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Spreading properties in Kermack–McKendrick models with nonlocal spatial interactions: a new look
[Une nouvelle approche de la propagation des fronts pour les modèles de Kermack–McKendrick avec interactions non locales]
Grégory Faye1 ; Jean-Michel Roquejoffre1  ; Mingmin Zhang2
1CNRS, UMR 5219, Institut de Mathématiques de Toulouse, 31062 Toulouse Cedex, France
2School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 381-429

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par : Henri Berestycki et al..  

In this paper, we revisit the famous Kermack–McKendrick model with nonlocal spatial interactions by shedding new light on associated spreading properties and we also prove the existence and uniqueness of traveling fronts. Unlike previous studies that have focused on integrated versions of the variable representing the susceptible population, we analyze the long time dynamics of the underlying age-structured model for the cumulative density of infected individuals and derive precise asymptotic estimates for the infected population. Our approach consists in studying the long time dynamics of an associated transport equation with nonlocal spatial interactions whose spreading properties are close to those of classical Fisher-KPP reaction-diffusion equations. Our study is self-contained and relies on comparison arguments.

Dans ce travail, nous portons un regard neuf sur les modèles de Kermack–McKendrick avec interactions non locales en proposant une nouvelle approche de la propagation des fronts et en démontrant l’existence et l’unicité des ondes progressives dans des cas non couverts auparavant. À rebours des études existantes, qui se concentrent sur la version intégrée des modèles, nous travaillons directement sur les équations structurées en âge de la densité cumulée de la population infectée, qui se présentent comme une équation de transport couplée à des conditions aux limites non linéaires et non locales. Nous analysons la dynamique en grand temps de leurs solutions et démontrons des asymptotiques précises. Ce travail auto-contenu repose sur des arguments de comparaison.

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Révisé le :
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DOI : 10.5802/crmath.830
Classification : 45K05, 35C07, 35B40, 92D30
Keywords: Kermack–McKendrick models, nonlocal interactions, spreading properties, traveling waves
Mots-clés : Modèles de Kermack–McKendrick, propagation de fronts, interactions non locales, ondes progressives

Grégory Faye  1   ; Jean-Michel Roquejoffre  1   ; Mingmin Zhang  2

1 CNRS, UMR 5219, Institut de Mathématiques de Toulouse, 31062 Toulouse Cedex, France
2 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Grégory Faye; Jean-Michel Roquejoffre; Mingmin Zhang. Spreading properties in Kermack–McKendrick models with nonlocal spatial interactions: a new look. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 381-429. doi: 10.5802/crmath.830
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