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Critical scaling for the SIS stochastic epidemic

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

R. G. Dolgoarshinnykh  and
Steven P. Lalley
R. G. Dolgoarshinnykh*
Affiliation:
Columbia University
Steven P. Lalley*
Affiliation:
University of Chicago
*
∗Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: regina@stat.columbia.edu
∗∗Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA. Email address: lalley@galton.uchicago.edu
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Abstract

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We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of infected and susceptible individuals, then when the time and the number currently infected are both scaled by , the resulting process converges, as N → ∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Löf (1998) and Aldous (1997) for the simple SIR epidemic.

Keywords

Stochastic epidemic modelSISSIRFeller diffusionOrnstein-Uhlenbeck process

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 892 - 898
Copyright
© Applied Probability Trust 2006 

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