# William J. Stewart

## A General Two Dimensional Markov Chain Model.

A completely general 2D model may be specified. The particular example chosen for the collection has been taken from:
An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains with Sparse Transition Structure. P.K. Pollett and D.E. Stewart, Advances in Applied Probability, Vol. 26, pp. 68-79, 1994.
See also: On a Stochastic Model of an Epidemic. C.J. Ridler-Rowe, Advances in Applied Probability, Vol. 4, pp. 19-33, 1967.

In the first dimension, the state variable assumes all values from 0 through Nx; in the second dimension the state variable takes on value from 0 through Ny. The state space is then as follows:

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This two dimensional Markov chain model allows for transitions from any nonboundary state to adjacent states in the North, South, East, West, North-East, North-West, South-East and South-West directions. The rates (or probabilities) of transition are set in the subroutine instant of the TwoD.f source code file.

In the 2D epidemic model specified in this collection, only transitions to the South, East and North-West are permitted. From any nonboundary state (u,v) , transitions to the South are assigned the value v ; transitions to the East are assigned the value 2025.0; and transitions to the North-West are assigned the value u.

In the data set corresponding to this model, TwoD_in, the values of Nx and Ny are as shown in the table below, along with the size of the matrix generated and the number of nonzeros in the matrix.

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Values of Nx, Ny, n and nz for the 10 datasets:

Nx      Ny           n            nz

10      10         121           441
64      16       1,105         4,257
64      64       4,225        16,641
100     100      10,201        40,401
256      64      16,705        66,177
256     256      66,049       263,169
512      64      33,345       132,225
512     128      66,177       263,425
512     256     131,841       525,825
512     512     263,169     1,050,625

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