中文摘要：

该文研究一类SEI传染病模型,其中病毒在潜伏期和感染期具有感染性.首先研究固定区域上SEI偏微分方程组,考虑平衡解的局部稳定性和全局稳定性.然后重点研究相应的自由边界问题,其中自由边界表示病毒的移动边沿。给出了该问题解的全局存在性、唯一性,讨论了自由边界的性质,证明了病毒要么蔓延,要么消退.还给出了蔓延和消退的充分条件,结果表明：当有效接触率很小或平均潜伏期较短,且初始染病区域小时,疾病消退;而当有效接触率大或平均潜伏期较长,且初始染病区域大时,疾病蔓延.

英文摘要：

This paper is concerned about an SEI model, in which the disease is infectious in the latent period and the infected period. We first consider the PDE system in a fixed domain, the local and global stabilities of equilibriums are given. More attention has been given to the free boundary problem, which describes the moving front. Global existence and uniqueness of the solution are first given and then the properties of the free boundary are discussed. We prove that either the disease spreads or vanishes. Sufficient conditions for spreading or extinction are given. Our results show that when the contact rate is very small or average incubation period is short, and the initial infected domain is small enough, then the disease vanishes; and when the contact rate is big or the average incubation period is long, and the initial infected domain is sufficiently large, then the disease spreads.