Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T14:37:43.685Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian approximation of some closed stochastic epidemic models

Published online by Cambridge University Press:  14 July 2016

Frank J. S. Wang
Frank J. S. Wang*
Affiliation:
University of Montana
Get access Rights & Permissions [Opens in a new window]

Abstract

A generalization of Bailey's general epidemic model is considered. In this generalized model, it is assumed that the probability of any particular susceptible becoming infected during the small time interval (t, t + Δt) is α(X(t))Δt + o(Δt), for some function a, where X(t) is the proportion of infected individuals in the entire population, the probability that an infected individual is infected for at least a length of time t is F(t), and recovered individuals are permanently immune from further attack. In this paper, central limit theorems are obtained for the proportion of infected individuals and the proportion of susceptibles in the entire population.

Keywords

EPIDEMIC MODELGAUSSIAN PROCESSCENTRAL LIMIT THEOREMBIVARIATE GAUSSIAN RANDOM FUNCTIONVOLTERRA INTEGRAL EQUATIONRESOLVENT KERNELPARTIAL DIFFERENTIAL EQUATIONSKOROHOD TOPOLOGY
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 2 , June 1977 , pp. 221 - 231
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. (1957) The Mathematical Theory of Epidemics. Hafner Publishing Co., New York.Google Scholar
[2] Barbour, A. (1974) On a functional central limit theorem for Markov population processes. Adv. Appl. Prob. 6, 21–39.Google Scholar
[3] Barlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
[4] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[5] Dietz, K. (1967) Epidemic and rumours: a survey. J. R. Statist. Soc. A 130, 505–528.Google Scholar
[6] Kendall, D. G. (1956) Deterministic and stochastic epidemic in a closed population. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
[7] Kermack, W. O. and Mckendrick, A. G. (1927) A contribution to the mathematical theory of epidemics. Proc. R. Soc. London A 115, 700–721.Google Scholar
[8] Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7, 49–58.Google Scholar
[9] Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential equations J. Appl. Prob. 8, 344–356.Google Scholar
[10] Kurtz, T. G. (1975) Limit theorems and diffusion approximations for density dependent Markov chains. Unpublished.Google Scholar
[11] Nagaev, A. V. and Startsev, (1970) The asymptotic analysis of a stochastic model of an epidemic. Theory Prob. Appl. 15, 98–107.Google Scholar
[12] Norman, M. F. (1972) A central limit theorem for Markov processes that move by small steps. Ann. Prob. 2, 1065–1074.Google Scholar
[13] Rosen, B. (1967) On the central limit theorem for sums of dependent random variables. Z. Wahrscheinlichkeitsth. 7, 48–82.Google Scholar
[14] Wang, F. J. S. (1975) Limit theorems for age and density dependent stochastic population models. J. Math. Biosci. 2, 373–400.Google Scholar
[15] Wang, F. J. S. (1977) A central limit theorem for age and density dependent population processes. Stoch. Proc. Appl. 5 (2).Google Scholar