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 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.     

A dynamical system for the approximate moments of nonlinear stochastic models of spiking neurons and networks

We consider multidimensional systems of coupled nonlinear stochastic differential equations suitable for the study of the dynamics of collections of interacting noisy spiking neurons. Assumptions based on the smallness of third and higher central order moments of membrane potentials and recovery variables are used to derive a system of ordinary differential equations for the approximate means, variances, and covariances. We show the usefulness of such a derivation for different cases of model neurons under the action of white noise currents.     

一类HIV/AIDS流行病模型

结合某市的艾滋病现状给出了相应的传染病动力学模型,研究了其平衡点的稳定性,讨论了流行病的阈值,并对不同的说服率、不同的因病死亡率、不同的传染率分别进行了数值模拟,对该市艾滋病的预防和控制给出了理论上的指导和建议.     

A fractional order epidemic model and simulation for avian influenza dynamics

We present a nonlinear fractional order epidemic model to investigate the spreading dynamical behavior of the avian influenza. The population of the model contains susceptible individuals, asymptomatic but infective latent individuals, and infective individuals. We first establish the existence, uniqueness, nonnegativity, and positive invariance of the solution, then we study the reproduction number of the model and the stability of the disease‐free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative ν. In terms of epidemics, this suggests that varying ν induces a change in the avian's epidemic status. Furthermore, we derive the sufficient conditions for the existence and the stability of the endemic equilibrium. Finally, we carry out some numerical simulations to validate the analytical results. We find from simulations that the solution of the fractional order model tends to a stationary state over a longer period of time with decreasing the value of the fractional derivative, and the size of epidemic decreases with decreasing ν.     

行为改变下的艾滋病流行预测

基于亚洲艾滋病流行模型(Asian Epidemic Model,AEM)基本思想,建立我国艾滋病流行预测模型.利用该模型,估计某地1989—2007年艾滋病流行状况,同时设计不同行为改变方案,预测该地2008—2020年艾滋病流行趋势.     

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.     

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.     

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

An HIV/AIDS epidemic model with different latent stages and treatment is constructed. The model allows for the latent individuals to have the slow and fast latent compartments. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are determined by the basic reproduction number under some conditions. If R₀ < 1, the disease free equilibrium is globally asymptotically stable, and if R₀ > 1, the endemic equilibrium is globally asymptotically stable for a special case. Some numerical simulations are also carried out to confirm the analytical results.     

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 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.     

Assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of HIV/AIDS in heterosexual settings

A deterministic compartmental sex-structured HIV/AIDS model for assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of the disease in heterosexual settings in which homosexuality and bisexuality issues have remained taboo is presented. The epidemic threshold and equilibria for the model are determined and stabilities are investigated. Comprehensive qualitative analysis of the model including invariance of solutions and permanence are carried out. The epidemic threshold known as the basic reproductive number suggests that heterosexuality, homosexuality, and bisexuality influence the growth of the epidemic in HIV/AIDS affected populations and the partial reproductive number (homosexuality induced or heterosexuality and bisexuality induced) with the larger value influences the overall dynamics of the epidemic in a setting. Numerical simulations of the model show that as long as one of the partial reproductive numbers is greater than unity, the disease will exist in the population. We conclude from the study that homosexuality and bisexuality enlarge the epidemic in a heterosexual setting. The theoretical study highlights the need to carry out substantial research to map homosexuals and bisexuals as it has remained unclear as to what extent this group has contributed to the epidemic in heterosexual settings especially in southern Africa, which has remained the epidemiological locus of the epidemic.     

Robust discrimination under a hierarchy on the scatter matrices

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.     

Stochastic models of HIV epidemic in homosexual populations—the effects of mixing patterns

Extending my previous work [1–3], in this paper I proceed to develop a stochastic model for HIV epidemic in a homosexual population under general conditions. Through computer generated data, I assess various deterministic models as compared with the expected numbers of the stochastic models. It is shown that different mixing patterns have significant impacts on the HIV epidemic except possibly restricted mixing. Thus, in populations with preferred mixing (mixing proportion less than 1) and with proportional mixing, the numbers of S people, L people, I people and A people differ significantly from the corresponding expected numbers of the stochastic models. For the L people, I people and A people, the numbers of the deterministic models first appear to be smaller and later appear to be larger than the corresponding mean numbers of the stochastic models, indicating that while in the short run the deterministic models would underestimate the true numbers, in the long run the deterministic models would overestimate the true numbers.     

A stochastic model for the HIV epidemic in homosexual populations involving age and race

In this paper, we derive a stochastic model for the HIV epidemic in homosexual populations involving age and race. To account for effects of different mixing patterns, a low risk selection rule is introduced. The model is then formulated in terms of chain multinomial distributions by means of which the means are derived. Some simulation studies by computer indicate clearly that age and race have a significant impact on the HIV epidemic.     

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.     

一类具有说服率的HIV/AIDS流行病模型

根据某市艾滋病出现的新特点,即外来人口对艾滋病的影响,给出了相应的传染病动力学模型,并进行了数值模拟.     

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.     

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.     

Conditionally heteroscedastic factorial HMMs for time series in finance

In this article, we develop a new approach within the framework of asset pricing models that incorporates two key features of the latent volatility: co‐movement among conditionally heteroscedastic financial returns and switching between different unobservable regimes. By combining latent factor models with hidden Markov chain models we derive a dynamical local model for segmentation and prediction of multivariate conditionally heteroscedastic financial time series. We concentrate more precisely on situations where the factor variances are modelled by univariate generalized quadratic autoregressive conditionally heteroscedastic processes. The expectation maximization algorithm that we have developed for the maximum likelihood estimation is based on a quasi‐optimal switching Kalman filter approach combined with a generalized pseudo‐Bayesian approximation, which yield inferences about the unobservable path of the common factors, their variances and the latent variable of the state process. Extensive Monte Carlo simulations and preliminary experiments obtained with daily foreign exchange rate returns of eight currencies show promising results. Copyright © 2007 John Wiley & Sons, Ltd.