Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of South Africa

HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested in fighting it. There is a need to integrate the series of control measures available to ensure a consistent reduction in the incidence of the disease pending the discovery of its cure. We present a deterministic model for controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapy as control measures. We formulate a fixed time optimal control problem subject to the model dynamics with the goal of finding the optimal combination of the two control measures that will minimize the cost of the control efforts as well as the incidence of the disease. We estimate the model state initial conditions and parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin's maximum principle to derive the optimality system and solve the system numerically. Compared with the practice in most resource-limited settings where ARV treatment is given only to patients with full-blown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress to the pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals' sexual habits reduces both the incidence and prevalence of the disease faster. In fact, the results predict that the implementation of the proposed strategy would drive new cases of the disease towards eradication in 10 years.     

广西靖西县1997—2012年HIV/AIDS流行病学特征分析

目的了解广西靖西县1997—2012年HIV/AIDS流行病学状况,分析流行趋势,为制定艾滋病防治措施提供科学依据。方法采用描述流行病学分析方法,对靖西县1997—2012年艾滋病疫情监测资料进行分析。结果靖西县1997—2012年累计报告HIV/AIDS 147例,其中HIV感染者60例,AIDS病人47例,死亡40例;男性95例,女性52例,男女性别比为1.83∶1;以20~49岁青壮年为主,这些年龄组发病率为3.24/10万;传播途径以性接触传播为主,占90.48%;其次为静脉注射毒品传播,占6.80%。结论靖西县艾滋病流行近年呈明显上升趋势,艾滋病已经从高危人群向普通人群扩散传播,性接触传播已经成为艾滋病的主要传播途径,应加强对高危人群性行为的干预,遏制艾滋病传播蔓延。     

HIV感染动力学模型概述

自1981年美国首次发现艾滋病以来,艾滋病在世界范围内广泛传播,引起医学专家、生物学家、数学家和物理学家等的极大关注。近年来,HIV动力学模型成为HIV治疗领域的研究热点。HIV基本动力学模型的研究有助于实现对未来疾病发展状况的描述与预测,HIV感染控制模型的研究有助于改善HIV病毒患者的治疗方案,对控制模型的优化有利于发现对HIV患者的有效治疗策略。本文概述了几种基本的HIV感染动力学模型,分析比较了它们的性能差异和各自存在的优缺点,介绍了HIV控制模型及其优化控制模型的计算机Matlab/simulink模拟。     

The effects of population structure on the spread of the HIV infection

A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.     

The transmission dynamics of human immunodeficiency virus (HIV)

The paper first reviews data on HIV infections and AIDS disease among homosexual men, heterosexuals, intravenous (IV) drug abusers and children born to infected mothers, in both developed and developing countries. We survey such information as is currently available about the distribution of incubation times that elapse between HIV infection and the appearance of AIDS, about the fraction of those infected with HIV who eventually go on to develop AIDS, about time-dependent patterns of infectiousness and about distributions of rates of acquiring new sexual or needle-sharing partners. With this information, models for the transmission dynamics of HIV are developed, beginning with deliberately oversimplified models and progressing--on the basis of the understanding thus gained--to more complex ones. Where possible, estimates of the model's parameters are derived from the epidemiological data, and predictions are compared with observed trends. We also combine these epidemiological models with demographic considerations to assess the effects that heterosexually-transmitted HIV/AIDS may eventually have on rates of population growth, on age profiles and on associated economic and social indicators, in African and other countries. The degree to which sexual or other habits must change to bring the 'basic reproductive rate', R0, of HIV infections below unity is discussed. We conclude by outlining some research needs, both in the refinement and development of models and in the collection of epidemiological data.     

A stochastic model for the development of an AIDS epidemic in a heterosexual population.

A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part on the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.     

Approaches to antiretroviral therapy in China

INTRODUCTION Advances in antiretroviral therapy (ART) have resulted in dramatic declines in hospitalizations and death rates in the U.S. and Europe, with concomitant steady increases in life expectancy. In China, the government has recog- nized the imminent threat that HIV/AIDS poses to its popu- lation and responded with a national antiretroviral (ARV) treatment program providing ARV drugs free to those most in need [1]. However, given the potentially catastrophic impact of HIV …     

The Potential for Elimination of Racial-Ethnic Disparities in HIV Treatment Initiation in the Medicaid Population among 14 Southern States

Objectives

The purpose of this study was to explore the racial and ethnic disparities in initiation of antiretroviral treatment (ARV treatment or ART) among HIV-infected Medicaid enrollees 18–64 years of age in 14 southern states which have high prevalence of HIV/AIDS and high racial disparities in HIV treatment access and mortality.

Methods

We used Medicaid claims data from 2005 to 2007 for a retrospective cohort study. We compared frequency variances of HIV treatment uptake among persons of different racial- ethnic groups using univariate and multivariate methods. The unadjusted odds ratio was estimated through multinomial logistic regression. The multinomial logistic regression model was repeated with adjustment for multiple covariates.

