Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional

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 λ0$$ {\lambda}_0 $$ corresponding to strictly positive eigenfunction which is expressed as λ0=R0−1$$ {\lambda}_0&#x0003D;{R}_0-1 $$. By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for R0<1$$ {R}_0&lt;1 $$. For R0>1$$ {R}_0&gt;1 $$, 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.