Currently, among the 20 individuals of a population, 2 have acertain infection that spreads as follows: Contacts between twomembers of the population occur in accordance with a Poissonprocess having rate ?. When a contact occurs, it is equally likelyto involve any of the possible pairs of individuals in thepopulation. If a contact involves an infected and a non-infectedindividual, then, with probability p the non-infected individualbecomes infected. Once infected, an individual remains infectedthroughout. Let ?(?) denote the number of infected members of thepopulation at time t. Considering the current time as t = 0, wewant to model this process as a continuous-time Markov chain.
(a) What is the state space of this process?
(b) What is the probability that an infected person contacts anon-infected person?
(c) What is the rate at which an infected person contacts anon-infected person (we denoted this type of contact by I-Ncontact) when there are X infected people in the population?
(d) Is the inter-contact time between two I-N contactsexponentially distributed? Why?
(e) Compute the expected time until all members of theconsidered population are infected.
