Epidemic models in measure spaces: persistence, concentration and oscillations

We investigate the long-time dynamics of a SIR epidemic model in the case of a population of pathogens infecting a single host population. The pathogen population is structured by a phenotypic 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. 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 many cases it is possible to compute the eventual distribution of the surviving competitors. 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

Travelling wave solutions for some models in phytopathology

In this work several models of fungal disease propagation are considered. They consist of reaction-diffusion systems coupled with ordinary differential equations with or without time delay as well as integro-differential system of equations. We derive some conditions that ensure the existence and uniqueness of travelling wave solutions for these various models. Our proof is based on a suitable re-formulation in the form of a nonlinear integral equation with measure kernel convolutions

Effect of Crop Growth and Susceptibility on the Dynamics of a Plant Disease Epidemic: Powdery Mildew of Grapevine

Modéliser les interactions entre développement et architecture de la plante et épidémies de maladies fongiques aériennes, pour une gestion durable des cultures

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 R0 of the pathogen can be defined in that case and corresponds to a threshold between the persistence (R0 > 1) and the extinction (R0 ≤ 1) of the pathogen. When R0 > 1 and the maximal fitness is attained by at least one variant, we show that the systems reaches an equilibrium state that can be explicitly determined from the initial data. When R0 > 1 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

Modeling of the Invasion of a fungal Disease over a vineyard

The spatiotemporal spreading of a fungal disease over a vineyard is investigated using a SEIR-type model coupled with a set of partial differential equations describing the dispersal of the spores. The model takes into account both short and long range dispersal of spores and growth of foliar surface. Results of numerical simulations are presented. A mathematical result for the asymptotic behavior of the solutions is given as well

Singular Perturbation Analysis of travelling Waves for a Model in Phytopathology

disponible sur http://www.ripublication.com/mmnp.htmInternational audienc

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 $\mathcal{R}_0$ of the pathogen can be defined in that case and corresponds to a threshold between the persistence ($\mathcal{R}_0>1$) and the extinction ($\mathcal{R}_0\leq 1$) of the pathogen. When $\mathcal{R}_0>1$ and the maximal fitness is attained by at least one variant, we show that the systems reaches an equilibrium state that can be explicitly determined from the initial data. When $\mathcal{R}_0>1$ 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.Comment: 37 pages, 5 color figure

Singular Perturbation Analysis of travelling Waves for a Model in Phytopathology

disponible sur http://www.ripublication.com/mmnp.htmInternational audienc