具有脉冲接种、可变传染性和免疫力减弱的随机流行病模型

中文摘要 AI 辅助

我们引入了一种完全随机、非马尔可夫的SIRS型流行病模型，该模型整合了可变传染性、免疫力减弱以及可能无法提供永久免疫力的脉冲接种策略。该模型在个体层面构建，其中每个人以随机传染性和易感性函数为特征，接种活动发生在具有任意强度的泊松随机测度的跳跃时刻。我们严格推导了作为相互作用随机粒子系统大种群极限的流行病动力学，得到了控制平均易感性和总感染力的非线性Volterra型积分方程组。我们为经验过程建立了函数大数定律（FLLN），并提供了极限隔间比例的显式表达式。我们分析了系统的长期行为：当基本再生数低于临界阈值时，证明无感染解是全局渐近稳定的；当该阈值超过时，疾病持续存在。该阈值由个体最大易感性的调和平均值给出，通过整合接种和记忆效应推广了先前的结果。我们的框架为经典确定性脉冲接种模型提供了基于概率的扩展，并为通过计划免疫政策控制流行病提供了新的见解。

英文摘要

We introduce a fully stochastic, non-Markovian SIRS-type epidemic model that incorporates varying infectivity, waning immunity and a pulse vaccination strategy that may not confer permanent immunity. The model is constructed at the individual level, where each person is characterized by random infectivity and susceptibility functions, and vaccination campaigns occur at the jump times of a Poisson random measure with arbitrary intensity. We rigorously derive the epidemic dynamics as the large-population limit of an interacting stochastic particle system, leading to a system of nonlinear Volterra-type integral equations governing the average susceptibility and total force of infection. We establish a functional law of large numbers(FLLN) for the empirical processes and provide explicit expressions for the limiting compartmental proportions. The long-term behavior of the system is analyzed: we prove that the infection-free solution is globally asymptotically stable when the basic reproduction number falls below a critical threshold, and that the disease persists when this threshold is exceeded. The threshold is given by the harmonic mean of the maximal susceptibility across individuals and generalizes previous results by incorporating vaccination and memory effects. Our framework provides a probabilistically grounded extension of classical deterministic pulse vaccination models and offers new insights into the control of epidemics through scheduled immunization policies.

发表机构

• Vanderbilt University（范德堡大学）

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