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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: [email protected]
∗∗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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References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Cavender, J. (1978). Quasistationary distributions of birth–death processes. Adv. Appl. Prob. 10, 570–586.CrossRefGoogle Scholar
Darroch, J., and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite {M}arkov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Ferrari, P., Kesten, H., Martínez, S., and Picco, P. (1995). Existence of quasi-stationary distributions. {A} renewal dynamic approach. Ann. Prob. 23, 501–521.CrossRefGoogle Scholar
Keilson, J., and Ramaswamy, R. (1984). Convergence of quasistationary distributions in birth–death processes. Stoch. Process. Appl. 18, 301–312.Google Scholar
Kijima, M., and Seneta, E. (1991). Some results for quasistationary distributions of birth–death processes. J. Appl. Prob. 28, 503–511.Google Scholar
Kryscio, R. and Lefèvre, C. (1989). On the extinction of the {S-I-S} stochastic logistic epidemic. J. Appl. Prob. 26, 685–694.Google Scholar
Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic {SIS} model. Adv. Appl. Prob. 28, 895–932.CrossRefGoogle Scholar
Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40.Google Scholar
Nåsell, I. (2001). Extinction and quasi-stationarity in the Verhulst logistic model. J. Theoret. Biol. 211, 11–27.Google Scholar
Van Doorn, E. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar