Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives

In this paper, we study the impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives on the transmission dynamics of the disease in a homogeneous population with constant immigration of susceptibles incorporating use of condom, screening of unaware infectives and treatment of the infected. Initially we consider constant controls and thereafter treat control measures as time dependent control parameters. In the constant controls case, we calculate the basic reproduction number and investigate the existence and stability of equilibria. The model is found to exhibit transcritical bifurcation. For the time dependent controls, we formulate the appropriate optimal control problem and investigate the necessary conditions for the disease control in order to determine the role of unaware infectives in the spread of HIV/AIDS. We observed that unawareness by infectives has a great cost impact on the community. We further investigate the impact of combinations of the strategies in the control of HIV/AIDS. Carrying out cost-effectiveness analysis, we found that the most cost-effective strategy is the combination of all the control strategies.     

Modeling sexual transmission of HIV/AIDS in Jiangsu province,China

HIV transmission by sexual activities exhibits a substantial increase and has become a primary transmission mode in China recently. A mathematical model is formulated so as to identify the key processes and parameters that could explain the quick increase in the proportion of heterosexual transmission and further to assist in suggesting control measures urgently. On the basis of surveillance data on a number of people living with HIV/AIDS in Jiangsu province, we parameterize the model and estimate the reproduction number by using the least squares method. The basic reproduction number was estimated to be R₀ = 3.52 for the therapy scenario of heterosexual transmission. The model predicts that the epidemic will peak in 2020. New infections are sensitive to the transmission coefficient, dependent on condom use rate, and the risky activities during the early period, whereas are sensitive to the recruitment rate in the late period of the transmission respectively. Antiviral therapy can either increase or decrease the new infections depending on both the extended life span of treated individuals and the infectiousness of the treated individuals. Hence, effective control measures during different transmission periods can be suggested, and antiretroviral therapy is a contentious issue for disease control. Copyright © 2012 John Wiley & Sons, Ltd.     

Global dynamics and cost-effectiveness analysis of HIV pre-exposure prophylaxis and structured treatment interruptions based on a multi-scale model

Structured treatment interruptions (STI) were proposed to reduce costs and side effects for HIV infected individuals, but whether the possible viral rebound within hosts after treatment interruption would lead to more new infections and additional costs among the population remains unknown. Oral pre-exposure prophylaxis (PrEP) is shown as an effective but expensive strategy to prevent the acquisition of HIV infection. To investigate the effectiveness and cost-effectiveness of STI and PrEP, we develop a multi-scale model linking within-host and between-host dynamics in the presence of drug resistance. Lyapunov functionals are constructed to analyze the global dynamics of the coupled system. We fit this model to the annual AIDS incidence and death data from 1980 to 2014 among men who have sex with men (MSM) in San Francisco and compare the impact of six various intervention scenarios (low, medium, high PrEP coverage with or without STI) on new infections and cost-effectiveness over the next 20 years. We estimate the PrEP efficacy needed to eliminate the disease for different fraction of acquired drug resistance under the above six scenarios. Numerical simulations show that expanding PrEP coverage is very cost-effective, but whether implementing STI is cost-saving depends on the efficacy of second-line drugs. That is to say, STI could always save money, but it may lead to more (less) new infections than continuous therapy and thus less (more) health benefit for high (low) efficacy of second-line drugs. These results provide insights about the long-term effect of STI and PrEP on the disease control and cost-effectiveness.     

SWEIA艾滋病毒传染模型及应用

人类免疫缺陷病毒(HIV)是一种严重威胁生命的病毒,感染艾滋病毒患者一般经历四个阶段:i)艾滋病毒阴性的窗口期(W);ii)阳性的无症状潜伏期(E);iii)有症状期(Ⅰ);以及iv)移除阶段(A).为深入研究艾滋病传播过程,建立SWEIA艾滋病毒传染模型,定义基本再生数,分析无病与地方病平衡点的存在性和局部稳定性,根据2004至2015年中国艾滋病患者数据,采用遗传算法对SWEIA模型中参数进行估计.通过对基本再生数敏感性分析以及模型数值随参数不同而产生的变化,揭示艾滋病窗口期的接触率是影响艾滋病流行的主要原因之一.     

