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

Abstract

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C_n with n vertices. We assign independent and identically distributed (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. There exists {{\displaystyle \lambda }}_c\rangle \mathrm{0} such that when \lambda \langle {{\displaystyle \lambda }}_c, the proportion {{\displaystyle r}}_{\infty } of vertices which have ever been infective converges to 0 weakly as n\to +\infty while when \lambda \rangle {{\displaystyle \lambda }}_c, there exist c(\lambda )\rangle \mathrm{0} and b(\lambda )\rangle \mathrm{0} such that for each n\ge \mathrm{1} with probability p\ge b(\lambda ), the proportion {{\displaystyle r}}_{\infty }\ge c(\lambda ). Furthermore, we prove that {{\displaystyle \lambda }}_c is the inverse of the production of the mean of ρ and the mean of the inverse of \xi.

Key words： Susceptible-infective-removed (SIR) model; complete graph; phase transition; random rate

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