Dynamics for a diffusive epidemic model with a free boundary: sharp asymptotic profile

TL;DR

This paper investigates the precise long-term spreading profiles of a diffusive epidemic model with a free boundary, revealing that boundary conditions at the fixed boundary influence the asymptotic behavior of the spreading front and solutions.

Contribution

It improves previous results by establishing sharp asymptotic profiles and showing boundary conditions at the fixed boundary lead to similar spreading behaviors.

Findings

Homogeneous Dirichlet and Neumann boundary conditions yield the same asymptotic spreading profiles.

Constructed upper and lower solutions to analyze the spreading behavior.

Determined the sharp asymptotic spreading speed and profile near the front.

Abstract

This paper concerns the sharp asymptotic profiles of the solution of a diffusive epidemic model with one free boundary and one fixed boundary which is subject to the homogeneous Dirichlet boundary condition and Neumann boundary condition, respectively. The longtime behaviors has been proved to be governed by a spreading-vanishing dichotomy in \cite{LL}, and when spreading happens, the spreading speed is determined in \cite{LLW}. In this paper, by constructing some subtle upper and lower solutions, as well as employing some detailed analysis, we improve the results in \cite{LLW} and obtain the sharp asymptotic spreading profiles, which show the homogeneous Dirichlet boundary condition and Neumann boundary condition imposed at the fixed boundary $x=0$ lead to the same asymptotic behaviors of $h(t)$ and $(u,v)$ near the spreading front $h(t)$.