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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: [email protected]
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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 

References

Ball, F. G. and Barbour, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479–490.Google Scholar
Barbour, A. D. and Pollett, P. K. (2010). Total variation approximation for quasi-stationary distributions. J. Appl. Prob. 47, 934–946.Google Scholar
Clancy, D. and Mendy, S. T. (2011). Approximating the quasi-stationary distribution of the SIS model for endemic infection. Methodology Comput. Appl. Prob. 13, 603–618.Google Scholar
Clancy, D. and Pollett, P. K. (2003). A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic. J. Appl. Prob. 40, 821–825.Google Scholar
Daniels, H. E. (1967). The distribution of the total size of an epidemic. In Proc. 5th Berkeley Symp. Math. Statist. Prob., Vol. 4, University of California Press, Berkeley, CA, pp. 281–293.Google Scholar
Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasistationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.Google Scholar
Keilson, J. and Ramaswamy, R. (1984). Convergence of quasi-stationary distributions in birth-death processes. Stoch. Process. Appl. 18, 301–312.Google Scholar
Kryscio, R. J. and Lefèvre, C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 27, 685–694.Google Scholar
Lefèvre, C. and Utev, S. (1995). Poisson approximation for the final state of a generalized epidemic process. Ann. Prob. 23, 1139–1162.Google Scholar
Lefèvre, C. and Utev, S. (1997). Mixed Poisson approximation in the collective epidemic model. Stoch. Process. Appl. 69, 217–246.Google Scholar
Matis, J. H. and Kiffe, T. R. (1996). On approximating the moments of the equilibrium distribution of a stochastic logistic model. Biometrics 52, 980–991.Google Scholar
Nåsell, I. (2001). Extinction and quasi-stationarity in the Verhulst logistic model. J. Theoret. Biol. 211, 11–27.Google Scholar
Nåsell, I. (2003). An extension of the moment closure method. Theoret. Pop. Biol. 64, 233–239.Google Scholar
Nåsell, I. (2007). Extinction and quasi-stationarity in the Verhulst logistic model: with derivations of mathematical results. Available at http://www.math.kth.se/∼ingemar/forsk/verhulst/ver10.pdf.Google Scholar
Nåsell, I. (2011). Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model. Springer, Heidelberg.Google Scholar
Nåsell, I. (2012). Extinction and quasi-stationarity in the stochastic logistic SIS model: a Maple module and corrections. Available at http://www.math.kth.se/∼ingemar/SIS/SIS.html.Google Scholar
Norden, R. H. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687–708.Google Scholar
Ovaskainen, O. (2001). The quasistationary distribution of the stochastic logistic model. J. Appl. Prob. 38, 898–907.Google Scholar
Pollett, P. K. and Vassallo, A. (1992). Diffusion approximations for some simple chemical reaction schemes. Adv. Appl. Prob. 24, 875–893.Google Scholar
Weiss, G. H. and Dishon, M. (1971). On the asymptotic behaviour of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.Google Scholar