Asymptotic behavior of an epidemic model with infinitely many variants.

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number [Formula: see text] of the pathogen can be defined in that case and corresponds to a threshold between the persistence ([Formula: see text]) and the extinction ([Formula: see text]) of the pathogen population. When [Formula: see text] and the maximal fitness is attained by at least one variant, we show that the systems reaches an endemic equilibrium state that can be explicitly determined from the initial data. When [Formula: see text] but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.

An epi-evolutionary model for predicting the adaptation of spore-producing pathogens to quantitative resistance in heterogeneous environments.

We have modeled the evolutionary epidemiology of spore-producing plant pathogens in heterogeneous environments sown with several cultivars carrying quantitative resistances. The model explicitly tracks the infection-age structure and genetic composition of the pathogen population. Each strain is characterized by pathogenicity traits determining its infection efficiency and a time-varying sporulation curve taking into account lesion aging. We first derived a general expression of the basic reproduction number R 0 for fungal pathogens in heterogeneous environments. We show that the evolutionary attractors of the model coincide with local maxima of R 0 only if the infection efficiency is the same on all host types. We then studied the contribution of three basic resistance characteristics (the pathogenicity trait targeted, resistance effectiveness, and adaptation cost), in interaction with the deployment strategy (proportion of fields sown with a resistant cultivar), to (i) pathogen diversification at equilibrium and (ii) the shaping of transient dynamics from evolutionary and epidemiological perspectives. We show that quantitative resistance affecting only the sporulation curve will always lead to a monomorphic population, whereas dimorphism (i.e., pathogen diversification) can occur if resistance alters infection efficiency, notably with high adaptation costs and proportions of the resistant cultivar. Accordingly, the choice of the quantitative resistance genes operated by plant breeders is a driver of pathogen diversification. From an evolutionary perspective, the time to emergence of the evolutionary attractor best adapted to the resistant cultivar tends to be shorter when resistance affects infection efficiency than when it affects sporulation. Conversely, from an epidemiological perspective, epidemiological control is always greater when the resistance affects infection efficiency. This highlights the difficulty of defining deployment strategies for quantitative resistance simultaneously maximizing epidemiological and evolutionary outcomes.

Switching from a mechanistic model to a continuous model to study at different scales the effect of vine growth on the dynamic of a powdery mildew epidemic.

BACKGROUND AND AIMS: Epidemiological simulation models coupling plant growth with the dispersal and disease dynamics of an airborne plant pathogen were devised for a better understanding of host-pathogen dynamic interactions and of the capacity of grapevine development to modify the progress of powdery mildew epidemics. METHODS: The first model is a complex discrete mechanistic model (M-model) that explicitly incorporates the dynamics of host growth and the development and dispersion of the pathogen at the vine stock scale. The second model is a simpler ordinary differential equations (ODEs) compartmental SEIRT model (C-model) handling host growth (foliar surface) and the ontogenic resistance of the leaves. With the M-model various levels of vine development are simulated under three contrasting climatic scenarios and the relationship between host and disease variables are examined at key periods in the epidemic process. The ability of the C-model to retrieve the main dynamics of the disease for a range of vine growth given by the M-model is investigated. KEY RESULTS: The M-model strengthens experimental results observed regarding the effect of the rate of leaf emergence and of the number of leaves at flowering on the severity of the disease. However, it also underlines strong variations of the dynamics of disease depending on the vigour and indirectly on the climatic scenarios. The C-model could be calibrated by using the M-model provided that different parameters before and after shoot topping and for various vigour levels and inoculation time are used. Biologically relevant estimations of the parameters that could be used for its extension to the vineyard scale are obtained. CONCLUSIONS: The M-model is able to generate a wide range of growth scenarios with a strong impact on disease evolution. The C-model is a promising tool to be used at a larger scale.