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The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

Published online by Cambridge University Press:  14 July 2016

J. Radcliffe
J. Radcliffe*
Affiliation:
Queen Mary College, London
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Abstract

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Let α be its Malthusian parameter, ß its velocity of propagation, Z(A, t) the number of individuals in the set A at time t, and A√(ßt) = [√(ßt) r: r ∈ A]. The mean square convergence of the random variable W(A, t)= e–αtZ(A√(ßt), t) to a limit variable W(A) is established.

Keywords

MEAN-SQUARE CONVERGENCEPOSITION-DEPENDENT BRANCHING PROCESSMEASLES EPIDEMIC
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 338 - 344
Copyright
Copyright © Applied Probability Trust 1976 

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References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Bartlett, M. S. (1954) Processus stochastiques ponctuels. Ann. Inst. H. Poincaré 14, 35–60.Google Scholar
Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 81–109.Google Scholar
Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Wiley, New York; Methuen, London.Google Scholar
Davies, A. W. (1965) On the theory of birth, death and diffusion processes. J. Appl. Prob. 2, 293–322.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta. Math., Stockh. 108, 1–31.Google Scholar
Radcliffe, J. (1973) The initial geographical spread of host-vector and carrier-borne epidemics. J. Appl. Prob. 10, 703–717.Google Scholar
Radcliffe, J. (1974) The effect of the length of incubation period on the velocity of propagation of an epidemic wave. Math. Biosci. 19, 257–262.Google Scholar
Srinivasan, S. K. (1969) Stochastic Theory and Cascade Processes. American Elsevier, New York.Google Scholar
Watanabe, S. (1965) On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto University 4, 385–398.Google Scholar