Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T13:39:06.474Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models

Published online by Cambridge University Press:  01 July 2016

Frank Ball
Frank Ball*
Affiliation:
University of Nottingham
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

Keywords

GENERAL EPIDEMICSWALD&S IDENTITYHOMOGENEOUS MIXINGMULTIPOPULATION EPIDEMICSPOISSON LIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 289 - 310
Copyright
Copyright © Applied Probability Trust 1986 

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

Abakuks, A. (1973) An optimal isolation policy for an epidemic. J. Appl. Prob. 10, 247–262.Google Scholar
Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.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
Ball, F. G. (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.CrossRefGoogle Scholar
Becker, N. G. (1979) The uses of epidemic models. Biometrics 35, 295–305.CrossRefGoogle ScholarPubMed
Cane, V. (1966) On the size of an epidemic and the number of people hearing a rumour. J.R. Statist. Soc., B 20, 487–490.Google Scholar
Daniels, H. E. (1967) The distribution of the total size of an epidemic. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 281–293.Google Scholar
Downton, F. (1967) A note on the ultimate size of a general stochastic epidemic. Biometrika 54, 314–316.Google Scholar
Downton, F. (1972) A correction to “The area under the infectives trajectory of the general stochastic epidemic”. J. Appl. Prob. 9, 873–876.CrossRefGoogle Scholar
Dunstan, R. (1980) A note on the moments of the final size of the general epidemic model. J. Appl. Prob. 17, 532–538.Google Scholar
Gani, J. and Jerwood, D. (1972) The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.Google Scholar
Lefevre, C. (1978) The expected ultimate size of a carrier-borne epidemic. J. Appl. Prob. 15, 414–419.Google Scholar
Ludwig, D. (1974) Stochastic Population Theories. Lecture Notes in Biomathematics 3, Springer-Verlag, Berlin.Google Scholar
Metz, J. A. J. (1977) The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biotheoretica 27, 75–123.CrossRefGoogle Scholar
Rushton, S. and Mautner, A. J. (1955) The deterministic model of a simple epidemic for more than one community. Biometrika 42, 126–132.CrossRefGoogle Scholar
ScAlia-Tomba, G. (1983) Extensions of the Reed-Frost Process. Ph.D. Thesis, Department of Mathematical Statistics, University of Stockholm.Google Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Wald, A. (1947) Sequential Analysis. Wiley, New York.Google Scholar
Watson, R. K. (1972) On an epidemic in a stratified population. J. Appl. Prob. 9, 659–666.CrossRefGoogle Scholar
Watson, R. K. (1980) On the size distribution for some epidemic models. J. Appl. Prob. 17, 912–921.CrossRefGoogle Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic–a note on Bailey&s paper. Biometrika 42, 116–122.Google Scholar