Near-critical SIR epidemic on a random graph with given degrees

作者：Janson Svante; Luczak Malwina^*; Windridge Peter; House Thomas

来源：Journal of Mathematical Biology, 2017, 74(4): 843-886.

DOI：10.1007/s00285-016-1043-z

• SIR epidemic
• Random graph with given degrees
• Configuration model
• Critical window

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is , with tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as . As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the 'sparse' regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

• 出版日期2017-3