| In this paper, a SIR epidemic model with standard incidence rate and time delay is investigated. By analyzing the dynamics properties of this model, the behavior of solution and the existence of the equilibrium points are proved. Then we discuss the local stability of the equilibrium points. Finally, we obtain the global asymptotic stability of the equilibrium points.The organization of this article is as follows:In chapter one, we introduces the significance of this research subject and the historical background of this issue. And we analyze the present research in both domestic and international, and propose the structure of this article and the research ideas.In chapter two, we introduce several basic conceptions and models of infectious diseases dynamics. Then we modify the traditional model and consider a SIR epidemic model with standard incidence and time delay.In chapter three, we analyze the behavior of solution and the existence of equilibrium points in the third chapter. And then, we discuss the local stability of an endemic equilibrium and a disease free equilibrium by analyzing the corresponding characteristic equations. It is proved that, if the basic reproductive number Ro<1, the disease free equilibrium E0of system (1.1) is locally stable. If/R0>1,2μ1,+k>β, the endemic equilibrium E*of system (1.1) is locally stable. Finally, we discuss the global stability of equilibrium points. We obtain the global stability of the disease free equilibrium by using Halanaly inequality, it is shown that if Ro<1, A/μ1<1, the disease free equilibrium Eo of system (1.1) is globally asymptotically stable. And by means of Lyapunov functional technique, we obtain the global stability of the endemic equilibrium, IfR0>1,2μ1+κ>β, the endemic equilibrium E*of system (1.1) is globally asymptotically stable.In chapter four, the main research results of this article are summarized. And points out the future research direction and the problems that need further study. |
PDF Full Text Request