Optimizing Prevention Strategies for Sneezles and Analyzing a Markov Model

Question 1

The King has appointed you responsible for deciding how to care for the 1,000 children in the realm who are at high risk of sneezles, a highly contagious infectious disease whose symptoms typically include fever, cough, runny nose, and opinions. You may assume that each of these children faces the same costs and risks. Your objective is to allocate a limited budget to minimize the expected number of sneezles cases in this population. You have three different prevention techniques at your disposal: vaccination, masking, and vitamin supplementation. These three techniques can be combined into a total of eight different

a. Suppose that The King has indicated a willingness to pay up to $500 to prevent a case of sneezles. What is the optimal strategy? If you implement this strategy across the 1000 children, what will be the total cost and how many cases of sneezles do you expect to observe? (20 points) Hint: Read the question carefully. The King is willing to pay up to $500 to prevent a case of sneezles.” That’s how much he values each case that can be averted. Not per child, not per population. It’s “per case of sneezles averted.”

b. How does your answer to part (a) change if The King is willing to pay up to $750 to prevent a case of sneezles? Be sure to report the optimal strategy, what it will cost, and how many cases of sneezles it will produce. (5 points)

c. How does your answer to part (a) change if The King is willing to pay up to $1,000 to prevent a case of sneezles? Again, what is the optimal strategy, what will it cost, and how many cases of sneezles will it produce? (5 points)

d. Suppose instead that The King has given you a budget of $10,000 and told you to allocate it across these prevention strategies in a manner that will minimize the expected number of sneezles cases among the 1,000 children. What is the optimal allocation and how many cases of sneezles do you expect to observe if you implement it? (15 points)

e. Suppose that, in addition to giving you a budget of $10,000, The King has also declared that every single child must receive exactly the same strategy. How does this change your answer to part (d)? Be sure to report both the optimal allocation and how many cases of sneezles you expect to observe if you implement it. (5 points).

Question 2

a. Write down the transition probability matrix for this Markov model. (5 points)

b. In the long run, what percentage of customers will be in each of the five categories?(15 points) An answer to the nearest percentage is accurate enough.

c. How does your answer to part (b) change if the annual probability of one or more accidental injuries falls to 75%? What is your answer to part (b) if the annual probability of one or more accidental injuries falls to 50%? (5 points) Hint: Your intuition should help you to confirm that you have obtained the correct answer to this question

d. Set the annual probability of one or more accidental injuries back to 90%. What is the long-run average annual premium the company should expect to receive from each of its customers? (5 points)