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International audience; 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 urn:x-wiley:17524571:media:eva13328:eva13328-math-0001 for fungal pathogens in heterogeneous environments. We show that the evolutionary attractors of the model coincide with local maxima of urn:x-wiley:17524571:media:eva13328:eva13328-math-0002 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.

In this work we study the travelling wave solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interaction. Here the mutation process is described using a non-local convolution operator in the phenotype space. Using dynamical system ideas coupled with refined estimates on the asymptotic behaviour of the profiles, we prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ode system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c ⋆ , provided some parameters condition expressed using the principle eigenvalue of some integral operator. It is also used to prove that any travelling wave solution connects two determined stationary states.

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.

International audience; This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next we more closely investigate the behaviour of the system in the presence of multiple evolutionary attractors. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. Using the variance of the dispersion in the phenotype trait space as a small parameter we provide a complete picture of the dynamical behaviour of the solutions of the problem. We show that the dynamics exhibits two main and long regimes – those durations are estimated – before the solution finally reaches its long time configuration, the endemic equilibrium. The analysis provided in this work rigorously explains and justifies the complex behaviour observed through numerical simulations of the system.

We investigate spreading properties of solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interactions. In this work the mutation process is described using a non-local convolution operator in the phenotype space. Initially equipped with a localized amount of infection, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed of the travelling wave solutions discussed in [ 1 ]. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable and the phenotype one are separated.

We model 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 describing its infection efficiency and a time-varying sporulation curve taking into account lesion ageing. We first derive a general expression of the basic reproduction number ℛ0for fungal pathogens in heterogeneous environments. We show that evolutionary attractors of the model coincide with local maxima of ℛ0only if the infection efficiency is the same on all host types. We then study how three basic resistance characteristics (pathogenicity trait targeted, resistance effectiveness, and adaptation cost) in interaction with the deployment strategy (proportion of fields sown with a resistant cultivar) (i) lead to pathogen diversification at equilibrium and (ii) shape the transient dynamics from evolutionary and epidemiological perspectives. We show that quantitative resistance impacting only the sporulation curve will always lead to a monomorphic population, while dimorphism (i.e. pathogen diversification) can occur with resistance altering infection efficiency, notably with high adaptation cost and proportion of R cultivar. Accordingly, the choice of quantitative resistance genes operated by plant breeders is a driver of pathogen diversification. From an evolutionary perspective, the emergence time of the evolutionary attractor best adapted to the R cultivar tends to be shorter when the resistance impacts infection efficiency than when it impacts sporulation. In contrast, from an epidemiological perspective, the epidemiological control is always higher when the resistance impacts infection efficiency. This highlights the difficulty of defining deployment strategies of quantitative resistance maximising at the same time epidemiological and evolutionary outcomes.

A mathematical model describing the propagation of fungal diseases in plants is proposed. The model takes into account both chronological age and age since infection. We investigate and fully characterize the large time behaviour of the solutions. Existence of a unique endemic stationary state is ensured by a threshold condition: \begin{document}$\mathcal R_0>1$\end{document} . Then using Lyapounov arguments, we prove that if \begin{document}$\mathcal R_0 ≤ 1$\end{document} the disease free stationary state is globally stable while when \begin{document}$\mathcal R_0>1$\end{document} , the unique endemic stationary state is globally stable with respect to a suitable set of initial data.

The aim of this article is to derive an asymptotic two-scale model for the propagation of a fungal disease over a large vineyard. The original model is based on a singularly perturbed system of two linear reaction-diffusion equations coupled with a set of nonlinear ordinary differential equations in a highly heterogeneous medium. We prove the well-posedness of the asymptotic model and obtain a convergence result confirmed by numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

International audience; 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.

SUMMARYSalmonellosis is a foodborne disease of humans and animals caused by infection with Salmonella. The aim of this paper is to improve a deterministic model (DM) and an individual-based model (IBM) with reference to Salmonella propagation in flocks of laying hens taking into account variations in hens housed in the same cage and to compare both models. The spatio-temporal evolution, the basic reproduction number, R0, and the speed of wave propagation were computed for both models. While in most cases the DM allows summary of all the features of the model in the formula for computation of R0, slight differences between individuals or groups may be observed with the IBM that could not be expected from the DM, especially when initial environmental contamination is very low and some cages may get rid of bacteria. Both models suggest that the cage size plays a role on the risk and speed of propagation of the bacteria, which should be considered when designing new breeding systems.

