Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-06T18:17:42.139Z Has data issue: false hasContentIssue false
  • English
  • Français

A Functional Limit Theorem for the Total Cost of a Multitype Standard Epidemic

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Håkan Andersson  and
Boualem Djehiche
Håkan Andersson*
Affiliation:
Stockholm University
Boualem Djehiche*
Affiliation:
Royal Institute of Technology, Stockholm
*
* Postal address: Department of Mathematics, Stockholm University, Box 6701, S-113 85 Stockholm, Sweden.
** Postal address: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

By imbedding the multitype version of the standard epidemic model in a multiparameter process, we derive a functional limit theorem for the total cost of the epidemics.

Keywords

EPIDEMIC PROCESSTOTAL COSTINFECTIOUS FORCEFUNCTIONAL LIMIT THEOREM

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 690 - 697
Copyright
Copyright © Applied Probability Trust 1994 

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. (1992) A threshold limit theorem for a multitype epidemic model. Research report No. 165, Institute of Actuarial Mathematics and Mathematical Statistics, Stockholm University, Sweden.Google Scholar
[2] Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
[3] Ball, F. (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.Google Scholar
[4] Ball, F. (1986) A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310.CrossRefGoogle Scholar
[5] Ball, F. and Clancy, D. (1993) The final size and severity of a generalized stochastic multitype epidemic model. Adv. Appl. Prob. 25, 721–736.CrossRefGoogle Scholar
[6] Bickel, P. J. and Wichura, M. J. (1971) Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42, 1656–1670.Google Scholar
[7] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[8] Korezlioglu, H. (1979) Two-parameter Gaussian Markov processes and their recursive linear filtering. Ann. Scient. Univ. Clermont 67, 69–93.Google Scholar
[9] Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.CrossRefGoogle Scholar
[10] Neuhaus, G. (1971) On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42, 1285–1295.Google Scholar
[11] Scalia-Tomba, G. (1985) Asymptotic final size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.Google Scholar
[12] Scalia-Tomba, G. (1986) Asymptotic final size distribution of the multitype Reed-Frost process. J. Appl. Prob. 23, 563–584.CrossRefGoogle Scholar
[13] Scalia-Tomba, G. (1988) On the asymptotic final size distribution of epidemics in heterogeneous populations. In Stochastic Processes in Epidemic Theory, ed. Gabriel, J.-P. et al., pp. 189–196. Lecture Notes in Biomathematics 86, Springer-Verlag, Berlin.Google Scholar
[14] Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
[15] Svensson, Å. (1992) Dynamics of an epidemic in a closed population. Preprint, Department of Statistics, Stockholm University.Google Scholar
[16] Walsh, J. B. (1986) Martingales with a multidimensional parameter and stochastic integrals in the plane. In Lecture Notes in Mathematics 1215, pp. 329–491. Springer-Verlag, Berlin.Google Scholar
[17] Watson, R. K. (1980) A useful random time-scale transformation for the standard epidemic model. J. Appl. Prob. 17, 324–332.Google Scholar