Longtime behaviors of an epidemic model with nonlocal diffusions and a free boundary: spreading-vanishing dichotomy

We propose a nonlocal epidemic model whose spatial domain evolves over time and is represented by $[0,h(t)]$ with $h(t)$ standing for the spreading front of epidemic. It is assumed that the agents can cross the fixed boundary $x=0$, but they will die immediately if they do it, which implies that the area $(-\mathrm{\infty },0)$ is a hostile environment for the agents. We first show that this model is well posed, then prove that the longtime behaviors are governed by a spreading-vanishing dichotomy and finally give some criteria determining spreading and vanishing. Particularly, we obtain the asymptotical behaviors of the principal eigenvalue of a cooperative system with nonlocal diffusions without assuming the related nonlocal operator is self-adjoint, and the steady state problem of such cooperative system on half space $[0,\mathrm{\infty })$ is studied in detail.