Quasistationary distributions for continuous-time Markov chains

Phil Pollett

Abstract: There are many stochastic systems arising in areas as diverse as wildlife management, chemical kinetics and reliability theory, which eventually ``die out'', yet appear to be stationary over any reasonable time scale. The notion of a quasistationary distribution has proved to be a potent tool in modelling this behaviour. In finite-state systems the existence of a quasistationary distribution is guaranteed. However, in the infinite-state case this may not always be so, and the question of whether or not quasistationary distributions exist is delicate.

The purpose of this paper is twofold: to present simple conditions for the existence of quasistationary distributions for continuous-time Markov chains, and, to describe an efficient computational procedure for evaluating them. The computational method I shall describe is particularly suited to problems where the transition-rate matrix is both large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. The methods will be illustrated with reference to a variety of examples, including birth-death processes, the birth-death and catastrophe process, and an epidemic model for which I shall compare the computed quasistationary distribution with an appropriate diffusion approximation.

Acknowledgement: This worked was funded by the Australian Research Council.

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Last modified: 1st March 1996