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Approximating Quasistationary Distributions of Birth–Death Processes

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Damian Clancy
Damian Clancy*
Affiliation:
University of Liverpool
*
∗ Postal address: Department of Mathematical Sciences, University of Liverpool, L69 7ZL, UK. Email address: d.clancy@liv.ac.uk
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Abstract

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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.

Keywords

Moment closurecumulant closureSIS epidemic modellogistic population growth

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D25: Population dynamics (general) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1036 - 1051
Copyright
© Applied Probability Trust 

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