BURIE, Jean-Baptiste; DUCROT, Arnaud; GRIETTE, Quentin

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BURIE, Jean-Baptiste
Institut de Mathématiques de Bordeaux [IMB]

DUCROT, Arnaud
Laboratoire de Mathématiques Appliquées du Havre [LMAH]

GRIETTE, Quentin
Laboratoire de Mathématiques Appliquées du Havre [LMAH]

Langue

Document de travail - Pré-publication

Résumé en anglais

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number R0 of the pathogen can be defined in that case and corresponds to a threshold between the persistence (R0 > 1) and the extinction (R0 ≤ 1) of the pathogen. When R0 > 1 and the maximal fitness is attained by at least one variant, we show that the systems reaches an equilibrium state that can be explicitly determined from the initial data. When R0 > 1 but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.< Réduire

Mots clés en anglais

Ordinary differential equations ODE

asymptotic behavior

Population dynamics

infinite dynamical system

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190551

Project ANR

Architecture génétique des caractères quantitatifs dans les interactions plante-virus: conséquences pour la gestion des variétés résistantes et/ou tolérantes à l'échelle du paysage. - ANR-18-CE32-0004
Dynamiques d'invasion et asymptotiques non triviales - ANR-21-CE40-0008

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251