Stability Analyses Of Deterministic And Stochastic Seiri Epidemic Models With Nonlinear Incidence Rates And Distributed Delay

Infective diseases dynamics is a very important branch of biomathematics.According to the quantitative and qualitative analyses as well as numerical simulations,the process of the disease propagation could be revealed,and then the trend of infective diseases could be anticipated.On the basis of the cognition of epidemiology,the disease control and prevention could be provided.It is of great significance to study the model of epidemic disease.Also,so far there are lots of scholars have paid great effort on studies of lots of kinds of epidemic models.In this paper,we firstly propose the single group and,respectively,multi-group epidemic models with distributed delays and nonlinear incidence rates which accounts for the relapse of recovered individuals.By using mathematical analyses,we obtain the existence and uniqueness of the endemic equilibrium,and then obtain the sufficient conditions for guaranteeing the global stability of the endemic equilibrium by applying the Lyapunov Functional.The situation in which the deterministic models are subject to perturbations of white noise type is also investigated.Some sufficient conditions for stochastic asymptotic stability are derived by constructing the Lyapunov Functional.Finally,numerical simulations are given to illustrate the mathematical results above.