ISE 505: Modeling for Health Policy and Medical Decision Making Homework 1 Due Feb 16, 2021 1. Decision trees and Markov models: the city of Waterville was subject to a freak water poisoning event...

Homework assignment on Markov Chains and Decision Trees.


ISE 505: Modeling for Health Policy and Medical Decision Making Homework 1 Due Feb 16, 2021 1. Decision trees and Markov models: the city of Waterville was subject to a freak water poisoning event where 1,000 healthy individuals were exposed to chemicals that can make some people sick or very sick over the course of the next five years (see Markov process below). However, the city has additional health policies it can implement. In policy 1, the probability of progression from healthy to sick can be reduced by half if exposed healthy individuals are monitored annually (annual monitoring cost = $5000 per person in addition to usual costs). The probability of death for a healthy individual remains at 0.1. Another option is to immediately provide an antidote to all individuals at a one-time cost of $0.20 per person. However, 20% of those who take the antidote will develop an allergic reaction and die immediately; the other 80% will remain healthy with no risk of ever becoming sick (but they will still die with annual probability of 0.1, and incur costs associated with being healthy). The city asks you which policy is cost-effective provided they only care about the next five years: do nothing, provide the antidote to all exposed individuals, or monitor exposed healthy individuals annually. a. Draw the decision tree and Markov model associated with this decision, where the “do nothing” Markov model is shown above. b. Implement the model in a software of your choice, and evaluate costs and QALYs associated with each policy. Assume no half-cycle correction is necessary and discount at 3%. Which policy would be preferred if the goal is to minimize costs? What is the cost? c. Plot the policies on the cost-QALY plane. Label the ICERs along the cost-effectiveness frontier. d. Which policies are cost-effective if the societal willingness-to-pay is $50,000/QALY gained? What if the WTP is $100,000/QALY gained? 2. Life tables can provide statistics around mortality. These tables are compiled by various health agencies. For instance, a US life table is available here: https://www.cdc.gov/nchs/data/nvsr/nvsr64/nvsr64_11.pdf In class we examined a healthy-sick-super sick-dead Markov model (see model in Question 1). We can run that model to find the proportion of the population alive every year (assuming a cycle length of one year). These have been provided to you in markovLifeTable_HW1.xlsx (Sheet “Life Table”) on Blackboard. Fill out the missing cells of the life table. (Some cells have been filled out for you; use them to guide you. Remember the half-cycle correction!) a. What is the equation relating d(x) to l(x)? Explain this relationship in words. b. What is the equation relating L(x) to d(x) and l(x)? Explain this relationship in words. c. What is the equation relating T(x) and L(x)? Explain this relationship in words. d. What is the equation relating e(x) to T(x) and l(x)? Explain this relationship in words. e. What is the expectation of life at birth (the value in the yellow cell)? 3. Show why the following two statements are true using the definitions of relative risk (RR) and hazard ratio (HR) along with the relationship between rates and probabilities. pc is the one-year probability of an adverse event for a control group, and rc is the annual rate of an adverse event for a control group.