1. Markov chains are used as models of disease processes. A non-fatal tropical disease takes three weeks to run its course. The chance of a healthy individual contracting the disease in any week is 0.1. Two drugs are available to shorten the length of the disease. The first drug can be used only during week 1, and it cures 50 percent of the patients immediately. The second drug must be used in week 2, when the cure rate is also 50 percent.

(i) What fraction of the population is infected at any time if no drugs are available? (Hint: First find the transition matrix, then calculate the stationary distribution)

(ii) If both drugs are used when appropriate, what fraction of the population is infected? What fraction of the infected population will have the length of the disease reduced by the drugs? (Hint: First find the transition matrix, then calculate the stationary distribution)

(iii) The cost of the drug program per week for each individual in the entire population is $5. If the cost of each week of work lost through the disease is assessed at $50, will the program pay for itself?