Concentration estimates in a multi-host epidemiological model structured by phenotypic traits

TL;DR

This paper analyzes the stationary states of a multi-host plant pathogen epidemic model, showing how pathogen distribution concentrates on fitness maxima as mutation width narrows, and establishing stability and uniqueness of these states.

Contribution

It introduces an asymptotic analysis of steady states in a multi-host epidemic model with phenotypic traits, revealing concentration phenomena and stability results.

Findings

Pathogen distribution converges to measures on fitness maxima.

Positive steady states are locally stable in narrow mutation regimes.

Uniqueness of stationary states is established via topological degree arguments.

Abstract

In this work we consider an epidemic system modelling the evolution of a spore-producing pathogen within a multi-host population of plants. Here we focus our analysis on the study of the stationary states. We first discuss the existence of such nontrivial states by using the theory of global attractors. Then we introduce a small parameter epsilon that characterises the width of the mutation kernel, and we describe the asymptotic shape of steady states with respect to epsilon. In particular, we show that the distribution of spores converges to the singular measure concentrated on the maxima of fitness of the pathogen in each plant population. This asymptotic description allows us to show the local stability of each of the positive steady states in the regime of narrow mutations, from which we deduce a uniqueness result for the nontrivial stationary states by means of a topological degree argument. These analyses rely on a careful investigation of the spectral properties of some non-local operators.