Results

Of the 23,801 Medicaid enrollees who met criteria for initiation of ARV treatment, only one third (34.6%) received ART consistent with national guideline treatment protocols, and 21.5% received some ARV medication, but with sub-optimal treatment profiles. There was no significant difference in the proportion of people who received ARV treatment between black (35.8%) and non-Hispanic whites (35.7%), but Hispanic/Latino persons (26%) were significantly less likely to receive ARV treatment.

Conclusions

Overall ARV treatment levels for all segments of the population are less than optimal. Among the Medicaid population there are no racial HIV treatment disparities between Black and White persons living with HIV, which suggests the potential relevance of Medicaid to currently uninsured populations, and the potential to achieve similar levels of equality within Medicaid for Hispanic/Latino enrollees and other segments of the Medicaid population.     

Epidemiological and statistical aspects of the AIDS epidemic: introduction

In 1988, a government working party studied estimates of incidence and prevalence of numbers of acquired immunodeficiency syndrome (AIDS) cases. They investigated a series of epidemiological, statistical and mathematical problems associated with predicting trends in incidences of AIDS. This paper introduces a series of papers that give a fuller and more technical exposition of the appendixes of that working party report. The papers provide a brief background to the current state of knowledge on the epidemiology of the infection and the disease; a deterministic model for human immunodeficiency virus (HIV) transmission in the male homosexual community in England and Wales is introduced. Back-projection methods are studied in two papers, following the distribution of the incubation period of the disease. The concept of minimum size of the epidemic is introduced. Mathematical functions to describe the spread of HIV infection are refined by using past trends in the incidence of AIDS to estimate values for some parameters. Survival times for AIDS patients from the point of diagnosis are considered and evidence for changes in male homosexual sexual behaviour is studied; lag-time from the point of diagnosis to the report of the case is also examined. There is a comparative analysis of the AIDS epidemic in various European countries. The incubation period of HIV in patients with haemophilia A and B infections and the problems associated with making predictions for different at-risk groups or small subgroups based on geographical area are discussed. Reasons for fluctuation between the number of reported cases from month to month are provided.     

HIV behind Bars: Human Immunodeficiency Virus Cluster Analysis and Drug Resistance in a Reference Correctional Unit from Southern Brazil

People deprived of liberty in prisons are at higher risk of infection by the human immunodeficiency virus (HIV) due to their increased exposure through intravenous drug use, unprotected sexual activity, tattooing in prison and blood exposure in fights and rebellions. Yet, the contribution of intramural HIV transmission to the epidemic is scarcely known, especially in low- and middle-income settings. In this study, we surveyed 1,667 inmates incarcerated at Presídio Central de Porto Alegre, located in southern Brazil, for HIV infection and molecular characterization. The HIV seroprevalence was 6.6% (110/1,667). Further analyses were carried out on 40 HIV-seropositive inmates to assess HIV transmission clusters and drug resistance within the facility with the use of molecular and phylogenetic techniques. The molecular epidemiology of HIV-1 subtypes observed was similar to the one reported for the general population in southern Brazil, with the predominance of HIV-1 subtypes C, B, CRF31_BC and unique BC recombinants. In particular, the high rate (24%) of URF_BC found here may reflect multiple exposures of the population investigated to HIV infection. We failed to find HIV-infected inmates sharing transmission clusters with each other. Importantly, the analysis of HIV-1 pol genomic fragments evidenced high rates of HIV primary and secondary (acquired) drug resistance and an alarming proportion of virologic failure among patients under treatment, unveiling suboptimal access to antiretroviral therapy (ARV), low ARV adherence and dissemination of drug resistant HIV strains in primary infections. Our results call for immediate actions of public authority to implement preventive measures, serological screening and, for HIV-seropositive subjects, clinical and treatment follow-up in order to control HIV infection and limit the spread of drug resistance strains in Brazilian prisons.     

Care of HIV-infected patients in China

Compared with high infection areas of the world, the total HIV infection rate in China is relatively low. Nonetheless, because of China's vast territory and large population, the potential infection risk must be taken seriously. In the next few years, needle sharing among injection drug users will remain the most common route of transmission for the HIV/AIDS epidemic in China. Unprotected sex is gradually becoming a major route of transmission. China began to implement HAART in 1999 according to international standards. Prior to 2003, there were only about 150 HIV/AIDS patients were treated with HAART in some clinical trials and about 100 HIV/AIDS patients were treated by private sources. Results of those treatments are the scientific basis for development of the therapeutic strategies in China. In March of 2003, the Chinese government initiated China CARES program. In November of 2003, the Chinese Ministry of Health announced a national policy of free ARV treatment to all HIV+ Chinese citizens who were in poverty and required ARV therapy. There are total of 19,456 HIV/AIDS patients received free ARV drugs to date in 159 regions and 441 towns. Current challenges are how to follow-up and evaluate those patients in the clinical settings. The longer the therapy is postponed, the more side effects and the higher probability of drug resistance are going to occur. It remains unclear, therefore, when HAART regimen should be started in the HIV/AIDS population in China.     

