传染性输入的动态人群流行病
An epidemic in a dynamic population with importation of infectives
摘要：
考虑一个大的均匀混合的动态种群，它具有常数的出生率和指数分布的寿命，平均总体规模$n$。分析了在该人群中发生的具有传染性输入的马尔可夫SIR（易感$\to$感染性$\to$恢复）传染病。处理的主要情况是其中$n\to\infty$，保持基本繁殖数$R_0$以及感染物的输入率固定，但假设平均感染期和平均寿命的商比$1/\log n$更快地趋于0。它示出，作为$ n \to \infty$，3维过程描述的人口是易感的，感染性的和恢复的分数的演变的行为，封装在一个1维再生过程$S=\{ S(t);t\ge 0\}$描述的人口是易感的限制部分。进程$S$确定性地增长，除了在每个再生周期的一个随机时间点处，在该时间点处，进程向下跳跃的大小完全由自上一次跳跃以来的等待时间确定。过程$S$的属性，包括跳跃大小和平稳分布，被确定。
英文摘要：
Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined.
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