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.

Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays

2014 ◽  
Vol 07 (05) ◽  
pp. 1450055 ◽  
Author(s):  
A. M. Elaiw ◽  
R. M. Abukwaik ◽  
E. O. Alzahrani
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Hiv Infection ◽  
Infection Model ◽  
Target Cells ◽  
Cell Mediated Immunity ◽  
Globally Asymptotically Stable ◽  
Global Properties ◽  
Hiv Infection Model ◽  
Ctl Immune Response

In this paper, we study the global properties of a human immunodeficiency virus (HIV) infection model with cytotoxic T lymphocytes (CTL) immune response. The model is a six-dimensional that describes the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages. The infection rate is given by saturation functional response. 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 infection reproduction number R0 and the immune response activation number [Formula: see text]. We have proven that if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if [Formula: see text], then the infected steady state without CTL immune response is GAS, and if [Formula: see text], then the infected steady state with CTL immune response is GAS.

scholarly journals Global Dynamics of an HIV Infection Model with Two Classes of Target Cells and Distributed Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. Elaiw
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hiv Infection ◽  
Infection Model ◽  
Target Cells ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Infection Model ◽  
Nonlinear Delay

We investigate the global dynamics of an HIV infection model with two classes of target cells and multiple distributed intracellular delays. The model is a 5-dimensional nonlinear delay ODEs that describes the interaction of the HIV with two classes of target cells, CD4+T cells and macrophages. The incidence rate of infection is given by saturation functional response. The model has two types of distributed time delays describing time needed for infection of target cell and virus replication. This model can be seen as a generalization of several models given in the literature describing the interaction of the HIV with one class of target cells, CD4+T cells. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of the model. We have proven that if the basic reproduction numberR0is less than unity then the uninfected steady state is globally asymptotically stable, and ifR0>1then the infected steady state exists and it is globally asymptotically stable.

scholarly journals Global Dynamics of HIV Infection of CD4+T Cells and Macrophages

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
A. M. Elaiw ◽  
A. S. Alsheri
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hiv Infection ◽  
Reproduction Number ◽  
Infection Model ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Virus Particles ◽  
Hiv Infection Model ◽  
Invariant Principle

We study the global dynamics of an HIV infection model describing the interaction of the HIV with CD4+T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected CD4+T cells and macrophages are given by general functions. We have incorporated two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (matures) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if the basic reproduction numberR0is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.

scholarly journals Mathematical analysis of an HTLV-I infection model with the mitosis of CD4+ T cells and delayed CTL immune response

2021 ◽  
Vol 26 (1) ◽  
pp. 1-20
Author(s):  
Chenwei Song ◽  
Rui Xu
Keyword(s):  
Immune Response ◽  
T Cells ◽  
Cd4 T Cells ◽  
Infection Model ◽  
Uniform Persistence ◽  
Type I ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Ctl Immune Response

In this paper, we consider an improved Human T-lymphotropic virus type I (HTLV-I) infection model with the mitosis of CD4+ T cells and delayed cytotoxic T-lymphocyte (CTL) immune response by analyzing the distributions of roots of the corresponding characteristic equations, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium when the CTL immune delay is zero is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. We define the immune-inactivated reproduction ratio R0 and the immune-activated reproduction ratio R1. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that if R0 < 1, the infection-free equilibrium is globally asymptotically stable; if R1 < 1 < R0, the immunity-inactivated equilibrium is globally asymptotically stable; if R1 > 1, the immunity-activated equilibrium is globally asymptotically stable when the CTL immune delay is zero. Besides, uniform persistence is obtained when R1 > 1. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Global Dynamics of Virus Infection Model with Antibody Immune Response and Distributed Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. M. Elaiw ◽  
A. Alhejelan ◽  
M. A. Alghamdi
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Virus Infection ◽  
Reproduction Number ◽  
Distributed Delays ◽  
Infection Model ◽  
Qualitative Behavior ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Virus Infection Model

We present qualitative behavior of virus infection model with antibody immune response. The incidence rate of infection is given by saturation functional response. Two types of distributed delays are incorporated into the model to account for the time delay between the time when uninfected cells are contacted by the virus particle and the time when emission of infectious (matures) virus particles. Using the method of Lyapunov functional, we have established that the global stability of the steady states of the model is determined by two threshold numbers, the basic reproduction numberR0and antibody immune response reproduction numberR1. We have proven that ifR0≤1, then the uninfected steady state is globally asymptotically stable (GAS), ifR1≤1<R0, then the infected steady state without antibody immune response is GAS, and ifR1>1, then the infected steady state with antibody immune response is GAS.

Asymptotic Analysis of an Age-Structured HIV Infection Model with Logistic Target-Cell Growth and Two Infecting Routes

2020 ◽  
Vol 30 (04) ◽  
pp. 2050059
Author(s):  
Dongxue Yan ◽  
Xianlong Fu
Keyword(s):  
Steady State ◽  
Hiv Infection ◽  
Logistic Growth ◽  
Infection Model ◽  
Target Cells ◽  
Cell Infection ◽  
Numerical Examples ◽  
Distribution Of Eigenvalues ◽  
Age Structured ◽  
Hiv Infection Model

This paper deals with an age-structured HIV infection model with logistic growth for target cells and both virus-to-cell and cell-to-cell infection routes. Based on the existence of the infection-free and infection equilibria and some rigorous analyses for the considered model, we study the asymptotic stability of these equilibria via determining the distribution of eigenvalues. We also address the persistence of the solution semi-flow by proving the existence of a global attractor. Furthermore, Hopf bifurcation occurring at the positive steady state is exploited. At last, some numerical examples are provided to illustrate the obtained results.

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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