Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan.
J Biol Dyn. 2008 Oct;2(4):449-64. doi: 10.1080/17513750802213581.
In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.
在本文中，我们报告了两个结构人口模型中的持久性的一些结果：一个慢性年龄结构的传染病模型和一个年龄持续时间结构的传染病模型。关于这些模型，我们观察到系统是一致强持续的，这意味着，大致来说，如果基本再生数大于 1，那么受感染亚群的比例是有界的，并且该界与初始数据无关，经过足够长的时间后。我们通过采用蒂梅的技术来证明这一点，该技术需要一些关于正定性和紧性的条件。虽然紧性条件在一般的无限维函数空间中很难证明，但我们可以将弗雷歇-科尔莫戈罗夫 L(1)-紧性准则应用于我们的模型。我们研究的两个例子阐明了在结构人口模型中显示持久性的一种有用方法。
