Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-24T02:53:09.012Z Has data issue: false hasContentIssue false
  • English
  • Français

Deterministic and stochastic epidemics with several kinds of susceptibles

Published online by Cambridge University Press:  01 July 2016

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

Abstract

We consider the spread of a general epidemic amongst a population consisting of invididuals with differing susceptibilities to the disease. Deterministic and stochastic versions of the basic model are described and analysed. For both versions of the model we show that assuming a uniform susceptible population, with average susceptibility, leads to an increased spread of infection. We also show how our results can be extended to the carrier-borne epidemic model of Weiss (1965).

Keywords

GENERAL EPIDEMICSSIZE OF EPIDEMICTHRESHOLD BEHAVIOURASYMPTOTIC SOLUTIONCOUPLINGSTOCHASTIC MAJORISATIONCARRIER-BORNE EPIDEMICS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 1 - 22
Copyright
Copyright © Applied Probability Trust 1985 

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 carried out in part at the University of Reading.

References

Armitage, P. (1975) Epidemiology and statistics. Bull. Internat. Statist. Inst. 45 (1), 258–264.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) The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.Google Scholar
Becker, N. G. (1973) Carrier-borne epidemics in a community consisting of different groups. J. Appl. Prob. 10, 491–501.Google Scholar
Becker, N. G. (1979) The uses of epidemic models. Biometrics 35, 295–305.Google Scholar
Billard, L. (1976) A stochastic general epidemic in m sub-populations. J. Appl. Prob. 13, 567–572.Google Scholar
Cane, V. and McNamee, R. (1982) The spread of infection in a heterogeneous population. In Essays in Statistical Science , ed. Gani, J. and Hannan, E. J., J. Appl. Prob. 19A, 173–184.Google Scholar
Cliff, A. D., Haggett, P., Ord, J. K. and Verfey, F. R. (1981) Spatial Diffusion: an Historical Geography of Epidemics in an Island Community. Cambridge University Press.Google Scholar
Gart, J. J. (1968) The mathematical analysis of an epidemic with two kinds of susceptibles. Biometrics 24, 557–566.CrossRefGoogle ScholarPubMed
Gart, J. J. (1972) The statistical analysis of chain-binomial epidemic models with several kinds of susceptibles. Biometrics 28, 921–930.CrossRefGoogle Scholar
Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Kendall, D. G. (1957) Contribution to the discussion of Google Scholar
Bartlett, M. S. (1957) Measles periodicity and community size. J. R. Statist. Soc. A 120, 48–70.Google Scholar
Kendall, W. S. and Saunders, I. W. (1983) Epidemics in competition II: The general epidemic. J. R. Statist. Soc. B 45, 238–244.Google Scholar
Lacayo, H. and Langberg, N. A. (1982) The exact and asymptotic formulas for the state probabilities in simple epidemics with m kinds of susceptibles. J. Appl. Prob. 19, 1–9.Google Scholar
Metz, J. A. J. (1978) 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.Google Scholar
Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google 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
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Severo, N. C. (1969) A recursion theorem on solving differential difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.Google Scholar
Shanbhag, D. N. (1972) On a vector-valued birth and death process. Biometrics 28, 417–425.Google Scholar
Watson, R. K. (1972) On an epidemic in a stratified population. J. Appl. Prob. 9, 659–666.Google Scholar
Watson, R. K. (1980) On the size distribution for some epidemic models. J. Appl. Prob. 17, 912–921.Google Scholar
Weiss, G. H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–490.Google Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic–a note on Bailey’s paper. Biometrika 42, 116–122.Google Scholar