Given the following info in a town where a virus spreads that turns humans into zombies.
What is going to happen eventually?
We can write the above in a following equation:
\[\left\{ \begin{array}{ll} h_t = 0.8h_{t-1} + 0.1z_{t-1}\\ z_t = 0.2h_{t-1} + 0.9z_{t-1}\\ \end{array}\right.\]
Can be organized as:
\[\begin{bmatrix} 0.8 & 0.1\\ 0.2 & 0.9 \end{bmatrix}\begin{bmatrix} h_{t-1}\\ z_{t-1} \end{bmatrix}= \begin{bmatrix} h_{t}\\ z_{t} \end{bmatrix}\]
The matrix is a Markov matrix and the eigenvalues are:
\[ \lambda = 0.7, 1 \]
Can calulate the eigenvectors are as eigenvalues are determined.
\[ v_1 = \begin{bmatrix} 0.5\\ 1 \end{bmatrix} \space v_2 = \begin{bmatrix} 1\\ -1 \end{bmatrix} \]
From the findings above, if there are 1500 people in town and if human = 500 and zombies = 1000: \[\begin{bmatrix} 0.8 & 0.1\\ 0.2 & 0.9 \end{bmatrix}\begin{bmatrix} 500\\ 1000 \end{bmatrix}= \begin{bmatrix} 500\\1000 \end{bmatrix}\] No matter how much progresses, the number of humans and zombies will remain the same.