3. We model here the evolution of an epidemic spreading among a group of 6 individuals. We shall use a compartmental model assuming rst that each individual can be in three possible states: they can be susceptible (of getting the epidemic). or infected. or removed, that is. not anymore in contact with the others. The epidemic is considered as over when there is no more infected individual left. The number of survivors is the number of sus- ceptibles left after the end of the epidemic. We make the following assumptions. 1- Any pair of individuals of the {non removed} population meet in average every three days. 2- When an infected individual meet a susceptible. the probability the susceptible gets the epidemic is g. 3- Each infected individual is removed from the population in average after 14 days. a) Give a continuous-time Markov chain modeling the situation specifying its Q-matrix. [4 marl: b) Give the Forward-Kolmogorov equations for this Markov chain. [5 mark c) Let S , and I, be respectively the number of susceptible and infected individuals. Show\" [9 mark that for any initial condition, for all r 2 l} d d IE[I.] = Elana] - nuts and IE[S.] = -E[E[S:L]- off n d: n for some parameters ,3, y and n to determine. d) Assume that initially there are 2 infected and 2 susceptible individuals. {i} What is the probability that there is one survivor after the epidemic? [4 marl: {ii} Given that two individuals survive. how long will it take in expectation before the epidemic is over? [5 marl: e) Let us modify the assumptions 1-2-3 in the following way. Infected individuals can be symptomatic or asymptomatic. 1- Any pair symptomatic-susceptible meets in average every 10 days. whereas any pair asymptomatic-susceptible meets in average every 3 days. 2- When a susceptible gets infected. there is a probability i that hefshe gets asympto