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A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Damian Clancy  and
Philip K. Pollett
Damian Clancy*
Affiliation:
University of Liverpool
Philip K. Pollett*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK. Email address: d.clancy@liv.ac.uk
∗∗Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia
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Abstract

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distribution ν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

Keywords

Likelihood ratio orderingstochastic orderinglimiting conditional distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 821 - 825
Copyright
Copyright © Applied Probability Trust 2003 

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