Epidemic models in measure spaces: persistence, concentration and oscillations

Jean-Baptiste Burie (1, 2) , Arnaud Ducrot (3, 4) , Quentin Griette (3, 4)

1 UB - Université de Bordeaux
2 IMB - Institut de Mathématiques de Bordeaux
3 ULH - Université Le Havre Normandie
4 LMAH - Laboratoire de Mathématiques Appliquées du Havre

We investigate the long-time dynamics of a SIR epidemic model in the case of a population of pathogens infecting a single host population. The pathogen population is structured by a phenotypic variable. When the initial mass of the maximal fitness set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and some other parameter of the model, and we provide a way to characterize when this unexpected extinction happens. Finally, we provide an example of a pathological situation in which the distribution never reaches a stationary distribution but oscillates forever around the set of fitness maxima.

Mots clés: ordinary differential equations; asymptotic behavior; population dynamics; Radon measure; evolution

Domaines: Systèmes dynamiques [math.DS]; Analyse classique [math.CA]; Equations aux dérivées partielles [math.AP]

Jean-Baptiste Burie, Arnaud Ducrot, Quentin Griette. Epidemic models in measure spaces: persistence, concentration and oscillations. 2022. ⟨hal-03286220v3⟩