Phase transition for the SIR model with random transition rates on complete graphs

TL;DR

This paper investigates a stochastic SIR epidemic model with random recovery and infection rates on complete graphs, revealing a phase transition at a critical infection rate related to the means of these random variables.

Contribution

It establishes a phase transition criterion for the SIR model with random rates on complete graphs, linking the critical point to the means of recovery and infection rate distributions.

Findings

Below critical rate, the epidemic dies out as network size grows.

Above critical rate, a positive fraction of the population is infected with high probability.

Critical infection rate is the inverse of the product of the means of infection and recovery rates.

Abstract

In this paper we are concerned with the Susceptible-Infective-Removed model with random transition rates on complete graphs $C_n$ with $n$ vertices. We assign i. i. d. copies of a positive random variable $\xi$ on each vertex as the recovery rates and i. i. d copies of a positive random variable $\rho$ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at $t=0$ one vertex is infective and others are susceptible. When $\lambda<\lambda_c$, the proportion of vertices which have ever been infective converges to $0$ weakly as $n\rightarrow+\infty$ while when $\lambda>\lambda_c$, there exist $c(\lambda)>0$ and $b(\lambda)>0$ such that for each $n\geq 1$ with probability at least $b(\lambda)$ the proportion of vertices which have ever been infective is at least $c(\lambda)$. Furthermore, we prove that $\lambda_c$ is the inverse of the production of the mean of $\rho$ and the mean of the inverse of $\xi$.