A linked risk group model for investigating the spread of HIV

This paper describes a model that simulates the spread of HIV and progression to AIDS. The model is based on classical models of disease transmission. It consists of six linked risk groups and tracks the numbers of infectives, AIDS cases, AIDS related deaths, and other deaths of infected persons in each risk group. Parametric functions are used to represent risk-group-specific and time-dependent average contact rates. Contacts are needle sharing, sexual contacts, or blood product transfers.

An important feature of the model is that the contact rate parameters are estimated by minimizing differences between AIDS incidence and reported AIDS cases adjusted for undercounting biases. This feature results in an HIV epidemic curve that is analogous to one estimated by backcalculation models but whose dynamics are determined by simulating disease transmission. The model exhibits characteristics of both the disease transmission and the backcalculation approaches, i.e., the model:

• reconstructs the historical behavior patterns of the different risk groups,

• includes separate effects of treatment and changes in average contact rates,

• accounts for other mortality risks for persons infected with HIV,

• calculates short-term projections of AIDS incidence, HIV incidence, and HIV prevalence,

• calculates cumulative HIV infections (the quantity calculated by backcalculation approaches) and HIV prevalence (the quantity measured by seroprevalence and sentinel surveys). This latter feature permits the validation of the estimates generated by two distinct approaches.

We demonstrate the use of the model with an application to U.S. AIDS data through 1991.     

Modeling and dynamics of HIV transmission among high-risk groups in Guangzhou city, China

In this paper, a multicompartmental model is formulated to study how HIV is transmitted among different HIV high-risk groups, including MSM (men who have sex with men), FRs (foreigner residents), FSWs (female sex workers), and IDUs (injection drug users). The explicit expression for the basic reproduction number is obtained via the next generation matrix approach. We show that the disease free equilibrium is locally as well as globally asymptotically stable (the disease goes to extinction) when the basic reproduction number is less than unity, and the disease is always present when the basic reproduction number is larger than unity. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct a case study where model parameters are estimated from the demographic and epidemiological data from Guangzhou. Using the parameter estimates, we predict the HIV/AIDS trend for each high-risk group. Furthermore, our study suggests that reducing the transmission routes of the disease and increasing condom use will be useful for control of HIV transmission.     

一类非线性动力系统的数值分析及控制策略研究

根据艾滋病在新疆的流行特点,建立了一个非线性动力系统的数学模型来研究艾滋病在新疆高危人群中传播的规律.通过查阅大量的统计数据和文献资料,确定了模型中部分参数的具体数值,然后通过数据拟合的方法得到了各个高危人群中的HIV病毒的传染性系数.在模型中,选择2004年底(2005年初)作为系统的初始点,预测了艾滋病未来几年内在新疆的流行趋势.最后,提出并比较遏止艾滋病传播的各项干预措施.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law

The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV-related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV-TB coinfected model in the Atangana-Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers-Ulam (HU) stability and numerical simulations with assumption of specific parameters.     

Use of discrete-event simulation to evaluate strategies for the prevention of mother-to-child transmission of HIV in developing countries

HIV/AIDS affects over 40 million people worldwide, and more than 70% of these people live in Africa. Mother-to-child transmission of HIV accounts for over 90% of all HIV infections in children under the age of 15 years. However, implementing HIV prevention policies in Africa is extremely difficult because of the poor medical and socio-economic infrastructure. In this paper, we present a discrete-event simulation model that evaluates the relative benefits of two potentially affordable interventions aimed at preventing mother-to-child transmission of HIV, namely anti-retroviral treatment at childbirth and/or bottlefeeding strategies. The model uses rural Tanzanian data and compares different treatment policies. Our results demonstrate that strategic guidelines about breastfeeding are highly dependent on the assumed increase in infant mortality due to bottlefeeding, the efficacy of anti-retroviral treatment at childbirth, and the maternal health stage. The cost of averted infections, though low by Western standards, may represent significant obstacles to policy implementation in developing countries.     

