Epidemic models in measure spaces: persistence, concentration and oscillations

TL;DR

This paper analyzes the long-term behavior of a measure-based SIR epidemic model with pathogen genotypic structure, revealing conditions for persistence, concentration on fitness maxima, coexistence, and oscillatory dynamics.

Contribution

It provides a detailed characterization of the asymptotic distributions and dynamics in a measure space epidemic model, including coexistence and oscillations, which were not previously understood.

Findings

Population persists on fitness maxima when initial mass is zero.

Multiple species can coexist if they maximize fitness.

Oscillatory behavior can occur, preventing convergence to a stationary distribution.

Abstract

We investigate the long-time dynamics of a SIR epidemic model in the case of a population of pathogens infecting a homogeneous host population. The pathogen population is structured by a genotypic 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. When this initial mass is zero, we show the persistence of the population of infected and the concentration of the population of pathogens on the set of genotypic traits that maximize the fitness. 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 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.