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In this work we consider an epidemic system modelling the evolution of aspore-producing pathogen within a multi-host population of plants. Here wefocus our analysis on the study of the stationary states. We first discuss theexistence of such nontrivial states by using the theory of global attractors.Then we introduce a small parameter epsilon that characterises the width of themutation kernel, and we describe the asymptotic shape of steady states withrespect to epsilon. In particular, we show that the distribution of sporesconverges to the singular measure concentrated on the maxima of fitness of thepathogen in each plant population. This asymptotic description allows us toshow the local stability of each of the positive steady states in the regime ofnarrow mutations, from which we deduce a uniqueness result for the nontrivialstationary states by means of a topological degree argument. These analysesrely on a careful investigation of the spectral properties of some non-localoperators.

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