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.

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

Global dynamics of a viral infection model with full logistic terms and antivirus treatments

2016 ◽  
Vol 10 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Lijuan Song ◽  
Cui Ma ◽  
Qiang Li ◽  
Aijun Fan ◽  
Kaifa Wang
Keyword(s):  
Viral Infection ◽  
Global Stability ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Global Dynamics ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Influx Rate ◽  
Viral Infection Model

In this paper, mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. Though the model is originally to study hepatitis C virus (HCV) dynamics in patients with high baseline viral loads or advanced liver disease, similar models still hold significance for other viral infection, such as hepatitis B virus (HBV) or human immunodeficiency virus (HIV) infection. By means of Volterra-type Lyapunov functions, we know that the basic reproduction number [Formula: see text] is a sharp threshold para-meter for the outcomes of viral infections. If [Formula: see text], the virus-free equilibrium is globally asymptotically stable. If [Formula: see text], the system is uniformly persistent, the unique endemic equilibrium appears and is globally asymptotically stable under a sufficient condition. Other than that, for the global stability of the unique endemic equilibrium, another sufficient condition is obtained by Li–Muldowney global-stability criterion. Using numerical simulation techniques, we further find that sustained oscillations can exist and different maximum de novo hepatocyte influx rate can induce different global dynamics along with the change of overall drug effectiveness. Finally, some biological implications of our findings are given.

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.

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.

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.

Dynamics of an infection model with two delays

2015 ◽  
Vol 08 (05) ◽  
pp. 1550068 ◽  
Author(s):  
Xinguo Sun ◽  
Junjie Wei
Keyword(s):  
Viral Infection ◽  
Asymptomatic Carrier ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Stability Switches ◽  
Globally Asymptotically Stable ◽  
Ctl Response ◽  
Reproduction Numbers ◽  
Two Delays ◽  
The Stability

In this paper, a HTLV-I infection model with two delays is considered. It is found that the dynamics of this model 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. If R1 < 1 < R0, the asymptomatic-carrier equilibrium P1 is globally asymptotically stable. If R1 > 1, there exists a unique HAM/TSP equilibrium P2. The stability of P2 is changed when the second delay τ2 varies, that is there exist stability switches for P2.

Analysis of a hepatitis B viral infection model with immune response delay

2017 ◽  
Vol 10 (02) ◽  
pp. 1750020 ◽  
Author(s):  
Jianhua Pang ◽  
Jing-An Cui
Keyword(s):  
Immune Response ◽  
Time Delay ◽  
Hepatitis B ◽  
Viral Infection ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Response Delay ◽  
The Stability ◽  
Hepatitis B Viral Infection ◽  
Viral Infection Model

In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Dynamics Analysis of a Viral Infection Model with a General Standard Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yu Ji ◽  
Muxuan Zheng
Keyword(s):  
Viral Infection ◽  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
The Basic Reproduction Number ◽  
General Standard

The basic viral infection models, proposed by Nowak et al. and Perelson et al., respectively, have been widely used to describe viral infection such as HBV and HIV infection. However, the basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection, which seems not to be reasonable. In this paper, we formulate an amended model with a general standard incidence rate. The basic reproduction number of the amended model is independent of total cells of the host’s organ. When the basic reproduction numberR0<1, the infection-free equilibrium is globally asymptotically stable and the virus is cleared. Moreover, ifR0>1, then the endemic equilibrium is globally asymptotically stable and the virus persists in the host.

scholarly journals Global Dynamics Behaviors of Viral Infection Model for Pest Management

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Chunjin Wei ◽  
Lansun Chen
Keyword(s):  
Small Amplitude ◽  
Critical Value ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Virus Particles ◽  
All Solutions ◽  
Pest Eradication ◽  
Viral Infection Model

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

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