一类一般传播模型的渐近行为
Asymptotic behavior for a general class of spreading models
AI总结：
提出统一框架建模感染与信息共演化的流体极限模型，在图论条件下推导渐近行为分类及稳定性，并引入依赖图揭示结构变化对轨迹的影响。
摘要：
关于流行病和谣言动力学的不断增长的文献表明，感染与信息是共同演化的。我们提出了一个统一的框架来建模感染和信息的传播：一类以交互为驱动、表示为耦合常微分方程的流体极限模型。该类模型包括SIR流行病模型、Daley-Kendall谣言模型以及许多扩展。对于这一一般类，我们推导了理论结果：在显式图论条件下，我们获得了渐近行为的分类，并激发了一个关于消失状态指数衰减的猜想。当这些条件被违反时，分类可能会失败，且衰减可能变为非指数的（例如，代数的）。在推导主要结果时，我们建立了状态变量的渐近稳定性和$L^1$可积性性质。伴随这些结果，我们引入了捕获流出依赖关系的依赖图，并为该模型类的结构提供了新视角。最后，我们用几个例子说明了结果，包括异质谣言模型和依赖谣言的SIR模型，展示了依赖图的微小改变如何翻转渐近行为并重塑流行病轨迹。
英文摘要：
Growing literatures on epidemic and rumor dynamics show that infection and information coevolve. We present a unified framework for modeling the spread of infection and information: a general class of interaction-driven fluid-limit models expressed as coupled ODEs. The class includes the SIR epidemic model, the Daley-Kendall rumor model, and many extensions. For this general class, we derive theoretical results: under explicit graph-theoretic conditions, we obtain a classification of asymptotic behavior and motivate a conjecture of exponential decay for vanishing states. When these conditions are violated, the classification can fail, and decay may become non-exponential (e.g., algebraic). In deriving the main result, we establish asymptotic stability and $L^1$-integrability properties for state variables. Alongside these results, we introduce the dependency graph that captures outflow dependencies and offers a new angle on the structure of this model class. Finally, we illustrate the results with several examples, including a heterogeneous rumor model and a rumor-dependent SIR model, showing how small changes to the dependency graph can flip asymptotic behavior and reshape epidemic trajectories.
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