AI 中文总结
该研究构建了异质性SIR流行病模型的随机框架，证明了相关大数定律，所得极限涵盖常见SIR系统，为对应确定性模型提供统一概率基础。
AI 中文摘要
我们为一类广泛的异质性SIR流行病模型构建了随机框架。在有限种群构造中，每个初始易感个体被赋予一个固定的非负易感性，当累积的种群水平感染压力超过个体随机阈值时发生感染。传染期相互独立且服从指数分布，具有共同的恢复率。对于任何具有有限均值的易感性分布，我们证明了易感者、感染者、移除者及累积感染压力过程的紧一致大数定律。在极限情况下，易感性为λ的个体在累积压力x下保持易感的概率为exp(-xλ)。由此可得，易感者比例由初始易感性分布的拉普拉斯变换给出，而发病率由保持易感者中的平均易感性决定。所得极限涵盖了几种常见的异质性SIR系统，包括经典幂律模型，还产生了其他闭合非线性发病率形式。因此，该框架为具有持续个体异质性的确定性流行病模型提供了统一的概率基础。
英文摘要
We develop a stochastic framework for a broad class of heterogeneous SIR epidemic models. In the finite-population construction, each initially susceptible individual is assigned a fixed nonnegative susceptibility, and infection occurs when the accumulated population-level infection pressure exceeds an individual random threshold. Infectious periods are independent and exponentially distributed with a common recovery rate. For any susceptibility distribution with finite mean, we prove a uniform-on-compact law of large numbers for the susceptible, infectious, removed, and cumulative infection-pressure processes. In the limit, an individual with susceptibility lambda remains susceptible under cumulative pressure x with probability exp(-x lambda). It follows that the susceptible fraction is given by the Laplace transform of the initial susceptibility distribution, while the incidence rate is governed by the mean susceptibility among those who remain susceptible. The resulting limits recover several familiar heterogeneous SIR systems, including the classical power-law model, and also yield other closed nonlinear incidence forms. The framework therefore provides a unified probabilistic foundation for deterministic epidemic models with persistent individual heterogeneity.
Comments27 pages, 1 figure
