A Class Of SIS Reaction-diffusion Equations With Nonlinear Saturation Incidence

In this thesis,we study an SIS reaction-diffusion epidemic model with nonlinear saturation incidence of S in a spatial heterogeneous environment,and give the existence,uniqueness,stability and asymptotic profile of the equilibrium solution of the model.First,we define the basic reproduction number R₀of the P DE model.By studying the properties of R₀,we conclude that if there is R₀<1,the disease-free equilibrium can always exist stably and the endemic equilibrium does not exist,if there is R₀>1,the disease-free equilibrium is unstable and there is an endemic equilibrium.Secondly,when a domain is high-risk,that is R₀>1,then an endemic equilibrium exists.Finally,we discuss the asymptotic profile of the endemic equilibrium when it exists.On the one hand,we consider that when the diffusion rate of the susceptible population tends to infinity,the disease will still exist and will not be eliminated.On the other hand,we consider that a domain is high-risk,then the disease will not be eliminated when the diffusion rate of susceptible population and the diffusion rate of infected population tend to infinity.If the diffusion rate of the susceptible population is fixed,the diffusion rate of the infected population tends to infinity,and then the diffusion rate of the susceptible population tends to zero,then the disease may disappear.