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

References

• Anderson , W.J. 1991 . Continuous-time Markov Chains: an Applications Oriented Approach , New York : Springer .
   Google Scholar
• Bartoszynski , R. and Puri , P.S. 1983 . On two classes of interacting stochastic processes arising in cancer modeling . Adv. Appl Probab , : 695 – 712 .
   Google Scholar
• Brandt , A. 1984 . Multigrid Techniques with Applications to Fluid Dynamics , The Weizmann Institute of Science .
   Google Scholar
• Cao , W.-L. and Stewart , W.J. 1985 . Iterative aggregation/disaggregation techniques for nearly uncoupled reducible systems . J. Assoc. Comp. Mach , 32 : 702 – 719 .
   Web of Science ®Google Scholar
• Chatelin , F. and Miranker , W.L. 1984 . Aggregation/disaggregation for eigenvalue Problems . SI AM J. Numer. Anal , 21 : 567 – 582 .
   Google Scholar
• Courtois , P.J. and Semal , P. 1986 . Block iterative algorithms for stochastic matrices . Lin. Alg. Appl , 76 : 59 – 70 .
   Web of Science ®Google Scholar
• Dambrine , S. and Moreau , M. 1981 . Note on the stochastic theory of a self-catalytic chemical reaction . I, Physica , 106A : 559 – 573 . II,Physica, 106A, 574-588
   Google Scholar
• Darroch , J.N. and Seneta , E. 1965 . On quasi-stationary distributions in absorbing discrete-time finite Markov chains . J. Appl Probab , 2 : 88 – 100 .
   Google Scholar
• Darroch , J.N. and Seneta , E. 1967 . On quasi-stationary distributions in absorbing continous-time finite Markov chains . J. Appl. Probab , 4 : 192 – 196 .
   Web of Science ®Google Scholar
• Van Doorn , E.A. 1991 . Quasi-stationary distributions and convergence to quasi-stationary of birth-death processes . Adv. Appl. Probab , 23 : 683 – 700 .
   Web of Science ®Google Scholar
• Flaspohler , D.C. 1974 . Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains . Ann. Inst. Statist. Math , 26 : 351 – 356 .
   Google Scholar
• George , A. , Heath , M. , Liu , J.W.H. and Ng , E. 1988 . Sparse Cholesky factorization on a local-memory multiprocessor . SI AM J. Sei. Stat. Comp , 9 : 327 – 340 .
   Web of Science ®Google Scholar
• Gray , P. 1988 . Instabilities and oscillations in chemical reactions in closed and open systems . Proc. Roy. Soc. Lond. A , 415 : 1 – 34 .
   Google Scholar
• Gray , P. and Scott , S.K. 1984 . Autocatalytic reactions in the CST R: oscillations and instabilities in the system . Chem. Eng. Sei , 39 : 1087 – 1097 .
   Web of Science ®Google Scholar
• Haviv , M. 1987 . Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain . SI AM J. Numer. Anal , 24 : 952 – 966 .
   Web of Science ®Google Scholar
• Haviv , M. 1992 . Aggregation/disaggregation algorithm for computing the stationary distribution of a large Markov chain . Commun. Stat. -Stochastic Models , 8 : 565 – 575 .
   Google Scholar
• Holling , C.S. 1973 . Resilience and stability of ecological systems . Ann. Rev. Ecol. Sys- tematics , 4 : 1 – 23 .
   Google Scholar
• Koury , J.R. , Mcallister , D.F. and Stewart , W.J. 1984 . Iterative methods for computing stationary distributions of nearly completely decomposable Markov chains . SIAM J. Discr. Alg. Methods , 5 : 164 – 186 .
   Google Scholar
• Mccormick , S.F. 1987 . Multigrid Methods , Philadelphia, PA : SIAM .
   Google Scholar
• Meyer , C.D. 1991 . Stochastic complementation, uncoupling Markov chains and the theory of nearly reducible systems . SIAM Rev , 31 : 187 – 207 .
   Google Scholar
• Minc , H. 1988 . Nonnegative Matrices , New York : Wiley .
   Google Scholar
• Nsell , I. 1991 . On the stationary distribution of the Ross malaria model . Math. Bio-sciences , 107 : 187 – 207 .
   Google Scholar
• Pakes , A.G. 1987 . Limit theorem for the population size of a birth and death process allowing catastrophes . J. Math. Biol , 25 : 307 – 325 .
   PubMed Web of Science ®Google Scholar
• Parsons , R.W. and Pollett , P.K. 1987 . Quasistationary distributions for some autocatalytic reactions . J. Statist. Phys , 46 : 249 – 254 .
   Google Scholar
• Pollett , P.K. 1988 . On the problem of evaluating quasistationary distributions for open reaction schemes . J. Statist. Phys , 53 : 1207 – 1215 .
   Google Scholar
• Pollett , P.K. and Roberts , A.J. 1990 . A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold . Adv. Appl. Probab , 22 : 111 – 128 .
   Google Scholar
• Pollett , P.K. and Stewart , D.E. 1994 . An efficient procedure for computing quasistationary distributions of Markov chains with sparse transition structure . Adv. Appl. Probab , 26 : 68 – 79 .
   Google Scholar
• Reuter , G.E.H. Competition processes . Proc. 4th Berkley Symp. Math. Statist. Prob . Vol. 2 , pp. 421 – 430 .
   Google Scholar
• Ridler-Rowe , C.J. 1967 . On a stochastic model of an epidemic . J. Appl. Probab , 4 : 19 – 33 .
   Google Scholar
• Ross , R. 1909 . Report on the Prevention of Malaria in Mauritius , London : Churchill .
   Google Scholar
• Ross , R. 1911 . The Prevention of Malaria , London : Murray .
   Google Scholar
• Seneta , E. 1981 . Non-negative matrices and Markov chains , New York : Springer-Verlag .
   Google Scholar
• Seneta , E. and Vere-Jones , D. 1966 . On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states . J. Appl. Probab , 3 : 403 – 434 .
   Web of Science ®Google Scholar
• Stewart , D.E. and Leyk , Z. 1994 . Meschach: Matrix Computations in C, CM A, School of Mathematical Sciences , Canberra : Australian National University .
   Google Scholar
• Stewart , G.W. and Zhang , G. 1991 . On a direct method for the solution of nearly uncoupled Markov chains . Numer. Math , 59 : 1 – 11 .
   Google Scholar
• Takahashi , Y. 1975 . A lumping method for numerical calculation of stationary distributions of Markov chains . Res. Rep., Dept. of Info. Sei, Tokyo Inst. Tech , B-18
   Google Scholar
• Tweedie , R.L. 1973 . The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties . J. Appl. Probab , 10 : 84 – 99 .
   Google Scholar
• Vere-Jones , D. 1969 . Some limit theorems for evanescent processes . Austral. J. Statist , 11 : 67 – 78 .
   Google Scholar