Bifurcation analysis and pattern formation of an epidemic model with diffusion

In this paper, the dynamics of a reaction-diffusion epidemic model with saturated incidence rate is investigated. First, the Turing instability of the equilibrium and a priori estimates of nonconstant positive steady-state solutions are described. Then, the nonexistence and existence of nonconstant positive steady-state solutions are obtained by the energy method and degree theory. Next, we investigate the local structure of the steady-state bifurcation at both simple and double eigenvalues. Meanwhile, some conditions to determine the bifurcation direction are derived and the global structure of the bifurcation from simple eigenvalues are established by the global bifurcation theorem. Finally, some numerical simulations are presented to enrich and support the analytical conclusions.