Dynamics in a reaction-diffusion epidemic model via environmental driven infection in heterogenous space.

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number is calculated. When the global asymptotical stability of the disease-free equilibrium is obtained, while when the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems.