{"title":"A SIR epidemic model on a refining spatial grid II-central limit theorem","authors":"Thierry Gallouët, Étienne Pardoux, Ténan Yeo","doi":"10.1007/s40072-024-00333-0","DOIUrl":null,"url":null,"abstract":"<p>A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein–Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.</p>","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":"151 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics and Partial Differential Equations-Analysis and Computations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40072-024-00333-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein–Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.
本文构建了一个考虑到空间环境异质性的随机 SIR 流行病模型。确定性模型由偏微分方程给出，随机模型由时空跃迁马尔可夫过程给出。两个模型的一致性由大数定律给出。本文通过函数中心极限定理研究了空间随机模型与确定性模型的偏差。极限是一个分布值的 Ornstein-Uhlenbeck 高斯过程，它是随机偏微分方程的温和解。
期刊介绍：
Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.
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