The initialization and sensitivity of multigroup models for the transmission of HIV

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression model, every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. We demonstrate the importance of choosing appropriate initial conditions, and define a new approach to distributing the initial population among the subgroups so as to minimize the artificial transients in the solutions due to unbalanced initial conditions. We demonstrate that the rate of removal in and out of a population is an important, yet often neglected, effect. We also illustrate the importance of distinguishing between the number of partners a person has and the number of contacts per partner. By assuming that people with many partners have fewer contacts per partner than people with few partners, we found that the epidemic is less sensitive to the partner acquisition rate than one might expect. However, because the probability of transmission of HIV per contact is low, the epidemic is very sensitive to the number of contacts per partner. Modeling this distinction is particularly important when estimating the impact of programs which encourage people to have fewer sexual partners.     

The general mixing of addicts and needles in a variable-infectivity needle-sharing environment

 In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into three distinct infectious stages. The impact of this assumption is to greatly increase the complexity of the HIV transmission mechanism. We begin the paper with a brief literature review followed by the derivation of a model which incorporates three classes of infectious addicts and three classes of infectious needles and where a general probability structure is used to represent the interaction of addicts and needles of varying levels of infectivity. We find that if the basic reproductive number is less than or equal to unity then there exists a globally stable disease free equilibrium. The model possesses an endemic equilibrium solution if the basic reproductive number exceeds unity. We then conduct a brief simulation study of our model. We find that the spread of disease is heavily influenced by the way addicts and needles of different levels of infectivity interact. Received: 20 September 2001 / Revised version: 21 December 2001 / Published online: 17 May 2002     

Three stage AIDS incubation period: a worst case scenario using addict-needle interaction assumptions

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. We start off with a brief literature survey and review; this is followed by the derivation of a model which allows addicts to progress through three distinct stages of variable infectivity prior to the onset of full blown AIDS and where the class of infectious needles is split into three according to the different levels of infectivity in addicts. Given the structure of this model we are required to make assumptions regarding the interaction of addicts and needles of different infectivity levels. We deliberately choose these assumptions so that our model serves as an upper bound for the prevalence of HIV under the assumption of a three stage AIDS incubation period. We then perform an equilibrium and stability analysis on this model. We find that there is a critical threshold parameter R(0) which determines the behaviour of the model. If R(0)< or =1, then irrespective of the initial conditions of the system HIV will die out in all addicts and all needles. If R(0)>1, then there is a unique endemic equilibrium which is locally stable if, as is realistic, the time scale on which addicts inject is much shorter than that of the other epidemiological and demographic processes. Simulations indicate that if R(0)>1, then provided that disease is initially present in at least one addict or needle it will tend to the endemic equilibrium. In addition we derive conditions which guarantee this. We also find that under calibration the long term prevalence of disease in our variable infectivity model is always greater than in an equivalent constant infectivity model. These results are confirmed and explored further by simulation. We conclude with a short discussion.     

Age and Sex Structured Model for Assessing the Demographic Impact of Mother-to-Child Transmission of HIV/AIDS

Age and sex structured HIV/AIDS model with explicit incubation period is proposed as a system of delay differential equations. The model consists of two age groups that are children (0–14 years) and adults (15–49 years). Thus, the model considers both mother-to-child transmission (MTCT) and heterosexual transmission of HIV in a community. MTCT can occur prenatally, at labour and delivery or postnatally through breastfeeding. In the model, we consider the children age group as a one-sex formulation and divide the adult age group into a two-sex structure consisting of females and males. The important mathematical features of the model are analysed. The disease-free and endemic equilibria are found and their stabilities investigated. We use the Lyapunov functional approach to show the local stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The basic reproductive number (ℛ₀) for the model shows that the adult population is responsible for the spread HIV/AIDS epidemic, thus up-to-date developed HIV/AIDS models to assess intervention strategies have focused much on heterosexual transmission by the adult population and the children population has received little attention. We numerically analyse the HIV/AIDS model to assess the community benefits of using antiretroviral drugs in reducing MTCT and the effects of breastfeeding in settings with high HIV/AIDS prevalence ratio using demographic and epidemiological parameters for Zimbabwe.     

