Abstract
We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue corresponding to strictly positive eigenfunction which is expressed as . By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for . For , and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function.
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