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.

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.

scholarly journals Effect of time delay on a detritus-based ecosystem

2006 ◽  
Vol 2006 ◽  
pp. 1-28 ◽  
Author(s):  
Nurul Huda Gazi ◽  
Malay Bandyopadhyay
Keyword(s):  
Time Delay ◽  
Equilibrium Points ◽  
Equation Model ◽  
Model System ◽  
Trophic Levels ◽  
Dynamical Analysis ◽  
Discrete Time Delay ◽  
Growth Equations ◽  
The Stability ◽  
Delay Differential

Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predator of detritivores. We have obtained the criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With delay differential equation model system we have studied the effect of time delay on the stability behaviour. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter.

scholarly journals Modelling the Drugs Therapy for HIV Infection with Discrete-Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Xueyong Zhou ◽  
Xiangyun Shi
Keyword(s):  
Time Delay ◽  
Hiv Infection ◽  
Discrete Time ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Discrete Time Delay ◽  
Form Theory ◽  
The Stability ◽  
Delay Differential ◽  
Two Equilibria

A discrete-time-delay differential mathematical model that described HIV infection of CD4+T cells with drugs therapy is analyzed. The stability of the two equilibria and the existence of Hopf bifurcation at the positive equilibrium are investigated. Using the normal form theory and center manifold argument, the explicit formulas which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.

scholarly journals Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kaushik Dehingia ◽  
Hemanta Kumar Sarmah ◽  
Yamen Alharbi ◽  
Kamyar Hosseini
Keyword(s):  
Time Delay ◽  
Immune Responses ◽  
Periodic Solutions ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Cancer Model ◽  
Delay Effect ◽  
Discrete Time Delay ◽  
The Stability ◽  
Vital Parameters

AbstractIn this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

scholarly journals Chaos in a Tumor Growth Model with Delayed Responses of the Immune System

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
M. Saleem ◽  
Tanuja Agrawal
Keyword(s):  
Immune System ◽  
Time Delay ◽  
Present Model ◽  
Type Model ◽  
Tumor Growth Model ◽  
Discrete Time Delay ◽  
Delay Effects ◽  
Equilibrium Level ◽  
The Stability ◽  
Growth Of Tumor

A simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed.

scholarly journals Stability analysis of an eco-epidemiological model incorporating a prey refuge

2010 ◽  
Vol 15 (4) ◽  
pp. 473-491 ◽  
Author(s):  
A. K. Pal ◽  
G. P. Samanta
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Predator Population ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Discrete Time Delay ◽  
The Stability

The present paper deals with the problem of a predator-prey model incorporating a prey refuge with disease in the prey-population. We assume the predator population will prefer only infected population for their diet as those are more vulnerable. Dynamical behaviours such as boundedness, permanence, local and global stabilities are addressed. We have also studied the effect of discrete time delay on the model. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.

Differential-Delay Equation Model of Gene Expression: A Continuum Approach

2008 ◽  
Author(s):  
Anael Verdugo ◽  
Richard H. Rand
Keyword(s):  
Analytical Study ◽  
Equation Model ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Differential Integral Equation ◽  
Gene Regulatory ◽  
The Stability ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Nonlinear Delay

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing instability is computed.

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