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The regulation of an age-structured population by a fatal disease. (English) Zbl 0812.92019
Summary: 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.

MSC:
 92D30 Epidemiology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92D25 Population dynamics (general) 35B35 Stability in context of PDEs