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The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Anders Martin-Löf
Anders Martin-Löf*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematical Statistics, Stockholm University, S-10691 Stockholm, Sweden. Email address: andersml@matematik.su.se
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Abstract

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

Keywords

Epidemic modelsize of epidemicnearly critical epidemicdiffusion approximationWiener processpassage timeparabolic barrierspectral theoryAiry functions

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 671 - 682
Copyright
Copyright © Applied Probability Trust 1998 

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