Erratum to: Andronov-Hopf bifurcation and sensitivity analysis of a time-delay HIV model with logistic growth and antiretroviral treatment

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.

Download Full-text

Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay

International Journal of Biomathematics ◽

10.1142/s179352451550059x ◽

2015 ◽

Vol 08 (05) ◽

pp. 1550059 ◽

Cited By ~ 4

Author(s):

M. Prakash ◽

P. Balasubramaniam

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Viral Infection ◽

Limit Cycles ◽

Vital Role ◽

Logistic Growth ◽

Infection Model ◽

Bifurcation Parameter ◽

Delay Differential ◽

Manifold Theory

Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state. Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E* and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.

Download Full-text

Stabilizing Effect of Prey Refuge and Predator’s Interference on the Dynamics of Prey with Delayed Growth and Generalist Predator with Delayed Gestation

International Journal of Ecology ◽

10.1155/2014/429086 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 7

Author(s):

Debaldev Jana

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Chaotic Behavior ◽

Stability Loss ◽

Critical Values ◽

Abnormal Behavior ◽

Logistic Growth ◽

Generalist Predator ◽

Multiple Time ◽

Predator Population

In the present paper, I study a prey-predator model with multiple time delays where the predator population is regarded as generalist. For this regard, I consider a Holling-Tanner prey-predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predator’s interference in predator-prey relationship provides better descriptions of predator's feeding over a range of prey-predator abundances, so the predator's functional response here is considered to be Type II ratio-dependent. In accordance with previous studies, it is observed that delay destabilizes the system, in general, and stability loss occurs via Hopf bifurcation. There exist critical values of delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when delay parameters cross their critical values. When delay parameters are large enough than their critical values, the system exhibits chaotic behavior and this abnormal behavior may be controlled by refuge. Numerical computation is also performed to validate different theoretical results. Lyapunov exponent, recurrence plot, and power spectral density confirm the chaotic dynamical behaviors.

Download Full-text

Dynamical analysis of a giving up smoking model with time delay

Advances in Difference Equations ◽

10.1186/s13662-019-2450-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 4

Author(s):

Zizhen Zhang ◽

Ruibin Wei ◽

Wanjun Xia

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Local Stability ◽

Center Manifold ◽

Dynamical Behavior ◽

Sufficient Conditions ◽

Dynamical Analysis ◽

Normal Form Theory ◽

Center Manifold Theorem ◽

Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

Download Full-text