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Susceptible-Infected Epidemics on Evolving Graphs at Critical Infection Rate

Wenze Chen, Haojie Hou, Ruibo Ma, Dong Yao

Abstract

Consider an SI process on a graph $G$ where each S--I connection becomes I--I at rate $λ$. Here S and I stand for ``susceptible'' and ``infected'' respectively. The evoSI model is a modification of the SI model in which S--I edges are broken at rate $ρ$ and the ``S'' connects to a randomly chosen vertex. It is proven in Durrett and Yao [2022, Electron. J. Probab.] that, for the supercritical evoSI process on the configuration model, there exists a quantity $Δ$ depending on the first three moments of the degree distribution such that the sign of $Δ$ governs the continuity of the phase transition of the final epidemic size near the critical infection rate $λ_c$. In this paper, we consider the critical evoSI model on the configuration model, i.e., $λ=λ_c$. We show that, if $Δ>0$, then the probability of a major outbreak starting from a single infected individual is $Cn^{-1/3}(1+o(1))$ for some explicit constant $C>0$, where $n$ is the size of the graph. On the contrary, if $Δ<0$, then this probability is $o(n^{-1/3})$. The case $Δ<0$ is reminiscent of the critical {\ER} graphs, where the probability for the size of the largest component to be of order $n$ decays exponentially in $n$.

Susceptible-Infected Epidemics on Evolving Graphs at Critical Infection Rate

Abstract

Consider an SI process on a graph where each S--I connection becomes I--I at rate . Here S and I stand for ``susceptible'' and ``infected'' respectively. The evoSI model is a modification of the SI model in which S--I edges are broken at rate and the ``S'' connects to a randomly chosen vertex. It is proven in Durrett and Yao [2022, Electron. J. Probab.] that, for the supercritical evoSI process on the configuration model, there exists a quantity depending on the first three moments of the degree distribution such that the sign of governs the continuity of the phase transition of the final epidemic size near the critical infection rate . In this paper, we consider the critical evoSI model on the configuration model, i.e., . We show that, if , then the probability of a major outbreak starting from a single infected individual is for some explicit constant , where is the size of the graph. On the contrary, if , then this probability is . The case is reminiscent of the critical {\ER} graphs, where the probability for the size of the largest component to be of order decays exponentially in .
Paper Structure (29 sections, 28 theorems, 289 equations)

This paper contains 29 sections, 28 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Results for random graphs with a random degree sequence
  4. Related work
  5. The size of components of Erdős-Rényi graphs in the critical window
  6. evoSI model in the supercritical regime
  7. SI-$\rho$ and SIR-$\rho$ model
  8. Proof of Theorem \ref{['surviprob']}: an outline
  9. Bounding evoSI by avoSI and AB-avoSI models
  10. The three-stage analysis of the avoSI model
  11. Organization
  12. Proof of Theorem \ref{['YZLimit']} and Corollary \ref{['phase1']}
  13. The number of jumps in $[0, qn ^ {-1/3}]$
  14. Construction of the comparison random walks
  15. Analysis for the upper bound $(Y_{\ell})$
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\{\mathbf{d}_n,n\geq 1\}$ be degree sequences that satisfy (H). Define Consider evoSI on the random graph $\mathbb{CM}(n,\mathbf{d}_n)$ with $\lambda=\lambda_c$.

Theorems & Definitions (56)

  • Remark 1: Bounds for the tail and maximum degree
  • Remark 2: On the verification of (H3)
  • Theorem 1.1
  • Remark 3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • Theorem 1.5
  • Theorem 1.6
  • ...and 46 more