# MARCA_Models:

TwoD --- A Two Dimensional Markov Chain.

# 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:

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.

***********************************************
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
***********************************************