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.     

HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission

HIV infection persists despite long-term administration of antiretroviral therapy. The mechanisms underlying HIV persistence are not fully understood. Direct viral transmission from infected to uninfected cells (cell-to-cell transmission) may be one of them. During cell-to-cell transmission, multiple virions are delivered to an uninfected cell, making it possible that at least one virion can escape HIV drugs and establish infection. In this paper, we develop a mathematical model that includes cell-to-cell viral transmission to study HIV persistence. During cell-to-cell transmission, it is assumed that various number of virus particles are transmitted with different probabilities and antiretroviral therapy has different effectiveness in blocking their infection. We analyze the model by deriving the basic reproduction number and investigating the stability of equilibria. Sensitivity analysis and numerical simulation show that the viral load is still sensitive to the change of the treatment effectiveness in blocking cell-free virus infection. To reduce this sensitivity, we modify the model by including density-dependent infected cell death or HIV latent infection. The model results suggest that although cell-to-cell transmission may have reduced susceptibility to HIV drugs, HIV latency represents a major reason for HIV persistence in patients on suppressive treatment.     

Global stability analysis and permanence for an HIV-1 dynamics model with distributed delays

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.     

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.     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Dynamics of an HIV‐1 virus model with both virus‐to‐cell and cell‐to‐cell transmissions,general incidence rate,intracellular delay,and CTL immune responses

Direct cell‐to‐cell transmission of HIV‐1 is a more efficient means of virus infection than virus‐to‐cell transmission. In this paper, we incorporate both these transmissions into an HIV‐1 virus model with nonlinear general incidence rate, intracellular delay, and cytotoxic T lymphocyte (CTL) immune responses. This model admits three types of equilibria: infection‐free equilibrium, CTL‐inactivated equilibrium, and CTL‐activated equilibrium. By using Lyapunov functionals and LaSalle invariance principle, it is verified that global threshold dynamics of the model can be explicitly described by the basic reproduction numbers.     

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.     

The global dynamics of a model about HIV-1 infection in vivo

A four dimension ODE model is built to study the infection of human immunodeficiency virus (HIV) in vivo. We include in this model four components: the healthy T cells, the latent-infected T cells, the active-infected T cells and the HIV virus. Two types of HIV transmissions in vivo are also included in the model: the virus-to-cell transmission, and the cell-to-cell HIV transmission. There are two possible equilibriums: the healthy equilibrium, and the infected steady state. The basic reproduction number R ₀ is introduced. When R ₀ < 1, the healthy equilibrium is globally stable and when R ₀ > 1, the infected equilibrium exists and is globally stable. Through simulations, we find that, the cell-to-cell HIV transmission is very important for the final outcome of the HIV attacking. Some important clinical observations about the HIV infection situation in lymph node are also verified.      

一类HIV病毒动力学模型的全局稳定性与周期解

建立了一类较广泛的HIV感染CD4+T细胞病毒动力学模型,给出了一个感染细胞在其整个感染期内产生的病毒的平均数(基本再生数)R0的表达式,运用Lyapunov原理和Routh-Hurwitz判据得到了该模型的未感染平衡点与感染平衡点的存在性与稳定性条件.同时也得到了模型存在轨道渐近稳定周期解和系统持续生存的条件,并通过数值模拟验证了所得到的结果.     

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.     

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.     

Dynamical behaviors of a discrete HIV‐1 virus model with bilinear infective rate

We establish a discrete virus dynamic model by discretizing a continuous HIV‐1 virus model with bilinear infective rate using ‘hybrid’ Euler method. We discuss not only the existence and global stability of the uninfected equilibrium but also the existence and local stability of the infected equilibrium. We prove that there exists a crucial value similar to that of the continuous HIV‐1 virus dynamics, which is called the basic reproductive ratio of the virus. If the basic reproductive ratio of the virus is less than one, the uninfected equilibrium is globally asymptotically stable. If the basic reproductive ratio of the virus is larger than one, the infected equilibrium exists and is locally stable. Moreover, we consider the permanence for such a system by constructing a Lyapunov function v_n. Copyright © 2013 John Wiley & Sons, Ltd.     

Global stability of an HIV-1 infection model with saturation infection and intracellular delay

In this paper, an HIV-1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable.     

Stability and Hopf Bifurcation of a Virus Infection Model with a Delayed CTL Immune Response

In this paper, a virus infection model with standard incidence rate and delayed CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the CTL-activated infection equilibrium are established, respectively. By means of comparison arguments, it is verified that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio is less than unity. By using suitable Lyapunov functional and LaSalle＇s invariance principle, it is shown that the CTL-inactivated infection equilibrium of the system is globally asymptotically stable if tile immune response reproduction ratio is less than unity and the basic reproduction ratio is greater than unity. Numerical simulations are carried out to illustrate the theoretical result.     

Global stability of delay-distributed viral infection model with two modes of viral transmission and B-cell impairment

This paper formulates a virus dynamics model with impairment of B-cell functions. The model incorporates two modes of viral transmission: cell-free and cell-to-cell. The cell-free and cell-cell incidence rates are modeled by general functions. The model incorporates both, latently and actively, infected cells as well as three distributed time delays. Nonnegativity and boundedness properties of the solutions are proven to show the well-posedness of the model. The model admits two equilibria that are determined by the basic reproduction number R₀. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions and time delays on the virus dynamics are studied. We have shown that if the functions of B-cell is impaired, then the concentration of viruses is increased in the plasma. Moreover, we have observed that increasing the time delay will suppress the viral replication.     

On the Bifurcations and Multiple Endemic States of a Single Strain HIV Model Dedicated to Professor Toshikazu Sunada on the Occasion of his 60th Birthday

The dynamics of a single strain HIV model is studied.The basic reproduction number R_0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations.The usual threshold unit value of R_0 does not completely determine the eradication of the disease in an HIV infected person.In particular,a sub-threshold value R_C is established which determines the system's number of endemic states:multiple if R_c Ro 1,only one if R_C = R_0 = 1,and none if R_0 R_C 1.     

Dynamics of virus infection models with density‐dependent diffusion and Beddington–DeAngelis functional response

In this work, we integrate both density‐dependent diffusion process and Beddington–DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single‐strain and multi‐strain viruses and find that, for the multi‐strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamics of an HIV Infection Model with Two Infection Routes and Evolutionary Competition between Two Viral Strains

The existing combination therapy of HIV antiretroviral drugs can lead to the emergence of drug-resistant viruses, and cannot effectively block direct cell-to-cell infections, these factors results in incomplete virus suppression and increased risk of disease progression. In this paper, we formulate an HIV model with two strains representing a drug-sensitive virus and a drug-resistant virus to study the joint mechanism of drug resistance. We first reduce the infection-age model to a system of integro-differential equations with infinite delays. Then the stability of the equilibria and the dynamics of competition between two viruses are studied to illuminate the joint effects of infection-age and two infection routes on the evolution of both drug-sensitive and drug-resistant strains before and during drug treatment. Applying a persistence theorem for infinite dimensional systems, we obtain that the disease is always present when the basic reproduction number is larger than unity. Numerical simulations confirm that the basic reproduction numbers and mutation coefficient are the key threshold parameters for determining the competition results of the two viral strains and indicate the cell-to-cell transmission increases the likelihood that HIV breaks out within the host. Finally, sensitivity analyses suggest that the available combination therapy should be taken once symptoms of resistance appear during drug treatment, and demonstrate that the presence of cell-to-cell transmission attenuates the efficacy of the existing antiretroviral drug treatments.