标题： 应用概率视角下的非线性流行病动力学独立跳跃研究 显示英文标题

标题： Applied Probability Insights into Nonlinear Epidemic Dynamics with Independent Jumps

摘要： 本文聚焦于一种随机 SAIRS 型流行病模型的分析，该模型明确纳入了无症状和有症状感染个体在疾病传播动力学中的作用。 无症状携带者由于缺乏症状而往往未被检测到，在包括新冠在内的许多传染性疾病传播中起着至关重要的作用。 我们的模型还考虑了疫苗接种，并利用 Lévy 过程考虑了环境和群体层面随机性的随机效应。 我们首先证明所提出的随机系统的全局正解的存在性和唯一性，确保模型的数学有效性。 随后，我们推导出足以使疾病要么灭绝要么随着时间持续的充分条件，这取决于参数和初始条件。 分析突出了随机扰动、无症状传播和疫苗策略对疾病动力学的影响。 最后，我们进行了全面的数值模拟以验证理论发现，并展示模型在各种随机性和参数设置场景下的行为。 这些结果为流行病爆发的随机动力学提供了有价值的见解，并为疾病管理和控制策略提供信息。 显示英文摘要

摘要： This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using L\'evy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.