Bifurcation and Spatiotemporal Patterns of a Diffusive SI Epidemic Model

This paper studies an SI spatio-temporal model with homogeneous Neumann boundary conditions. First, the nonexistence of nonconstant positive steady states is proved by the energy method. Second, a priori estimate of nonconstant positive solutions is given, and the existence of nonconstant positive steady states is proved by using the Leray–Schauder degree theory. Then, the Turing instability of the unique positive constant steady-state is discussed. Furthermore, the local and global structures of the steady-state bifurcation are established using simple eigenvalues by applying the bifurcation theory. Finally, some analysis results are supplemented by numerical simulations.