Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T22:40:05.929Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes

Published online by Cambridge University Press:  01 July 2016

Philippe Picard  and
Claude Lefevre
Philippe Picard*
Affiliation:
Université de Lyon 1
Claude Lefevre*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne, France.
∗∗Postal address: Institut de Statistique C. P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
Get access Rights & Permissions [Opens in a new window]

Abstract

An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.

Keywords

MARTINGALESFAMILY OF POLYNOMIALS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990 , pp. 269 - 294
Copyright
Copyright © Applied Probability Trust 1990 

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.)

Footnotes

Research supported by the Institut National de la Santé et de la Recherche Médicale under contract n° 898014.

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
Anderson, R. M. (1988) The epidemiology of HIV infection: variable incubation plus infectious periods and heterogeneity in sexual activity. J. R. Statist. Soc. A 151, 66–93.CrossRefGoogle Scholar
Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Bailey, N. T. J. (1988) Simplified modelling of the population dynamics of HIV/AIDS. J. R. Statist. Soc. A 151, 31–34.Google Scholar
Ball, F. (1983). The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Ball, F. (1986a) A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310.Google Scholar
Ball, F. (1986b) A note on the total size distribution of epidemic models. J. Appl. Prob. 23, 832–836.Google Scholar
Berg, S. (1983) Random contact processes, snowball sampling and factorial series distributions. J. Appl. Prob. 20, 31–46.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
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.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Gani, J. and Jerwood, D. (1972) The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.CrossRefGoogle Scholar
Isham, V. (1988) Mathematical modelling of the transmission dynamics of HIV infection and AIDS: a review. J. R. Statist. Soc. A 151, 5–30.Google Scholar
Lefevre, Cl. and Picard, Ph. (1989) On the formulation of discrete-time epidemic models. Math. Biosci. 95, 27–35.CrossRefGoogle ScholarPubMed
Lefevre, Cl. and Picard, Ph. (1990) A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.Google Scholar
Ludwig, D. (1975) Final size distributions for epidemics. Math. Biosci. 23, 33–46.Google Scholar
Malice, M.-P. and Lefevre, Cl. (1985) On the general epidemic model in discrete time. Lecture Notes in Biomathematics 57, Springer-Verlag, Heidelberg, 164–170.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
May, R. M. and Anderson, R. M. (1984) Spatial heterogeneity and the design of immunization programs. Math. Biosc. 72, 83–111.Google Scholar
Pettigrew, H. M. and Weiss, G. H. (1967) Epidemics with carriers: the large population approximation. J. Appl. Prob. 4, 257–263.Google Scholar
Picard, Ph. (1980) Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.CrossRefGoogle Scholar
Scalia-Tomba, G. (1986) Asymptotic final size distribution of the multitype Reed-Frost process. J. Appl. Prob. 23, 563–584.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
Weiss, G. H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–491.Google Scholar