Study on a HIV/AIDS model with application to Yunnan province,China

This paper presents an epidemic model aiming at the prevalence of HIV/AIDS in Yunnan, China. The total population in the model is restricted within high risk population. By the epidemic characteristics of HIV/AIDS in Yunnan province, the population is divided into two groups: injecting drug users (IDUs) and people engaged in commercial sex (PECS) which includes female sex workers (FSWs), and clients of female sex workers (C). For a better understanding of HIV/AIDS transmission dynamics, we do some necessary mathematical analysis. The conditions and thresholds for the existence of four equilibria are established. We compute the reproduction number for each group independently, and show that when both the reproduction numbers are less than unity, the disease-free equilibrium is globally stable. The local stabilities for other equilibria including two boundary equilibria and one positive equilibrium are figured out. When we omit the infectivity of AIDS patients, global stability of these equilibria are obtained. For the simulation, parameters are chosen to fit as much as possible prevalence data publicly available for Yunnan. Increasing strength of the control measure on high risk population is necessary to reduce the HIV/AIDS in Yunnan.     

Modelling the Spread of HIV/AIDS Epidemic

In this paper, a deterministic mathematical model for the spread of HIV/AIDS in a variable size population through horizontal transmission is considered. The existence of a threshold parameter, the basic reproduction number, is established, and the stability of both the disease-free equilibrium and the endemic equilibrium is discussed in terms of $R_0$.     

The population‐level impact of HBV and its vaccination on HIV transmission dynamics

Hepatitis B virus (HBV) and its vaccination strategy may affect human immunodeficiency virus (HIV) transmission dynamics because both viruses have synergistic effects. To quantitatively assess the potential impact of HBV and its vaccination strategy on HIV transmission dynamics at the population level, in this paper, we formulate a deterministic compartmental model that describes the joint dynamics of HBV and HIV. We first derive the explicit expressions for the basic reproduction numbers of HIV and HBV and analyze the dynamics of HIV and HBV subsystems, respectively. Then a systematic qualitative analysis of the full system is also provided, which includes the local and global behavior. By using a set of reasonable parameter values, the full system is numerically investigated to assess the potential impact of HBV and its vaccination strategy on HIV transmission. The direct and indirect population level impact of HBV on HIV is demonstrated by calculating the fraction of HIV infections attributable to HBV and the difference between HIV prevalence in the presence and absence of HBV, respectively. The findings imply that the increase of HBV vaccination rate may unusually accelerate HIV epidemics indirectly, although the direct effect of HBV on HIV transmission decreases as HBV vaccination rate increases. Finally, the potential impact of HIV on HBV transmission dynamics is investigated by way of parenthesis. Copyright © 2016 John Wiley & Sons, Ltd.     

具有商业性行为的女性吸毒者对HIV/AIDS传播的影响

利用数学模型,研究了具有商业性行为的女性吸毒者对HIV/AIDS传播的影响.通过理论分析,讨论了系统的一致持续性和地方病平衡点的存在性,从理论上揭示了女性吸毒者的商业性行为可加强HIV/AIDS的传播和流行.特别地,若无商业性行为且吸毒人群和一般男性人群中均无疾病流行时,商业性行为的存在将会导致两类人群中的疾病均流行起来.这为防控工作的开展提供了重要参考.     

Bayesian segmental growth mixture Tobit models with skew distributions

This paper presents an extension of the standard Tobit to simultaneously address segmental phases, subpopulation heterogeneity, lower limit of detection, and skewness in outcomes of human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) longitudinal data. A major problem often encountered in an HIV/AIDS research is the development of drug resistance to antiretroviral (ARV) drug or therapy. For dealing with drug resistance problem, estimating the time at which drug resistance would develop is usually sought. Following ARV treatment, the profile of each subject’s viral load tends to follow a ‘broken stick’ like growth trajectory, indicating multiple phases of decline and increase in viral loads. Such multiple phases with multiple change-points are captured by subject-specific random parameters of growth curve models. To account subpopulation heterogeneity of drug resistance among patients, the turning-points are also allowed to differ by latent classes of patients on the basis of trajectories of observed viral loads. These features of viral longitudinal data are jointly modeled in a unified framework of segmental growth mixture Tobit mixed-effects models with skew distributions for a response variable with left censoring and skewness under the Bayesian approach. The proposed methods are illustrated using real data from an AIDS clinical study.     

