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Exercise 5.3.6

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Let $x$ , $y$ , $z$ , and $w$ denote the percentage of the healthy, ambulatory, bedridden, and dead patients. And we know that

(\begin{matrix} x \\ y \\ z \\ w \end{matrix}) = (\begin{matrix} 0 \\ 0.3 \\ 0.7 \\ 0 \end{matrix})

and the percentage of each type would be described by the vector

(\begin{matrix} 1 & 0.6 & 0.1 & 0 \\ 0 & 0.2 & 0.2 & 0 \\ 0 & 0.2 & 0.5 & 0 \\ 0 & 0 & 0.2 & 1 \end{matrix}) (\begin{matrix} 0 \\ 0.3 \\ 0.7 \\ 0 \end{matrix}) = (\begin{matrix} 0.25 \\ 0.2 \\ 0.41 \\ 0.14 \end{matrix})

in the same order. So this answer the first question. For the second question we should calculate the limit of power of that transition matrix, say $A$ . It would be

L = \lim_{m \to \infty} A^{m} = (\begin{matrix} 1 & \frac{8}{9} & \frac{5}{9} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \frac{1}{9} & \frac{4}{9} & 1 \end{matrix}) .

So the limit of percentage of each type would be described by the vector

L (\begin{matrix} 0 \\ 0.3 \\ 0.7 \\ 0 \end{matrix}) = (\begin{matrix} \frac{59}{90} \\ 0 \\ 0 \\ \frac{31}{90} \end{matrix}) .

jephianlin

2011-06-27 00:00

Exercise 5.3.6

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