Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T19:34:54.976Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Operations research and management science Graph theory 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: a.j.e.m.janssen@tue.nl
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Abramowitz, M. and Stegun, I. A. (eds) (1984). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. John Wiley, New York.Google Scholar
Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812–854.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2009). Scaling limits for critical inhomogeneous random graphs with finite third moments. Preprint. Available at http://arxiv.org/abs/0907.4279v2.Google Scholar
Bollobás, B. (2001). Random Graphs (Camb. Stud. Adv. Math. 73), 2nd edn. Cambridge University Press.Google Scholar
De Bruijn, N. G. (1981). Asymptotic Methods in Analysis, 3rd edn. Dover, New York.Google Scholar
Doney, R. A. and Yor, M. (1998). On a formula of Takács for Brownian motion with drift. J. Appl. Prob. 35, 272–280.Google Scholar
Gordillo, L. F., Marion, S. A., Martin-Löf, A. and Greenwood, P. E. (2008). Bimodal epidemic size distributions for near-critical SIR with vaccination. Bull. Math. Biol. 70, 589–602.Google Scholar
Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Prob. Theory Relat. Fields 81, 79–109.CrossRefGoogle Scholar
Janson, S. (2007). Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Prob. Surveys 4, 80–145.Google Scholar
Janson, S. and Spencer, J. (2007). A point process describing the component sizes in the critical window of the random graph evolution. Combinatorics Prob. Comput. 16, 631–658.Google Scholar
Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.CrossRefGoogle Scholar
Lalley, S. P. (2007). Critical scaling of stochastic epidemic models. In Asymptotics: Particles, Processes and Inverse Problems (IMS Lecture Notes Monogr. Ser. 55), Institute for Mathematical Statistics, Beachwood, OH, pp. 167–178.Google Scholar
Łuczak, T., Pittel, B. and Wierman, J. C. (1994). The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341, 721–748.CrossRefGoogle Scholar
Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.Google Scholar
Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671–682.Google Scholar
Nachmias, A. and Peres, Y. (2007). The critical random graph, with martingales. Preprint. Available at http://arxiv.org/abs/math/0512201v4.Google Scholar
Pittel, B. (2001). On the largest component of the random graph at a nearcritical stage. J. Combinatorial Theory B 82, 237–269.Google Scholar
Startsev, A. N. (2001). Asymptotic analysis of the general stochastic epidemic with variable infectious periods. J. Appl. Prob. 38, 18–35.Google Scholar
Takács, L. (1989). Ballots, queues and random graphs. J. Appl. Prob. 26, 103–112.Google Scholar
Turova, T. S. (2009). Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1. Preprint. Available at http://arxiv.org/abs/0907.0897v2.Google Scholar
Van der Hofstad, R., Kager, W. and Müller, T. (2009). A local limit theorem for the critical random graph. Electron. Commun. Prob. 14, 122–131.Google Scholar