摘要
We are concerned with the susceptible-infective-removed （SIR） model with random transition rates on complete graphs Cn with n vertices. We assign independent and identically distributed （i.i.d.） copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d, copies of a positive random variable ρ 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 λc 〉 0 such that when λ 〈 λc, the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → ＋∞ while when λ 〉 λc, there exist c（λ） 〉 0 and b（λ） 〉 0 such that for each n ≥ 1 with probability p ≥ b（λ）, the proportion r∞ ≥ c（λ）. Furthermore, we prove that Ac is the inverse of the production of the mean of p and the mean of the inverse of ξ.
We are concerned with the susceptible-infective-removed （SIR） model with random transition rates on complete graphs Cn with n vertices. We assign independent and identically distributed （i.i.d.） copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d, copies of a positive random variable ρ 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 λc 〉 0 such that when λ 〈 λc, the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → ＋∞ while when λ 〉 λc, there exist c（λ） 〉 0 and b（λ） 〉 0 such that for each n ≥ 1 with probability p ≥ b（λ）, the proportion r∞ ≥ c（λ）. Furthermore, we prove that Ac is the inverse of the production of the mean of p and the mean of the inverse of ξ.
基金
Acknowledgements The author is grateful to the reviewers. Their comments is a great help for him to improve this paper. In the original version, the author only proved that the main result holds under Assumption （7）. According to the reviewers＇ comments, he learned how to show that the main result holds under Assumption （1）. This work was supported by the National Natural Science Foundation of China （Grant No. 11501542） and the financial support from Beijing Jiaotong University （Grant No. KSRC16006536）.
