title: Near-critical SIR epidemic on a random graph with given degrees.
abstract: 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 [Formula: see text], with [Formula: see text] 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 [Formula: see text]. 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.
authors.affiliation: Department of Mathematics, Uppsala University, PO Box 480, 751 06, Uppsala, Sweden.
authors.forename: Svante
authors.initials: S
authors.lastname: Janson
authors.affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK. m.luczak@qmul.ac.uk.
authors.forename: Malwina
authors.identifier: 0000-0002-9451-8825
authors.initials: M
authors.lastname: Luczak
authors.affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK.
authors.forename: Peter
authors.initials: P
authors.lastname: Windridge
authors.affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, UK.
authors.forename: Thomas
authors.initials: T
authors.lastname: House
mesh_terms: D058872:Epidemics; D006801:Humans; D008962:Models, Theoretical; D011156:Population Density; D011336:Probability
keywords: Configuration model; Critical window; Random graph with given degrees; SIR epidemic