Analysis On A Diffusive SIS Epidemic Model With Mass Action Incidence Rate And Spontaneous Infection In A Heterogeneous Environment

In this paper,we mainly investigate two reaction-diffusion SIS epidemic model with mass action and spontaneous infection mechanism in a spatially heterogeneous environment.On the one hand,we take no account of the influence of the advective environment on the model.Firstly,we prove that the solutions of parabolic system are uniformly bounded and investigate the existence of the endemic equilibrium（EE）.Then,we find that the endemic steady state is globally asymptotically stable in a spatially homogeneous environment.Finally,with the movement rate of the susceptible and infected population large,the asymptotic profiles of positive steady state are studied.On the other hand,we consider the influence of the advective environment on the model.Firstly,we similarly prove the uniform boundedness of parabolic system and the existence of the endemic equilibrium（EE）in a advective environment.Then,with the movement rate of the susceptible and infected population small and the advective velocity large,the asymptotic profiles of positive steady state are studied.These results will help us to better prevent and control the disease.