Fear effect on the mobility of individuals in a spatially heterogeneous environment: a delayed diffusive SPIR epidemic model

Abstract

Abstract As the fear of infection is a crucial factor in the progress of the disease in the population. We aim, in this study, to investigate a susceptible-protected-infected-recovered (SPIR) epidemic model with mixed diffusion modeled by local and nonlocal diffusions. These types of diffusion are used to model the fear effect of being infected by the population. The model is shown to be well-posed; the solution exists, is positive, and is unique. The variational expression is obtained to determine threshold role of $$\mathfrak {R}_0$$ , also known as the basic reproduction number. Indeed, for $$\mathfrak {R}_0<1$$ , we show that the epidemic will extinct, corresponding to the global asymptotic stability of the infection-free equilibrium state. However, when $$\mathfrak {R}_0>1$$ , the existence of the infection equilibrium state and the uniform persistence of the solution have been proved. The Lyapunov function have been used to show the global asymptotic stability of the infection equilibrium state. Moreover, we compared the obtained results with the classical SIR epidemic model for determining the required protection function for stopping the disease, which can be obtained by reducing $$\mathfrak {R}_0$$ below one.