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December 2012 Approximating quasistationary distributions of birth--death processes
Damian Clancy
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J. Appl. Probab. 49(4): 1036-1051 (December 2012). DOI: 10.1239/jap/1354716656


For a sequence of finite state space birth--death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth--death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl--Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.


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Damian Clancy. "Approximating quasistationary distributions of birth--death processes." J. Appl. Probab. 49 (4) 1036 - 1051, December 2012.


Published: December 2012
First available in Project Euclid: 5 December 2012

zbMATH: 1275.60061
MathSciNet: MR3058987
Digital Object Identifier: 10.1239/jap/1354716656

Primary: 60J28
Secondary: 60J80 , 92D25 , 92D30

Keywords: cumulant closure , logistic population growth , Moment closure , SIS epidemic model

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 4 • December 2012
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