Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention

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• CONTENTS

In this paper, we propose a survey of mathematical models for the spread of HIV/AIDS through shared drug injection equipment in groups of injecting drug users. We extend the expression for the force of infection by including the mechanism of exchange of a drug injecting equipment (DIE) in a friendship group. The classical models based on the law of mass action for direct transmission are obtained as limiting cases. We analyze models for a homogeneous population; for multiple stages of infectivity of HIV-infected individuals, and for stratified populations and multiple stages of infectivity by providing threshold theorems and possible stability results for nontrivial endemic states.

To estimate the probabilities of HIV infection and HIV seroconversion and to compare HIV seroconversions from different populations, in this paper we have developed some statistical models and state space models for HIV infection and seroconversion. By combining these models with the multi-level Gibbs sampling procedures, in this paper we have developed some efficient methods to estimate simultaneously these probabilities and the state variables as well as other unknown parameters. By using the complete likelihood function, we have also developed a generalized likelihood ratio test for comparing several HIV seroconversion distributions. As an illustration, we have applied the models and the methods to some data generated by the cooperative study on HIV under IDU and cocaine crack users by the National Institute of Drug Abuse/NIH. Our results show that there are significant differences in HIV seroconversion and HIV infection between populations of IDU, homosexuals and individuals with both IDU and homosexual behavior. For homosexuals, IDU and homosexuals with IDU, the probability density functions of times to HIV infection and HIV seroconversion are bi-model curves with two peaks and with heavy weights on the right. The average window period is about 2.75 months. Also, there are significant differences between the death and retirement rates of S (susceptible) people and I (infected but not seroconverted) people in all populations.

Whenever a Bayesian procedure that involves the numerical evaluation of multi-dimensional integrals is contemplated, a decision as to whether proceed with a computer implementation of the procedure will often depend on the ease with which the software may be written. The ease with which the software may be written will, in turn, depend not only programming language chosen but also a programmer's knowledge of the language. Because the author has had rather extensive experience in using the programming language, APL, which, among other things, is known for ease with which succinct code may be written to process complex arrays, it was known at the outset that the software to program Bayesian structure outlined in this paper could written and validated with relative ease. Furthermore, it seems likely that any other programming language with array processing capabilities could also be used to write code the implement the ideas presented in this paper. Consequently, a decision was made to organize the ideas in a precise mathematical form as a first step to writing computer code in any programming language chosen by an investigator. One of the outstanding problems in using the Bayesian paradigm to estimate unknown parameters and make statistical inferences, using stochastic models of HIV/AIDS epidemics, was that of developing a methodology for drawing sample from the joint conditional posterior distribution of the parameters and the stochastic process, given the data. Chapter 6 in the recent book by (Tan, 2000) may be consulted for technical details. Among other things, this paper contains a novel procedure, depending on the ease with which large arrays may be processed in a computer, for drawing random samples from the posterior of the parameters and the process, given a sample of data.

A class of four linear and nonlinear differential equations models is given to describe the detection of HIV-positive individuals in Cuba through random screening and contact tracing. The basic reproduction number is obtained for each of the four models. Cuban HIV data from 1986 to 2002 are used to fit the models for the purpose of comparison. We also use the models to gauge the difference in detection time through random screening and contact tracing. Remarks on the implications for intervention measures and treatment of people living with HIV in Cuba are also given.

There is an urgent need for a preventive HIV vaccine. Several candidate vaccines currently under development are designed to suppress HIV viral load and slow HIV progression post acquisition of HIV. In randomized, blinded efficacy trials of such a vaccine, concerns about the unreliability of viral load and CD4 cell counts as surrogate endpoints for AIDS-defining events and HIV transmission to others, and the use of antiretroviral therapies (ARTs) by infected trial participants, complicate the choice of post infection endpoints and analytic approaches for judging the vaccine's effectiveness. This chapter reviews some endpoint and analytic issues involved in assessing post infection vaccine effects, and evaluates in simulations (based on VaxGen's Phase 3 trial in Thailand) the use of time-to-event composite endpoints defined as the first event of ART initiation or biomarker failure (either high viral load or low CD4 count). Methods are presented for hypothesis testing and for constructing confidence bands about efficacy parameters corresponding to the composite endpoints; the tests and confidence bands are simultaneous over multiple clinically relevant biomarker failure thresholds. The simulation experiments support that simultaneous inferences of the viralload/ART and CD4/ART composite endpoints is a useful component of an analysis plan to evaluate post-infection vaccine effects.

