scholarly journals Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Mengye Chen ◽  
Liang You ◽  
Jie Tang ◽  
Shasha Su ◽  
Ruiming Zhang
Keyword(s):  
Immune Response ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Infection Model ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Virus Infection Model ◽  
Viral Infection Model

We investigate the dynamical behavior of a virus infection model with delayed nonlytic immune response. By analyzing corresponding characteristic equations, the local stabilities of two boundary equilibria are established. By using suitable Lyapunov functional and LaSalle’s invariance principle, we establish the global stability of the infection-free equilibrium. We find that the infection free equilibriumE0is globally asymptotically stable whenR0⩽1, and the infected equilibrium without immunityE1is local asymptotically stable when1<R0⩽1+bβ/cd. Under the conditionR0>1+bβ/cdwe obtain the sufficient conditions to the local stability of the infected equilibrium with immunityE2. We show that the time delay can change the stability ofE2and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions are studied and numerical simulations to our theorems are provided.

Stability of a CD4+ T cell viral infection model with diffusion

2018 ◽  
Vol 11 (05) ◽  
pp. 1850071 ◽  
Author(s):  
Zhiting Xu ◽  
Youqing Xu
Keyword(s):  
T Cell ◽  
Viral Infection ◽  
Lyapunov Functions ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

Global Dynamics of a HIV Infection Model with Delayed CTL Response and Cure Rate

2013 ◽  
Vol 791-793 ◽  
pp. 1322-1327
Author(s):  
Yan Yan Yang ◽  
Hui Wang ◽  
Zhi Xing Hu ◽  
Wan Biao Ma
Keyword(s):  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Cure Rate ◽  
Globally Asymptotically Stable ◽  
Ctl Response ◽  
Stable Periodic ◽  
Hiv Infection Model ◽  
The Stability ◽  
Viral Infection Model

In this paper, we have considered a viral infection model with delayed CTL response and cure rate. For this model, we have researched the stability of these three equilibriums depend on two threshold parameters and , that is, if , the infected-free equilibrium is locally asymptotically stable; if , the infected equilibrium without CTL response is globally asymptotically stable; and if , the infected equilibrium exists, at he same time, we have found that the time delay can lead to Hopf bifurcations and stable periodic solutions when the is unstable.

Global Stability of a HBV Infection Model with Saturated Infection Rates and Time Delay

2013 ◽  
Vol 791-793 ◽  
pp. 1314-1317
Author(s):  
Wei Juan Pang ◽  
Zhi Xing Hu ◽  
Fu Cheng Liao
Keyword(s):  
Global Stability ◽  
Sufficient Conditions ◽  
Hbv Infection ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Infection Rates ◽  
Globally Asymptotically Stable ◽  
Viral Infection Model

This paper investigates the global stability of a viral infection model of HBV infection of hepatocytes with saturated infection rate and intracellular delay. we obtain if the basic reproductive number is less than or equal to one, the infection-free equilibrium is globally asymptotically stable. If its greater than one, we obtain the sufficient conditions for the global stability of the infected equilibrium.

scholarly journals Globale Stability in a Viral Infection Model with Beddington-DeAngelis Functional Response

2015 ◽  
Vol 9 (1) ◽  
pp. 27-29
Author(s):  
Wang Zhanwei ◽  
He Xia
Keyword(s):  
Lyapunov Function ◽  
Viral Infection ◽  
Functional Response ◽  
Reproduction Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model ◽  
The Basic Reproduction Number

The stability of a mathematical model for viral infection with Beddington-DeAngelis functional response is considered in this paper. If the basic reproduction number R ≤1, by the Routh-Hurwitz criterion and Lyapunov function, the uninfected equilibrium E is globally asymptotically stable. Then, the global stability of the infected equilibrium E is obtained by the method of Lyapunov function

scholarly journals Stability Analysis for Viral Infection Model with Multitarget Cells, Beddington-DeAngelis Functional Response, and Humoral Immunity

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Xinxin Tian ◽  
Jinliang Wang
Keyword(s):  
Immune Response ◽  
Viral Infection ◽  
Incidence Rate ◽  
Functional Response ◽  
Humoral Immunity ◽  
Infection Model ◽  
Target Cells ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Viral Infection Model

