Hopf Bifurcation Analysis Of A HIV-1 System With Time Delay

There is a method regarding the spread of the virus as a di?erential equation model and it belongs to the dynamics. We can more deeply understand the process of the spread of the virus through the models. Time delay always appears in the process of the spread of the virus. The emergence of time delay will make the dynamic behavior of the system more complex. So the study of time delay system has very important practical signi?cance. In this paper, according to the latent period of the virus, production cycle we have studied a mathematical model for HIV-1 infection with time delay and a mixing propagation model of computer viruses with time delay system. The main contents are as follows:First, a mathematical model for HIV-1 infection with multiply delays is considered.By applying the theorem of Hopf bifurcation and regarding the delay τ1, τ2, τ3as the bifurcation parameter. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability the positive equilibrium is investigated and Hopf bifurcation are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional di?erential equations.Finally, some numerical simulations are carried out for illustrating the theoretical results.Moreover, the e?ect multi-time delays is considered as an important factor, which will be more close to really nature. Comparing to the case of single time delay or the case without delay, research multi-time delays will need to more complicated and technique mass. Moreover, all the three state variables are in?uenced by time delay, multi-time delays can better impact the economic problems.Second, a mixing propagation model of computer viruses with time delay is considered.By applying the theorem of Hopf bifurcation and regarding the delay τ as the bifurcation parameter, the stability of positive equilibrium and existence of Hopf bifurcation are demonstrated. Then the direction of the Hopf bifurcation and stability of the bifurcating periodic solution is considered. Finally, numerical simulations are given.