Asymptotic behavior for a general class of spreading models

TL;DR

This work unifies epidemic and rumor spreading into a single fluid-limit framework of interaction-driven dispersion, modeled by coupled ODEs for state fractions $y_i(t)$ with inflows $I_i(t)$ and outflows $O_i(t)$. By introducing transition and dependency graphs, the authors derive a DAG-based asymptotic stability result and, under the DAG$^{-}$ condition, a complete classification of vanishing versus persistent states via integrability of trajectories, plus a conjecture that vanishing states decay exponentially. When the graph conditions fail, the dynamics can exhibit non-exponential decay, as illustrated by self-loops and cycles. The paper further demonstrates the framework on heterogeneous rumor models and SIR-DK hybrids, showing how small structural changes in the graphs can drastically alter long-run trajectories, providing structural guidance for understanding and controlling coupled spreading processes in social and biological networks.

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.

Figures (15)

• Figure 1: Transition and Dependency graph representation of the SIR and DK models
• Figure 2: Dependency graphs of the SIR and DK models with highlighted cycles in blue, red and green
• Figure 3: Example of a symmetric and asymmetric dependency graph (with transition graph identical to their dependency graph) and its asymptotic characterization. The asymmetric graph in (b) is construed by removing the $7 \to 9$ edge. Blue: persistent states. Red: vanishing states.
• Figure 4: Transition and dependency graph of the model in Chen2020, with highlighted cycles in blue and red.
• Figure 5: Modified dependency graph and asymptotic characterization of the model in Chen & Wang Chen2020. The dashed edges are removed in order to arrive at the modified dependency graph. The complete characterization then follows from Corollary \ref{['cor:complete_char']} by applying Theorem \ref{['thm:vanish_persist_basic']} iteratively starting with the sink state $5$. Blue: asymptotically persistent states. Red: asymptotically vanishing states.

Theorems & Definitions (38)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Remark 1
• Lemma 1: DAG has a sink and a source
• Lemma 2: Confinement of state variables