Suppose that a contagious virus spreads among three individuals that are all neighbors with each other as:

Whether or not an individual is sick on day n+1 depends only on how many sick neighbors she had on day n . At the end of day n, if an individual has zero, one, or two sick neighbors, then she is sick at the beginning of day n+1 with probability p0,p1 or p2 , respectively. The condition (sick/well) of an individual does not change throughout a day.

Suppose that X=(Xn:n0) is a Markov chain for which Xn denotes the number of sick individuals during day n . Find the transition matrix of X.

(8 points. Getting each row right is worth 2 points.)