Global dynamics of a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments

In this paper, we study a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments under Neumann boundary conditions. Firstly, the existence and boundedness of the solution of the system are addressed. Then, on the basis of the differential inclusions theory, we analysis the existence of endemic equilibrium. Furthermore, by constructing different appropriate Lyapunov functions, we investigate the global asymptotic stability of the disease free equilibrium(DFE) and the endemic equilibrium(EE), respectively. Additionally, numerical simulations are given to confirm the correctness of theorem. Finally, we give a brief conclusion and discussion in the end of the paper.