This chapter first summarizes results of recent vaccine trials. Then we present structures and results of selected mathematical models for prophylactic (non-live-attenuated and live-attenuated) and therapeutic HIV/AIDS vaccines. Finally, important modeling considerations and discussions are provided. The purpose of this chapter is not to emphasize the usefulness of mathematical modeling, but to provide modelers a good collection of various types of deterministic models and discussions concerning differences among the models.

In this chapter, we have developed stochastic models for subsets of CD4⁽⁺⁾ T cells, CD8⁽⁺⁾ T cells and B cells under various conditions, including AIDS vaccination and HIV infection. We have used these models to generate some Monte Carlo data to assess both the prophylactic effects and the therapeutic effects of AIDS vaccines. Our Monte Carlo results indicated that in all cases, the immune system under AIDS vaccination will reach a steady state condition in at most a few months. At this steady state condition, the total number of HIV was kept below 100/mL whereas the total number of CD4 T cells is 670/mL in all cases. These results are consistent and comparable with the findings of clinical trials of AIDS vaccines on rhesus monkeys by Amara et al. (2001) and Barouch et al. (2000). These results seemed to indicate that the AIDS vaccines were very effective as both prophylactic and therapeutic agents if they can stimulate cytotoxic responses and humoral responses although it cannot completely prevent HIV infection. The Monte Carlo results also implied that the prophylactic effects of AIDS vaccines derive mainly from the cytotoxic immune responses through CTL cells (the primed CD8⁽⁺⁾ T cells) although the humoral responses through HIV antibodies might also be important albeit with a lesser degree; on the other hand, as therapeutic agents, the humoral vaccines appeared to be more important than the cytotoxic vaccines although the difference may not be very big.

Faced with an ongoing HIV virus pandemic, a deeper understanding of the interactions between this virus and an adaptive immune system is as essential to develop efficient counteractive measures. This review introduces models that aim to understand the course of disease as an emergent from the complex network of interactions of virus particles and immune cells at the microscopic scale, though not focusing on the details of the individual interactions. This allows the derivation of constraints on the global dynamics within cellular automata models and an approach based on stochastic processes. Special attention is devoted to a better understanding of the origin of the incubation period distribution. Moreover, vaccination strategies and recent developments in drug design based on fusion inhibition are discussed in the light of the presented theoretical models.

HIV can mutate and thereby escape immune recognition — an event that may precipitate AIDS. In a theoretical discussion of escape, it is essential to treat the infected-cell kinetics, and not just the mutations, as a random process. Deterministic modeling of infection dynamics exaggerates the rate-of-escape and obscures scenarios that decrease it. Using stochastic models, we show how extra-Poisson variability in virus production, possibly related to the viral “effective population size” in vivo, can lower the escape rate by orders-of-magnitude. If such variation is not operating, our simulations indicate that escape is very fast even if a CTL vaccine lowers viral load two logs. We combine loss-of-fitness and killing-rate data into a formula quantifying the influence of a broad response. Finally, we describe experiments that could reveal mechanisms that delay escape and discuss the relevance of our analysis to CTL vaccine design.

During HIV-1 infection, interactions between immune cells and virus yield three distinct stages of infection: high viral levels in acute infection, immune control in the chronic stage, and AIDS, when CD4⁺ T cells fall to extremely low levels. The immune system consists of many players that have key roles during infection. In particular, CD8⁺ T cells are important for killing of virally infected cells as well as inhibition of cellular infection and viral production. Activated CD8⁺ T cells, or cytotoxic T cells (CTLs) have unique functions during HIV-1, most of which are thought to be compromised during HIV-1 infection progression. Controversy exists regarding priming of CTLs, and our work attempts to address the dynamics occurring during HIV-1 infection. To explore the influence of CD8⁺ T cells as determinants in disease progression and issues relating to their priming and activation, we develop a two-compartment ordinary differential equation model describing cellular interactions that occur during HIV-1 infection. We track CD4⁺ T cells, CD8⁺ T cells, dendritic cells, infected cells, and virus, each circulating between blood and lymphatic tissues. Using parameter estimates from literature, we simulate commonly observed infection patterns. Our results indicate that CD4⁺ T cells as well as dendritic cells likely play a significant role in successful activation of CD8⁺ T cells into CTLs. Model simulations correlate with clinical data confirming a quantitative relationship between CD4⁺ T cells and CD8⁺ T-cell effectiveness.

