Dynamics analysis of a spatiotemporal SI model

In this article, we investigate a susceptible-infected (SI) model with the saturated treatment, the non-monotonic incidence rate, the logistic growth, and the homogeneous Neumann boundary conditions. The global existence and the uniform boundedness of the parabolic system are performed. After that, we investigate the global stability of the disease free equilibrium (DFE) and the endemic equilibrium (EE), respectively. In the end, we give a priori estimates, some propositions of the nonconstant steady states to the elliptic system. Meanwhile, we find that the diffusion rates of the susceptible and the infected population can affect the nonexistence of the nonconstant steady states. An interesting finding is DFE and basic reproduction number do not exist when the intrinsic growth rate of the susceptible class less than the rate of susceptible individuals being vaccinated.