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.     

Dynamical analysis of public health education on HIV/AIDS transmission

An Human Immunodeficiency Virus/Acquired Immuno‐Deficiency Syndrome (HIV/AIDS) epidemic model for sexual transmission with asymptomatic and symptomatic phase is proposed as a system of differential equations. The threshold and steady state for the model are determined and stabilities of disease free steady state is investigated. We use the model and study the effect of public health education on the spread of HIV/AIDS as a single‐strategy in HIV prevention. The education, including basic reproduction number for the model with public health education, is compared with the basic reproduction number for the HIV/AIDS in the absence of public health education. By comparing these two values, influence of public health education appears. According to property of , threshold proportion of educated adolescents, education rate for susceptible individuals and education efficacy is obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Quantifying the strength and durability of induced immunity to HIV infection in women engaging in unprotected sexual contacts with infected men

Evidence is accumulating that exposure to human immunodeficiency virus (HIV) can lead to an increased resistance or immunity to subsequent infection. A multirisk model that permits either induced immunity or infection to develop after heterosexual inoculation with HIV is shown to be compatible with a wide spectrum of disparate male-to-female transmission data.When the model is applied to time-dependent, HIV-seroprevalence data, the probability that an unexposed woman would remain unexposed after an unprotected contact with an infected man was estimated to be greater than 0.95 on the average. Thus, it would require at least 14 unprotected sexual contacts with HIV-infected men for 50% of an unexposed cohort of women to become exposed to the virus. This suggests that there is a low probability that HIV virions will be found to have penetrated the mucosal barriers of the reproductive tract after a contact.The model also predicts, that the average woman whose mucosal barriers have been breached by HIV has a significant probability of developing immunity to the virus rather than infection. Modelling data for a cohort of unexposed Nairobi women leads to the prediction that the probability of acquiring induced immunity per contact is about 60% of the probability of acquiring the disease per contact.The modelling results also predict that those who had developed resistance to HIV run the small, but significant risk of becoming infected nonetheless by continuing high-risk behavior. For the common contact rate of ten per month, the modelling predicts that the HIV-transmission risk per contact for unexposed women in the Nairobi cohort is 1/178 while the transmission risk for the cohort's immunized women is 1/1548. These numbers suggest that HIV infection is difficult to transmit through heterosexual intercourse on the average and that male-to-female HIV-transmission risk per contact for African women lies between 1/178 and 1/1548.Direct confirmation of the predictions in the last paragraph has been subsequently observed in two completely independent studies. The Nairobi research team recently reported that a notable number of Nairobi prostitutes previously identified to be members of the HIV-resistant group became infected nonetheless. Second, in a study of 174 sexually monogamous, discordant couples in Rakai, Uganda reporting contacts rates of nine to ten per month, the male-to-female HIV-transmission risk per contact was found to be 1/769 by direct measurement, a value that falls between the above limits of 1/178 and 1/1548 predicted by the modelling. Thus, a second major prediction of this paper has been directly confirmed, and induced immunity to HIV is limited and not absolutely protective.Circumstantial evidence suggests that the induced immunity to HIV predicted by the model could be generated and/or initiated by nonspecific innate immune responses, specific immunological responses, including IgA-mediated mucosal immunity and cytotoxic T lymphocytes (CTL) immunity, or some combination of the above. It is suggested here, that a decrease in the ability of HIV virions to penetrate the protective mucus layer of the reproductive tract may be a prerequisite, cofactor, or the principle cause of the induced immunity or resistance demonstrated to exist in this paper. The value of the probability that induced immunity to HIV will develop after a contact is shown to be a sensitive function of the woman's human leucocyte antigen (HLA) supertype profile.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.     

Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.     

A fractional order HIV/AIDS model based on the effect of screening of unaware infectives

In this paper, a fractional order model for the spread of human immunodeficiency virus (HIV) infection is proposed to study the effect of screening of unaware infected individuals on the spread of the HIV virus. For this purpose, local asymptotic stability analysis of the disease‐free equilibrium is investigated. In addition, the model is studied for different values of the fractional order to show the relation between the variations of the reproduction number and the order of the proposed model. Finally, numerical solutions are simulated by using a predictor‐corrector method to illustrate the dynamics between susceptible individuals and unaware infected individuals.     

Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection

A susceptible‐infected‐susceptible (SIS) epidemic reaction‐diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed‐point theorem. The global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. When the saturated incidence rate tends to infinity, the susceptible and infective distributes in the habitat in a nonhomogeneous way; this result is in strong contrast with the case of no spontaneous infection, where the endemic equilibrium tends to the disease free equilibrium. Our analysis shows that the spontaneous infection can enhance the persistence of an infectious disease and may provide some useful implications on disease control.     

Effect of a preventive vaccine on the dynamics of HIV transmission

A deterministic mathematical model for the transmission dynamics of HIV infection in the presence of a preventive vaccine is considered. Although the equilibria of the model could not be expressed in closed form, their existence and threshold conditions for their stability are theoretically investigated. It is shown that the disease-free equilibrium is locally–asymptotically stable if the basic reproductive number (thus, HIV disease can be eradicated from the community) and unstable if (leading to the persistence of HIV within the community). A robust, positivity-preserving, non-standard finite-difference method is constructed and used to solve the model equations. In addition to showing that the anti-HIV vaccine coverage level and the vaccine-induced protection are critically important in reducing the threshold quantity , our study predicts the minimum threshold values of vaccine coverage and efficacy levels needed to eradicate HIV from the community.     

The coupled within- and between-host dynamics in the evolution of HIV/AIDS in China

In this work, we develop and analyze mathematical models for thecoupled within-host and between-host dynamics caricaturing the evolutionof HIV/AIDS. The host population is divided into susceptible, the infectedwithout receiving treatment and the infected receiving ART treatment in accordancewith China’s Four-Free-One-Care Policy. The within-host model is a typicalODE model adopted from literatures. The between-host model incorporatesage-since-infection described by a system of integrodifferential equations. Thetwo models are coupled via the viral load and number of CD4+ T cells ofwithin the hosts. For the between-host model with an arbitrarily selectedHIV infected individual, we focus on the analyses of the basic reproductionnumber R0 and the stabilities of equilibria. Through simulations we also findthat the within-host dynamics does influence the between-host dynamics, andthe nesting of within-host and between-host play a very important role in theHIV/AIDS evolution.     

Modelling the dynamics of avian influenza with nonlinear recovery rate and psychological effect

In this paper, a SI-SEIR type avian influenza epidemic model with psychological effect, nonlinear recovery rate and saturation inhibition effect is formulated to study the transmission and control of avian influenza virus. By setting the basic reproductive number as the threshold parameter and constructing Lyapunov function, Dulac function and using the Li-Muldowney''s geometry approach, we prove the local and global stability of disease-free equilibria and endemic equilibrium. Theoretical analysis are carried out to show the role of the saturation inhibition effect, psychological effect and effective medical resources in this model, and numerical simulations are also given to verify the results.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.