Markov Processes

The authors of this paper had the overarching goal of trying to find out how valuable in accuracy it was to screen both pre and post-menopausal women for breast cancer, and how often screenings would occur at the most optimal rate. This is important because breast cancer is a relatively common cause of death in the health industry, and it is moral to minimize risk through proper screening, while mitigating the unnecessary financial costs that come with screening. In order to find these results, the authors formulated a partially observed discrete-time Markov chain that captured some of the age-based inputs that weren?t previously considered. They then used sample path enumeration to evaluate various policies and understand their effects on the value of screenings as well as to provide recommendations for optimal screening methods.  
The results indicated that regardless of screening intervals, screening should start early in life and continue through relatively later life periods. Additionally, policy types that changed the intervals between early and late life stages were found to have patterns correlated with screening value.  
The markov chain contained five states: {0 - no breast cancer, I - early breast cancer, II - later/advanced breast cancer, III - breast cancer induced death, IV - non breast cancer induced death}. These states were chosen to be consistent with the stages described in the medical field, and also were convenient for determining factors such as lymph node involvement.  
Transition probability matrices were created for each 5 year age interval, from [25,29] to [80,85], then topping off with [85,100]. Each of these probabilities were found by assuming six-month transitions and then solving a series of equations to integrate the remaining data sources. The data was pulled from sources including Bloom, Tabar, CDC and SEER mortality, and Elber. Each data source modeled a different parameter and had variation in age groups. When untreated data was unavailable, the authors used the National Cancer Institute?s Surveillance, Epidemiology, and End Results (SEER) Statistics Review.  
This real-world example is most similar to an example in class that involved a patient coming back to a doctor every year, in which the doctor would decide whether to perform surgery or not depending on the state that the patient was currently in. This decision would influence the probabilities of which state the patient would be in for the checkup the following year, with the goal to maximize the patient?s overall quality of life. It is different because it includes more inputs such as relative age and had a slightly different goal equation, with preventing risk more important than treating current patients to maximize their life quality.