Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-08T02:05:13.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Interparticle correlation in death processes with application to variability in compartmental models

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Peter Donnelly
Frank Ball*
Affiliation:
University of Nottingham
Peter Donnelly*
Affiliation:
University College London
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Postal address: Department of Statistical Science, University College, Gower Street, London WC1E 6BT, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with a pure death process, starting with N individuals, with death rates μ n, n = 1, 2, …, N. It is shown that the fates of distinct individuals are positively correlated if μ n/n decreases with n, and negatively correlated if μ n/n increases with n. The application of this result to the problem of variability in compartmental models is elaborated and in particular a conjecture of Faddy (1985) is settled. Further applications to well-known death processes are also briefly described.

Keywords

CONCAVE AND CONVEX DEATH RATESFKG INEQUALITYDETERMINISTIC AND STOCHASTIC MODELSSIMPLE EPIDEMIC
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 755 - 766
Copyright
Copyright © Applied Probability Trust 1987 

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

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Barbour, A. D. (1974) On a functional central limit theorem for Markov population processes. Adv. Appl. Prob. 6, 21–39.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, New York.Google Scholar
Coleman, B. D. (1957) Time dependence of mechanical breakdown in bundles of fibres I. Constant total load. J. Appl. Phys. 28, 1058–1064.Google Scholar
Donnelly, P. (1985) Contribution to the discussion, “Symposium on stochastic networks”. J. R. Statist. Soc. B 47, 422–423.Google Scholar
Donnelly, P. and Welsh, D. J. A. (1983) Finite particle systems and infection models. Math. Proc. Camb. Phil. Soc. 94, 167–182.CrossRefGoogle Scholar
Faddy, M. J. (1983) Compartmental models with restricted movement. Bull. Math. Biol. 45, 401–408.Google Scholar
Faddy, M. J. (1985) Non-linear stochastic compartmental models. IMA J. Math. Appl. Med. Biol. 2, 287–297.Google Scholar
Faddy, M. J., Gosden, R. G. and Edwards, R. G. (1983) Ovarian follicle dynamics in mice: a comparative study of three inbred strains and an F1-hybrid. J. Endocr. 96, 23–33.CrossRefGoogle Scholar
Feller, W. (1939) Die grundlagen der volterraschen theorie des kampfes ums dasein in wahrscheinlichkeitstheoretischer behandlung. Acta Biotheoretica 5, 11–40.Google Scholar
Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.Google Scholar
Harris, T. E. (1977) A correlation inequality for Markov processes in partially ordered state spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, London.Google Scholar
Matis, J. H. and Wehrly, T. E. (1979) Stochastic models of compartmental systems. Biometrics 35, 199–220.CrossRefGoogle Scholar
Morgan, B. T. J. (1976) Stochastic models of grouping changes. Adv. Appl. Prob. 8, 30–57.Google Scholar
Purdue, P. (1979) Stochastic compartmental models: a review of the mathematical theory with ecological applications. In Compartmental Analysis of Ecosystem Models, ed. Matis, J. H., Patten, B. C. and White, G. C.. International Cooperative Publishing House, Fairland, Maryland, 223–260.Google Scholar
Purdue, P. (1981) Variability in a single compartment system: a note on S. R. Bernard's model. Bull. Math. Biol. 43, 111–116.CrossRefGoogle Scholar
Seymour, P. D. and Welsh, D. J. A. (1975) Combinatorial applications of an inequality from statistical mechanics. Math. Proc. Camb. Phil. Soc. 77, 485–495.CrossRefGoogle Scholar
Tavaré, S. (1984) Line-of-descent and geneological processes, and their applications in population genetic models. Theoret. Popn Biol. 26, 119–164.Google Scholar
Taylor, H. M. and Karlin, S. (1984) An Introduction to Stochastic Modelling. Academic Press, Orlando, Florida.Google Scholar
Welsh, D. J. A. (1986) Correlated percolation and repulsive particle systems. In Stochastic Spatial Processes, ed. Tauta, P., Lecture Notes in Mathematics 1212, Springer-Verlag, Berlin, 300–311.Google Scholar