Dynamics for a diffusive epidemic model with a free boundary: spreading-vanishing dichotomy

Abstract

This paper involves a diffusive epidemic model whose domain has one free boundary with the Stefan boundary condition, and one fixed boundary subject to the usual homogeneous Dirichlet or Neumann condition. By using the standard upper and lower solutions method and the regularity theory, we first study some related steady state problems which help us obtain the exact longtime behaviors of solution component $(u,v)$. Then we prove there exists the unique classical solution whose longtime behaviors are governed by a spreading-vanishing dichotomy. Lastly, the criteria determining when spreading or vanishing happens are given with respect to the basic reproduction number $\mathcal{R}_0$, the initial habitat $[0,h_0]$, the expanding rates $\mu_1$ and $\mu_2$ as well as the initial function $(u_0,v_0)$. The criteria reveal the effect of the cooperative behaviors of agents and humans on spreading and vanishing.