Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-08-02T03:11:06.850Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
Coddington, E., and Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill, New York.Google Scholar
Dhlakama, R. (1994). Research Report no. 178, Department of Mathematical Statistics, Stockholm University.Google Scholar
Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Prob. Theory. Rel. Fields 81, 79–109.Google Scholar
Jeffreys, H., and Jeffreys, B. (1956). Methods of Mathematical Physics. Cambridge University Press, Cambridge.Google Scholar
Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.Google Scholar
Nåsell, I. (1995). The threshold concept in stochastic epidemic and endemic models. In Epidemic Models: Their Structure and Relation to Data. Ed Mollison, D. Publications of the Newton Institute, Cambridge University Press, Cambridge.Google Scholar
Salminen, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Adv. Appl. Prob. 20, 411–426.CrossRefGoogle Scholar