AI 中文总结
研究网络上高阶SIS流行病模型，提出自下而上推导方法，核心是依赖网络的封闭算子，可恢复现有模型为特殊情况，揭示启发式方法隐含假设，为建模提供基础和新途径。
AI 中文摘要
网络上的易感-感染-易感（SIS）流行病模型由分层矩方程控制，较小子系统的动力学依赖于较大子系统的状态。矩封闭近似通过用低阶状态概率表示高阶状态概率来截断这种层次结构，对获得可处理的简化系统至关重要。高阶网络引入了组合爆炸式的封闭配置，使系统推导变得困难。现有高阶SIS模型是启发式推导的，其封闭的结构和动力学假设并不总是从公式中明显看出。我们开发了一种自下而上的高阶SIS动力学推导方法，从节点级方程系统地构建到对、三元组和三体相互作用。我们方法的核心是一个依赖网络的封闭算子，它从局部对和三元组结构生成拓扑上合适的近似。使用这个框架，我们恢复了三个现有的高阶SIS模型——Burgio等人的最大团模型、Malizia等人的基于对和阶间模型——作为特殊情况，每个模型都在特定的拓扑和动力学假设下出现。我们的推导揭示了启发式方法中不可见的假设：例如，Malizia等人的阶间重叠参数单独不足以在我们的框架内表达该模型，尽管它在模拟中表现良好，原始推导隐含地调用了额外的结构假设。我们的框架为高阶流行病建模提供了基础，为理解启发式推导的平均场封闭中隐含的假设提供了建设性途径，并提供了一种生成新模型的原则性方法。
英文摘要
Susceptible-infected-susceptible (SIS) epidemic models on networks are governed by hierarchical moment equations where the dynamics of smaller subsystems depend on the state of larger ones. Moment closure approximations, which truncate this hierarchy by expressing higher-order state probabilities in terms of lower-order ones, are essential for obtaining tractable reduced systems. Higher-order networks, which extend the pairwise structure to include group interactions, introduce a combinatorial explosion of closure configurations, making systematic derivation harder. Consequently, existing higher-order SIS models are derived heuristically, where structural and dynamical assumptions underpinning their closures are not always apparent from the formulation alone. We develop a bottom-up derivation of higher-order SIS dynamics, building systematically from node-level equations to pairs, triplets, and three-body interactions. Central to our approach is a network-dependent closure operator that generates topologically appropriate approximations from local pairwise and triadic structure. Using this framework, we recover three existing higher-order SIS models--Burgio et al.'s maximal clique, Malizia et al.'s pair-based and inter-order models--as special cases, each arising under specific topological and dynamical assumptions. Our derivation reveals assumptions that are invisible from heuristic approaches: for instance, Malizia et al.'s inter-order overlap parameter is insufficient alone to express the model within our framework despite performing well against simulations, with the original derivation implicitly invoking additional structural assumptions. Our framework offers both a foundation for higher-order epidemic modeling and a constructive pathway for understanding the assumptions implicit in heuristically derived mean-field closures and provides a principled method of generating new models.
Comments52 pages, 4 figures
