Non-constant positive steady states of the epidemic model with non-monotonic incidence rate

In this paper, the positive steady states of the epidemic model with non-monotonic incidence rate are considered. Firstly, it is proved that the unique positive constant steady state is stable for the ODE system and the PDE system. Secondly, a priori estimate of positive steady states is given, and the non-existence of non-constant positive steady states is established by using Poincare inequality and Young inequality. Finally, the existence and bifurcation of non-constant positive steady states are studied by using the degree theory and the global bifurcation theorem.