Exact sensitivity analysis of Markov reward processes via algebraic geometry

TL;DR

This paper presents an algebraic geometry-based method using cylindrical algebraic decomposition for exact sensitivity analysis of Markov reward processes, enabling precise multi-way analysis in health economics.

Contribution

It introduces a novel algebraic approach with a specialized algorithm for tractable exact sensitivity analysis of Markov reward processes, including software implementation.

Findings

Successfully applied to synthetic and real case studies

Demonstrates advantages over standard sensitivity analysis techniques

Provides an open-source software tool for the community

Abstract

We introduce a new approach for deterministic sensitivity analysis of Markov reward processes, commonly used in cost-effectiveness analyses, via reformulation into a polynomial system. Our approach leverages cylindrical algebraic decomposition (CAD), a technique arising from algebraic geometry that provides an exact description of all solutions to a polynomial system. While it is typically intractable to build a CAD for systems with more than a few variables, we show that a special class of polynomial systems, which includes the polynomials arising from Markov reward processes, can be analyzed much more tractably. We establish several theoretical results about such systems and develop a specialized algorithm to construct their CAD, which allows us to perform exact, multi-way sensitivity analysis for common health economic analyses. We develop an open-source software package that implements our algorithm. Finally, we apply it to two case studies, one with synthetic data and one that re-analyzes a previous cost-effectiveness analysis from the literature, demonstrating advantages of our approach over standard techniques. Our software and code are available at: \url{https://github.com/mmaaz-git/markovag}.