Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T12:58:48.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic compartmental models as approximations to more general stochastic systems with the general stochastic epidemic as an example

Published online by Cambridge University Press:  01 July 2016

M. J. Faddy
M. J. Faddy*
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Abstract

A general time-dependent stochastic compartmental model is considered, where particles may move between a set of m compartments or out of the system, and new particles may be introduced into the compartments by immigration. A simple argument not relying on generating function techniques is given for the solution of such a system. It is then demonstrated that this class of compartmental models can be used as approximations to more complex stochastic systems by replacing dependence on certain stochastic variables by dependence on corresponding deterministic variables, with some recent examples being discussed. A compartmental approximation to the general stochastic epidemic is then constructed which appears on the basis of some simulations to compare favourably with the true process, particularly after the infection has got established in the population.

Keywords

COMPARTMENTAL SYSTEMSGENERAL TIME-DEPENDENT STOCHASTIC MODELAPPROXIMATING STOCHASTIC SYSTEMSGENERAL STOCHASTIC EPIDEMIC
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 448 - 461
Copyright
Copyright © Applied Probability Trust 1977 

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

[1] Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
[2] Cardenas, M. and Matis, J. H. (1974) On the stochastic theory of compartments: solution for n-compartment systems with irreversible time-dependent transition probabilities. Bull. Math. Biol. 36, 489–504.Google Scholar
[3] Cardenas, M. and Matis, J. H. (1975) On the time-dependent reversible stochastic compartmental model — II. A class of n-compartment systems. Bull. Math. Biol. 37, 555–564.Google Scholar
[4] Faddy, M. J. (1976) A note on the general time-dependent stochastic compartmental model. Biometrics 32, 443–448.Google Scholar
[5] Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
[6] Kermack, W. O. and McKendrick, A. G. (1927) Contributions to the mathematical theory of epidemics. Proc. R. Soc. London A 115, 700–721.Google Scholar
[7] Lewis, T. (1975) A model for the parasitic disease bilharziasis. Adv. Appl. Prob. 7, 673–704.Google Scholar
[8] McClean, S. I. (1976) A continuous-time population model with Poisson recruitment. J. Appl. Prob. 13, 348–354.Google Scholar
[9] Raman, S. and Chiang, C. L. (1973) On a solution of the migration process and the application to a problem in epidemiology. J. Appl. Prob. 10, 718–727.Google Scholar
[10] Renshaw, E. (1973) Interconnected population processes. J. Appl. Prob. 10, 1–14.Google Scholar