Modeling the AIDS epidemic in Mexico City

To study the future course of the AIDS epidemic in Mexico City, we use an open compartmental model to forecast new AIDS cases among homosexual and bisexual males and among heterosexual males and females. For each group three compartments are defined: uninfected persons, infected but asymptomatic persons, and persons diagnosed with AIDS. It is assumed that the AIDS epidemic will follow the propagation of infectious disease model, where spread of infection is proportional to the product of the number of healthy persons and the number of infected ones. The compartmental model is represented by a system of nonlinear differential equations describing the rate of change in the number of persons in each compartment. The impact of preventive measures is explored by decreasing the probability of HIV transmission, which is one of the model parameters representing behavioral patterns. By April 1989, 491 AIDS cases had been reported in Mexico City and classified as sexually related. Our model predicts that the AIDS incidence will continue to rise in Mexico City for the foreseeable future and will spread among the heterosexual population. Decreasing the transmission probability by 10% in all groups (through education programs) will result in a decrease of 18.1% in the number of accumulated cases over a 5-year period. A 20% decrease would prevent more than 31% of the cases. We conclude that mathematical models can be valuable in predicting the spread of the AIDS epidemic and the impact of behavioral change on its spread.     

Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe

We present a sex-structured model for heterosexual transmission of HIV/AIDS in a community. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with a time delay due to incubation period. The sex-structured HIV/AIDS model divides the population into a two sex-structure consisting of females and males. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to focus on the effects of condom use as a single-strategy approach in HIV prevention in the absence of any treatment. Initially we model the use of male condoms and further extend the model to incorporate the use of both female and male condoms. The model includes two primary factors in condom use to control HIV that are condom efficacy and compliance. The exposure risk of infection after each intervention is obtained. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male and female condom use in a community. The models are numerically analysed to assess the effectiveness of condom use on the transmission dynamics of HIV/AIDS using demographic and epidemiological parameters for Zimbabwe. The study demonstrates the use of sex-structured HIV/AIDS models in assessing the effectiveness of female and male condom use as a preventative strategy in a heterosexually active population. Z. Mukandavire would like to acknowledge financial support given by the National University of Science and Technology through a Staff Development Scholarship. The authors are grateful to Eagle Insurance Company of Zimbabwe for financial support.     

Injection drug use and HIV/AIDS transmission in China

After nearly three decades of being virtually drug free, use of heroin and other illicit drugs has re-emerged in China as a major public health problem. One result is that drug abuse, particularly heroin injection, has come to play a predominant role in fueling China's AIDS epidemic. The first outbreak of HIV among China's IDUs was reported in the border area of Yunnan province between China and Myanmar where drug trafficking is heavy. Since then drug-related HIV has spread to all 31 provinces, autonomous regions and municipalities. This paper provides an overview to HIV/AIDS transmission through injection drug use in China. It begins with a brief history of the illicit drug trade in China, followed by a discussion of the emergence of drug related AIDS, and a profile of drug users and their sexual partners who have contracted the virus or who are vulnerable to infection. It ends by summarizing three national strategies being used by China to address both drug use and AIDS as major health threats.     

Modeling AIDS as a function of other sexually transmitted disease.

The spread of AIDS, as with any sexually transmitted disease, will depend on the pattern of sexual activity. Both the proportion of the population who have high partner exchange rates and the extent to which that proportion interacts with the remainder of the population are likely to be important determinants of the AIDS epidemic. However, it does not seem likely that surveys could obtain sufficiently reliable information of this nature for use in an accurate model of the AIDS epidemic. On the other hand, such information is implicitly contained in the epidemiology of other sexually transmitted diseases (STDs). Therefore a method is suggested of calculating the parameters of a model of the AIDS epidemic by comparing it with the epidemiology of another STD. The result is a model that predicts the likelihood of infection by the AIDS virus as a function of time and an individual's history of STD. It is suggested that further work along these lines may lead to a quantitative approach to assessing the importance of various STDs as cofactors in the spread of AIDS.     

Insulin Degrading Enzyme Induces a Conformational Change in Varicella-Zoster Virus gE,and Enhances Virus Infectivity and Stability

Varicella-zoster virus (VZV) glycoprotein E (gE) is essential for virus infectivity and binds to a cellular receptor, insulin-degrading enzyme (IDE), through its unique amino terminal extracellular domain. Previous work has shown IDE plays an important role in VZV infection and virus cell-to-cell spread, which is the sole route for VZV spread in vitro. Here we report that a recombinant soluble IDE (rIDE) enhances VZV infectivity at an early step of infection associated with an increase in virus internalization, and increases cell-to-cell spread. VZV mutants lacking the IDE binding domain of gE were impaired for syncytia formation and membrane fusion. Pre-treatment of cell-free VZV with rIDE markedly enhanced the stability of the virus over a range of conditions. rIDE interacted with gE to elicit a conformational change in gE and rendered it more susceptible to proteolysis. Co-incubation of rIDE with gE modified the size of gE. We propose that the conformational change in gE elicited by IDE enhances infectivity and stability of the virus and leads to increased fusogenicity during VZV infection. The ability of rIDE to enhance infectivity of cell-free VZV over a wide range of incubation times and temperatures suggests that rIDE may be useful for increasing the stability of varicella or zoster vaccines.     

