In a small remote neighbourhood consisting of k : 5 individuals, one individual is infected with a virus at time 0. Suppose that each pair of individuals in this neighbourhood meets at times of a Poisson process of rate 1 [independent of other meetings). Meetings between an infected individual and an uninfected individual result in the uninfected individual being infected with probability p E [Us l}. Infected individuals remain infected for an Exponential amount of time with mean a, independent of everything else. Noone interacts with anyone from outside the town. Let N: denote the munber of infected individuals at time t (so Nu : 1). For the CTMC (Ntlt'aui {a} Draw the transition diagram. {b} Find the generator matrix. {c} Find all stationary distributions. {d} Find the probability that at some time everyone in the neighbourhood is (simultaneously) infected ifp : p. : 1