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A stochastic model for two interacting populations

Published online by Cambridge University Press:  14 July 2016

Niels G. Becker
Niels G. Becker*
Affiliation:
Cornell University and University of Sheffield
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Extract

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 3 , December 1970 , pp. 544 - 564
Copyright
Copyright © Applied Probability Trust 1970 

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References

Bailey, N. T. J. (1964) The Elements of Stochastic Processes with Applications to the Natural Sciences. Wiley, New York.Google Scholar
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Dietz, K. (1969) Carrier-borne epidemics with immigration II–Immigration of susceptibles and birth of new carriers. Unpublished.Google Scholar
Dietz, K. and Downton, F. (1968) Carrier-borne epidemics with immigration I–Immigration of both susceptibles and carriers. J. Appl. Prob. 5, 31–42.Google Scholar
Downton, F. (1967) Epidemics with carriers: A note on a paper of Dietz. J. Appl. Prob. 4, 264–270.Google Scholar
Weiss, G. H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–490.CrossRefGoogle ScholarPubMed