Louie K, Roberts M G, Wake G C
Department of Mathematics, Massey University, Palmerston North, New Zealand.
IMA J Math Appl Med Biol. 1994;11(4):229-44. doi: 10.1093/imammb/11.4.229.
A model describing the effect of a fatal disease on an age-structured population which would otherwise grow is presented and analysed. If the disease is capable of regulating host numbers, there is an endemic steady age distribution (SAD), for which an analytic expression is obtained under some simplifying assumptions. The ability of the disease to regulate the population depends on a parameter R(alpha), which is defined in terms of the given age-dependent birth and death rates, and where alpha is the age-dependent disease-induced death rate. If R(alpha) < 1 the endemic SAD is attained, while R(alpha) > 1 means the disease cannot control the population's size. The number R(0) is the expected number of offspring produced by each individual in the absence of the disease; for a growing population we require R(0) > 1. A stability analysis is also performed and it is conjectured that the endemic SAD is locally asymptotically stable whenever it is attained. This is demonstrated explicitly for a very simple example where all rates are taken as constant.
本文提出并分析了一个模型，该模型描述了一种致命疾病对原本会增长的年龄结构人口的影响。如果该疾病能够调节宿主数量，则存在一个地方病稳定年龄分布（SAD），在一些简化假设下可得到其解析表达式。疾病调节种群的能力取决于参数R(α)，它是根据给定的年龄依赖性出生率和死亡率定义的，其中α是年龄依赖性疾病诱导死亡率。如果R(α) < 1，则达到地方病SAD，而R(α) > 1意味着疾病无法控制种群规模。数量R(0)是在没有疾病的情况下每个个体产生的后代预期数量；对于增长的种群，我们要求R(0) > 1。还进行了稳定性分析，并推测只要达到地方病SAD，它就是局部渐近稳定的。对于一个非常简单的例子，即所有速率都视为常数的情况，这一点得到了明确证明。
