In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number $ R_0 $ obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if $ R_0>1 $, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

中文翻译：

---

具有自由边界的扩散流行病模型的长期动力学

在本文中，我们确定了具有自由边界的反应扩散系统的长期动力学行为，该系统对以自由边界表示的流行病的传播进行了建模。该系统简化为Capasso和Maddalena的流行模型[5当边界固定时，它简化为Ahn等人的模型。[1个如果忽略感染性宿主群体的扩散。我们证明了消失的二分法，并确切地确定了每一个替代方案何时发生。如果从相应的ODE模型获得的复制数$ R_0 $不大于1，则此处建模的流行病将消失，而如果$ R_0> 1 $，则该流行病可能会消失或扩散，取决于其初始大小，取决于二分法标准。此外，当发生传播时，我们表明该流行病的传播前沿具有渐近的传播速度，这取决于相关的半波问题。