Historically, medicine has seen many applications of mathematics and statistics, with examples including the validation of the effectiveness of new drugs, estimation of survival rates for patients undergoing treatments, and medical imaging (CT scans and MRIs). Going forward, health care data analytics, which bridges the fields of computer science, engineering, statistics and medicine, will improve the delivery of health care, increase the precision of medicine, improve the quality of life, and extend the human lifespan. For example, mining of genetic data could inform the creation of an early warning system to predict the onset of diseases, such as cancer or diabetes, in at-risk patients.
There is a dire need to develop new methods to improve health and fight disease using data science, computational modeling, and machine learning. Some of the most pressing issues are the identification of medical procedures that are most likely to be effective based on an individual’s specific genetic makeup, the understanding of how the changing environment will affect the emergence and spread of infectious diseases, and streamlining health care delivery to improve both cost and patient outcomes.
Currently, a great deal of effort is being put into collecting and interpreting data on health and medical care and, in turn, building better models for diseases, responses to treatment, prognostic plans, diagnostic regimes, vaccinations, and mental health. However, as better and more complex models are developed and back-tested, there is a pressing need to develop equally sophisticated decision making models for their subsequent use. This disparity both requires and motivates the development of richer models for complex health and medical care settings, allowing, for example, for high-dimensions (multi-attribute clinical profiles), path dependence (medical history and response to recent treatments), state and control constraints (morbidity factors, coexisting medical conditions), optimal stopping (best time to induce labor, stop treatment, start prophylactic medication, etc.), optimal pasting across random horizons (corresponding to conditional responses to sequential treatments), and more.
There is also need to develop sophisticated and reliable models to quantify and assess risk related to various decisions in health and medical care. Methods from financial mathematics and quantitative finance can play a major role here, as they are directly related to the concepts of risk quantification, risk management, project valuation, contract design, and more. Indeed, there is a plethora of direct analogies between valuing financial derivatives and contracts and pricing a medical treatment, risk assessment of vaccines, optimal funding of a health-care activity, design of contracts between stakeholders, etc.
Another direction, currently very much overlooked, is related to how individuals think about their health, the extent to which they fear disease and its treatment, how they react to information and uncertainty, perceive risks, follow regimes, and more. There is currently very little work in this direction, but it would be possible to borrow fundamental concepts from decision analysis and economics to build meaningful models of how patients react to uncertainty while making decisions about their medical treatment, life-style, insurance plan, choice of medical facilities and specialized doctors, and so on.
Mathematical sciences have also played a significant role in studying a wide range of issues including, for example, oncology (tumor growth), cardiology (blood flow, cardiovascular system and heart conditions), and neurology (brain disorders). There remains room for progress in these directions as well.
The topics indicated above are by no means exhaustive. Rather they provide an indication of the wide range of health and medicine-related issues which can be tackled by mathematical and statistical techniques, and have the potential to inspire the development of new tools and techniques in the mathematical sciences. With ever increasing advances in medicine and health care, on one hand, and mathematical and statistical modeling and computational methods on the other, this role may be strengthened significantly further through fruitful and multifaceted scientific interactions.