We investigate the structure of travelling waves for a model of a fungal disease propa- gating over a vineyard. This model is based on a set of ODEs of the SIR-type coupled with two reaction-diffusion equations describing the dispersal of the spores produced by the fungus inside and over the vineyard. An estimate of the biological parameters in the model suggests to use a singular perturbation analysis. It allows us to compute the speed and the profile of the travelling waves. The analytical results are compared with numerical simulations.

International audience; A PDEs–ODEs model was developed to describe the spread of an airborne fungal pathogen on grapevine. The model was able to retrieve the main characteristics of the system: (1) a host growing during the whole season with time evolution in susceptibility, (2) a crop highly structured in rows with potential heterogeneities of plant growth and susceptibility within and between plots. These characteristics are modified by cultural management. Simulations were performed to test the effect of crop spatial heterogeneities, within and between plots, on the disease spread. Heterogeneities considered were the plant growth (vigour, earliness), susceptibility (susceptible vs resistant, treated vs untreated) and the spatial arrangements (patches vs rows). The main effect on disease reduction was obtained by arrangement in rows of susceptible and fully resistant plants.

International audience; As any epidemic on plants is driven by the amount of susceptible tissue, and the distance between organs, any modification in the host population, whether quantitative or qualitative, can have an impact on the epidemic dynamics. In this paper we examine using examples described in the literature, the features of the host plant and the use of crop management which are likely to decrease diseases. We list the pathogen processes that can be affected by crop growth and architecture modifications and then determine how we can highlight the principal ones. In most cases, a reduction in plant growth combined with an increase in plant or crop porosity reduces infection efficiency and spore dispersal. Experimental approaches in semi-controlled conditions, with concomitant characterisation of the host, microclimate and disease, allow a better understanding and analysis of the processes impacted. Afterwards, the models able to measure and predict the effect of plant growth and architecture on epidemic behaviour are reviewed

International audience; The spread of a pathogen within an anthropized crop plot depends on many factors acting at contrasted spatio-temporal scales. This is of paramount importance for vine and powdery mildew, one of its airborne pathogen. We aimed at developing a coupled PDEs-ODEs model for plant-pathogen interactions at the plot scale level in order to assess the effects of various host heterogeneities on the epidemic spread.

Chantier qualité GA; International audience; We propose a mathematical spatial and age structured model to describe the spatial spread of Salmonella among laying hens in industrial hen houses. We provide a mathematical study of traveling pulses of infection and describe a minimal speed property for such a problem. The dependence with respect to some heterogeneities of the medium is also discussed. Finally, based on biological data, the parameters of the model are estimated to provide some information on the propagation speed of the bacteria.

International audience; *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.

International audience; For a better understanding of host / pathogen dynamical interactions and of the capacity of host development to modify disease progress, we devised epidemiological simulation models coupling plant growth with the dispersal and disease dynamics of an airborne plant pathogen. Our first model is a complex discrete deterministic model that explicitly incorporates the dynamics of host growth (e.g., distance between organs and their susceptibility) and the development and dispersion of the pathogen. The second one is a much simpler compartmental SEIR-like model. It takes into account the host growth with the foliar surface as unit of host growth and the ontogenetic resistance of the leaves. We compare the output of both models when applied to the specific system formed by the powdery mildew, Erysiphe necator, of grapevine.

The spatiotemporal spreading of a fungal disease over a vineyard is investigated using a susceptible-exposed-infected-removed (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 the foliar surface. Results of numerical simulations are presented. A mathematical result for the asymptotic behavior of the solutions is given as well.

International audience; A PDE-ODE model was developed to describe the spread of powdery mildew on grapevine. The model was able to retrieve the main characteristics of the system: 1) a host growing during the whole season with time evolution in susceptibility, 2) a crop highly structured in rows with potential heterogeneities of plant growth and susceptibility within and between plots. These characteristics are modified by cultural management. Simulations were performed to test the effect of grapevine spatial heterogeneities, within and between plots, on the disease spread. Heterogeneities considered were the plant growth (vigour, earliness), susceptibility (susceptible vs resistant, treated vs untreated) and the spatial arrangements (patches vs rows). The main effect on disease reduction was obtained by arrangement in rows of susceptible and fully resistant plants.

