Turing patterns in a diffusive epidemic model with saturated infection force

In this paper, we investigate the complex dynamics of a reaction-diffusion epidemic model with a saturated infection force analytically and numerically. We give the stability of the constant positive steady-states and the nonexistence/existence of nonconstant positive steady-states of the model which shows that, if the diffusion coefficients are properly chosen, the model exhibits stationary Turing pattern as a result of diffusion. Via numerical simulations, we present the evolutionary processes that involve organism distribution and the interaction of spatially distributed infection with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth of holes, stripes and spots pattern replication. In the viewpoint of epidemiology, we must regulate and control the parameters in the special range to control the disease.