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Critical epidemics, random graphs, and Brownian motion with a parabolic drift

Part of: Graph theory Operations research and management science Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Remco van der Hofstad ,
A. J. E. M. Janssen  and
Johan S. H. van Leeuwaarden
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
A. J. E. M. Janssen*
Affiliation:
EURANDOM and Eindhoven University of Technology
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗∗∗ Postal address: Department of Electrical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

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We investigate the final size distribution of the SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of Martin-Löf (1998) for the hitting time of a Brownian motion with parabolic drift, we derive asymptotic expressions for the final size distribution that capture the effect of the initial number of infectives and the closeness of the reproduction number to zero. These asymptotics shed light on the bimodularity of the limiting density of the final size observed in Martin-Löf (1998). We also discuss the connection to the largest component in the Erdős-Rényi random graph, and, using this connection, find an integral expression of the Laplace transform of the normalized Brownian excursion area in terms of Airy functions.

Keywords

SIR epidemic modelfinal size distributioncritical random graphlargest connected componentphase transitionAiry functionBrownian motion with parabolic drift

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1187 - 1206
Copyright
Copyright © Applied Probability Trust 2010 

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