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Foutel-Rodier F, Blanquart F, Courau P, Czuppon P, Duchamps JJ, Gamblin J, Kerdoncuff É, Kulathinal R, Régnier L, Vuduc L, Lambert A, Schertzer E. From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID-19 dynamics. J Math Biol 2022; 85:43. [PMID: 36169721 PMCID: PMC9517997 DOI: 10.1007/s00285-022-01794-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2021] [Revised: 02/21/2022] [Accepted: 05/11/2022] [Indexed: 11/25/2022]
Abstract
We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual’s infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type i is simply obtained by integrating the probability of being in state i at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.
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Affiliation(s)
- Félix Foutel-Rodier
- Département de Mathématiques, Université du Québec á Montréal, Montréal, QC, Canada.
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France.
| | - François Blanquart
- Infection, Antimicrobials, Modeling, Evolution UMR 1137, Université de Paris, INSERM, Paris, France
| | - Philibert Courau
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
| | - Peter Czuppon
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institute for Evolution and Biodiversity, University of Münster, 48149, Münster, Germany
| | - Jean-Jil Duchamps
- Laboratoire de mathématiques de Besançon UMR 6623, Université Bourgogne Franche-Comté, CNRS, F-25000, Besançon, France
| | - Jasmine Gamblin
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
| | - Élise Kerdoncuff
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institut de Systématique, Biodiversité, Évolution UMR 7205, Muséum National d'Histoire Naturelle, CNRS, Paris, France
- Department of Molecular and Cell Biology, University of California, Berkeley, California, USA
| | - Rob Kulathinal
- Department of Biology, Temple University, Philadelphia, PA, USA
| | - Léo Régnier
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Laboratoire de Physique Théorique de la Matiére Condensée, CNRS/Sorbonne University, Paris, France
| | - Laura Vuduc
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Université Paris-Saclay, Centrale Supélec, MICS Lab Gif-sur-Yvette, Berkeley, France
| | - Amaury Lambert
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institut de Biologie de l'ENS, École Normale Supérieure, CNRS UMR 8197 INSERM U 1024, Paris, France
| | - Emmanuel Schertzer
- Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria
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Foutel-Rodier F, Lambert A, Schertzer E. Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1641] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Félix Foutel-Rodier
- Laboratoire de Probabilités, Statistiques & Modélisation, Sorbonne Université, and Center for Interdisciplinary Research in Biology, Collège de France
| | - Amaury Lambert
- Laboratoire de Probabilités, Statistiques & Modélisation, Sorbonne Université, and Center for Interdisciplinary Research in Biology, Collège de France
| | - Emmanuel Schertzer
- Laboratoire de Probabilités, Statistiques & Modélisation, Sorbonne Université, and Center for Interdisciplinary Research in Biology, Collège de France
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