Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T15:09:28.308Z Has data issue: false hasContentIssue false
  • English
  • Français

The threshold behaviour of epidemic models

Published online by Cambridge University Press:  14 July 2016

Frank Ball
Frank Ball*
Affiliation:
University of Sussex
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Keywords

EPIDEMICSSIZE OF EPIDEMICTHRESHOLD THEOREMSBIRTH-AND-DEATH PROCESSESSTRONG CONVERGENCEBRANCHING PROCESSESGEOGRAPHICAL SPREAD
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 227 - 241
Copyright
Copyright © Applied Probability Trust 1983 

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

Anderson, D. A. and Watson, R. K. (1980) On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67, 191–198.Google Scholar
Von Bahr, and Martin-Löf, A, (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.Google Scholar
Bailey, N. T. J. (1964) Some stochastic models for small epidemics in large populations. Appl. Statist. 13, 9–19.Google Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Ball, F. G. (1983) A threshold theorem for the Reed-Frost chain-binomial epidemic. J. Appl. Prob. 20, 153–157.Google Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, London.Google Scholar
Foster, F. G. (1955) A note on Bailey's and Whittle's treatment of a general stochastic epidemic. Biometrika 42, 123–125.Google Scholar
Griffiths, D. A. (1972) A bivariate birth-death process which approximates to the spread of a disease involving a vector. J. Appl. Prob. 9, 65–75.Google Scholar
Griffiths, D. A. (1973) Multivariate birth-and-death processes as approximations to epidemic processes. J. Appl. Prob. 10, 15–26.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Kendall, D. G. (1948) On the generalised ‘birth-and-death’ process. Ann. Math. Statist. 19, 1–15.Google Scholar
Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google Scholar
Otter, R. (1949) The multiplicative process. Ann. Math. Statist. 20, 206–224.Google Scholar
Rajarshi, M. B. (1981) Simpler proofs of two threshold theorems for a general stochastic epidemic. J. Appl. Prob. 18, 721–724.Google Scholar
Watson, R. K. (1980) On the size distribution for some epidemic models. J. Appl. Prob. 17, 912–921.Google Scholar
Williams, T. (1971) An algebraic proof of the threshold theorem for the general stochastic epidemic (abstract). Adv. Appl. Prob. 3, 223.Google Scholar