彩色流行病模型：泛函大数定律与混沌传播
Coloured Epidemic Models: Functional Law of Large Numbers and Propagation of Chaos
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中文总结 AI 辅助
本文提出一个带个体协变量的随机SIR流行病模型，通过相互作用粒子系统建立泛函大数定律和混沌传播，为异质性群体的参数估计提供理论基础。
中文摘要 AI 辅助
在本文中，我们研究了一个随机易感-感染-移除（SIR）模型，其中感染率和恢复率取决于个体协变量，这些协变量分别反映感染者和被感染者的易感性与传染性。此类模型允许在发病率项中引入显式非线性。它们从实践角度也很重要，因为它们允许将个体异质性纳入流行病过程。当数学模型忽略群体异质性时，关键流行病学参数（如基本再生数、群体免疫阈值）的统计估计可能差异巨大，甚至产生偏差。我们将流行病模型描述为由泊松随机测度驱动的随机微分方程（SDEs）的相互作用粒子系统（IPS）。我们的主要数学贡献是泛函大数定律（FLLN），该定律通过确定性测度值函数近似IPS的经验随机测度；以及混沌传播现象，该现象确立了当群体规模趋于无穷时粒子的渐近独立性，并显式构造了McKean-Vlasov型Kac的“非线性过程”。我们还简要提及混沌传播现象如何导致乘积形式的似然函数，这构成了基于稀疏数据进行参数推断的所谓动态生存分析（DSA）方法的基础。
英文摘要
In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean--Vlasov type Kac's ``nonlinear process''. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.
发表机构
- University of Nottingham（诺丁汉大学）
- The Ohio State University（俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。
