Hopf Bifurcation Analysis of a Two-Delay HIV-1 Virus Model with Delay-Dependent Parameters

Yu Xiao,Yunxian Dai,Jinde Cao,Mohammad Yaghoub Abdollahzadeh Jamalabadi

doi:10.1155/2021/2521082

Abstract

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

Highlights

AIDS is a very dangerous infectious disease caused by HIV-1 virus which attacks the human immune system
Dong et al [6] studied the dynamics of the tumor immune system interaction model and investigated the existence of Hopf bifurcation with two time delays as bifurcation parameters
In [7], the authors discussed the influence of awareness coverage and time delays on infectious diseases and found that the endemic equilibrium existed a Hopf bifurcation in both delayed and nondelayed system

Summary

Introduction

AIDS is a very dangerous infectious disease caused by HIV-1 virus which attacks the human immune system. Dong et al [6] studied the dynamics of the tumor immune system interaction model and investigated the existence of Hopf bifurcation with two time delays as bifurcation parameters. Where e− mτ is the probability of surviving the time period from t − τ1 to t in case m is the death rate of infected but not yet virus-producing cells. E authors took the time delay as the parameter to carry out the dynamic analysis of the system, including the equilibrium stability and Hopf bifurcation existence by using the method in [14]. E luminescent spots are as follows: (1) in this paper, two different spreads of HIV-1 virus are studied, and we will introduce time delay into the coefficients to be more realistic; (2) the conditions of the local stability of positive equilibrium and the existence of.

Stability of the Positive Equilibrium and Existence of Hopf Bifurcation

Direction and Stability of the Hopf Bifurcation

Conclusion