Dynamics for a diffusive epidemic model with a free boundary: spreading speed

TL;DR

This paper investigates the spreading speed of a diffusive epidemic model with a free boundary, providing exact spreading speed and asymptotic behavior of solutions when spreading occurs.

Contribution

It derives the exact spreading speed and sharp estimates for the asymptotic behavior of the epidemic model with a free boundary, advancing understanding of epidemic spread dynamics.

Findings

Exact spreading speed of the epidemic front is obtained.

Sharp estimates on the asymptotic behavior of solution components are derived.

The analysis relies on detailed study of semi-wave and steady state problems.

Abstract

We study the spreading speed of a diffusive epidemic model proposed by Li et al. \cite{LL}, where the Stefan boundary condition is imposed at the right boundary, and the left boundary is subject to the homogeneous Dirichlet and Neumann condition, respectively. A spreading-vanishing dichotomy and some sharp criteria were obtained in \cite{LL}. In this paper, when spreading happens, we not only obtain the exact spreading speed of the spreading front described by the right boundary, but derive some sharp estimates on the asymptotical behavior of solution component $(u,v)$. Our arguments depend crucially on some detailed understandings for a corresponding semi-wave problem and a steady state problem.