Study On A Kind Of SIS Epidemic Reaction-diffusion Model In A Spatially Heterogeneous Environment

In this paper,we study two susceptible-infected-susceptible epidemic reaction-diffusion models.The first is the SIS epidemic reaction-diffusion model with the saturation inci-dence rate.In the spatial heterogeneous environment,the model focuses on the existence,uniqueness and stable asymptotic distribution.We give the definition and properties of the basic reproduction number R0 under the model.Our conclusion shows that if R0<1 The unique disease-free equilibrium is globally asymptotic stable;if R0>1,the disease-free equilibrium is unstable,and there is a unique endemic equilibrium.Furthermore,we discuss the global attractivity of the endemic equilibrium for two cases,that is,when the disease transmission rate and disease recovery rate are constants,or when the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals,our conclusion shows that,In two cases,when R0>1,the endemic equilibrium is globally attractive.In the second model,the spontaneous infection term is added on the basis of the first model.We discuss the linear stability of the unique endemic equilibrium when the spatial environment is homogeneous.When the spatial environment is heterogeneous,we prove the existence of the endemic equilibrium and then analyze their asymptotic profile when the movement rate of susceptible or infected individuals is large or small.Compared with the situation without spontaneous infection,our study shows that spontaneous infection can enhance the persistence of infectious diseases,thus making the diseases more threatening.