An SIR epidemic model with free boundary

TL;DR

This paper analyzes a reaction-diffusion SIR epidemic model with a free boundary, establishing conditions for disease spread or vanishing based on the basic reproduction number and initial infected area size.

Contribution

It introduces a free boundary SIR model and provides rigorous conditions for disease spreading or vanishing, including existence and uniqueness of solutions.

Findings

Disease does not spread if R0<1 or initial infected radius is small.

Disease spreads if R0>1 and initial infected radius is large.

Existence and uniqueness of solutions are established.

Abstract

An SIR epidemic model with free boundary is investigated. This model describes the transmission of diseases. The behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain is investigated. The existence and uniqueness of the global solution are given by the contraction mapping theorem. Sufficient conditions for the disease vanishing or spreading are given. Our result shows that the disease will not spread to the whole area if the basic reproduction number $R_{0}<1$ or the initial infected radius $h_0$ is sufficiently small even that $R_{0}>1$. Moreover, we prove that the disease will spread to the whole area if $R_{0}>1$ and the initial infected radius $h_0$ is suitably large.