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SIR Epidemics on Evolving Erdős-Rényi Graphs

Wenze Chen, Yuewen Hou, Dong Yao

TL;DR

This work analyzes SIR epidemics on evolving Erdős-Rényi graphs (SIR-ω) where S-I edges break at rate ω and are either rewired by probability α or dropped by 1−α, with evoSIR recovered as the α=1 case. Using a time-change and a coupled construction, the authors derive a four-dimensional ODE system (ŝ, î, î_E, ŵ) that captures the limit behavior and exhibits a discontinuity barrier at t_* when î_E hits zero. They establish a necessary-and-sufficient condition for a discontinuous phase transition in the final epidemic size on ER graphs: ω(2α−1) > γ and μ > 2ωα /(ω(2α−1)−γ); in all other parameter regimes the phase transition is continuous and the scaled final size converges to the corresponding ODE limit. The results resolve a long-standing gap between prior sufficient conditions and provide a rigorous description of the final-size behavior near the critical threshold, with implications for understanding how network evolution influences epidemic outcomes. The methodology combines a precise stochastic construction, a time-change to render the process amenable to deterministic approximation, and detailed tightness/uniqueness proofs to justify the ODE limit and final-size conclusions.

Abstract

In the standard SIR model, infected vertices infect their neighbors at rate $λ$ independently across each edge. They also recover at rate $γ$. In this work we consider the SIR-$ω$ model where the graph structure itself co-evolves with the SIR dynamics. Specifically, $S-I$ connections are broken at rate $ω$. Then, with probability $α$, $S$ rewires this edge to another uniformly chosen vertex; and with probability $1-α$, this edge is simply dropped. When $α=1$ the SIR-$ω$ model becomes the evoSIR model. Jiang et al. proved in \cite{DOMath} that the probability of an outbreak in the evoSIR model converges to 0 as $λ$ approaches the critical infection rate $λ_c$. On the other hand, numerical experiments in \cite{DOMath} revealed that, as $λ\to λ_c$, (conditionally on an outbreak) the fraction of infected vertices may not converge to 0, which is referred to as a discontinuous phase transition. In \cite{BB} Ball and Britton give two (non-matching) conditions for continuous and discontinuous phase transitions for the fraction of infected vertices in the SIR-$ω$ model. In this work, we obtain a necessary and sufficient condition for the emergence of a discontinuous phase transition of the final epidemic size of the SIR-$ω$ model on \ER\, graphs, thus closing the gap between these two conditions.

SIR Epidemics on Evolving Erdős-Rényi Graphs

TL;DR

This work analyzes SIR epidemics on evolving Erdős-Rényi graphs (SIR-ω) where S-I edges break at rate ω and are either rewired by probability α or dropped by 1−α, with evoSIR recovered as the α=1 case. Using a time-change and a coupled construction, the authors derive a four-dimensional ODE system (ŝ, î, î_E, ŵ) that captures the limit behavior and exhibits a discontinuity barrier at t_* when î_E hits zero. They establish a necessary-and-sufficient condition for a discontinuous phase transition in the final epidemic size on ER graphs: ω(2α−1) > γ and μ > 2ωα /(ω(2α−1)−γ); in all other parameter regimes the phase transition is continuous and the scaled final size converges to the corresponding ODE limit. The results resolve a long-standing gap between prior sufficient conditions and provide a rigorous description of the final-size behavior near the critical threshold, with implications for understanding how network evolution influences epidemic outcomes. The methodology combines a precise stochastic construction, a time-change to render the process amenable to deterministic approximation, and detailed tightness/uniqueness proofs to justify the ODE limit and final-size conclusions.

Abstract

In the standard SIR model, infected vertices infect their neighbors at rate independently across each edge. They also recover at rate . In this work we consider the SIR- model where the graph structure itself co-evolves with the SIR dynamics. Specifically, connections are broken at rate . Then, with probability , rewires this edge to another uniformly chosen vertex; and with probability , this edge is simply dropped. When the SIR- model becomes the evoSIR model. Jiang et al. proved in \cite{DOMath} that the probability of an outbreak in the evoSIR model converges to 0 as approaches the critical infection rate . On the other hand, numerical experiments in \cite{DOMath} revealed that, as , (conditionally on an outbreak) the fraction of infected vertices may not converge to 0, which is referred to as a discontinuous phase transition. In \cite{BB} Ball and Britton give two (non-matching) conditions for continuous and discontinuous phase transitions for the fraction of infected vertices in the SIR- model. In this work, we obtain a necessary and sufficient condition for the emergence of a discontinuous phase transition of the final epidemic size of the SIR- model on \ER\, graphs, thus closing the gap between these two conditions.
Paper Structure (32 sections, 15 theorems, 265 equations, 1 figure)

This paper contains 32 sections, 15 theorems, 265 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous work
  4. SIR on static graphs
  5. SI on evolving graphs
  6. SIR on evolving graphs
  7. Proof strategies and organization of the paper
  8. Construction of the SIR-$\omega$ process
  9. Existence and uniqueness of the solution to \ref{['hats']}-\ref{['hatw']}
  10. Existence and uniqueness for positive initial condition
  11. Existence and uniqueness for zero initial condition
  12. Proof of the existence
  13. Proof of the uniqueness
  14. Approximations of the SIR-$\omega$ process by differential equations
  15. Proof of \ref{['eq:hatsw']}: convergence of $\widehat{S}^{(n)}(t)/n$ and $\widehat{W}^{(n)}(t)/n$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Consider the SIR-$\omega$ model starting with one infected individual and the rest being susceptible. Suppose then for any $\epsilon>0$, there exists $\lambda_1>\lambda_c$ such that Combining cont and discont, the phase transition for the final epidemic size of the SIR-$\omega$ model near $\lambda_c$ is discontinuous if and only if cond_discont holds.

Figures (1)

  • Figure 1.1: Simulation of the final epidemic size of the evoSIR model on an Erdős-Rényi graph with $\mu=5$, $\omega=4$ and $n\in [10^4,10^5]$. In this case $\lambda_c=1.25$. The top curve is the simulation of final size of evoSIR. The bottom curve is the final size of the delSIR (SIR-$\omega$ with $\alpha=0$) epidemic with the same parameters. Other two curves are approximations of evoSIR, which are discussed in DOMath.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 2.1
  • Lemma 2.2
  • ...and 11 more