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Non-constant positive steady states of the epidemic model with non-monotonic incidence rate. (English) Zbl 1333.35116

Summary: 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 Poincaré 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.

MSC:

35K57 Reaction-diffusion equations
35B09 Positive solutions to PDEs
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B32 Bifurcations in context of PDEs
92D30 Epidemiology
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