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May 8th, 2017, 04:32 PM   #1
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Markov Chain Discrete Dynamical System

Hey guys, so my teacher told us to look at the book and find a very similar problem and to just do it. I don't typically ask for the answer but I just need help setting up this problem.

A drug is administered @ 100 units of the drug into the blood.
Every 10 minutes, 50% of the drug goes to bloodstream & 50% to liver
During this same time period 75% of the drug in the liver goes to the bloodstream while 25% stays in the liver.
Initially injection gets process started 100 units of the drug directly to blood stream(0 to liver). How to model this system given
x sub k = amount in blood stream after 10 min time interval.
y sub k = amount of drug in the liver after k 10 min time intervals have passed.

Using matrix algebra: X(k+1) = A * X(k)[X is a vector] where X(k)= Matrix x sub k over y sub k. Then there is another matrix showing A = (2x2) matrix top 2 values are 1/2 & 3/4. Bottom 2 values are 1/2 and 1/4. Can any one tell me what the hell is going on? I think I can use the markov chain or the matrix algebra. I'm super lost and a step by step is appreciated.
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May 8th, 2017, 04:56 PM   #2
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if I understand this what you have is

$\begin{bmatrix}b_{n+1} \\l_{n+1}\end{bmatrix}=\begin{bmatrix}0.5 &0.75\\0.5 &0.25\end{bmatrix}\begin{bmatrix}b_n \\ l_n\end{bmatrix}$

$\begin{bmatrix}b_0 \\ l_0\end{bmatrix}=\begin{bmatrix}100 \\ 0\end{bmatrix}$
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May 8th, 2017, 05:35 PM   #3
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Quote:
Originally Posted by romsek View Post
if I understand this what you have is

$\begin{bmatrix}b_{n+1} \\l_{n+1}\end{bmatrix}=\begin{bmatrix}0.5 &0.75\\0.5 &0.25\end{bmatrix}\begin{bmatrix}b_n \\ l_n\end{bmatrix}$

$\begin{bmatrix}b_0 \\ l_0\end{bmatrix}=\begin{bmatrix}100 \\ 0\end{bmatrix}$

That is correct. It's a bit confusing saying find all the eigen values & bases for all eigenspaces of the matrix A. I get eigen-values and all. But it's confusing in the form of a word problem. I think you help me set it up tho.
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