Global stability and Hopf bifurcation of an HIV-1 infection model with saturation incidence and delayed CTL immune response

2014 ◽  
Vol 237 ◽  
pp. 146-154 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Global Stability ◽  
Infection Model ◽  
Ctl Immune Response ◽  
Hiv 1

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.

scholarly journals Global Stability of HIV Infection of CD4+T Cells and Macrophages with CTL Immune Response and Distributed Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
A. M. Elaiw ◽  
R. M. Abukwaik ◽  
E. O. Alzahrani
Keyword(s):  
Immune Response ◽  
T Cells ◽  
Steady State ◽  
Hiv Infection ◽  
Global Stability ◽  
Reproduction Number ◽  
Infection Model ◽  
Target Cells ◽  
Globally Asymptotically Stable ◽  
Ctl Immune Response

We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describes the interaction of the HIV with two classes of target cells, CD4+T cells and macrophages. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction numberR0and the immune response reproduction numberR0∗. We have proven that, ifR0≤1, then the uninfected steady state is globally asymptotically stable (GAS), ifR0*≤1<R0, then the infected steady state without CTL immune response is GAS, and, ifR0*>1, then the infected steady state with CTL immune response is GAS.

scholarly journals Global Stability of Delayed Viral Infection Models with Nonlinear Antibody and CTL Immune Responses and General Incidence Rate

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Hui Miao ◽  
Zhidong Teng ◽  
Zhiming Li
Keyword(s):  
Immune Response ◽  
Immune Responses ◽  
Antibody Response ◽  
Viral Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Cytotoxic T Lymphocyte ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Ctl Immune Response

The dynamical behaviors for a five-dimensional viral infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses, and nonlinear incidence rate are investigated. The threshold values for viral infection, antibody response, CTL immune response, CTL immune competition, and antibody competition, respectively, are established. Under certain assumptions, the threshold value conditions on the global stability of the infection-free, immune-free, antibody response, CTL immune response, and interior equilibria are proved by using the Lyapunov functionals method, respectively. Immune delay as a bifurcation parameter is further investigated. The numerical simulations are performed in order to illustrate the dynamical behavior of the model.

scholarly journals Stability and Hopf Bifurcation in an HIV-1 Infection Model with Latently Infected Cells and Delayed Immune Response

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Haibin Wang ◽  
Rui Xu
Keyword(s):  
Immune Response ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Latently Infected ◽  
Infected Cells ◽  
Ctl Immune Response ◽  
Hiv 1

An HIV-1 infection model with latently infected cells and delayed immune response is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria is established and the existence of Hopf bifurcations at the CTL-activated infection equilibrium is also studied. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infectionR0≤1; if the basic reproduction ratio for viral infectionR0>1and the basic reproduction ratio for CTL immune responseR1≤1, the CTL-inactivated infection equilibrium is globally asymptotically stable. If the basic reproduction ratio for CTL immune responseR1>1, the global stability of the CTL-activated infection equilibrium is also derived when the time delayτ=0. Numerical simulations are carried out to illustrate the main results.

scholarly journals Global Stability of HIV-1 Infection Model with Two Time Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hui Miao ◽  
Xamxinur Abdurahman ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Immune Response ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Infection Model ◽  
Global Dynamics ◽  
Response Model ◽  
Delay Differential ◽  
Ctl Immune Response ◽  
Hiv 1

We investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two time delays describing time needed for infection of cell and CTLs generation. Our model admits three possible equilibria: infection-free equilibrium, CTL-absent infection equilibrium, and CTL-present infection equilibrium. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied.

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