A numerical approach to solve the model for HIV infection of CD4+T cells

In this study, we will obtain the approximate solutions of the HIV infection model of CD4⁺T by developing the Bessel collocation method. This model corresponds to a class of nonlinear ordinary differential equation systems. Proposed scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are computed using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed approach are demonstrated in the different time intervals by a numerical example. All computations have been made with the aid of a computer code written in Maple 9.     

具有感染年龄结构的CD4+ T-细胞感染HIV病毒模型分析

本文建立和研究一类具有感染年龄结构的CD4+ T-细胞感染HIV病毒的动力学模型.得到决定该模型的未感染平衡点和感染平衡点的存在性和局部渐近稳定性条件,即当一个感染细胞在其整个感染期间产生病毒的总数不超过某-个阈值时,系统总存在局部渐近稳定的未感染平衡点;当-个感染细胞在其整个感染期间产生病毒的总数超过这一阈值时,未感染平衡点不稳定,此时存在局部渐近稳定的感染平衡点.     

A Delayed HIV Infection Model with the Homeostatic Proliferation of CD4+ T Cells

In this paper, we investigate a delayed HIV infection model that considers the homeostatic proliferation of CD4⁺ T cells. The existence and stability of uninfected equilibrium and infected equilibria(smaller and larger ones) are studied by analyzing the characteristic equation of the system. The intracellular delay does not affect the stability of uninfected equilibrium, but it can change the stability of larger positive equilibrium and Hopf bifurcation appears inducing stable limit c...     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

一类带有治愈率的HIV感染的CD4＋T细胞模型的动力学性质

本文主要介绍一类带有治愈率的HIV感染的CD4 T细胞模型的动力学性质,同时证明了如果基本再生数R0＜1,HIV感染消失;如果R0＞1,HIV感染持续.然后进行数值模拟,给出了地方性平衡点E·全局稳定的参数域,得到了地方性平衡点E·不稳定时周期解存在.     

A nested model on HIV/AIDs, antiretroviral therapy and drug resistance

A coupled within- (immunological) and between-host (epidemiological) dynamic model was developed which is about the spreading of drug-sensitive HIV strain and drug-resistant HIV strain in men who have sex with men (MSM) population. The within-host model was nested within the between-host model by linking the dynamics of the within-host model to the additional host mortality and transmission rate of the infection. The existences of equilibria and their stabilities were found, as well as the thresholds $\mathcal {R}_S$ and $\mathcal {R}_R$ for the two different strains of the nested model. Some simulations about the spreading of the two HIV strains in Beijing MSM population were given. Our results show that the drug-resistant strain will increase quite fast in this population and both strains can coexist, which will make a big pressure for China''s ``Four-Free-One-Care Policy''.     

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.     

A fractional-order model of HIV infection with drug therapy effect

In this paper, the fractional-order model that describes HIV infection of CD4⁺ T cells with therapy effect is given. Generalized Euler Method (GEM) is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.     

基于CD4细胞计数的HIV仓室模型与数据分析

迄今我国每年统计艾滋病的新增人数与死亡人数仍呈上升趋势,由于潜伏期长和初期无明显症状等原因,还存在大量未发现的HIV携带者,这给HIV的防控带来巨大挑战.旨在利用中国疾控中心网报数据和深圳市患者随访数据,结合传染病动力学和统计分析方法与临床知识,建立依微观CD4细胞计数划分的宏观HIV仓室数学模型,通过数值计算方法与MCMC参数估计方法实现稳健的参数拟合,进而利用不确定性和敏感性以及随机森林方法进行灵敏度分析.研究结果表明:2013年广东省未确诊HIV携带者约为13.1061万人,且该地区HIV疫情传播的基本再生数为2.8133.敏感性分析揭示艾滋病疫情防控最优方法是控制患者有效接触人数与沉默系数,由此建议制定针对控制艾滋病传播中一些现象的法律法规,在艾滋病高发地区实施清洁针具交换工作等,对疫情防控提出指导性建议.     

A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis

In this paper, we propose a novel time delayed HIV/AIDS mathematical model and further analyze the effect of vaccination and ART (antiretroviral therapy) on this time delayed model, in which the time delay is due to the strong immune response to AIDS for the HIV-infected-aware because of the good physical conditions. We introduce the different stages of the period of AIDS infection having different abilities of transmitting disease, which reflects the developing progress of AIDS infection more realistically. By using suitable Lyapunov functionals and the LaSalle invariant principle, we obtain the basic reproduction number _R0${\mathfrak{R}}_0$ and derive that if _R0<1${\mathfrak{R}}_0<1$ and some parameters satisfy a given condition, the disease-free equilibrium is globally asymptotically stable, while the disease will be died out. Numerical simulations are carried out to verify the obtained stability criteria and demonstrate the effect of the vaccination rate and _R0${\mathfrak{R}}_0$ and the ART on the infective individuals.     

