At the beginning of every time period, each of N individuals is in one of three possible conditions: infectious, infected but not infectious, or noninfected. If a noninfected individual becomes infected during a time period then he or she will be in an infectious condition during the following time period, and from then on will be in an infected (but not infectious) condition. During every time period each of the pairs of individuals are independently in contact with probability p. If a pair is in contact and one of the members of the pair is infectious and the other is noninfected then the noninfected person becomes infected (and is thus in the infectious condition at the beginning of the next period). Let X and Y, denote the number of infectious and the number of noninfected individuals, respectively, at the beginning of time period n.

(a) If there are i infectious individuals at the beginning of a time period, what is the probability that a specified noninfected individual will become infected in that period?

(b) Is {X,, n 0} a Markov chain? If so, give its transition probabilities.

(c) Is {Y,, n 0} a Markov chain? If so, give its transition probabilities.

(d) Is {(X, Y), n 0} a Markov chain? If so, give its transition probabil- ities.