## Adaptive dynamics of spores-producing pathogens

Evolutionary epidemiology dynamics, particularly for spore producing pathogens, is here describe by integro-differential equations with nonlocal mutation terms. Parasites reproduce clonally and each strain is characterized by several pathogenicity traits corresponding to the basic infection steps (eg., infection efficiency, latent period, sporulation capacity depending on the age of infection). In a host homegenous environement, here the basic reproduction number (R0) can be used to study the spread of a new mutant strain in a host population already infected by a resident strain, and to characterize pathogen’s evolutionary attractors among a large number of pathogen strains. The joint epidemiological and evolutionary dynamics illustrates the concentration of the pathogen population around a unique evolutionary attractor (which is characterized by the R0). Before reaching this phenotype, the pathogen population lives during certain time around the initial dominant phenotype and then shifts by mutation and lives for a relatively long time around a local maximum of the R0. (Djidjou-Demasse, Ducrot & Fabre, Burie, Djidjou-Demasse & Ducrot, Burie, Djidjou-Demasse & Ducrot)

The monomorphic concentration phenomenon at the evolutionary equilibrium illustrates above is not generaly verify in the case of host heteregeneous environment. That is because the the optimization principle holds only for some configurations, and the shape of R0 function does not allow to characterize evolutionary attractors. The Pairwise Invasibility Plot (PIP) must be used instead. As illustrate below, we have a polymorphic pathogen population at equilibrium in different proportions (panels d,e) but only one local maximum for the R0 as confirmed by the gradient (panels b,c). However, the PIP predicts a polymorphic population (panel a). Indeed, singular strategies, muS and muR, are respectively branching point and evolutionarily stable (no nearby mutant can invade) (Djidjou-Demasse et al.).