There is a need to accurately estimate all viral, host cell and immune response specific parameters. Before such an extensive parameter estimation exercise can be carried out, an identifiability analysis needs to be done to investigate whether or not it is possible to determine all the parameters. If it is found to be possible, then the conditions or restrictions that apply need to be known beforehand. The issues to address are the variables to be measured, the minimal number of measurements for a complete determination of all parameters, the frequency and when, during the course of the viral infection, such measurements can be taken. It is important to know this in advance, especially where budgets are concerned. The measured variable combination and conditions that result in the least number of measurements or cost less could then be selected. All the foregoing will be investigated in this article using a well-established nonlinear system identifiability theory, where identifiability is a basic system property of whether all model parameters can be calculated from the measured system outputs and known inputs.

Incorporation of descriptions of drug pharmacokinetics in models of viral dynamics is important in predicting the long-term outcome of therapy and in the analysis of short-term viral load decay from which crucial parameters that govern viral replication kinetics are determined. In this chapter, we discuss recent developments in this area, focusing on the role of drug pharmacokinetics in the suppression of viral load and the emergence of drug resistance in an effort to establish optimal treatment protocols for HIV infection.

The interactions between the HIV RNA and the CD4⁺ T cells are well explained by a system of differential equations and predictions about growth of cell populations can be made. Reasonable parameter value estimation enabled meaningful analysis of system behavior, and Mathematica was used to simulate model dynamics. The model accounts for many observed behaviors, including the undetectability of the latent class of uninfected CD4⁺ T cells. A better understanding of virus-immune dynamics are obtained, allowing for improved research on treatment.

In this chapter we have developed an individual-based state space model for HIV pathogenesis in HIV-infected individuals treated with various drug regimens including HA ART. For monitoring effects of anti-viral drugs to search for optimal treatment regimens, we have used this model to develop a general procedure via multi-level Gibbs sampling to estimate the unknown parameters and the state variables. The unknown parameters include the infection rates, turn-over rates of HIV and CD4 T cells, and rates measuring effects of various drugs. The state variables include the numbers of infectious and non-infectious free HIV in blood over time. As an illustration, we have applied the method of this paper to the data of a patient from the St. Jude Children Hospital treated with various types of anti-retroviral drugs including HAART. The results indicated that for this individual, monotherapies using (Zidovudine or Abacavir or Stavudine) and 2 drugs therapies using (Zidovudine-Didanosine, or Abacavir-Stavudine, or Lamivudine-Zidovudine) were not very effective in suppressing HIV replication. On the other hand, the HAART regimens involving 2 NRTI drugs and 1 PI drug (Lamivudine-Zidovudine Nelfinavir, or Lamivudine-Stavudine-Indinavir) were very effective in suppressing HIV replication. In a few days it can bring down the infectious virus load from ≥146,000/mL to ≤1000/mL and stays at this low level during the whole treatment period under HAART. Also, using the total virus loads may be misleading and erroneous. It appeared that in some time periods, the number of infectious HIV can be very low (≤1000/mL) indicating that the drugs were in fact effective but the total number of HIV were very high reaching ≥45,000/mL.

There have been substantial interests in investigating HIV dynamics for understanding the pathogenesis of HIV-1 infection and antiviral treatment strategies. However, it is difficult to establish a relationship between pharmacokinetics (PK) and antiviral response due to too many confounding factors related to antiviral response during the treatment process. In this article, a mechanism-based dynamic model for HIV infection with intervention by antiretroviral therapies is proposed. In this model, we directly incorporate drug concentration, adherence and drug susceptibility into a function of treatment efficacy defined as an inhibition rate of virus replication. In order to focus our attention on estimating dynamic parameters for all subjects, we investigate a Bayesian approach under a framework of the hierarchical Bayesian (mixed-effects) model. The proposed methods and models not only can help to alleviate the difficulty in identifiability, but also can flexibly deal with sparse and unbalanced longitudinal data. The viral dynamic parameters estimated from the proposed method are, thus, more accurate since the variations in PK, adherence and drug resistance have been considered in the model.

Despite the tremendous progress that has been made in elucidating the molecular biology, virology, and immunology of HIV-1 infection, some of the most basic questions about how this virus causes disease remain unanswered. In this review, I outline seven problems relating to HIV-1 infection that might be addressed by computational biologists. Solutions to these problems would contribute significantly to our understanding of HIV-1 pathogenesis and improved treatments for HIV-1 infections.

In recent years, mathematical models have been used to examine the effects of adherence and/or structured treatment interruptions in HIV therapy and the resulting changes in population dynamics of HIV-1, the immune system and the emergence of drug resistance. These models have contributed to a better understanding of the effects of missed doses on the evolution of drug-resistant HIV strains. This chapter is a review of these studies; the methods and most important results are presented. We outline the significance of these results and discuss some of the drawbacks of the various modeling approaches.

