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

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##### Author

Janson, S; Luczak, M; Windridge, P; House, T##### Date

2017-03-01##### Source Title

Journal of Mathematical Biology##### Publisher

SPRINGER HEIDELBERG##### University of Melbourne Author/s

Luczak, Malwina##### Affiliation

School of Mathematics and Statistics##### Metadata

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Journal Article##### Citations

Janson, S., Luczak, M., Windridge, P. & House, T. (2017). Near-critical SIR epidemic on a random graph with given degrees. JOURNAL OF MATHEMATICAL BIOLOGY, 74 (4), pp.843-886. https://doi.org/10.1007/s00285-016-1043-z.##### Access Status

**Open Access**

##### 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.

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