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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 requires delicate mathematical analysis. 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 a variant of Arnoldi's algorithm and it 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 analytical and computational 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.
AMS 1991 Subject Classification: 60J27; 65F15.
Keywords: Stochastic modelling; Markov chains; quasistationary distributions; Arnoldi Algorithm.
Acknowledgement: This worked was funded by the Australian Research Council.
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Last modified: 26th December 1995