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 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 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.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

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 MODEL OF IMMUNE EFFECTOR RESPONSES TO VIRAL INFECTIONS

2007 ◽  
Vol 10 (04) ◽  
pp. 495-503 ◽  
Author(s):  
XIA WANG ◽  
XINYU SONG
Keyword(s):  
Immune Response ◽  
Immune Responses ◽  
Lyapunov Functions ◽  
Viral Infections ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Immune Effector ◽  
Global Properties ◽  
Ctl Immune Response

This article proposes a mathematical model which has been used to investigate the importance of lytic and non-lytic immune responses for the control of viral infections. By means of Lyapunov functions, the global properties of the model are obtained. The virus is cleared if the basic reproduction number R0 ≤ 1 and the virus persists in the host if R0 > 1. Furthermore, if R0 > 1 and other conditions hold, the immune-free equilibrium E0 is globally asymptotically stable. The equilibrium E1 exists and is globally asymptotically stale if the CTL immune response reproductive number R1 < 1 and the antibody immune response reproductive number R2 > 1. The equilibrium E2 exists and is globally asymptotically stable if R1 > 1 and R2 < 1. Finally, the endemic equilibrium E3 exists and is globally asymptotically stable if R1 > 1 and R2 > 1.

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.

scholarly journals Threshold dynamics of a general delayed HIV model with double transmission modes and latent viral infection

2021 ◽  
Vol 7 (2) ◽  
pp. 2456-2478
Author(s):  
Xin Jiang ◽  
Keyword(s):  
Time Delay ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Model ◽  
Transmission Modes ◽  
Latently Infected ◽  
Vital Effect ◽  
Infected Cells

<abstract><p>In this paper, a general HIV model incorporating intracellular time delay is investigated. Taking the latent virus infection, both virus-to-cell and cell-to-cell transmissions into consideration, the model exhibits threshold dynamics with respect to the basic reproduction number $ \mathfrak{R}_0 $. If $ \mathfrak{R}_0 &lt; 1 $, then there exists a unique infection-free equilibrium $ E_0 $, which is globally asymptotically stable. If $ \mathfrak{R}_0 &gt; 1 $, then there exists $ E_0 $ and a globally asymptotically stable infected equilibrium $ E^* $. When $ \mathfrak{R}_0 = 1 $, $ E_0 $ is linearly neutrally stable and a forward bifurcation takes place without time delay around $ \mathfrak{R}_0 = 1 $. The theoretical results and corresponding numerical simulations show that the existence of latently infected cells and the intracellular time delay have vital effect on the global dynamics of the general virus model.</p></abstract>

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.

INTRINSIC TRANSMISSION GLOBAL DYNAMICS OF TUBERCULOSIS WITH AGE STRUCTURE

2011 ◽  
Vol 04 (03) ◽  
pp. 329-346 ◽  
Author(s):  
JUN-YUAN YANG ◽  
XIAO-YAN WANG ◽  
XUE-ZHI LI ◽  
FENG-QIN ZHANG
Keyword(s):  
Steady State ◽  
Disease Transmission ◽  
Death Rate ◽  
Transmission Dynamics ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured

An age-structured epidemiological model for the disease transmission dynamics of TB is studied. We show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number is below one, and in this case, the disease always dies out. We prove that the endemic steady state exists when the basic reproductive number is above one. In addition, the endemic steady state is globally asymptotically stable if the basic reproductive number is above one and death rate due to TB is zero.

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