A CA-based epidemic model for HIV/AIDS transmission with heterogeneity

The complex dynamics of HIV transmission and subsequent progression to AIDS make the mathematical analysis untraceable and problematic. In this paper, we develop an extended CA simulation model to study the dynamical behaviors of HIV/AIDS transmission. The model incorporates heterogeneity into agents’ behaviors. Agents have various attributes such as infectivity and susceptibility, varying degrees of influence on their neighbors and different mobilities. Additional, we divide the post-infection process of AIDS disease into several sub-stages in order to facilitate the study of the dynamics in different development stages of epidemics. These features make the dynamics more complicated. We find that the epidemic in our model can generally end up in one of the two states: extinction and persistence, which is consistent with other researchers’ work. Higher population density, higher mobility, higher number of infection source, and greater neighborhood are more likely to result in high levels of infections and in persistence. Finally, we show in four-class agent scenario, variation in susceptibility (or infectivity) and various fractions of four classes also complicates the dynamics, and some of the results are contradictory and needed for further research.     

寨卡病毒传播潜力与控制策略有效性分析

目前寨卡病毒已在超过65个国家和地区传播, 为了估计新加坡寨卡病毒的传播潜力和有关控制策略的有效性, 首先采用经典的传染病模型并结合累计报告病例数, 借助最小二乘法和MCMC方法进行模型参数估计, 寻求拟合累计病例数最佳的参数集合及其相应的置信区间.进而根据再生矩阵法求得的基本再生数公式,得到了新加坡寨卡爆发的阈值参数R₀的估计值和置信区间, 通过对比分析验证了新加坡寨卡病毒传播基本再生数的可靠性.之后, 分析了累计病例数对各个关键参数的敏感性, 探讨针对寨卡病毒传播控制策略的有效性.结果表明: 在对新加坡寨卡病毒的控制中, 需要通过增加检疫次数和检疫率、对患者进行隔离以及有效地灭蚊, 并且通过减少疫区的游客数量达到控制疫情的效果.     

Global results for an HIV/AIDS model with multiple susceptible classes and nonlinear incidence

In this paper, an HIV/AIDS epidemic model is proposed in which there are two susceptible classes. Two types of general nonlinear incidence functions are employed to depict the scenarios of infection among cautious and incautious individuals. Qualitative analyses are performed, in terms of the basic reproduction number $\R_0$, to gain the global dynamics of the model: the disease-free equilibrium is of global asymptotic stability when $\R_0\leq 1$; a unique endemic equilibrium exists and globally asymptotically stable $\R_0> 1$. The introduction of cautious susceptible and the resulting multiple transmission functions has positive effect on HIV/AIDS prevalence. Numerical simulations are carried out to illustrate and extend the obtained analytical results.     

Modelling circumcision and condom use as HIV/AIDS preventive control strategies

We present a sex-structured model for heterosexual transmission of HIV/AIDS with explicit incubation period for modelling the effect of male circumcision as a preventive strategy for HIV/AIDS. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with delay due to incubation period. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to incorporate the effects of condom use as another preventive strategy for controlling HIV/AIDS. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male circumcision and condom use in a community. The models are numerically analysed to assess the effects of the two preventive strategies on the transmission dynamics of HIV/AIDS. We conclude from the study that in the continuing absence of a preventive vaccine or cure for HIV/AIDS, male circumcision is a potential effective preventive strategy of HIV/AIDS to help communities slow the development of the HIV/AIDS epidemic and that it is even more effective if implemented jointly with condom use. The study provides insights into the possible community benefits that male circumcision and condom use as preventive strategies provide in slowing or curtailing the HIV/AIDS epidemic.     

Modelling treatment effects in the HIV/AIDS epidemic

Advances in recent treatments for HIV/AIDS patients have shown dramatic outcomes in extending the incubation period and AIDS survival time, while also providing significant improvements in the quality of patients' lives. A compartmental model is proposed to analyse the effects of the various treatment regimens which have been introduced. The results produced are in good agreement with routinely collected data relating to levels of HIV/AIDS incidence and prevalence in the UK homosexual population. Some parameter values within the model are obtained from surveys, census results, etc, but others are derived using a maximum likelihood estimation procedure. Finally, the model is used to project levels of incidence and prevalence over the next few years, and to investigate several possible scenarios.