一类具有自由边界的反应扩散对流的SI传染病模型

A Reaction Diffusion-advection SI Epidemic Model with Free Boundaries

本文研究了一类具有双自由边界的SI模型,引入两个自由边界来描述疾病传播的边沿.首先,讨论了该问题全局解的存在性和唯一性.其次,分别定义了相应于该模型下的常微分方程系统和在固定域上的系统的基本再生数R0与R0^D.进而,定义了该模型在自由边界条件下的基本再生数R0^F,并获得了疾病消退或蔓延的充分条件,结果表明：当R0〈1时,无论染病者的初始值多少,疾病都不会蔓延到整个区域.而当R0-F〈1且染病者的初始值‖I0（x）‖C（[-h0,h0]）充分小时,疾病将消退;当R0-F〉1时,疾病将蔓延.

A reaction-diffusion-advection SI epidemic model is investigated, Two free boundaries are introduced to describe the spreading frontiers of the disease. First, the existence and uniqueness of the global solution are given. Secondly, the basic reproduction number Ro and R0^D are defined for the corresponding ODE system and the corresponding system in the fixed domain of the free boundary problem, meanwhile, the basic reproduction number R0^F is defined for the free boundary problem. Sufficient conditions for vanishing or spreading of the disease are obtained. Our results show that the disease will not spread to the whole area if R0〈1 no matter what the initial date are, and vanishing of the disease if R^F（0）〈1 and ‖I0（x）‖C（[-h0,h0]） is sufficiently small. Spreading of the disease occurs if R0^F（0）〉1.