Question

We are interested in the movement of patients within a hospital. For purposes of our analysis, we consider the hospital to have three different types of rooms: general care, special care, and intensive care. Based on past data, 60% of arriving patients are initially admitted into the general care category, 30% in the special care category, and 10% in intensive care. A “general care” patient has a 55% chance of being released healthy the following day, a 30% chance of remaining in the general care room, and a 15% chance of being moved to the special care facility. A "special care” patient has a 10% chance of being released the following day, a 20% chance of being moved to general care, a 10% chance of being upgraded to intensive care, and a 5% chance of dying during the day. An "intensive care" patient is never released from the hospital directly from the intensive care unit (ICU), but is always moved to another facility first. The probabilities that the patient is moved to general care, special care, or remains in intensive care are 5%, 30%, or 55%, respectively. Let X be a Markov chain where Xo is the type of room that an admitted patient initially uses, and Xn is the room category of that patient at the end of day n. transition probability

Answer 1

a) Answer: ---- Find the Date: --- 27/9/2019 matrix for X i 0.55 0.1 0 0 0 0 0.3 0.15 0.2 0.55 0.05 0.3 0 0 0 0 0.1 0.55 0 0 ] 0 0.05 0.1 1 Where the treats are {Being released, General care, Special care, Intensive care, Dead) b) Either you can calculate the matrix to a high power, say mº or solve the eigenvalue problem. Let the matrix containing all eigenvectors of M and V be the diagonal matrix of eigenvectors. Now define V by setting all eigenvalues in V equal to zero except those equal to 1 V =" when n approaches infinity (1.0000 0 0 0 0 ] 0.9542 0 0 0 0.0458 U XVU-= 0.7864 0 0 0 0.2136 0.6303 0 0 0 0.3697 LO 000 1.0000 Therefore, 63.03% is the probability of leaving healthy if the patient was admitted in intensive care. The probability of the patient stay in ICU for 1 day is 0.55, 2 days 0.552.... So the expected value of number of days is, En(0.55) * = (1-0.55) 0.55 0.2055 2.676