With a town of 20 people, 2 have a certain disease that spreads as follows: Contacts...

With a town of 20 people, 2 have a certain disease that spreads as follows: Contacts between two members of the town occurred in accordance with a Poisson process having rate ?. When contact occurs, it is equally likely to involve any of the possible pairs of people in the town. If a diseased and non-diseased person interect, then, with probability p the non-diseased person becomes diseased. Once infected, a person remains infected throughout. Let ?(?) denote the number of diseased people of the town at time t. Considering the current time as t = 0, we want 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 a diseased person contacts a non-diseased person?

(c) What is the rate at which a diseased person contacts a non-diseased person (we denoted this type of contact by I-N contact) when there are X diseased people in the town?

(d) Is the inter-contact time between two I-N contacts exponentially distributed? Why?

(e) Compute the expected time until all people of the town are infected by the disease.

Expert Solution

(a) X(t) ia a continuous time Markov chain because the individual have contact involes only infected and an noninfected.

This a pure birth process, when a person is contract for infected individual, this person is also infected. So, the individual is increase continuously.

rest parts

orchestra answered 2 years ago

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