SIR model with time dependent infectivity parameter : approximating the epidemic attractor and the importance of the initial phase.

Abstract : We consider a SIR model with birth and death terms and time-varying infectivity parameter β (t). In the particular case of a sinusoidal parameter, we show that the average Basic Reproduction Number ¯ R o , introduced in [Bacaër & Guernaoui, 2006], is not the only relevant parameter and we emphasize the rôle played by the initial phase, the amplitude and the period. For a (general) periodic infectivity parameter β (t) a periodic orbit exists, as already proved in [Katriel, 2014]. In the case of a slowly varying β (t) an approximation of such a solution is given, which is shown to be asymptotically stable under an extra assumption on the slowness of β (t). For a non necessarily periodic β (t) , all the trajectories of the system are proved to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given.
Type de document :
Pré-publication, Document de travail
2018
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Soumis le : lundi 8 janvier 2018 - 16:27:29
Dernière modification le : jeudi 22 novembre 2018 - 14:04:34

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  • HAL Id : hal-01677886, version 1

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Stefanella Boatto, Catherine Bonnet, Bernard Cazelles, Frédéric Mazenc. SIR model with time dependent infectivity parameter : approximating the epidemic attractor and the importance of the initial phase.. 2018. 〈hal-01677886〉

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