Stability And Hopf Bifurcation On Some Delayed Dynamic Epidemic Models

By the differential equations model, the propagation law of infectious diseases is investigated, which can reveal the effect of the various factors in the propagation law, then we can achieve the ultimate purpose of controlling infectious diseases finally, which is an important issue in mathematical biology. Taking into account the fact that the propagation of infectious diseases is only depend on the current status, we study the epidemic model by the ordinary differential equation. If we consider the fact that the propagation of infectious diseases is also depend on the previous status, then we can analyze the delayed epidemic model.In this paper, using the method of constructing Lyapunov functional, LaSalle’s invariance principle and the uniform persistence, combining Hopf bifurcation theory and center manifold theory to study the delayed dynamic model of infectious diseases, the stability and Hopf bifurcation are analyzed. The main contents are as follows:First, we incorporate the nonlinear incidence rate into epidemic model, and the delayed epidemic model with relapse phenomenon and nonlinear incidence rate is investigated. We prove that the system is uniformly persistent. Constructing suitable Lyapunov functional and using LaSalle’s invariance principle, the global attractiveness of the disease-free equilibrium and endemic equilibrium of the system is established respectively. The disease-free equilibrium is globally attractive when the basic reproduction number is less than or equal to one, and the endemic equilibrium is globally attractive when the basic reproduction number is greater than one.Secondly, by incorporating the heterogeneities in the host population and heterogeneous stages of infection we consider the multi-group model with nonlinear incidence rate and infection delay, and the multistage model with nonlinear incidence rate and infection delay respectively. We utilize the graph theory to the calculation of Lyapunov functionals, and use LaSalle’s invariance principle, and then we prove that the global dynamics for each of the two models are determined completely by the corresponding basic reproduction number. If the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than one, the endemic equilibrium is globally asymptotically stable. Then we conclude that the heterogeneity does not alter the global dynamics of the SIR model when the incidence rate is a general nonlinear function.Thirdly, we investigate the two SIV strains infection model with immune response delay and infection delay. The global stability of infection-free equilibrium and the local stability of single infection equilibrium are obtained, and we analyze the Hopf bifurcation of system. We show that the infection delay can destabilize the single infection equilibrium leading to Hopf bifurcation and periodic oscillations. The results show that both virus strains can not coexist without immune responses, and the infection delay can change the outcome of the competition.Finally, by incorporating an extra logistic growth for T-cells and immune response delay, we study the HIV model with both logistic growth for T-cells and immune response delay. Constructing suitable Lyapunov functional and using LaSalle’s invariance principle, we obtain that the infection-free steady state is globally asymptotically stable. We prove that the system is uniformly persistent. we show that both the immune response delay and the intracellular delay may destabilize the infected steady state leading to Hopf bifurcation, on which we analyze the direction of the Hopf bifurcation as well as the stability of the bifurcating periodic orbits by normal form and center manifold theory. Our results suggest that the logistic growth for T-cells, the infection delay and the immune delay may be responsible for the rich virus dynamics. We can control the viral load by adjusting the logistic growth for T-cells, the intracellular delay and the immune delay.