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Spatiotemporal dynamics of a reaction–diffusion epidemic model with nonlinear incidence rate

In this paper, we investigate the complex dynamics of a reaction–diffusion I–R model with a nonlinear rate of incidence of saturated mass action under zero-flux boundary conditions. We give an analysis of the boundedness, dissipation, and local and global stability of the positive equilibria. And we show the conditions for Turing instability and determine the Turing space in the parameter space. On the basis of these results, we present the evolutionary processes that involve organism distribution and the interaction of spatially distributed infection with local diffusion, and we find that the model dynamics exhibits a diffusion-controlled formation growth of spot, stripe–spot, stripe, stripe–hole and hole pattern replication. Furthermore, we indicate that the speed of disease spreading increases with the parameter A or the diffusion of infection increasing.