Dynamics of reaction–diffusion equations for modeling CD4+ T cells decline with general infection mechanism and distinct dispersal rates

In this paper, we are concerned with a diffusive viral infection dynamical model with general infection mechanism and distinct dispersal rates. In a general setting in which the model parameters are spatially heterogeneous, it is shown that if ${ℛ}_0\le 1$, the infection-free steady state is globally asymptotically stable; while if ${ℛ}_0>1$, the model is uniformly persistent. The asymptotic profiles of the infection steady state are discussed as the dispersal rate of uninfected CD4${}^{+}$ T cells approaches zero by means of the persistence theory of semidynamical systems and the eigenvalue theory of elliptic equations.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T-cells

In this paper, a mathematical model for HILV-I infection of CD4⁺ T-cells is investigated. The force of infection is assumed be of a function in general form, and the resulting incidence term contains, as special cases, the bilinear and the saturation incidences. The model can be seen as an extension of the model [Wang et al. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci. 179 (2002) 207-217; Song, Li, Global stability and periodic solution of a model for HTLV-I infection and ATL progression, Appl. Math. Comput. 180(1) (2006) 401-410]. Mathematical analysis establishes that the global dynamics of T-cells infection is completely determined by a basic reproduction number _R0${\mathfrak{R}}_0$. If _R0?1${\mathfrak{R}}_0?1$, the infection-free equilibrium is globally stable; if _R0>1${\mathfrak{R}}_0>1$, the unique infected equilibrium is globally stable in the interior of the feasible region.     

The impact of zidovudine (AZT) therapy on the survivability of those with the progressive HIV infection

An analysis of the data regarding the impact of zidovudine therapy on the Survivability of those with progressive HIV disease demonstrates that this therapy extends longevity for perhaps 5.5 months on the average but does not prevent the disease from eventually being fatal. All of the benefits of zidovudine therapy in extending survivability appear to accrue within a relatively short treatment period, perhaps within a few months, but the effectiveness of this drug wanes in time, suggesting that zidovudine therapy could eventually be stopped without influencing survivability. Since the efficacy of zidovudine therapy in extending survivability appears to be independent of the stage of the HIV infection in infecteds whose CD4 T-cell densities fall below 200 cells/mm³, zidovudine therapy should be initiated when the patient's infection reaches a potentially fatal stage. Zidovudine therapy may be viewed as causing the HIV infection to regress to a previous stage of the disease, but the infection's progression promptly resumes and follows a course similar to one uninfluenced by zidovudine.     

The Modeling Dynamics of HIV and CD4$$^{}$$ T-cells During Primary Infection in Fractional Order: Numerical Simulation

This article is devoted to introduce a numerical treatment using Adams–Bashforth–Moulton method of the fractional model of HIV-1 infection of CD4\(^{+}\) T-cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. The fractional derivative is described in Caputo sense. Special attention is given to present the local stability of the proposed model using fractional Routh–Hurwitz stability criterion. Qualitative results show that the model has two equilibria: the disease-free equilibrium and the endemic equilibrium points. We compare our numerical solutions with those numerical solutions using fourth-order Runge–Kutta method (RK4). The obtained numerical results of the proposed model show the simplicity and the efficiency of the proposed method.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T cells with delayed CTL response

Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4⁺ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8⁺ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R₀ and R₁, basic reproduction numbers for viral infection and for CTL response, respectively. If R₀≤1, the infection-free equilibrium P₀ is globally asymptotically stable, and the HTLV-I viruses are cleared. If R₁≤1<R₀, the asymptomatic-carrier equilibrium P₁ is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R₁>1, a unique HAM/TSP equilibrium P₂ exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.     

Qualitative behavior of systems of tumor–CD4+–cytokine interactions with treatments

Immunotherapies are important methods for controlling and curing malignant tumors. Based on recent observations that many tumors have been immuno‐selected to evade recognition by the traditional cytotoxic T lymphocytes, we propose mathematical models of tumor–CD4⁺–cytokine interactions to investigate the role of CD4⁺ on tumor regression. Treatments of either CD4⁺ or cytokine are applied to study their effectiveness. It is found that doses of treatments are critical in determining the fate of the tumor, and tumor cells can be eliminated completely if doses of cytokine are large. Bistability is observed in models with either of the treatment strategies, which signifies that a careful planning of the treatment strategy is necessary for achieving a satisfactory outcome. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission

Recent studies reveal that cell-to-cell transmission via formation of virological synapses can contribute significantly to virus spread, and hence, may play a more important role than virus-to-cell infection in some situations. Age-structured models can be employed to study the variations w.r.t. infection age in modeling the death rate and virus production rate of infected cells. Considering the above characteristics for within-host dynamics of HIV, in this paper, we formulate an age-structured hybrid model to explore the effects of the two infection modes in viral production and spread. We offer a rigorous analysis for the model, including addressing the relative compactness and persistence of the solution semiflow, and existence of a global attractor. By subtle construction and estimates of Lyapunov functions, we show that the global attractor actually consists of an singleton, being either the infection free steady state if the basic reproduction number is less than one, or the infection steady state if the basic reproduction number is larger than one.