There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process having rate λ. When a contact occurs, it is equally likely to involve any of the  N 2

pairs of individuals in the population. If a contact involves an infected and a noninfected individual, then with probability p the noninfected individual becomes infected. Once infected, an individual remains infected throughout. Let X(t) denote the number of infected members of the population at time t.

(a) Is {X(t), t 0} a continuous-time Markov chain?

(b) Specify its type.

(c) Starting with a single infected individual, what is the expected time until all members are infected?