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On quasi-stationary distributions for stochastic rumor models

Iddo Ben-Ari, Elcio Lebensztayn, Lucas Sousa Santos

TL;DR

The paper studies quasi-stationary behavior in stochastic rumor processes, focusing on the Maki–Thompson model and extending to Daley–Kendall and SIR dynamics. Using van Doorn–Pollett theory, it shows that standard absorption yields a trivial QSD for MT and DK, and introduces a non-return conditioning to obtain non-trivial QSDs with explicit path-based formulas; analogous constructions are developed for DK and contingent conditions are found for SIR (μ>βN). The ratio-of-expectations approach is discussed as an alternative pre-absorption descriptor. Overall, the work provides rigorous characterization of long-term behavior before absorption and offers practical tools for comparing QSD and RE-distributions in finite rumor-epidemic systems.

Abstract

This paper examines the quasi-stationary behavior of stochastic rumor processes. Using the results by van Doorn and Pollett (2008), we first prove that the continuous-time Maki--Thompson model has a unique quasi-stationary distribution (QSD) given by the point mass at the state \((0, 1)\). To obtain a non-trivial QSD, we modify the absorption set by conditioning the process on not returning to the level \(y=1\) after leaving the initial state \((N, 1)\). For this modified model, we establish the existence and uniqueness of a non-trivial QSD that assigns positive probability to all transient states, and then derive an explicit formula for this QSD in terms of paths and transition rates. We also discuss the ratio of expectations distribution as an alternative approach to describe the long-term behavior before absorption. The analysis is further extended to the Daley--Kendall rumor model and the stochastic SIR epidemic model.

On quasi-stationary distributions for stochastic rumor models

TL;DR

The paper studies quasi-stationary behavior in stochastic rumor processes, focusing on the Maki–Thompson model and extending to Daley–Kendall and SIR dynamics. Using van Doorn–Pollett theory, it shows that standard absorption yields a trivial QSD for MT and DK, and introduces a non-return conditioning to obtain non-trivial QSDs with explicit path-based formulas; analogous constructions are developed for DK and contingent conditions are found for SIR (μ>βN). The ratio-of-expectations approach is discussed as an alternative pre-absorption descriptor. Overall, the work provides rigorous characterization of long-term behavior before absorption and offers practical tools for comparing QSD and RE-distributions in finite rumor-epidemic systems.

Abstract

This paper examines the quasi-stationary behavior of stochastic rumor processes. Using the results by van Doorn and Pollett (2008), we first prove that the continuous-time Maki--Thompson model has a unique quasi-stationary distribution (QSD) given by the point mass at the state \((0, 1)\). To obtain a non-trivial QSD, we modify the absorption set by conditioning the process on not returning to the level after leaving the initial state \((N, 1)\). For this modified model, we establish the existence and uniqueness of a non-trivial QSD that assigns positive probability to all transient states, and then derive an explicit formula for this QSD in terms of paths and transition rates. We also discuss the ratio of expectations distribution as an alternative approach to describe the long-term behavior before absorption. The analysis is further extended to the Daley--Kendall rumor model and the stochastic SIR epidemic model.

Paper Structure

This paper contains 12 sections, 9 theorems, 80 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Rumor models
  3. Quasi-stationary distributions
  4. Reducible case
  5. Ratio of expectations approach
  6. Quasi-stationary behavior of the MT model
  7. QSDs for the MT model
  8. Explicit formula for the non-trivial QSD
  9. RE-distribution for the MT model
  10. DK model and SIR model
  11. DK model
  12. SIR model

Key Result

Theorem 3.3

If $\nu$ is a QSD, then there exists $\lambda(\nu) \geq 0$ such that i.e., starting from $\nu$, $\tau_{\Delta}$ is exponentially distributed with parameter $\lambda(\nu)$.

Figures (9)

  • Figure 3.1: Example of an initial distribution and the support of the QSD according to the partial order.
  • Figure 4.1: Possible transitions for the MT model.
  • Figure 4.2: The modified MT model.
  • Figure 4.3: All paths from $(9,1)$ to $(3,5)$ for the modified MT model with $N=9$.
  • Figure 4.4: QSD $\nu^{\ast}$ for the MT model with $(X_0=200, Y_0=1)$. Values smaller than $10^{-10}$ are omitted.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Exponential killing time
  • Theorem 3.5: van08, Theorem 3
  • Theorem 3.6: van08, Theorem 5
  • Definition 3.7
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 10 more