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Travelling Wave Solutions and Propagation Properties for a Non-Local Evolutionary-Epidemic System Lara Abi Rizk

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Travelling Wave Solutions and Propagation Properties for a Non-Local Evolutionary-Epidemic System Lara Abi Rizk Travelling wave solutions and propagation properties for a non-local evolutionary-epidemic system Lara Abi Rizk To cite this version: Lara Abi Rizk. Travelling wave solutions and propagation properties for a non-local evolutionary- epidemic system. Analysis of PDEs [math.AP]. Université de Bordeaux, 2020. English. NNT : 2020BORD0244. tel-03158763 HAL Id: tel-03158763 https://tel.archives-ouvertes.fr/tel-03158763 Submitted on 4 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE présentée pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ DE BORDEAUX École Doctorale Mathématiques et Informatique Spécialité Mathématiques appliquées par Lara Abi Rizk Ondes progressives et propriétés de propagation pour un problème d’épidémiologie évolutive non-local. Soutenue le 15 décembre 2020 devant le jury composé de : M. Matthieu ALFARO Professeur Université de Rouen Rapporteur M. Jean-Baptiste BURIE Maître de Conférence Université de Bordeaux Directeur de thèse M. Arnaud DUCROT Professeur Université du Havre Co-directeur de thèse M. Pierre MAGAL Professeur Université de Bordeaux Examinateur Mme. Martine MARION Professeure École centrale de Lyon Présidente du jury Mme. Sepideh MIRRAHIMI Chargée de Recherche Université Paul Sabatier Rapporteure Ondes progressives et propriétés de propagation pour un problème d’épidémiologie évolutive non-local. Résumé. 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 sim- ple 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 pro- gressives 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. Mots-clés. Évolution, ondes progressives, vitesse minimale, système de réaction-diffusion non local, dynamique des populations, comportement asymptotique, épidémiologie, vitesse de propagation. Financement. Financement de la Région Aquitaine Limousin Poitou-Charentes et la chair Idex ACIDDS de l’Université de Bordeaux de M. Tucsnak. Laboratoire d’accueil. Institut de Mathématiques de Bordeaux, UMR CNRS, 5251. 351, cours de la Libération, 33405, Talence, France. Travelling wave solutions and propagation properties for a non-local evolutionary-epidemic system Abstract. 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. Keywords. Evolution, travelling wave solutions, minimal wave speed, non-local diffu- sive epidemic system, population dynamics, long time behaviour, epidemiology, spreading speed. Funding. We received the support of Région Aquitaine Limousin Poitou-Charentes and of M. Tucsnak’s Idex Chair ACIDDS of Université de Bordeaux. Institute. Institut de Mathématiques de Bordeaux, UMR CNRS, 5251. 351, cours de la Libération, 33405, Talence, France. Remerciements Je voudrais tout d’abord remercier grandement mes directeurs de thèse Jean-Baptiste Burie et Arnaud Ducrot, qui m’ont soutenu et surtout supporté dans tout ce que j’ai entrepris. Je suis ravie d’avoir travaillé en leurs compagnies. Outre leurs appuis scien- tifiques, ils ont toujours été là pour me soutenir et me conseiller au cours de l’élaboration de cette thèse. Je les remercie aussi de m’avoir transmis leurs connaissances et leurs savoirs précieux. Je leur suis reconnaissante pour le temps qu’ils ont consacré à ma recherche, mais aussi pour leur gentillesses, leurs disponibilités permanentes et pour leurs nombreux encouragements. Enfin, ils ont toujours répondu avec calme et patience a mes questions quotidiennes. Les rapporteurs de cette thèse, Sepideh Mirrahimi, chargée de recherche CNRS à l’Institut de mathématiques de Toulouse et Matthieu Alfato, Professeur à l’Université de Rouen. Je les remercie pour le temps consacré à la lecture de cette thèse, leurs rapports détaillés et leurs remarques. Je remercie Professeur Martine Marion pour l’honneur qu’elle ma fait d’être la prési- dente de mon jury de thèse. Je tiens à remercier également Professeur Pierre Magal pour avoir accepté de participer à mon jury de thèse et particulièrement pour toutes nos discussions et ses conseils. J’exprime ma gratitude au Professeur Frédéric Hérau, qui m’a accompagné pendant ma première année d’étude en France à l’Université de Nantes et a tout fait pour m’aider. Il m’a non seulement initié à la recherche lorsque j’étais une de ses élèves, mais il m’a aussi prodigué de nombreux conseils pour bien débuter le troisième cycle universitaire, dont cette thèse est l’accomplissement. Hicham Gebran my favorite professor at the Lebanese University. Thank you for be- lieving in me and thank you for being the example of a great teacher. Merci également à toute l’équipe administrative de l’IMB, en particulier Carole, Ida, Muriel, Karine, Cathy, Estelle. Je tiens à remercier aussi l’équipe de cellule informatique en particulier Thomas. Je remercie toutes les personnes avec qui j’ai partagé ces années de thèse qui sont de très belles rencontres inattendues: "Les copaings Bordelais." Commençant par les Aristocrates du bureau 203 : Thomas et Corentin, vous avez rendu les journées de travail et la rédaction plus facile. Merci Thomas de m’avoir toujours soutenu face aux méchants et les cochons, j’ai passé de très belles journées et soirées avec toi, merci aussi de m’avoir supporter, tu es le meilleur co-bureau. Corentin, toi tu es le synonyme du respect, merci de m’avoir encouragé à écouter toujours de la bonne musique, tu resteras le meilleur Dj de nos soirée <3 Tu manges taboulé maintenant. Alexandre, la zbélé de la team! Merci pour les soirées inoubliables, pendant lesquelles tu 7 as bouffé tout mon pain et tu as hurlé "Lara, au secoure au secoure ça tourne!" Antonin le seul beau du Labri, merci pour toutes les bières cochon rose que j’adore, merci de me juger quand je bois du café soluble :| Bon courage pour les cours du chinois! Merci Jonathan pour l’oreiller très confortable et surtout merci pour la raclette c’était très bon! Merci Nicolas pour ta gentillesse :D j’espère qu’on fera une randonnée un jour dans les montagnes pour voir les mouton(e)s. Niko, maintenant merci à toi de m’avoir toujours dit que je ne faisais pas des vrais Maths. Sami, merci de m’avoir permis d’ajouter plein de paillettes et de l’élégance à ton anniversaire, merci pour le Jack, le Liban et Feyrouz te remercient aussi :D Merci Guillaume et Manon pour les soirées sympas. Merci Aurore pour les meilleurs recettes, merci d’avoir adopté Thomas, tu as ajouté ton grain de féminité dans cette équipe de gars "Woman’s Power!!" Elsa, la merveilleuse, merci pour ta bonne humeur, ton rire et ta positivité. Merci Roxanne pour toutes les soirées, j’espère qu’on pourra bientôt utiliser ton code promo : OthmaXanne-ServiceClient-15. Merci Chloé de ne pas laisser Coco en vue cela nous à permis de te rencontrer! Merci Léa et Nobel pour les bons moments à la plage. Merci Marco d’être le Méchanta le plus gentil.
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