Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-29T15:16:36.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes  and
A. C. Trajstman
Anthony G. Pakes*
Affiliation:
University of Western Australia
A. C. Trajstman*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
∗∗ Postal address: CSIRO Division of Mathematics and Statistics, Private Bag 10, Clayton, VIC 3168, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift.

This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

Keywords

CONTINUOUS BRANCHING PROCESSLÉVY PROCESSLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 23 - 41
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.)

References

Athreya, K. B. and Karlin, S. (1967) Limit theorems for the split times of branching processes. J. Math. Mech. 17, 257–278.Google Scholar
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Bartoszynski, R. (1975a) On the risk of rabies. Math. Biosci. 24, 355–377.Google Scholar
Bartoszynski, R. (1975b) A model for the risk of rabies. ISI Invited Paper No. 9, ISI 1-9, IX Warsaw.Google Scholar
Bartoszynski, R. (1976) A stochastic model of development of rabies. In Proceedings of the IASPS Symposium to honour Jerzy Neyman , ed. Klonecki, W.. PWN, Warsaw.Google Scholar
Bingham, N. H. (1976) Continuous branching processes and spectral positivity. Stoch. Proc. Appl. 4, 217–242.Google Scholar
Bühler, W. J. and Keller, G. (1978) Some remarks on Bartoszynski’s rabies model. Math. Biosci. 39, 273–279.Google Scholar
Grey, D. R. (1974) Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Prob. 11, 669–677.Google Scholar
Helland, I. S. (1978) Continuity of a class of random time transformations. Stoch. Proc. Appl. 7, 79–99.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Lamperti, J. (1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.Google Scholar
Pakes, A. G. (1979) Some limit theorems for Jirina processes. Periodica Math. Hung. 10, 55–66.Google Scholar
Puri, P. S. (1969) Some limit theorems on branching processes and certain related processes. Sankhya 31A, 57–74.Google Scholar
Rogozin, B. A. (1976) Relatively stable walks. Theory Prob. Appl. 21, 375–379.CrossRefGoogle Scholar
Schuh, H-J. (1982) Sums of i.i.d. random variables and an application to the explosion criterion for Markov branching processes. J. Appl. Prob. 19, 29–38.Google Scholar
Seneta, E. (1976a) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
Seneta, E. (1976b) Some supplementary notes on one-type continuous-state branching processes. Z. Wahrscheinlichkeitsth. 34, 87–89.Google Scholar
Seneta, E. and Vere-Jones, D. (1968) On the asymptotic behaviour of subcritical branching processes with continuous state space. Z. Wahrscheinlichkeitsth. 10, 212–225.Google Scholar
Shurenkov, V. M. (1973) Some limit theorems for branching processes with the continuous space of states. (in Russian). Teor. Verojatnost. i Mat. Statist. (Kiev) 9, 167–172.Google Scholar
Silverstein, M. L. (1968) A new approach to local times. J. Math. Mech. 17, 1023–1054.Google Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
Trajstman, A. C. and Tweedie, R. L. (1982) Techniques for estimating parameters in Bartoszynski’s virus model. Math. Biosci. 58, 277–307.CrossRefGoogle Scholar