An iterative aggregation/disaggregation procedure for modelling the long-term behaviour of continuous-time evanescent random processes

We describe a multilevel aggregation/disaggregation method for finding the quasistationary distribution and decay parameters for continuous-time Markov chains. This complements existing techniques dealing with the discrete-time case, or for finding the stationary distribution. Finding the quasistationary distribution is equivalent to calculating the eigenvector corresponding to the smallest eigenvalue of the q-matrix restricted to the non-absorbing class. The method presented here is similar to an algebraic multigrid, with restriction operators that depend on the current approximation to the solution. The smoothers are short Arnoldi iterations or Gauss-Seidel iterations. Numerical results are presented for a variety of models of differing character, including simple epidemic, bivariate SIS, predator-prey, and the cubic auto-catalator. These indicate that the number of cycles required grows only very slowly with the size of the problem.

[1]  D. E. Stewart,et al.  An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains by Sparse Transition Structure , 1994, Advances in Applied Probability.

[2]  Moshe Haviv,et al.  An aggregation/disaggregation algorithm for computing the stationary distribution of a large markov chain , 1992 .

[3]  I. Nåsell.  On the quasi-stationary distribution of the Ross malaria model. , 1991, Mathematical biosciences.

[4]  G. Stewart,et al.  On a direct method for the solution of nearly uncoupled Markov chains , 1991 .

[5]  W. J. Anderson.  Continuous-Time Markov Chains: An Applications-Oriented Approach , 1991 .

[6]  W. J. Anderson.  Continuous-Time Markov Chains , 1991 .

[7]  E. V. Doorn,et al.  Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes , 1991 .

[8]  P. K. Pollett,et al.  A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold , 1990, Advances in Applied Probability.

[9]  Carl D. Meyer,et al.  Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems , 1989, SIAM Rev..

[10]  R. Ross,et al.  Prevention of malaria. , 2012, BMJ.

[11]  Philip K. Pollett,et al.  On the problem of evaluating quasistationary distributions for open reaction schemes , 1988 .

[12]  Michael T. Heath,et al.  Sparse Cholesky factorization on a local-memory multiprocessor , 1988 .

[13]  Review Lecture - Instabilities and oscillations in chemical reactions in closed and open systems , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[14]  M. Haviv.  Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain , 1987 .

[15]  A. Pakes.  Limit theorems for the population size of a birth and death process allowing catastrophes , 1987, Journal of mathematical biology.

[16]  P. Pollett,et al.  Quasistationary distributions for autocatalytic reactions , 1987 .

[17]  P. Courtois,et al.  Block iterative algorithms for stochastic matrices , 1986 .

[18]  William J. Stewart,et al.  Iterative aggregation/disaggregation techniques for nearly uncoupled markov chains , 1985, JACM.

[19]  W. Stewart,et al.  ITERATIVE METHODS FOR COMPUTING STATIONARY DISTRIBUTIONS OF NEARLY COMPLETELY DECOMPOSABLE MARKOV CHAINS , 1984 .

[20]  Françoise Chatelin,et al.  Aggregation/Disaggregation for Eigenvalue Problems , 1984 .

[21]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C , 1984 .

[22]  R. Bartoszynski,et al.  On two classes of interacting stochastic processes arising in cancer modeling , 1983, Advances in Applied Probability.

[23]  David C. Flaspohler.  Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains , 1974 .

[24]  C. S. Holling.  Resilience and Stability of Ecological Systems , 1973 .

[25]  R. Tweedie.  The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties , 1973, Journal of Applied Probability.

[26]  David Vere-Jones.  SOME LIMIT THEOREMS FOR EVANESCENT PROCESSES , 1969 .

[27]  On a stochastic model of an epidemic , 1967 .

[28]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

[29]  E. Seneta,et al.  On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states , 1966, Journal of Applied Probability.

[30]  E. Seneta,et al.  On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains , 1965, Journal of Applied Probability.