Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-28T06:20:29.651Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate birth-and-death processes as approximations to epidemic processes

Published online by Cambridge University Press:  14 July 2016

D. A. Griffiths
D. A. Griffiths*
Affiliation:
University of Oxford
*
∗ Now at: C. S. I. R. O. 60 King Street, Newtown, N. S. W., Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

Keywords

CARRIER-BORNE EPIDEMICSMULTIVARIATE LINEAR BIRTH-AND-DEATH PROCESSLARGE POPULATION EPIDEMIC MODEL
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 15 - 26
Copyright
Copyright © Applied Probability Trust 1973 

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

Adke, S. R. (1964) Multidimensional birth and death process. Biometrics 20, 212–216.CrossRefGoogle Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Becker, N. G. (1968) The spread of an epidemic to fixed groups within the population. Biometrics 24, 1007–1014.Google Scholar
Bellman, R. (1953) Stability Theory of Differential Equations. McGraw-Hill, New York.Google Scholar
Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
Denton, Gillian M. (1971) On the time-dependent solution of Downton's carrier-borne epidemic. Manchester-Sheffield School of Probability and Statistics Research Report.Google Scholar
Dietz, K. (1966) On the model of Weiss for the spread of epidemics by carriers. J. Appl. Prob. 3, 375–382.Google Scholar
Downton, F. (1967) Epidemics with carriers; a note on a paper of Dietz. J. Appl. Prob. 4, 264–270.Google Scholar
Downton, F. (1968) The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.Google Scholar
Gani, J. (1967) On the general stochastic epidemic. Proc. 5th Berkeley Symp. Math. Statist. and Prob. 4, 271–279.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
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Haskey, H. W. (1957) Stochastic cross-infection between two otherwise isolated groups. Biometrika 44, 193–204.CrossRefGoogle Scholar
Kendall, D. G. (1948) On the generalised “birth-and-death” process. Ann. Math. Statist. 19, 1–15.Google Scholar
Mode, C. J. (1962) Some multi-dimensional birth and death processes and their application in population genetics. Biometrics 18, 543–567.Google Scholar
Otter, R. (1949) The multiplicative process. Ann. Math. Statist. 20, 206–224.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
Rushton, S. and Mautner, A. J. (1955) The deterministic model of a simple epidemic in more than one community. Biometrika 42, 126–132.Google Scholar
Weiss, G. H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–490.Google Scholar
Wilson, E. B. and Worcester, J. (1945) The spread of an epidemic. Proc. Nat. Acad. Sci. U.S.A. 31, 327–332.CrossRefGoogle ScholarPubMed