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An application of the Jordan canonical form to the epidemic problem

Published online by Cambridge University Press:  14 July 2016

Linda P. Gilbert  and
A. M. Johnson
Linda P. Gilbert*
Affiliation:
Louisiana Tech University
A. M. Johnson*
Affiliation:
University of Arkansas at Little Rock
*
∗Postal address: Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, U.S.A.
∗∗Postal address: Department of Mathematics and Computer Science, University of Arkansas at Little Rock, 33rd and University, Little Rock, AR 72204, U.S.A.
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Abstract

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

Keywords

SIMPLE EPIDEMICGENERAL EPIDEMICRIGHT-SHIFT PROCESSJORDAN CANONICAL FORM
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 313 - 323
Copyright
Copyright © Applied Probability Trust 1980 

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References

Bailey, N. T. J. (1950) A simple stochastic epidemic. Biometrika 37, 193–202.CrossRefGoogle ScholarPubMed
Bailey, N. T. J. (1964) The Elements of Stochastic Processes. Wiley, New York.Google Scholar
Birkhoff, G. and Maclane, S. (1953) A Brief Survey of Modern Algebra. MacMillan, New York.Google Scholar
Gani, J. (1965) On a partial differential equation of epidemic theory. I. Biometrika 52, 617–622.CrossRefGoogle ScholarPubMed
Gilbert, J. D. (1970) Elements of Linear Algebra. International Textbook Company, Scranton, Pennsylvania.Google Scholar
Kryscio, R. J. and Severo, N. C. (1975) Computational and estimation procedures in multidimensional right-shift processes and some applications. Adv. Appl. Prob. 7, 349–382.CrossRefGoogle Scholar
Murdoch, G. H. (1977) A note on Bailey's analysis of the simple epidemic. Biometrical J. 19, 9–22.CrossRefGoogle Scholar
Severo, N. C. (1969a) A recursion theorem of solving differential-difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.CrossRefGoogle Scholar
Severo, N. C. (1969b) Right-shift processes. Proc. Nat. Acad. Sci. U.S.A. 64, 1162–1164.Google Scholar
Siskind, V. (1965) A solution of the general stochastic epidemic. Biometrika 52, 613–616.Google Scholar