Dynamics of a nonlocal SIR epidemic model with free boundaries

Abstract

This paper is devoted to a nonlocal SIR type epidemic model, with free boundary denoting the expanding front. We aim to examine the full information about the dynamical behaviors of the involved disease that caused by the nonlocal diffusion and growing habitat. We obtain a classification for the spreading and vanishing cases, and show that those are determined by the new proposed threshold function R0F(t) and especially the corresponding initial data R0F(0). We prove that R0F(t)≤R0 and R0F(t)→R0 as t tends to ∞, R0 here representing the basic reproductive number of the associated ordinary differential equation. Also, the result show that the introduce of the nonlocal diffusion makes the disease becomes more easier to spread. In addition, we present some numerical simulations to confirm the theoretical results, as well as the impacts of R0 and the expanding capability.

Keywords

• Nonlocal diffusion
• SIR epidemic model
• Free boundary
• Stefan condition
• Spreading–vanishing dichotomy
• Basic reproductive number