Asymptotic behavior for a general class of spreading models

TL;DR

This paper develops a unified mathematical framework for analyzing the long-term behavior of various epidemic and rumor spreading models, revealing how graph structure influences decay rates and stability.

Contribution

It introduces a general class of coupled ODE models, derives conditions for asymptotic behavior, and highlights the impact of dependency graph structures on spreading dynamics.

Findings

Classification of asymptotic behaviors under graph conditions

Exponential decay conjecture for vanishing states

Small graph changes can alter epidemic trajectories

Abstract

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.