Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T01:18:13.623Z Has data issue: false hasContentIssue false

Limit theorems for the total size of a spatial epidemic

Published online by Cambridge University Press:  14 July 2016

Håkan Andersson*
Affiliation:
Stockholm University
Boualem Djehiche*
Affiliation:
Royal Institute of Technology, Stockholm
*
Postal address: Department of Mathematics, Stockholm University, S-106 91 Stockholm, Sweden.
∗∗Postal address: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden.

Abstract

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1997 

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] Andersson, H. and Djehiche, B. (1994) A functional limit theorem for the total cost of a multitype standard epidemic. Adv. Appl. Prob. 26, 690697.CrossRefGoogle Scholar
[2] Andersson, H. and Djehiche, B. (1995) Limit theorems for multitype epidemics. Stoch. Proc. Appl. 56, 5775.Google Scholar
[3] Aronson, D. G. (1977) The asymptotic speed of propagation of a simple epidemic. In Nonlinear Diffusion. ed. Fitzgibbon, W. E. III and Walker, H. F. Pitman, London.Google Scholar
[4] Ball, F. (1983) The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227241.Google Scholar
[5] Griffiths, D. A. (1973) Multivariate birth-and-death processes as approximations to epidemic processes. J. Appl. Prob. 10, 1526.Google Scholar
[6] Harris, T. E. (1963) The Theory of Branching Processes. Dover, New York.CrossRefGoogle Scholar
[7] Kendall, D. G. (1965) Mathematical models of the spread of infections. Mathematics and Computer Science in Biology and Medicine. Medical Research Council, London.Google Scholar
[8] Kurtz, T. G. (1981) Approximation of population processes. CBMS-NSF Regional Conf. Series in Appl. Math. 36, SIAM, Philadelphia, PA.Google Scholar
[9] Liptser, R. Sh. and Shiryayev, A. N. (1989) Theory of Martingales. Kluwer, Dordrecht.Google Scholar
[10] Svensson, A. (1995) On the simultaneous distribution of size and costs of an epidemic in a closed multigroup population. Math. Biosci. 127, 167180.Google Scholar