirus and its RNA carries the genetic information. The HIV has a dense cylindrical core encasing two molecules of the viral genome. Virus-encoded enzymes required for efficient multiplication, such as reverse transcriptase and integrase, are also incorporated into the virus particle. After attaching itself to the cell wall of the host T4 cell, the virus injects its RNA together with the enzymes reverse transcriptase and integrase into the cytoplasm of the host cell. The viral reverse transcriptase enzyme first synthesises a single complementary, negative-sense DNA copy to the HIV RNA; next the RNA is denatured; and then a complementary positive-sense DNA copy is synthesised to create double-stranded proviral DNA.

**TABLE OF CONTENT :**

- TITLE
- DECLARATION
- ACKNOWLEDGEMENT
- DEDICATION
- SUMMARY
- TABLE OF CONTENT
- TABLES
- FIGURES
- CHAPTER ONE: INTRODUCTION
- 1.1 OVERVIEW
- 1.2 ACRONYMS AND TERMINOLOGIES
- 1.3 ROLE OF MATHEMATICAL AND STATISTICAL MODELS IN HIV/AIDS EPIDEMIC
- 1.4 HIV MODELING
- 1.5 THESIS OUTLINE

- CHAPTER TWO: PATHOGENESIS OF HIV
- 2.1 INTRODUCTION
- 2.2 HIV PATHOGENESIS
- 2.2.1 HIV Genomic Structure
- 2.2.2 Genes and Enzymes in HIV Entry and Replication
- 2.2.3 HIV-1 Strains
- 2.2.4 HIV Co-receptors
- 2.2.5 HIV-1 Subgroups, Recombination and Epidemiological Structure

- 2.3 RECENT DEVELOPMENTS AND PROBLEMS

**CHAPTER THREE: A STOCHASTIC POINT PROCESS MODEL OF THE INCUBATION PERIOD OF A HIV INFECTED INDIVIDUAL**- 3.1 INTRODUCTION
- 3.2 A STOCHASTIC MODEL OF THE PERIOD OF THE GAY-LIFE
- 3.2.1 The Probability Distribution Function of the Gay-life
- 3.2.2 The Moments of Seroconversion Time
- 3.2.3 Estimation of the Parameters of q(t)

- 3.3 A STOCASTIC MODEL OF THE HIV INCUBATION PERIOD
- 3.3.1 The Probability Distribution of the Incubation Period
- 3.3.2 The Moments of Incubation Period
- 3.3.3 Estimation of the Parameter of pn
- 3.3.3.1 The Method of Moments
- 3.3.3.2 The Method of Maximum Likelihood
- 3.3.3.3 The Method of Median

- 3.3.4 A Numerical Example
- 3.4 THE BACK-CALCULATION AND THE INFECTION RATE
- 3.5 CONCLUSION

**CHAPTER FOUR: STOCHASTIC MODEL OF THE GROWTH OF HIV IN AN INFECTED INDIVIDUAL**- 4.1 INTRODUCTION
- 4.2 MODEL I: THE MUTATION MODEL
- 4.2.1 The Probability Generating Functions
- 4.2.2 A Particular Case
- 4.3 MODEL II: THE MULTIPLICATION PROCESS INSIDE A T4 CELL
- 4.3.1 The Life Cycle of HIV
- 4.3.2 The Model Formulation
- 4.3.3 The Probability Generating Function of (X(t), N(t))
- 4.3.4 The Moment of (X(t), N(t))
- 4.3.5 A Numerical Illustration

- 4.4 CONCLUSION

**CHAPTER FIVE: THE T4 CELL COUNT AS A MARKER OF HIV PROGRESSION IN THE ABSENCE OF ANY MECHANISM**- 5.1 INTRODUCTION
- 5.2 THE CATASTROPHE MODEL
- 5.3 THE PROBABILITY GENERATING FUNCTION
- 5.4 THE MOMENT STRUCTURE OF (X(t),Y(t),Z(t))
- 5.5 THE AMOUNT OF TOXIN PRODUCED
- 5.6 NUMERICAL ILLUSTRATION
- 5.7 CONCLUSION

**CHAPTER SIX: A STOCHASTIC MODEL OF THE DYNAMICS OF HIV UNDER A COMBINATION THERAPEUTIC INTERVENTION**- 6.1 INTRODUCTION
- 6.2 THE FORMULATION OF THE MODEL
- 6.3 PROBABILITY GENERATING FUNCTION
- 6.4 THE MOMENT STRUCTURE OF (X(t),V(t),D(t))
- 6.5 NUMERICAL ILLUSTRATION OF MODEL
- 6.6 CONCLUSION
- REFERENCES

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Stochastic analysis of AIDS Epidemiology