We formulate a (2n+2)-dimensional viral infection model with humoral immunity,nclasses of uninfected target cells and  nclasses of infected cells. The incidence rate of infection is given by nonlinear incidence rate, Beddington-DeAngelis functional response. The model admits discrete time delays describing the time needed for infection of uninfected target cells and virus replication. By constructing suitable Lyapunov functionals, we establish that the global dynamics are determined by two sharp threshold parameters:R0andR1. Namely, a typical two-threshold scenario is shown. IfR0≤1, the infection-free equilibriumP0is globally asymptotically stable, and the viruses are cleared. IfR1≤1<R0, the immune-free equilibriumP1is globally asymptotically stable, and the infection becomes chronic but with no persistent antibody immune response. IfR1>1, the endemic equilibriumP2is globally asymptotically stable, and the infection is chronic with persistent antibody immune response.

scholarly journals Threshold dynamics and threshold analysis of HIV infection model with treatment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zhimin Chen ◽  
Xiuxiang Liu ◽  
Liling Zeng
Keyword(s):  
Hiv Infection ◽  
Time Delays ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Immunodeficiency Virus ◽  
Hiv Infection Model

Abstract In this paper, a human immunodeficiency virus (HIV) infection model that includes a protease inhibitor (PI), two intracellular delays, and a general incidence function is derived from biologically natural assumptions. The global dynamical behavior of the model in terms of the basic reproduction number $\mathcal{R}_{0}$ R 0 is investigated by the methods of Lyapunov functional and limiting system. The infection-free equilibrium is globally asymptotically stable if $\mathcal{R}_{0}\leq 1$ R 0 ≤ 1 . If $\mathcal{R}_{0}>1$ R 0 > 1 , then the positive equilibrium is globally asymptotically stable. Finally, numerical simulations are performed to illustrate the main results and to analyze thre effects of time delays and the efficacy of the PI on $\mathcal{R}_{0}$ R 0 .

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

Global Stability and Hopf Bifurcation in a Delayed Viral Infection Model with Cell-to-Cell Transmission and Humoral Immune Response

2019 ◽  
Vol 29 (12) ◽  
pp. 1950161 ◽  
Author(s):  
Jinhu Xu ◽  
Yan Geng ◽  
Suxia Zhang
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Viral Infection ◽  
Humoral Immune Response ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Humoral Immune ◽  
Cell Transmission ◽  
Viral Infection Model ◽  
Intracellular Delays

We have developed a class of viral infection model with cell-to-cell transmission and humoral immune response. The model addresses both immune and intracellular delays. We also constructed Lyapunov functionals to establish the global dynamical properties of the equilibria. Theoretical results indicate that considering only two intracellular delays did not affect the dynamical behavior of the model, but incorporating an immune delay greatly affects the dynamics, i.e. an immune delay may destabilize the immunity-activated equilibrium and lead to Hopf bifurcation, oscillations and stability switches. Our results imply that an immune delay dominates the intracellular delays in the model. We also investigated the direction of the Hopf bifurcation and the stability of the periodic solutions by applying normal form and center manifold theory, and investigated the existence of global Hopf bifurcation by regarding the immune delay as a bifurcation parameter. Numerical simulations are carried out to support the analytical conclusions.

The theorem of the carrying simplex for competitive system defined on the n-rectangle and its application to a three-dimensional system

2014 ◽  
Vol 07 (06) ◽  
pp. 1450063 ◽  
Author(s):  
Jifa Jiang ◽  
Lei Niu
Keyword(s):  
Dynamical Behavior ◽  
Three Dimensional ◽  
Linear Structure ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Dimensional System ◽  
Asymptotically Stable ◽  
Competitive System ◽  
Globally Asymptotically Stable ◽  
Three Dimensional System

First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle: {x ∈ Rn : 0 ≤ xi ≤ ki, i = 1, …, n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique (n - 1)-dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.

Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750070 ◽  
Author(s):  
A. M. Ełaiw ◽  
A. A. Raezah ◽  
Khalid Hattaf
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Dynamical Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Saturated Incidence ◽  
Ctl Immune Response ◽  
Hiv 1

This paper studies the dynamical behavior of an HIV-1 infection model with saturated virus-target and infected-target incidences with Cytotoxic T Lymphocyte (CTL) immune response. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate and the effect of CTL immune response. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number ([Formula: see text]) and the CTL immune response activation number ([Formula: see text]). By using suitable Lyapunov functionals, we show that if [Formula: see text], then the infection-free steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text] [Formula: see text], then the CTL-inactivated infection steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text], then the CTL-activated infection steady state [Formula: see text] is globally asymptotically stable. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown.

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