R_0 in continuous-time ecological models structured by age and space
Abstract
In this work we study the asymptotic behaviour in linear models of population dynamics by means of the basic reproduction number R_0. Our aim is to give a practical approach to the computation of the reproduction number in continuous-time population models structured by age and/or space. The traditional approach to the study of linear continuous-time population dynamics is the computation of the Malthusian parameter, i.e. the exponential growth rate of the population. Yet, another equivalent approach is possible which takes the generational viewpoint, [2], [3], [5]. For each system, firstly one has to distinguish between birth terms and the other ones like mortality and transition terms. Then, the basic reproduction number is computed as the spectral radius of the next-generation operator. However, different interpretations of what is a birth event give rise to different expressions and results, [2] and [1].В For infinite-dimensional systems (e.g. PDE), the second approach is always related to an eigenvalue problem with its corresponding eigenfunction [4] whereas, in general, this is not the case for the first approach.References
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