Modeling the impact of random screening and contact tracing in reducing the spread of HIV

Mathematical models can help predict the effectiveness of control measures on the spread of HIV and other sexually transmitted diseases (STDs) by reducing the uncertainty in assessing the impact of intervention strategies such as random screening and contact tracing. Even though contact tracing is one of the most effective methods used for controlling treatable STDs, it is still a controversial strategy for controlling HIV because of cost and confidentiality issues. To help estimate the effectiveness of these control measures, we formulate two models with random screening and contact tracing based on the differential infectivity (DI) model and the staged-progression (SP) model. We derive formulas for the reproductive numbers and the endemic equilibria and compare the impact that random screening and contact tracing have in slowing the epidemic in the two models. In the DI model the infected population is divided into groups according to their infectiousness, and HIV is largely spread by a small, highly infectious, group of superspreaders. In this model contact tracing is an effective approach to identifying the superspreaders and has a large effect in slowing the epidemic. In the SP model every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. In this model random screening is more effective than for the DI model, and contact tracing is less effective. Thus the effectiveness of the intervention strategy strongly depends on the underlying etiology of the disease transmission.     

Statistical analysis of HIV infectivity based on partner studies

Partner studies produce data on the infection status of partners of individuals known or assumed to be infected with the human immunodeficiency virus (HIV) after a known or estimated number of contacts. Previous studies have assumed a constant probability of transmission (infectivity) of the virus at each contact. Recently, interest has focused on the possibility of heterogeneity of infectivity across partnerships. This paper develops parametric and nonparametric procedures based on partner data in order to examine the risk of infection after a given number of contacts. Graphical methods and inference techniques are presented that allow the investigator to evaluate the constant infectivity model and consider the impact of heterogeneity of infectivity, error in measurement of the number of contacts, and regression effects of other covariates. The majority of the methods can be computationally implemented easily with use of software to fit generalized linear models. The concepts and techniques are closely related to ideas from discrete survival analysis. A data set on heterosexual transmission is used to illustrate the methods.     

HIV/AIDS: in search of an animal model

AIDS is among the most devastating diseases of our time, claiming the lives of approximately 3 million people per year. The primary cause of AIDS, human immunodeficiency virus type 1 (HIV-1), is a pathogen that is highly specific for humans and generally does not infect or cause disease in other species. This property complicates the generation of animal models that are urgently needed to test new antiretroviral therapies and vaccines. The most practical animal models developed to date consist of infection of rhesus macaques with a simian immunodeficiency virus (SIV) or chimeric HIV/SIV viruses. Although these models are useful for particular applications, the fact that SIV is a distinct virus compared with HIV-1 represents a significant limitation to their use. Here, we discuss the uses and limitations of existing models and recent advances that might lead to better animal models for HIV/AIDS.     

Estimation of HIV infection and incubation via state space models

By using the state space model (Kalman filter model) of the HIV epidemic, in this paper we have developed a general Bayesian procedure to estimate simultaneously the HIV infection distribution, the HIV incubation distribution, the numbers of susceptible people, infective people and AIDS cases. The basic approach is to use the Gibbs sampling method combined with the weighted bootstrap method. We have applied this method to the San Francisco AIDS incidence data from January 1981 to December 1992. The results show clearly that both the probability density function of the HIV infection and the probability density function of the HIV incubation are curves with two peaks. The results of the HIV infection distribution are clearly consistent with the finding by Tan et al. [W.Y. Tan, S.C. Tang, S.R. Lee, Estimation of HIV seroconversion and effects of age in San Francisco homosexual populations, J. Appl. Stat. 25 (1998) 85]. The results of HIV incubation distribution seem to confirm the staged model used by Satten and Longini [G. Satten, I. Longini, Markov chain with measurement error: estimating the 'true' course of marker of the progression of human immunodeficiency virus disease, Appl. Stat. 45 (1996) 275].     

The prevalence of viral hepatitides among youth

The study of 632 teenagers has revealed that young people are the main group of risk with respect to the spread of virus hepatitides B and C, HIV infection. In accordance with risk factors, the teenagers have been divided into 4 groups: using drugs by injection; leading sexual life; using drugs and leading sexual life; having no risk factor indicated. The presence of hepatitis B and hepatitis C virus infection has been established in 1.42 and 1.27% of cases respectively. This is due to the frequent change of partners, the use of narcotic drugs since school age, the absence of necessary information and contacts with parents. When considering this problem, mass media have been found to play an important role. The presence of reliable information breaking old stereotypes may help parents pay greater and better attention to the upbringing and health of their children.     

