Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T05:55:28.896Z Has data issue: false hasContentIssue false
  • English
  • Français

The advancing wave in a spatial birth process

Published online by Cambridge University Press:  14 July 2016

H. E. Daniels
H. E. Daniels*
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that, in the deterministic model of a one-dimensional spatial linear birth process, a finite initial distribution of individuals develops into an advancing wave with the minimum velocity possible for such waves. In this paper, the spatial covariance density of the process is studied in the vicinity of a point moving outwards with a given velocity. Its behaviour casts doubt on the applicability of the deterministic result to individual realisations of the process.

Keywords

SPATIAL BIRTH PROCESSADVANCING WAVESADDLEPOINT APPROXIMATIONDIFFUSION APPROXIMATIONCOVARIANCE DENSITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 689 - 701
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.)

Footnotes

An earlier version of this paper was presented to the Sixth Conference on Stochastic Processes and their Applications, Tel Aviv, June 1976.

References

[1] Atkinson, C. and Reuter, G. E. H. (1976) Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc. 80, 315–330.Google Scholar
[2] Bartlett, M. S. (1954) Processus stochastiques ponctuels. Ann. Inst. H. Poincaré 14, 35–60.Google Scholar
[3] Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
[4] Daniels, H. E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–650.Google Scholar
[5] Daniels, H. E. (1975) The deterministic spread of a simple epidemic. In Perspectives in Probability and Statistics, ed. Gani, J., Applied Probability Trust, Sheffield; Academic Press, London, 373–386.Google Scholar
[6] Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369.Google Scholar
[7] Kendall, D. G. (1965) Mathematical models of the spread of infection. In Mathematics and Computer Sciences in Biology and Medicine, Medical Research Council, London, 213–225.Google Scholar
[8] Kolmogoroff, A. N., Petrovsky, I. G. and Piscounoff, N. S. (1937) Étude de l'équation de diffusion avec croissance de la quantité de matière et son application à une problème biologique. Bull. Univ. d'Etat Moscou (ser. Intern.) A 1(6), 1–25.Google Scholar
[9] McKean, H. P. (1975) Application of Brownian motion to the equation of Kolmogoroff, Petrovsky and Piscounoff. Comm. Pure Appl. Maths 28, 321–331.Google Scholar
[10] Mollison, D. (1972) Possible velocities for a simple epidemic. Adv. Appl. Prob. 4, 233–258.Google Scholar
[11] Mollison, D. and Daniels, H. E. (1977) The deterministic simple epidemic unmasked.Google Scholar
[12] Mollison, D. (1977) Spatial contact models for ecological and epidemiological spread. J. R. Statist. Soc. B. 39, 283–326.Google Scholar
[13] Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input–output streams. J. R. Statist. Soc. B 30, 321–333.Google Scholar