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A Limit Theorem for a Weiss Epidemic Process

Part of: Markov processes Special processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

A. V. Kalinkin  and
A. V. Mastikhin
A. V. Kalinkin*
Affiliation:
Bauman Moscow State Technical University
A. V. Mastikhin*
Affiliation:
Bauman Moscow State Technical University
*
∗ Postal address: Department of Higher Mathematics, Bauman Moscow State Technical University, 2nd Bauman St., 5, 105005 Moscow, Russia.
∗∗∗ Email address: [email protected]
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Abstract

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For a Markov two-dimensional death-process of a special class we consider the use of Fourier methods to obtain an exact solution of the Kolmogorov equations for the exponential (double) generating function of the transition probabilities. Using special functions, we obtain an integral representation for the generating function of the transition probabilities. We state the expression of the expectation and variance of the stochastic process and establish a limit theorem.

Keywords

Markov epidemic processtransition probabilitiesexponential generating functionclosed solutionbranching property

MSC classification

Primary: 60F99: None of the above, but in this section 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 247 - 257
Copyright
© Applied Probability Trust 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Hafner, New York.Google Scholar
Bartlett, M. S. (1955). An Introduction to Stochastic Processes, With Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Ditkin, V. A. and Prudnikov, A. P. (1962). Operational Calculus in Two Variables and Its Applications. Pergamon, New York.Google Scholar
Gani, J. (1965). On a partial differential equation of epidemic theory. I. Biometrika 52, 617–622.Google Scholar
Hazewinkel, M. (ed.) (1989). Epidemic process. In Encyclopaedia of Mathematics, Vol. 3, Kluwer, Dordrecht.Google Scholar
Kalinkin, A. V. (1999). Final probabilities for a branching process with interaction of particles and an epidemic process. Theory Prob. Appl. 43, 633–640.Google Scholar
Kalinkin, A. V. (2002). Markov branching processes with interaction. Russian Math. Surveys 57, 241–304.Google Scholar
Mastikhin, A. V. (2007). Final distribution for the Markov process of the Gani epidemic. Math. Notes 82, 787–797.Google Scholar
Mastikhin, A. V. (2012). Final probabilities for Becker epidemic Markov process. Theory Prob. Appl. 56, 521–527.Google Scholar
Sakino, S. (1968). On the solution of the epidemic equations. Ann. Inst. Statist. Math. Suppl. 5, 9–19.Google Scholar
Sevastyanov, B. A. (1971). Vetvyashchiesya Protsessy (Branching Processes). Nauka, Moscow. (In Russian.) (Also available in German: Sewastjanow, B. A. (1974). Verzweigungsprozesse. Akademie-Verlag, Berlin.)Google Scholar
Sevastyanov, B. A. and Kalinkin, A. V. (1982). Random branching processes with interaction of particles. Soviet Math. Dokl. 25, 644–646.Google Scholar
Siskind, V. (1965). A solution of the general stochastic epidemic. Biometrics 52, 613–616.Google Scholar
Weiss, G. H. (1965). On the spread of epidemics by carriers. Biometrics 21, 481–490.CrossRefGoogle ScholarPubMed