Suppose that there is a contagious disease. An individual can be in one of three possiblestates: susceptible to the disease; infected; or immune. An individual who is susceptiblehas a 10% chance of becoming infected by next week; otherwise they remain susceptible tothe disease. If an individual has become infected, then they recover a week later. Whenthe individual recovers, they recover with permanent immunity with probability12; otherwisethey recover and return to the susceptible class, where they can become infected again. Oncean individual is immune, they stay immune for life.

(a) Model this using an absorbing Markov chain with three states (susceptible, infected, andimmune) and give the transition matrix in canonical form.

(b) Calculate the fundamental matrix (IQ)1for the chain.

(c) Use your answer from part (b) to calculate the expected number of weeks until anindividual becomes immune to the disease, given that they start in the susceptible class.

(d) Use your answer from part (b) to calculate the expected number of times a susceptibleindividual becomes infected before they become immune to the disease.

(e) Do you think this is a suitable model for a disease like COVID-19? Why or why not?