Stability Analysis Of Two Kinds Of Nonlinear Epidemic Model With Time Delay

A predator-prey model in the eco epidemiological model is a very important class of models, has been concerned by a mathematical biology field all the time, but the research most concentrated in the infected prey, a small number considering disease in the predator. Meanwhile, in recent years, consider the time lag factor, the stability of the model, asymptotic, cycle and all kinds of bifurcation phenomena research has increasingly become an important research topic. Introducing delay on ecosystem stability will produce great influence, delay in the interaction of population is unavoidable, therefore with delay of eco epidemiological model close to ecological reality. Based on the above In this paper, we study the stability of two kinds of nonlinear epidemic model with time delay, and the delay can be found to have an important effect on the stability of the model. This paper is divided into five chapters:The first chapter mainly introduceintroduces the research background and significance of the infectious disease model as well as the current domestic and international research status.The second chapter is the preparation of knowledge, respectively, the stability of the relevant theoretical knowledge, Hurwitz discriminant rule, Hopf bifurcation theory and the method of the stability of the relevant knowledge.Chapter three discusses a class of nonlinear delay SIRS infectious disease model with, first determine the basic reproductive number of the model, and by means of linearization, Hurwitz theorem and asymptotic stability theorem and the stability of the disease-free equilibrium are analyzed. For, and then use the delay differential equation stability theory the local stability of the positive equilibrium point are discussed, thus infer that reach a sufficient condition for the corresponding equilibrium state. The results show that in the propagation process of introducing time delays can destroy the system stability, when the delay as bifurcation parameter periodic solutions will also have a Hopf bifurcation, and then use the standard form theory and the center manifold theorem to derive the determining peripheral branchesThe fourth chapter mainly discusses the predator with global dynamics of an infectious disease of the predator- prey model. Among them, the predator can infect each other, delay said predator and prey during the gestation period, through the analysis of the characteristic equations of the equilibrium point corresponding to discuss the boundary equilibrium and the positive equilibrium point exists and Hopf bifurcation of the existence. By constructing function using invariance principle respectively, the sufficient conditions for the global asymptotic stability of the equilibrium point and the global stability of the discussion.The fifth chapter summarizes the content of this paper, points out the future research direction and the need to improve some places.