Nonconstant positive steady states and pattern formation of a diffusive epidemic model

It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zero-flux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey's functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.