Determination of the spread of HIV from the AIDS incidence history

The relation between the incidence of HIV in the general population, the number of AIDS cases, and the incubation period for the disease is examined. The number of AIDS cases can be expressed in terms of a convolution integral over the incubation period distribution and the temporal history of HIV incidence. In order to determine the level of HIV incidence it is necessary to invert the convolution. In this manner, it is possible to determine the spread of HIV up to the present time from knowledge of the AIDS incidence history and the incubation period. We describe the inversion of the convolution in terms of a Laplace transform technique that is applicable for any given incubation period distribution. Substantial simplifications in the technique are found in the case of an Erlang distribution for the probability density. The spread of HIV infections in the United States is charted through 1988 using AIDS incidence data that are corrected for both the revised AIDS case definition and reporting time delays. The results are consistent with current estimates of the HIV incidence in the United States and show no evidence of saturation in the rate of new infections. Indeed, the rate of new infections still appears to be climbing as of that date. While the technique is unable to predict the future course of the epidemic, it may provide a useful benchmark for comparison with mathematical models of the epidemic. The techniques are conceptually applicable to diseases other than AIDS.     

The differential infectivity and staged progression models for the transmission of HIV

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transmission studies and suggest that a small subset of infected people may be responsible for a disproportionate number of infections. We use two simple models to study the impact of variations in infectiousness. In the first model, we account for different levels of virus between individuals during the chronic phase of infection, and the increase in the average time from infection to AIDS that goes along with a decreased viral load. The second model follows the more standard hypothesis that infected individuals progress through a series of infection stages, with the infectiousness of a person depending upon his current disease stage. We derive and compare threshold conditions for the two models and find explicit formulas of their endemic equilibria. We show that formulas for both models can be put into a standard form, which allows for a clear interpretation. We define the relative impact of each group as the fraction of infections being caused by that group. We use these formulas and numerical simulations to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease. The acute stage and the most infectious group both appear to have a disproportionate effect, especially on the early epidemic. Contact tracing to identify super-spreaders and alertness to the symptoms of acute HIV infection may both be needed to contain this epidemic.     

The HIV epidemic: medical and social challenges

The sudden appearance, rapid spread, and devastating clinical impact of HIV infection in Africa, Europe and North America has created a medical problem unprecedented in the modern era. HIV is sexually transmitted, afflicts sexual and racial minorities in developed countries, and appears likely to be fatal and incurable in a majority of infected people. Its epidemiology (transmission and natural history) and clinical manifestations have been well described, but treatment of HIV remains minimally effective, creating only a short respite from progressive deterioration. In the absence of effective vaccination, HIV will continue to spread, abetted by a long period of asymptomatic carriage during which carriers are infectious. It has spread internationally to most undeveloped countries aided by fear and ignorance. The problem will resist simple technological solutions and adversely impact the lives of tens of millions of people in these areas over the next several decades. In developed countries HIV will strain medical resources and kill several million people before the end of the century. Despite the tremendous problems created by the AIDS epidemic, it has driven a remarkable expansion of virologic and immunologic understanding which promises to ultimately lead to control of not only AIDS, but a variety of other serious diseases. The following reviews of pivotal issues in AIDS research document this progress.     

A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation.

A model is formulated for the spread of the human immunodeficiency virus (HIV) and the subsequent development of acquired immunodeficiency syndrome (AIDS) in the population of homosexual men in San Francisco. The dynamic simulation model includes sexually very active and active subpopulations, migration, and a staged progression of HIV-infected persons to AIDS and death. Numerous data sources are used to estimate parameter values in the model. In a companion paper, simulations using the model and parameter estimates are found that are consistent with HIV and AIDS incidence data.     

HIV epidemiology: past, present, and future

The worldwide pandemic of acquired immunodeficiency syndrome (AIDS) has the potential to cause catastrophic medical and social effects that will influence world health well into the 21st century. The causative agent, a lentiretrovirus called human immunodeficiency virus (HIV-1), is spread by intimate exposure to blood and bodily fluids through sexual, parenteral, and mother-to-infant exposure. The natural history from exposure to disease has a median incubation period of 8-10 years and is characterized by progressive depletion of CD-4 positive T lymphocytes as well as effects on other immune and central nervous system cell populations. The World Health Organization (WHO) estimates that between 8 and 10 million persons are currently infected with the virus worldwide, with 8 to 10 times this level projected by some estimates into the 21st century. Recent leveling off of AIDS incidence in the U.S. appears to represent the positive benefits of antiretroviral therapy, and considerable benefit could be seen if such therapies were made more widely available to medically underserved populations. With prolonged survival, however, other long-term sequelae such as cancer and lymphoma may emerge as significant complications of prolonged immunodeficiency. Furthermore, the large pool of already infected persons and continued spread of the virus make the development of additional therapies and an effective anti-HIV vaccine priorities of medical research.