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Disease extinction versus persistence in discrete-time epidemic models. (English) Zbl 1437.92141

In this programmatic paper, the authors provide a theoretical framework for the computation of the basic reproduction number \(\mathcal{R}_0\) for a wide class of discrete-time populations that are governed by constant, geometric, Beverton-Holt or Ricker demographic equations (that is, the demographic of the population dynamics is either asymptotically constant, or under geometric growth), while also discussing the extinction of the disease or its persistence in terms of \(\mathcal{R}_0\).
To this purpose, they use a version of the next generation method tailored for discrete-time models to compute \(\mathcal{R}_0\), subsequently finding verifiable conditions for the global asymptotic stability of the disease-free equilibrium, provided that it is unique, when \(\mathcal{R}_0<1\) and proving the uniform persistence of the disease when \(\mathcal{R}_0>1\).
This theoretical framework is then applied to discuss the dynamics of a wealth of concrete, relevant models (a SEIR model for which the recruitment function can take several distinct forms, a cholera spread model which accounts both water-to-human and human-to-human transmission and an anthrax epidemic model in animal populations). The abstract stability and persistence results are also supplemented by means of numerical simulations using real-life epidemic data which, among others, suggest that the endemic equilibria of those models are asymptotically stable provided that they are unique.

MSC:

92D30 Epidemiology
93C55 Discrete-time control/observation systems
34D23 Global stability of solutions to ordinary differential equations

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References:

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