scholarly journals Hopf Bifurcation for a Model of HIV Infection ofCD4+T Cells with Virus Released Delay

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Jun-Yuan Yang ◽  
Xiao-Yan Wang ◽  
Xue-Zhi Li
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Hiv Infection ◽  
Numerical Simulations ◽  
Period Solution ◽  
The Stability

A viral model of HIV infection ofCD4+T-cells with virus released period is formulated, and the effect of this released period on the stability of the equilibria is investigated. It is shown that the introduction of the viral released period can destabilize the system, and the period solution may arise. The direction and stability of the Hopf bifurcation are also discussed. Numerical simulations are presented to illustrate the results.

scholarly journals Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4+T-cells

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
P. Balasubramaniam ◽  
M. Prakash ◽  
Fathalla A. Rihan ◽  
S. Lakshmanan
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Hiv Infection ◽  
Logistic Growth ◽  
Differential Model ◽  
Discrete Time Delay ◽  
The Stability ◽  
Delay Differential

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection ofCD4+T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

Stability and Hopf Bifurcation for a HIV Infection Model with Delayed Immune Response

2013 ◽  
Vol 641-642 ◽  
pp. 808-811
Author(s):  
Xiao Zhang ◽  
Dong Wei Huang ◽  
Yong Feng Guo
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Hiv Infection ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Infection Model ◽  
Immune State ◽  
Hiv Infection Model ◽  
The Stability

In this paper, a class of HIV infection model with delayed immune response has been studied. We analyze the global asymptotic stability of the viral free equilibrium, and the stability and Hopf bifurcation of the infected equilibrium have been studied. Numerical simulations are carried out to explain the results of the analysis, and the change of the immune response of CTLs infects stability of system. These results can explain the complexity of the immune state of AIDs.

scholarly journals Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

2018 ◽  
Vol 28 (09) ◽  
pp. 1850109 ◽  
Author(s):  
Xiangming Zhang ◽  
Zhihua Liu
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Hiv Infection ◽  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Positive Equilibrium ◽  
Semilinear Equations ◽  
Age Structured ◽  
Hiv Infection Model

We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD[Formula: see text] T cells by a logistic function and the infected CD[Formula: see text] T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some parameter values such that this HIV infection model has a nontrivial periodic solution which bifurcates from the positive equilibrium. The numerical simulations are also carried out.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

Dynamical analysis of tumor-immune-help T cells system

2019 ◽  
Vol 12 (07) ◽  
pp. 1950075
Author(s):  
Huixia Li ◽  
Shaoli Wang ◽  
Fei Xu
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Tumor Cells ◽  
Tumor Antigens ◽  
Critical Value ◽  
Helper T Cells ◽  
Dynamical Analysis ◽  
Bistable Behavior ◽  
The Stability

In this paper, we construct a mathematical model to investigate the interaction between the tumor cells, the immune cells and the helper T cells (HTCs). We perform mathematical analysis to reveal the stability of the equilibria of the model. In our model, the HTCs are stimulated by the identification of the presence of tumor antigens. Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells, which leads to the elimination of the bistable behavior of the tumor-immune system, i.e. the equilibrium corresponding to the high steady state of tumor cells is destabilized. Choosing immune intensity [Formula: see text] as bifurcation parameter, there exists saddle-node bifurcation. Besides, there exists a critical value [Formula: see text], at which a Hopf bifurcation occurs. The stability and direction of Hopf bifurcation are discussed.

scholarly journals Stability Analysis for a Fractional HIV Infection Model with Nonlinear Incidence

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Linli Zhang ◽  
Gang Huang ◽  
Anping Liu ◽  
Ruili Fan
Keyword(s):  
Population Dynamics ◽  
Stability Analysis ◽  
Hiv Infection ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Nonnegative Solution ◽  
Infection Model ◽  
Nonlinear Incidence ◽  
Hiv Infection Model ◽  
The Stability

We introduce the fractional-order derivatives into an HIV infection model with nonlinear incidence and show that the established model in this paper possesses nonnegative solution, as desired in any population dynamics. We also deal with the stability of the infection-free equilibrium, the immune-absence equilibrium, and the immune-presence equilibrium. Numerical simulations are carried out to illustrate the results.

scholarly journals A Differential Equation Model of HIV Infection ofCD4+T-Cells with Delay

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang ◽  
Fengqin Zhang
Keyword(s):  
T Cells ◽  
Time Delay ◽  
Hiv Infection ◽  
Equation Model ◽  
Feasible Region ◽  
Delay Model ◽  
Cure Rate ◽  
Differential Delay ◽  
Discrete Time Delay ◽  
The Stability

An epidemic model of HIV infection ofCD4+T-cells with cure rate and delay is studied. We include a baseline ODE version of the model, and a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction numberR0<1. IfR0<1, the disease-free equilibrium is asymptotically stable and the disease dies out. IfR0>1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region. In the DDE model, the delay stands for the incubation time. We prove the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the virus-to-healthy cells transmission term can destabilize the system, and periodic solutions can arise through Hopf bifurcation.

Bifurcation Analysis in a Lotka-Volterra Model with Delay

2012 ◽  
Vol 594-597 ◽  
pp. 2693-2696
Author(s):  
Chang Jin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Volterra Model ◽  
The Stability ◽  
Theoretical Results

In this paper, a Lotka-Volterra model with time delay is considered. The stability of the equilibrium of the model is investigated and the existence of Hopf bifurcation is proved. Numerical simulations are performed to justify the theoretical results. Finally, main conclusions are included.

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