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 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

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 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 of a Delayed HIV-1 Infection Model with CTL Immune Response

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Yunfei Li ◽  
Rui Xu ◽  
Zhe Li ◽  
Shuxue Mao
Keyword(s):  
Immune Response ◽  
Viral Infection ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Ctl Immune Response ◽  
Hiv 1

A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.

Stability analysis for delayed viral infection model with multitarget cells and general incidence rate

2015 ◽  
Vol 09 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Jinliang Wang ◽  
Xinxin Tian ◽  
Xia Wang
Keyword(s):  
Viral Infection ◽  
Incidence Rate ◽  
Time Delays ◽  
Infection Model ◽  
Target Cells ◽  
Uniform Persistence ◽  
Nonlinear Incidence Rate ◽  
General Incidence Rate ◽  
Infected Cells ◽  
Viral Infection Model

In this paper, the sharp threshold properties of a (2n + 1)-dimensional delayed viral infection model are investigated. This model combines with n classes of uninfected target cells, n classes of infected cells and nonlinear incidence rate h(x, v). Two kinds of distributed time delays are incorporated into the model to describe the time needed for infection of uninfected target cells and virus replication. Under certain conditions, it is shown that the basic reproduction number is a threshold parameter for the existence of the equilibria, uniform persistence, as well as for global stability of the equilibria of the model.

scholarly journals Global stability of a distributed delayed viral model with general incidence rate

2018 ◽  
Vol 16 (1) ◽  
pp. 1374-1389
Author(s):  
Eric Ávila-Vales ◽  
Abraham Canul-Pech ◽  
Erika Rivero-Esquivel
Keyword(s):  
Immune Response ◽  
Viral Infection ◽  
Global Stability ◽  
Numerical Simulations ◽  
Incidence Rate ◽  
Existence And Uniqueness ◽  
Lyapunov Functional ◽  
Infection Model ◽  
General Incidence Rate ◽  
Viral Infection Model

AbstractIn this paper, we discussed a infinitely distributed delayed viral infection model with nonlinear immune response and general incidence rate. We proved the existence and uniqueness of the equilibria. By using the Lyapunov functional and LaSalle invariance principle, we obtained the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. Numerical simulations are given to verify the analytical results.

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.

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.

Dynamics of a Delayed HIV-1 Infection Model with Saturation Incidence Rate and CTL Immune Response

2016 ◽  
Vol 26 (14) ◽  
pp. 1650234 ◽  
Author(s):  
Ting Guo ◽  
Haihong Liu ◽  
Chenglin Xu ◽  
Fang Yan
Keyword(s):  
Immune Response ◽  
Equilibrium Point ◽  
Incidence Rate ◽  
Virus Production ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Globally Asymptotically Stable ◽  
Ctl Immune Response ◽  
Response Formula

In this paper, we investigate the dynamics of a five-dimensional virus model incorporating saturation incidence rate, CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model by proving the positivity and boundedness of the solutions. Our model admits three possible equilibrium solutions, namely the infection-free equilibrium [Formula: see text], the infectious equilibrium without immune response [Formula: see text] and the infectious equilibrium with immune response [Formula: see text]. Moreover, by analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcation at the equilibrium point [Formula: see text] are established, respectively. Further, by using fluctuation lemma and suitable Lyapunov functionals, it is shown that [Formula: see text] is globally asymptotically stable when the basic reproductive numbers for viral infection [Formula: see text] is less than unity. When the basic reproductive numbers for immune response [Formula: see text] is less than unity and [Formula: see text] is greater than unity, the equilibrium point [Formula: see text] is globally asymptotically stable. Finally, some numerical simulations are carried out for illustrating the theoretical results.

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