On the uniqueness of endemic equilibria of an HIV/AIDS transmission model for a heterogeneous population

In this paper, we examine an HIV/AIDS transmission model which has been widely used for studying the spread of HIV/AIDS through sexual contact. Some sufficient conditions are obtained for the uniqueness of endemic equilibrium. We also present a two-group model whose no-disease equilibrium is unstable but it has at least three positive endemic equilibria.     

Analysis of recruitment and industrial human resources management for optimal productivity in the presence of the HIV/AIDS epidemic

The aim of this paper is to analyze the recruitment effects of susceptible and infected individuals in order to assess the productivity of an organizational labor force in the presence of HIV/AIDS with preventive and HAART treatment measures in enhancing the workforce output. We consider constant controls as well as time-dependent controls. In the constant control case, we calculate the basic reproduction number and investigate the existence and stability of equilibria. The model is found to exhibit backward and Hopf bifurcations, implying that for the disease to be eradicated, the basic reproductive number must be below a critical value of less than one. We also investigate, by calculating sensitivity indices, the sensitivity of the basic reproductive number to the model’s parameters. In the time-dependent control case, we use Pontryagin’s maximum principle to derive necessary conditions for the optimal control of the disease. Finally, numerical simulations are performed to illustrate the analytical results. The cost-effectiveness analysis results show that optimal efforts on recruitment (HIV screening of applicants, etc.) is not the most cost-effective strategy to enhance productivity in the organizational labor force. Hence, to enhance employees’ productivity, effective education programs and strict adherence to preventive measures should be promoted.     

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.     

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.     

Hierarchical Bayesian inference for HIV dynamic differential equation models incorporating multiple treatment factors

Studies on HIV dynamics in AIDS research are very important in understanding the pathogenesis of HIV‐1 infection and also in assessing the effectiveness of antiretroviral (ARV) treatment. Viral dynamic models can be formulated through a system of nonlinear ordinary differential equations (ODE), but there has been only limited development of statistical methodologies for inference. This article, motivated by an AIDS clinical study, discusses a hierarchical Bayesian nonlinear mixed‐effects modeling approach to dynamic ODE models without a closed‐form solution. In this model, we fully integrate viral load, medication adherence, drug resistance, pharmacokinetics, baseline covariates and time‐dependent drug efficacy into the data analysis for characterizing long‐term virologic responses. Our method is implemented by a data set from an AIDS clinical study. The results suggest that modeling HIV dynamics and virologic responses with consideration of time‐varying clinical factors as well as baseline characteristics may be important for HIV/AIDS studies in providing quantitative guidance to better understand the virologic responses to ARV treatment and to help the evaluation of clinical trial design in response to existing therapies.     

Evidence for recent changes in sexual behaviour in homosexual men in England and Wales

Over 80% of cases of Acquired Immune Deficiency Syndrome (AIDS) in England and Wales have occurred in homosexual men. Changes in sexual behaviour in this group may have a substantial influence on the incidence of Human Immunodeficiency Virus (HIV) infection and will therefore be crucial in determining future cases of AIDS. This paper critically weighs the indirect and direct evidence for changes in behaviour in homosexual men since the advent of the AIDS epidemic. The paper reports on falling incidence of gonorrhoea, hepatitis B and syphilis in homosexual men, the changes being most marked from 1985 onwards. Data on temporal trends in HIV prevalence and incidence in homosexual men are reviewed. These suggest that the maximum incidence of HIV infection occurred in 1982-84 and may have fallen since then. Evidence for a concomitant change in sexual behaviour is reported from several sources. This points towards a recent change in sexual behaviour characterized by reduction in the numbers of partners and adoption of safer sexual practices. In some places change may have occurred as early as 1983. A change became apparent generally in 1985 and this appears to have been sustained in 1986-87. Nevertheless, a substantial proportion of homosexual men studied continue to practice high risk sexual practices, such as anal intercourse, including relationships with casual partners.     

The effect of density-dependent treatment and behavior change on the dynamics of HIV transmission

In this work, we propose a model for heterosexual transmission of HIV/AIDS in a population of varying size with an intervention program in which treatment and/or behavior change of the infecteds occur as an increasing function of the density of the infected class in the population. This assumption has socio-economic implications which is important for public health considerations since density-dependent treatment/behavior change may be more cost-saving than a program where treatment/behavior change occurs linearly with respect to the number of infecteds. We will make use of the conservation law of total sexual contacts which enables us to reduce the two-sex model to a simpler one-sex formulation. Analytical results will be given. Unlike a similar model with linear treatment/behavior change in Hsieh (1996) where conditions were obtained for the eradication of disease, we will show that density-dependent treatment/behavior change cannot eradicate the disease if the disease is able to persist without any treatment/behavior change. This work demonstrates the need to further understand how treatment/behavior change occurs in a society with varying population.