Analysis Of Epidemic Dynamical Models With Nonlinear Rate And Time Delay

The thesis mainly considers some factors which influence the epidemic law of infectious diseases. By introducing nonlinear incidence rate, three classes of epidemic models were established and the dynamics behavior of the models were analyzed.Firstly, some infectious diseases have both horizontal and vertical mode of transmission and infectious disease may affect the patient’s fertility. In addition, after the patient recover, they may be loss of immunity and become susceptible again. Considered the above factors, a delayed SIRS epidemic model was established by using the nonlinear incidence rate. It was calculated the existence of equilibrium and get the basic reproductive number R^ of the model. The local asymptotical stability of the disease-free equilibrium and endemic equilibrium were discussed by using Jacobi matrix, Hurwtiz theorem, LaSalle invariance principle and intermediate value theorem.Secondly, considering people prevent and control infectious diseases from vaccination, therefore some susceptible people are become the recovered people. And some immunity to disease is not permanent, which makes the part of the recovered will become the susceptible. It should be better to describe the immune period of the vaccinators by using delay. Therefore, in chapter 4, the case of continuous vaccination was considered.By introducing the nonlinear incidence rate (βSI)/(1+α1S+α2I), a class of delayed epidemic model with continuous vaccination was established. Furthermore, we analyzed the existence of equilibrium of the model, discussed the local asymptotic stability of the model by linearizing the model at the equilibrium point and proved the global asymptotic stability of endemic equilibrium by constructing Liapunov function.Finally, considering the delay in the infected period and the probability of survival situation, we assume that the recovered people has permanent immunity. A class of delayed SIR epidemic model with nonlinear incidence rate were established. We proved global asymptotic stability of the endemic equilibrium by constructing Liapunov function. Furthermore, we illustrate the validity of the theoretical results for the above models by numerical simulations.