Spatiotemporal dynamics of a reaction–diffusion epidemic model with nonlinear incidencerate

Abstract

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