具有潜伏和警惕效应的单纯形SAEIS模型

Simplicial SAEIS Model with Latent and Alert Effect

摘要: 在传染病动力学研究中，高阶传播、个体行为影响与潜伏感染虽已被广泛研究，但三者融合的数学模型较少。本研究构建单纯形SAEIS传染病模型，纳入高阶相互作用、个体警惕行为及潜伏感染特征。通过理论推导，明确了模型解的有界性、不变集结构，以及平衡点的存在与稳定条件；借助数值模拟，揭示其不连续相变和双稳态动力学行为。研究发现，高阶感染率与警惕丧失率的提升会加速传染病扩散，而警惕获得率、潜伏个体感染转化率和恢复个体警惕转移比例的增强可有效抑制疫情。特别地，恢复率对传染病传播的影响与感染密度相关：低感染密度时，提升恢复率利于控制疫情；高感染密度下，其增加反而扩大感染范围。本研究以理论与数值分析结合，解析模型动力学与参数机制，为复杂场景下传染病防控提供理论依据。

Abstract: In the realm of infectious disease dynamics research, although high-order transmission, the influence of individual behaviors, and latent infection have each been extensively studied, mathematical models that integrate these three elements are relatively scarce. In this study, a simplicial SAEIS infectious disease model was constructed, incorporating high-order interactions, individual vigilance behaviors, and the characteristics of latent infection. Through theoretical derivation, the boundedness of the model solutions, the structure of the invariant set, as well as the existence and stability conditions of the equilibrium points were determined. By means of numerical simulations, the discontinuous phase transitions and bistable dynamic behaviors of the model were revealed. The study has found that an increase in the high-order infection rate and the rate of vigilance loss will accelerate the spread of infectious diseases. Conversely, an enhancement in the rate of vigilance acquisition, the conversion rate of latent individuals to infected individuals, and the proportion of recovered individuals transitioning to the vigilance state can effectively curb the epidemic. Notably, the impact of the recovery rate on the spread of infectious diseases is related to the infection density. When the infection density is low, increasing the recovery rate is beneficial for controlling the epidemic. However, under conditions of high infection density, an increase in this rate will instead expand the scope of infection. This study combines theoretical and numerical analyses to elucidate the dynamics of the model and the parameter mechanisms, providing a theoretical basis for the prevention and control of infectious diseases in complex scenarios.