Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T09:45:47.656Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of martingale theory to some epidemic models

Published online by Cambridge University Press:  14 July 2016

Philippe Picard
Philippe Picard*
Affiliation:
Université de Lyon I
*
∗Postal address: Mathématiques Appliquées, 43, Boulevard du 11 Novembre 1918, 69621 Villeurbanne, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here Xt and Yt respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.

Keywords

EPIDEMICSEND AND DIMENSION OF EPIDEMICMARTINGALESTOPPING TIMEAREA UNDER TRAJECTORY OF CARRIERSPARAMETER ESTIMATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 583 - 599
Copyright
Copyright © Applied Probability Trust 1980 

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

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
Daniels, H. E. (1972) An exact relation in the theory of carrier-borne epidemics. Biometrika 59, 211–213.CrossRefGoogle Scholar
Downton, F. (1968) The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.CrossRefGoogle Scholar
Downton, F. (1972) The area under the infectives trajectory of the general stochastic epidemic. J. Appl. Prob. 9, 414–417; correction J. Appl. Prob. 9, 873–876.CrossRefGoogle Scholar
Gani, J. and Jerwood, D. (1972) The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.CrossRefGoogle Scholar
Gontcharoff, W. (1937) Détermination des fonctions entières par interpolation. Hermann et Cie, Paris.Google Scholar
Jerwood, D. (1974) The cost of a carrier-borne epidemic. J. Appl. Prob. 11, 642–651.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. ?. (1975) A First Course in Stochastic Processes , 2nd edn. Academic Press, New York.Google Scholar
Kryscio, R. J. and Saunders, R. (1976) A note on the cost of carrier-borne, right-shift, epidemic models. J. Appl. Prob. 13, 652–661.CrossRefGoogle Scholar
Lefevre, C. (1978) The expected ultimate size of a carrier-borne epidemic. J. Appl. Prob. 15, 414–419.CrossRefGoogle Scholar
Picard, Ph. (1978) Test et estimation pour une classe de processus bidimensionnels. Statistique et Analyse des Données 3, 45–55.Google Scholar