Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction–diffusion–absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015, France.

The simultaneous estimation of multiple quantiles is a crucial statistical task that enables a thorough understanding of data distribution for robust analysis and decision-making. In this study, we adopt a Bayesian approach to tackle this critical task, employing the asymmetric Laplace distribution (ALD) as a flexible framework for quantile modeling. Our methodology implementation involves the Hamiltonian Monte Carlo (HMC) algorithm, building on the foundation laid in prior work, where the error term is assumed to follow an ALD. Capitalizing on the interplay between two distinct quantiles of this distribution, we endorse a straightforward and fully Bayesian method that adheres to the non-crossing property of quantiles. Illustrated through simulated scenarios, we showcase the effectiveness of our approach in quantile estimation, enhancing precision via the HMC algorithm. The proposed method proves versatile, finding application in finance, environmental science, healthcare, and manufacturing, and contributing to sustainable development goals by fostering innovation and enhancing decision-making in diverse fields.

Inférer et prédire les dynamiques d'espèces invasives : focus sur Xylella fastidiosa: La thèse porte sur la recherche d’une méthodologie générique permettant d'améliorer les prédictions d’une invasion biologique pour laquelle on ne dispose pas de modèle spécifique et dont les conditions initiales sont inconnues. Pour atteindre cet objectif, on procède suivant deux axes de recherche complémentaires. Dans le premier axe, on s’intéresse à l’inférence des invasions biologiques à partir d’un modèle spatio-temporel de propagation et de données collectées, en suivant une approche mécanistico-statistique. Elle repose sur (i) une équation aux dérivées partielles (EDP) offrant une représentation concise d’une dynamique qui envahit un domaine hétérogène, (ii) un modèle stochastique représentant le processus d’observation et (iii) une méthode d’inférence Bayésienne pour estimer les paramètres du modèle. Un modèle dérivé des processus de Markov déterministes par morceaux est proposé pour remplacer l'EDP permettant un compromis entre réalisme du modèle et facilité d’estimation. Dans le deuxième axe, on propose une approche prenant en compte les incertitudes entourant des modèles en compétition. La technique du Bayesian model-averaging combine les prédictions de ces modèles pour obtenir une prédiction unifiée améliorée. Cette technique a souvent été utilisée en sciences environnementales. Toutefois, elle n’est pas répandue dans le domaine de l’épidémiologie. L’un des buts méthodologiques de la thèse est d’en évaluer l’intérêt pour l’épidémiologie prédictive. Le cas d’étude est celui de Xylella fastidiosa, bactérie phytopathogène ayant le potentiel de causer en France une crise sanitaire majeure en santé végétale à l’image de celle qu’elle cause depuis 2013 en Italie

Fragmentation processes arise in many physical systems, including droplet breakup, aerosols, sprays, comminution, and granular media. A central difficulty in fragmentation modeling is the identification of the breakup law from observed particle-size distributions. In this work, we propose a physics-constrained neural framework for the inverse identification of fragmentation laws. Starting from a size-structured binary fragmentation equation, the kernel is decomposed into a breakup rate $B\left(s\right)$, depending on the parent size s, and a daughter-size distribution $\kappa \left(z\right)$, depending on the normalized daughter-size fraction z. The unknown functions B and $\kappa$ are represented by neural networks designed to satisfy physical constraints, including non-negativity of the breakup rate, non-negativity and normalization of the daughter distribution, and first-moment conservation in the symmetric binary setting. The direct fragmentation equation is solved using a first-moment-conserving quadrature discretization, and the inverse problem is formulated as a regularized optimization problem constrained by the fragmentation dynamics. Synthetic experiments, generated independently of the inverse solver, show that the proposed method can recover B and $\kappa$ from simulated particle-size distributions, with accurate recovery in the unimodal case and reasonable performance in the more challenging bimodal case. Robustness tests indicate stability with respect to moderate observational noise and highlight the importance of multiple observation times. For experimental droplet-breakup measurements without time-resolved particle-size distributions, the same constrained neural daughter-density representation is used to estimate effective marginal daughter-volume-fraction distributions from normalized daughter-to-parent volume fractions. The resulting distributions distinguish rim and node fragments and different breakup modes, while stratified cluster-bootstrap confidence bands quantify their sampling uncertainty. The experimental analysis is therefore interpreted as constrained density estimation from final fragment measurements rather than as full validation of the PDE-constrained inverse recovery.