Bifurcation and Instability of a Spatial Epidemic Model

This paper is concerned with a spatial S - I epidemic model with nonlinear incidence rate. First, the existence of the equilibrium is discussed in different conditions. Then the main criteria for the stability and instability of the constant steady-state solutions are presented. In addition, the effect of diffusion coefficients on Turing instability is described. Next, by applying the normal form theory and the center manifold theorem, the existence and direction of Hopf bifurcation for the ordinary differential equations system and the partial differential equations system are given, respectively. The bifurcation diagrams of Hopf and Turing bifurcations are shown. Moreover, a priori estimates and local steady-state bifurcation are investigated. Furthermore, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results demonstrate that the intrinsic growth rate, denoted as r, has significant influence on the spatial pattern. Specifically, different patterns appear, with the increase of r. The obtained results greatly expand on the discovery of pattern formation in the epidemic model.