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.     

An age-structured within-host HIV model with T-cell competition

Recent studies demonstrate that resource competition is an essential component of T-cell proliferation in HIV progression, which can contribute instructively to the disease development. In this paper, we formulate an age-structured within-host HIV model, in the form of a hyperbolic partial differential equation (PDE) for infected target cells coupled with two ordinary differential equations for uninfected T-cells and the virions, to explore the effects of both the T-cell competition and viral shedding variations on the viral dynamics. The basic reproduction number is derived for a general viral production rate which determines the local stability of the infection-free equilibrium. Two special forms of viral production rates, which are extensively investigated in previous literature, the delayed exponential distribution and a step function rate, are further investigated, where the original system can be reduced into systems of delay differential equations. It is confirmed that there exists a unique positive equilibrium for two special viral production rates when the basic reproduction number is greater than one. However, the model exhibits the phenomenon of backward bifurcation, where two positive steady states coexist with the infection-free equilibrium when the basic reproduction number is less than one.     

Global stability for an HIV-1 infection model including an eclipse stage of infected cells

We consider the mathematical model for the viral dynamics of HIV-1 introduced in Rong et al. (2007) [37]. One main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class. The viral dynamics is described by four nonlinear ordinary differential equations. In Rong et al. (2007) [37], the stability of the infected equilibrium has been analyzed locally. Here, we perform the global stability analysis using two techniques, the Lyapunov direct method and the geometric approach to stability, based on the higher-order generalization of Bendixson?s criterion. We obtain sufficient conditions written in terms of the system parameters. Numerical simulations are also provided to give a more complete representation of the system dynamics.     

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.     

Global stability of the virus dynamics model with intracellular delay and Crowley–Martin functional response

In this paper, a virus dynamics model with intracellular delay and Crowley–Martin functional response is discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle for delay differential equations, we established the global stability of uninfected equilibrium and infected equilibrium; it is proved that if the basic reproductive number is less than or equal to one, the uninfected equilibrium is globally asymptotically stable; if the basic reproductive number is more than one, the infected equilibrium is globally asymptotically stable. We also discuss the effects of intracellular delay on global dynamical properties by comparing the results with the stability conditions for the model without delay. Copyright © 2013 John Wiley & Sons, Ltd.     

Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

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.     

Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load, and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.     

Dynamics of an HIV-1 infection model with cell mediated immunity

In this paper, we study the dynamics of an improved mathematical model on HIV-1 virus with cell mediated immunity. This new 5-dimensional model is based on the combination of a basic 3-dimensional HIV-1 model and a 4-dimensional immunity response model, which more realistically describes dynamics between the uninfected cells, infected cells, virus, the CTL response cells and CTL effector cells. Our 5-dimensional model may be reduced to the 4-dimensional model by applying a quasi-steady state assumption on the variable of virus. However, it is shown in this paper that virus is necessary to be involved in the modeling, and that a quasi-steady state assumption should be applied carefully, which may miss some important dynamical behavior of the system. Detailed bifurcation analysis is given to show that the system has three equilibrium solutions, namely the infection-free equilibrium, the infectious equilibrium without CTL, and the infectious equilibrium with CTL, and a series of bifurcations including two transcritical bifurcations and one or two possible Hopf bifurcations occur from these three equilibria as the basic reproduction number is varied. The mathematical methods applied in this paper include characteristic equations, Routh–Hurwitz condition, fluctuation lemma, Lyapunov function and computation of normal forms. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.     

Global properties for virus dynamics model with mitotic transmission and intracellular delay

In this paper we study the basic model of viral infections with mitotic transmission and intracellular delay discrete. The delay corresponds to the time between infection of uninfected cells and the emission of virus on a cellular level. By means of Volterra-type Lyapunov functionals, we provide the global stability for this model. Let η be the number of virus produced per infected cell. If ηcrit, the critical number, satisfies η?ηcrit, then the virus-free steady state is globally asymptotically stable. On the contrary if η>ηcrit, then the infected steady state is globally asymptotically stable if a sufficient condition is satisfied.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

Global properties for virus dynamics model with Beddington–DeAngelis functional response

This paper investigates the global stability of virus dynamics model with Beddington–DeAngelis infection rate. By constructing Lyapunov functions, the global properties have been analysed. If the basic reproductive ratio of the virus is less than or equal to one, the uninfected steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is more than one, the infected steady state is globally asymptotically stable. The conditions imply that the steady states are always globally asymptotically stable for Holling type II functional response or for a saturation response.     

Stabilization of HIV/AIDS model by receding horizon control

This work concerns the stabilization of uninfected steady state of an ordinary differential equation system modeling the interaction of the HIV virus and the immune system of the human body. The control variable is the drug dose, which, in turn, affects the rate of infection of CD4⁺ T cells by HIV virus. The feedback controller is constructed by a variant of the receding horizon control (RHC) method. Simulation results are discussed.     

The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model

In this paper, generalized Euler method (GEM) and homotopy analysis method (HAM) are performed to solve the problem of the population dynamics of the human immunodeficiency type 1 virus (HIV-1). We introduce fractional orders to the model of HIV-1 whose components are plasma densities of uninfected CD4⁺ T-cells, the infected such cells and the free virus. The effect of the drug treatment of HIV-1 will be discussed in this paper.     

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.     

Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number $R_{0}$ to characterize the viral dynamics. We prove that when $R_{0}<1$, the uninfected equilibrium of the proposed model is globally asymptotically stable. When $R_{0}>1$, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control.     

Asymptotic properties of a HIV-1 infection model with time delay

Based on some important biological meanings, a class of more general HIV-1 infection models with time delay is proposed in the paper. In the HIV-1 infection model, time delay is used to describe the time between infection of uninfected target cells and the emission of viral particles on a cellular level as proposed by Herz et al. [A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247-7251]. Then, the effect of time delay on stability of the equilibria of the HIV-1 infection model has been studied and sufficient criteria for local asymptotic stability of the infected equilibrium and global asymptotic stability of the viral free equilibrium are given.     

Global dynamics of an age‐structured virus model with saturation effects

An infection‐age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values R ₀, the basic reproductive number for viral infection, and R ₁, the viral reproductive number at the immune‐free infection steady state (R ₁<R ₀). If R ₀<1, the uninfected steady state E ⁰ is globally asymptotically stable; if R ₀>1 > R ₁, the immune‐free infected steady state E ^? is globally asymptotically stable; while if R ₁>1, the antibody immune infected steady state is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.     

Threshold dynamics for compartmental epidemic models with impulses

The basic reproductive number and its calculation for general impulsive compartmental epidemic models, with pulses on both the infected and the uninfected compartments, are established. Theoretical results show that the basic reproductive number serves as a threshold parameter: the disease dies out if the basic reproductive number is smaller than unity, and breaks out if it is larger than unity. The global dynamics of a viral dynamical model with impulsive immune response is analyzed to study how the vaccination strength and the vaccination interval affect the basic reproductive number and virus progression.