Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T22:28:34.714Z Has data issue: false hasContentIssue false
  • English
  • Français

Recent investigations involving stochastic population models

Published online by Cambridge University Press:  01 July 2016

M. S. Bartlett
M. S. Bartlett*
Affiliation:
Emeritus Professor of Biomathematics, University of Oxford
*
∗Postal address: Priory Orchard, Priory Avenue, Totnes, Devon TQ9 5HR, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed further, viz. spatial nearest-neighbour lattice models and doubly stochastic models.

In Section 1 of the paper, the first type of model is considered primarily for its relevance to recent work by the author and others affecting the practical design and analysis of replicated field experiments.

In Section 2, doubly stochastic processes are discussed more theoretically, particularly models investigated recently by the author involving infinitesimal transition operators in continuous time linear in the (variable) parameters.

Some new numerical results on extinction and other ‘absorption' probabilities are presented; these are intended to throw further light on the extent to which the assumption of ‘white-noise' variability of the parameters can be a useful approximation to more realistic models.

Keywords

NEAREST NEIGHBOURPAPADAKIS METHODDOUBLY STOCHASTIC PROCESSESWHITE NOISESWITCHING MODELSEXTINCTION PROBABILITIESEPIDEMIC SIZE
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 449 - 470
Copyright
Copyright © Applied Probability Trust 1984 

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

This paper was originally presented as a keynote address at the ORSA/TIMS Special Interest Meeting on Applied Probability in Biology and Engineering, University of Kentucky, Lexington, 18–20 July 1983.

References

Anderson, R. M., (Ed.) (1982) The Population Dynamics of Infectious Diseases. Chapman and Hall, London.Google Scholar
Anderson, R. M. and May, R. M. (1983) Vaccination against rubella and measles: quantitative investigation of different policies. J. Hyg. Camb. 90, 1–67.Google Scholar
Atkinson, A. C. (1969) The use of residuals as a concomitant variable. Biometrika 56, 33–41.CrossRefGoogle Scholar
Bailey, N. T. J. (1957) The Mathematical Theory of Epidemics. Griffin, London.Google Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases. 2nd edn of Bailey (1957).Google Scholar
Bartlett, M. S. (1938) Approximate recovery of information from replicated field experiments with large blocks. J. Agric. Sci. 28, 418–427.CrossRefGoogle Scholar
Bartlett, M. S. (1940) The present position of mathematical statistics. J. R. Statist. Soc. A 103, 1–19.Google Scholar
Bartlett, M. S. (1960) Stochastic Population Models. Methuen, London.Google Scholar
Bartlett, M. S. (1975) The Statistical Analysis of Spatial Pattern. Chapman and Hall, London.Google Scholar
Bartlett, M. S. (1978a) Introduction to Stochastic Processes, 3rd edn. Cambridge University Press.Google Scholar
Bartlett, M. S. (1978b) Nearest neighbour models in the analysis of field experiments. J. R. Statist. Soc. B 40, 140–174.Google Scholar
Bartlett, M. S. (1981a) A further note on the use of neighbouring plot values in the analysis of field experiments. J. R. Statist. Soc. B 43, 100–102.Google Scholar
Bartlett, M. S. (1981b) Population and community structure and interaction. In The Mathematical Theory of the Dynamics of Biological Populations II, ed. Hiorns, R. W. and Cooke, P. Academic Press, London.Google Scholar
Bartlett, M. S. (1982a) The development of population models. In Statistics and Probability, ed. Kallianpur, G., Krishnaiah, P. R. and Ghosh, J. K. North-Holland, Amsterdam.Google Scholar
Bartlett, M. S. (1982b) Some stochastic models in biology. Utilitas Math. 31A, 291–309.Google Scholar
Bartlett, M. S. (1983) On doubly-stochastic population processes. In Probability, Statistics and Analysis, ed. Kingman, J. F. C. and Reuter, G. E. H. Cambridge University Press.Google Scholar
Cox, D. R. (1982) Randomization and concomitant variables. In Statistics and Probability, ed. Kallianpur, G., Krishnaiah, P. R. and Ghosh, J. K. North-Holland, Amsterdam.Google Scholar
Kaplan, H. (1973) A continuous time Markov branching model with random environments. Adv. Appl. Prob. 5, 37–54.Google Scholar
Martin, R. J. (1982) Some aspects of experimental design and analysis when errors are correlated. Biometrika 69, 597–612.Google Scholar
Matern, B. (1971) Doubly stochastic Poisson processes in the plane. In Statistical Ecology, Vol. I, Pennsylvania State University Press, 195–213.Google Scholar
Papadakis, J. S. (1937) Méthode statistique pour des expériences sur champ. Bull. Inst. Amél. Plantes à Salonique No. 23.Google Scholar
Pearce, S. C. and Moore, C. S. (1976) Reduction of experimental error in perennial crops, using adjustment by neighbouring plots. Exp'l Agric. 12, 267–272.Google Scholar
Slade, N. A. and Levenson, H. (1982) Estimating population growth rates from stochastic Leslie matrices. Theoret. Popn Biol. 22, 299–308.Google Scholar
Wilkinson, G. N. and Mayo, O. (1982) Control of variability in field trials: an essay on the controversy between “Student” and Fisher, and a resolution of it. Utilitas Math. 21A, 169–188.Google Scholar
Wilkinson, G. N., Eckert, S. R., Hancock, T. W. and Mayo, O. (1983) Nearest neighbour (NN) analysis of field experiments. J. R. Statist. Soc. B 45, 151–211.Google Scholar
Williams, R. M. (1952) Experimental designs for serially correlated observations. Biometrika 39, 151–167.CrossRefGoogle Scholar