In a small remote neighbourhood consisting of l: = 5 individuals, one individual is infected with a virus at time U. 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 (U, 1}- 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 mm'iber of infected individuals at time t (so N0 = 1}. For the CTMC (\"them {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)I infected ifp = p. = 1. {e} Find the expected time until noone in the neighbourhood is infected by the virus when HZPZL {f} How would your answers to (d) and {e} above change as p. /' so or p. \\4 U