Assignment Task
 

Question
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 ? (0, 1). Infected individuals remain infected for an Exponential amount of time
with mean µ, independent of everything else. Noone interacts with anyone from outside the
town. Let Nt denote the number of infected individuals at time t (so N0 = 1). For the CTMC
(Nt)t?0:
(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 if p = µ = 1.
(e) Find the expected time until noone in the neighbourhood is infected by the virus when
µ = p = 1.
(f) How would your answers to (d) and (e) above change as µ % ? or µ & 0?
(g) If µ = p = 1, what happens to the answers to (d) and (e) as k (the number of individuals
in the neighbourhood) goes to infinity? (Note that for (d), the state k that you are trying
to reach is growing as k ? ?, but so is e.g. p1

    


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  • Uploaded By : Roman
  • Posted on : October 16th, 2019

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