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.

scholarly journals Stability of Virus Infection Models with Antibodies and Chronically Infected Cells

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Mustafa A. Obaid ◽  
A. M. Elaiw
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Virus Infection ◽  
Incidence Rate ◽  
Lyapunov Functions ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Infection Models ◽  
Infected Cells

Two virus infection models with antibody immune response and chronically infected cells are proposed and analyzed. Bilinear incidence rate is considered in the first model, while the incidence rate is given by a saturated functional response in the second one. One main feature of these models is that it includes both short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate. Our mathematical analysis establishes that the global dynamics of the two models are determined by two threshold parametersR0andR1. By constructing Lyapunov functions and using LaSalle's invariance principle, we have established the global asymptotic stability of all steady states of the models. We have proven that, the uninfected steady state is globally asymptotically stable (GAS) ifR0<1, the infected steady state without antibody immune response exists and it is GAS ifR1<1<R0, and the infected steady state with antibody immune response exists and it is GAS ifR1>1. We check our theorems with numerical simulation in the end.

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 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 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.

Global dynamics for a live-virus-blood model with distributed time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750067 ◽  
Author(s):  
Ding-Yu Zou ◽  
Shi-Fei Wang ◽  
Xue-Zhi Li
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Live Virus ◽  
Number Formula ◽  
Distributed Time Delay ◽  
The Basic Reproduction Number

In this paper, the global properties of a mathematical modeling of hepatitis C virus (HCV) with distributed time delays is studied. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states. It is shown that if the basic reproduction number [Formula: see text] is less than unity, then the uninfected steady state is globally asymptotically stable. If the basic reproduction number [Formula: see text] is larger than unity, then the infected steady state is globally asymptotically stable.

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.

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