Bifurcation and Spatiotemporal Patterns of SI Epidemic Model with Diffusion

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations.