Deriving the HIV incubation period distribution for AIDS using immunological markers and devising infection treatment strategies

Survey data suggest that it is impossible for HIV infecteds to develop AIDs if the values of their CD4+ T-cell densities are above a critical threshold. An infected whose CD4+ T-cell density falls below 200 cells per microliter is now automatically regarded as having AIDS by the CDC. Using the CD4+ T-cell density as a surrogate marker of disease progression, a model that is consistent with the data is developed and applied to the homosexual/bisexual and IVDU risk groups. Assuming that the critical CD4+ T-cell density for these risk groups are identical, it is found that their progression towards AIDS during the incubation period is identical, suggesting that the dynamics of the HIV infection may be independent of risk group. The different incubation period distributions obtained from this modelling for these two risk groups is shown to be entirely due to their different normal seronegative CD4+ T-cell density distributions. Using IFN-γ as a surrogate marker is shown to give similar results.The impact of the HIV infection on the immune system is reviewed, and immunological infection models are developed. The data suggest to this author that Homo sapiens have generally lost the ability to generate T-cells and B-cells with the specificity necessary to neutralize HIV as they evolved from the primates. It is plausible that a legacy of primate immunity to HIV still remains in the 10% of Homo sapiens who show no immune system deterioration in the first 10 years of the HIV infection. New HIV infection treatment strategies based on this model are devised and discussed.     

Accurately simulating the growth in the size of the HIV infected population in any AIDS epidemic country: Computing the U.S.A. HIV infection curve

A reliable approach to the simulation of the time-dependent growth of the size of a country's HIV population is described in detail and applied to the USA epidemic. The simulation depends on a knowledge of AIDS incidence data and the HIV incubation period distribution but is independent of any model regarding how the disease was spread. Using the Centers for Disease Control's December 31, 1991 update of the reported AIDS incidence data, a cumulative total of 645,445 Americans was calculated to be HIV infected as of January 1, 1991.The HIV infection curves for the USA risk groups were separately computed, and they indicate that the current rates of the spread of the infection in all of the risk groups are small fractions of what they were in the early phase of the epidemic. In fact, the calculated increase in the cumulative number of USA HIV infecteds from January 1, 1990 to January 1, 1991 was only 1.44%. These results suggest that the annual number of AIDS cases to be obtained in the next few years will not be substantially different from what it was in 1991. Since the calculated HIV infection curves for the transfusion and hemophiliac risk groups are currently growing at a particularly low rate, the modelling results confirm the great safety of the nation's blood and blood product supplies.     

香港艾滋病近期传播特征与动态预测

首次研究出香港艾滋病目前传播势态为一元线性回归方程,并由方程的预测值和斜率的变化动态监测香港艾滋病近期发病趋势,艾滋病病毒感染者年平均发病率为255至271个,艾滋病病患者年平均发病率为56个左右.本文研究方法简单易行,对香港政府近期预防艾滋病有重要的理论和实用价值.     

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.     

Modeling the dynamical impact of HIV on the immune system: Viral clearance,infection, and AIDS

A phenomenological model simulating the time-dependent consequences of the HIV challenge on the immune system is presented. One of the important features of the model is its ability to handle T helper cell production and apoptosis (genetically determined suicide). The values of the independent, generally time-dependent, model parameters were chosen to be compatible with known experimental data. A new approach to the numerical solution of the resulting coupled, nonlinear model equations is presented, and simulations of a typical viral challenge that is cleared and one that leads to infection and AIDS are exhibited.It is shown that a change in the saturated value of a single model parameter is sufficient to change a simulated challenge on its way to being cleared into one that leads to infection instead (and vice versa). If the saturated values of all of the independent model parameters are known at the beginning of a challenge, the outcome of the challenge can be predicted in advance. If the virulence of the HIV strain (defined in this paper) is above a critical threshold at inoculation, infection will result regardless of the initial viral load. This latter result could explain why accidental HIV contaminated needle sticks sometime result in infection regardless of the counter-measures undertaken.A model simulating the time evolution of the collapse of the T helper cell density leading to AIDS is introduced. This model consists of immunological and mathematical parts and is compatible with experimental data. The immediate cause of the beginning of this collapse is postulated to be a spontaneous mutation of the virus into a more virulent form that not only leads to an explosion in the viral load but also to a dramatic increase in the level of induced apoptosis of T helper cells. The results of this model are consistent with the known experimental behavior of the viral load and T helper cell densities in the final stage of HIV infection.     

Asymptotic properties of an HIV/AIDS model with a time delay

A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.     

带有发病年龄的HIV/AIDS传播的数学建模及分析

所建的模型是用含有常微分方程和偏微分方程的方程组来描述的,考虑了垂直感染,发病年龄,AIDS病人有希望恢复到潜伏期以及不同患者接受治疗的能力不同等诸多因素.利用导出的方程,直接推出:当AIDS引起死亡率增加时,社会总人口衰减.利用泛函分析方法和有界线性算子半群理论分析了系统的适定性,并证明了系统方程存在正解.     

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.     

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.     

Estimating HIV/AIDS incidence: Adjusting case reports for changes in definition

The purpose of this paper is to assess different procedures used in the modeling of the AIDS epidemic that account for changes in the definition of AIDS. Specifically, we assess three methods that adjust for the effect of definition changes on the modeling of HIV reconstructions. Our results suggest that how we account for definition changes is a critical influence on the severity and the shape of the estimated HIV/AIDS epidemic. Also, the scope of the recent (1993) definition change is sufficiently broad to limit approaches based on adjusting to a single, consistent definition. The most realistic approach appears to be one based on a dynamic disease progression model that accounts for AIDS definition changes via changes in disease incubation.     

A nonlinear HIV/AIDS model with contact tracing

A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical simulation. The model analysis shows that contact tracing may be of immense help in reducing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction.     

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.     

一类具有双线性传染率的HIV/AIDS病毒动力学改进模型

针对HIV/AIDS传播的具有常数移民和指数出生的SI型模型,为了更加符合实际意义,对具有双线性传染率的模型进行局部改进,并对改进后的动力学模型进行了简化.对于改进后的模型,证明了平衡点的存在与局部稳定性,并证明了传染病毒的灭绝与持续性,得到了传染病毒的基本再生数.结果表明:当单位时间内从外界迁入人口中染病者的比例系数c近似等于零时,基本再生数小于1时,传染病毒最终灭绝;当基本再生数大于1时,模型存在唯一的正平衡点,且是局部渐近稳定的,说明传染病毒一致持续存在.     

Asymptotics for a Curve with a Given Distribution Density of the First Exit Position for a Wiener Process

The following inverse first exit problem for a Wiener process is considered: to find an upper class with a given distribution of the first exit point from the domain bounded by this curve. Some estimates are obtained for a curve with a given density at zero. Bibliography: 2 titles.     

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.     

The persistence of HIV-1 spreading in MSM population in China

In this work, we develop and analyze mathematical models for the dynamics about the evolution of HIV/AIDS in men who have sex with men in China. We focus on the analyses of the basic reproduction number $R_0$ and the persistence of infection. Through simulations we also find that the interventions including antiviral therapy, condom using and potential vaccinations play very important roles in the HIV/AIDS spreading in MSM population in China.     

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.     

The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

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.     

一类SIQA模型解的全局稳定性

讨论用脉冲隔离的方案控制HIV的传播.假定艾滋病感染者发展成艾滋病人和感染年龄有关,我们建立了带脉冲隔离类和感染年龄的HIV模型.在一定条件下证明该模型的无病平衡态是全局稳定的.