Drug resistance is one of the major barriers to the successful control of HIV infections. Despite numerous effective medications and increasingly tolerable regimens, resistance is a persistent and pervasive problem. Knowing when resistant mutants are present and where the mutants that cause treatment failure originate are important steps to understanding and ultimately defeating treatment failure. In this chapter, we explore a branching process model for the evolution of resistant mutants before and after treatment initiation. We find that simple resistant mutants with 1 or 2 mutations appear very early during the acute stage of infection, leaving only a small window of opportunity to treat before any resistant mutations are present. While mutants with single point mutations are much more likely to appear before treatment than from residual replication during treatment, double, triple and more complex mutants are often produced, on average, in greater numbers on treatment than before treatment. However, the amount of replication on treatment in this simple model is insufficient to explain most treatment failure, suggesting that most mutants, other than single mutants and some double mutants, neither pre-exist nor evolve in well-suppressed systems. It is much more likely that lack of replication control, due to faulty adherence or sanctuary replication, is the cause of complex resistance patterns. Additional research is needed to understand the evolution of resistance under such partially controlled replication.

In this chapter, we have assessed the time to development of drug resistance in HIV-infected individuals treated with antiviral drugs by using longitudinal viral load HIV-1 counts. Through log transformed data of HIV virus counts over time, we have assumed a linear changing-point model and developed procedures to estimate the unknown parameters by using the Bayesian approach. We have applied the method and procedure to the data generated by the ACTG 315 involving treatment by the drug combination (3TC, AZT and Ritonavir). Our analysis showed that the mean time to the first changing point (i.e. the time the macrophage and long-lived cells began to release HIV particles) was around 15 days whereas the time to development of drug resistance by HIV was around 75 days. The Bayesian HPD intervals for these changing points are given by (8.7,21.3) and (42,108) respectively. This analysis indicated that if we use the combination of three drugs involving 2 NRTI inhibitors (3TC and AZT) and 1 PI inhibitor (Ritonavir) to treat HIV-infected individuals, in about two and half months it would be beneficial to change drugs to avoid the problem of drug resistance.

The use of a less sensitive or detuned test for recent human immunodeficiency virus (HIV) infection requires some modification to the standard incidence estimator. We use a stochastic simulation to demonstrate the details of using a detuned test to estimate incidence. We calculate the probabilities of the outcomes and give the result of the simulation as an example. Two estimators are compared and recommendations are made for calculating confidence intervals. We also show that the window period should be chosen to be half the collection period in order to minimize bias in the incidence estimator.

This chapter reviews design issues for studies of HIV dynamics that use a nonlinear mixed-effects model. Both the choice of sampling times and also the trade-off between the number of sampling times and the number of subjects are discussed. A method based on a first-order approximation, used for similar design issues in pharmacokinetics and pharma-codynamics, is reviewed and discussed. Limitations of this method are discussed and other methods are described, including methods based on the exact calculation of the Fisher information matrix and Bayesian methods, which, although computationally intensive, provide alternatives. A computationally intensive Bayesian method is recommended.

This chapter provides a brief summary of the use of mathematical models and associated statistical analysis which have contributed to research on the pathogenesis of HIV infection and etiology. A review of some associated statistical methods is included with an example hypothesis testing using a mathematical model.

In recent years, the incidence of infection with HIV among young homosexual men has been on the rise in developed countries. It is, therefore, of interest to study stochastic models of HIV/AIDS in which HIV infections become endemic in a population of homosexuals. Within a stochastic paradigm, the idea of an endemic equilibrium corresponds to the mathematical concept of a stochastic process converging in distribution to a stationary distribution. In Mode and Sleeman (2002), it was conjectured that in a Monte Carlo simulation experiment convergence to a stationary distribution of a Markov chain was being observed. In this paper, a proof of this conjecture is constructed under a reasonable sufficient condition within an abstract framework that resembles a general sub-critical branching process. The stationary distribution turns out to be a mixture of conditionally independent Poisson densities and the mixing measure is that underlying the population process. It is often difficult to decide in a Monte Carlo simulation experiment whether convergence to a stationary distribution is actually being observed, which sometime leads to controversies. Knowing that convergence to a stationary distribution does indeed occur, under a plausible sufficient in the class of models considered in the paper, will provide a firm mathematical basis for avoiding controversies. In the literature on stochastic models of HIV/AIDS, the class of models considered in this paper is often referred to as chain multinomial models.

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