Exercise 1. You have been hired as a consultant by the human resources department of a meat processing company. Human resources is trying to persuade management to introduce new ways to help essential workers who are exposed to a novel coronavirus. It needs your help in making its case. Model current trends with a Markov process. Predict what will happen if the trends continue, that is, if no changes are introduced. After analyzing statistics, you find that the current trends indicate the following. Of the essential workers that are well, 96% remain well after a 14-day period has passed, and 4% become infected with the novel coronavirus. Of the essential workers that are infected, 35% produce antibodies that protect them after a 14-day period, 50% remain in an infected state, and 15% are put into ventilator-equipped facilities. Of the essential workers in ventilator-equipped facilities, 10% return to an (ordinary) infected state (leaving the facility) after a 14-day period, 72% remain in a ventilatorequipped facility, and 18% die. Of the essential workers that produce antibodies that protect them, 100% continue to produce antibodies after a 14-day period. Of the essential workers that are dead, 100% remain dead after a 14-day period. (a) Draw a transition diagram representing the Markov process with the probabilities given by the current trends. Use W, I, V, A, and D to represent the states of being well, infected, in a ventilator-equipped facility, producing antibodies, and dead, respectively. (b) Write the standard form of the transition matrix for this Markov process. With two absorbing states, it should have I2 in the upper left corner. (c) Write down the 23 matrix S that represents the portion of the matrix from (b) which is to the right of I2. (d) Write down the 33 matrix R that represents the portion of the matrix from (b) which is below S. (e) Write down the matrix I3 R, where R is your matrix from part (d). (f) Compute the fundamental matrix F = (I3 R) 1 by calculating the inverse of I3 R with the Gauss-Jordan method. Do not round. (g) Compute the stable matrix for this Markov process. Recall that the 2 3 portion of the stable matrix to the right of the I2 in the upper left will be the product S(I3 R) 1 . (h) Human resources estimates that among its essential workers in South Dakota, 1,000 are well, 125 are infected but not in ventilator-equipped facilities, and 25 are in ventilator-equipped facilities. According to your stable matrix from part (g), if current trends continue, how many of these 1,150 people will eventually produce antibodies? How many will eventually die? (i) Now management has a question. Using the fundamental matrix F from part (f), find how many expected 14-day periods before an essential worker who starts in the state of being well ends up either dead or producing antibodies. How many days is this, rounded to the nearest whole number of days? Recall that the sum of the entries of the jth column of the stable matrix F is the expected number of time periods before the process enters an absorbing state, when the process begins in the state corresponding to column j. Exercise 2 (Optional). We have now completed about half of our course. If you wish, you may provide feedback to your instructor on whichever aspects of this class that you want, such as the syllabus, homeworks, in-class activities, problem sets, tests, lectures, and office hours. What works well for you? What could be improved?