Stability Analysis Of HIV-1Infection Model With Two Time Delays

AIDS is a chronic infectious disease of high mortality rate, which is caused by humanimmunodefciency virus (HIV) infection. From being infected with HIV to developingAIDS can last10years. After the body infected with HIV, the body’s immune system isin the destruction. Since the frst AIDS case was discovered in1981, there are60millionpeople infected with HIV/AIDS in the world. Malignant spread of AIDS already hassignifcant impacts on the population age structure in some regions and countries. Inview of the high spread speed of AIDS infection, the importance of the research of itspathogenesis and preventive treatment is increasingly prominent. The current research onthe mathematical models of AIDS mainly has two directions, at the macro level, to studypopulation infection; at a micro level, to research HIV transmission mechanism in thebody. Considering delay factors often can be more realistic, it can refect the incubationperiod of disease infection, immune period of the recovered and other practical phenomena.In this paper, we studied two diferent types of mathematical models to illustrate the efectof delay on the spread of HIV/AIDS disease. Its main contents can be summarized asfollows:In the frst section, we developed a HIV/AIDS model with intracellular delay andimmune delay. We prove local stability and global stability of the disease-free equilib-rium and infected equilibrium by using Routh-Hurwitz criterion and Lyapunov-Lasalleinvariance principle, respectively. CTL-response delay, that is, the time between anti-genic stimulation and generating CTLs. Using this delay as a bifurcation parameter, weobtain that the model undergoes a Hopf bifurcation. Finally, some numerical simulationsby Matlab are presented to illustrate the results obtained.In the second section, the delay τ2, which describes the maturation time of the newlyproduced viruses as a parameter. We considered a virus dynamics model with Beddington-DeAngelis functional response and two time delays. We also obtain some sufcient condi- tions for the existence to the three positive equilibrium. By using Routh-Hurwitz criterion,we prove local stability of the disease-free equilibrium and infected equilibrium. ThroughLyapunov-Lasalle invariance principle, we investigated the global asymptotical stabilityof the equilibrium. So we conclude that the delay has no efect to the stability of theequilibrium.