In this thesis we study the existence of a travelling wave solutions for an integro-differential system of equations from evolutionary epidemiology. We use ideas from dynamical system ideas theory coupled with estimates of the asymptotic behaviour of profiles. We prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ODE system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c?, provided that the epidemic threshold R0, which is expressed as a function of the principal eigenvalue of a certain integral operator, is strictly greater than 1. This same threshold condition is also used to prove that any travelling wave connects two determined stationary states. In the second part, we study the propagation properties of the solutions for the same spatially distributed system of equations, when the initial density of infected plants is a compactly supported function with the space variable x. When R0 > 1, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed c? of the travelling wave solutions discussed in the first part. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable x and the phenotype one y are separated.; Dans cette thèse nous étudions l’existence d’une onde progressive pour un système d’équations intégro-différentiels provenant de l’épidémiologie évolutive. Nous utilisons des idées issues de la théorie des systèmes dynamiques couplées à des estimations sur le comportement asymptotique des profils. Nous prouvons que les ondes progressives ont une structure assez simple découplant les variables de propagation spatio-temporelle des variables de trait phénotypique. Cette analyse nous permet de réduire le système d’équations des profils d’ondes progressives à dimension infinie à un système d’EDO à quatre dimensions. Nous prouvons l’existence d’ondes progressives pour toute vitesse d’onde supérieure à une vitesse minimale c?, pourvu que le seuil épidémique R0, qui s’exprime en fonction de la valeur propre principale d’un certain opérateur intégral, soit strictement supérieur à 1. Cette même condition de seuil est également utilisée pour démontrer que toute onde progressive relie deux états stationnaires déterminés. Dans une deuxième partie, nous étudions les propriétés de propagation des solutions pour le même système d’équations spatialement distribué, avec une densité initiale de plantes infectées à support compact spatialement en x. Lorsque R0 > 1, nous prouvons que la propagation se produit avec une vitesse de propagation qui coïncide avec la vitesse minimale c? des ondes progressives étudiées dans la première partie. De plus, la solution du problème de Cauchy converge asymptotiquement vers une fonction spécifique pour laquelle la variable x du repère mobile et celle du phénotype y sont séparées.

In this thesis we study the existence of a travelling wave solutions for an integro-differential system of equations from evolutionary epidemiology. We use ideas from dynamical system ideas theory coupled with estimates of the asymptotic behaviour of profiles. We prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ODE system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c?, provided that the epidemic threshold R0, which is expressed as a function of the principal eigenvalue of a certain integral operator, is strictly greater than 1. This same threshold condition is also used to prove that any travelling wave connects two determined stationary states. In the second part, we study the propagation properties of the solutions for the same spatially distributed system of equations, when the initial density of infected plants is a compactly supported function with the space variable x. When R0 > 1, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed c? of the travelling wave solutions discussed in the first part. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable x and the phenotype one y are separated.; Dans cette thèse nous étudions l’existence d’une onde progressive pour un système d’équations intégro-différentiels provenant de l’épidémiologie évolutive. Nous utilisons des idées issues de la théorie des systèmes dynamiques couplées à des estimations sur le comportement asymptotique des profils. Nous prouvons que les ondes progressives ont une structure assez simple découplant les variables de propagation spatio-temporelle des variables de trait phénotypique. Cette analyse nous permet de réduire le système d’équations des profils d’ondes progressives à dimension infinie à un système d’EDO à quatre dimensions. Nous prouvons l’existence d’ondes progressives pour toute vitesse d’onde supérieure à une vitesse minimale c?, pourvu que le seuil épidémique R0, qui s’exprime en fonction de la valeur propre principale d’un certain opérateur intégral, soit strictement supérieur à 1. Cette même condition de seuil est également utilisée pour démontrer que toute onde progressive relie deux états stationnaires déterminés. Dans une deuxième partie, nous étudions les propriétés de propagation des solutions pour le même système d’équations spatialement distribué, avec une densité initiale de plantes infectées à support compact spatialement en x. Lorsque R0 > 1, nous prouvons que la propagation se produit avec une vitesse de propagation qui coïncide avec la vitesse minimale c? des ondes progressives étudiées dans la première partie. De plus, la solution du problème de Cauchy converge asymptotiquement vers une fonction spécifique pour laquelle la variable x du repère mobile et celle du phénotype y sont séparées.