Question

Currently, among the 20 individuals of a population, 2 have a certain infection that spreads as follows: Contacts between two members of the population occur in accordance with a Poisson process having rate ?. When a contact occurs, it is equally likely to involve any of the possible pairs of individuals in the population. If a contact involves an infected and a non-infected individual, then, with probability p the non-infected individual becomes infected. Once infected, an individual remains infected throughout. Let ?(?) denote the number of infected members of the population 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 an infected person contacts a non-infected person?

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

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

(e) Compute the expected time until all members of the considered population are infected.

Answer 1

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(a) The state space X was derived as the number of infected persons in the population

The state space diagram and the explanation is provided below

(b) The probability that an infected person "contacts" a non-infected person was calculated as:

(c) The rate at which an infected person contacts a non-infected person was calculated as:

The derivation is given below in Page 2 and Page 3:

(d) The inter-contact time between two I-N contacts was derived to be exponentially distributed
with

  ,
  

(e) The Expected time untill all members of the considered population are infected was derived as:

The explanation is given below: