Evaluating Semi-Markov Processes and Other Epidemiological Time-to-Event Models by Computing Disease Sojourn Density as Partial Differential Equations

Introduction Epidemiological models benefit from incorporating detailed time-to-event data to understand how disease risk evolves. For example, decompensation risk in liver cirrhosis depends on sojourn time spent with cirrhosis. Semi-Markov and related models capture these details by modeling time-to-event distributions based on published survival data. However, implementations of semi-Markov processes rely on Monte Carlo sampling methods, which increase computational requirements and introduce stochastic variability. Explicitly calculating the evolving transition likelihood can avoid these issues and provide fast, reliable estimates. Methods We present the sojourn time density framework for computing semi-Markov and related models by calculating the evolving sojourn time probability density as a system of partial differential equations. The framework is parametrized by commonly used hazard and models the distribution of current disease state and sojourn time. We describe the mathematical background, a numerical method for computation, and an example model of liver disease. Results Models developed with the sojourn time density framework can directly incorporate time-to-event data and serial events in a deterministic system. This increases the level of potential model detail over Markov-type models, improves parameter identifiability, and reduces computational burden and stochastic uncertainty compared with Monte Carlo methods. The example model of liver disease was able to accurately reproduce targets without extensive calibration or fitting and required minimal computational burden. Conclusions Explicitly modeling sojourn time distribution allows us to represent semi-Markov systems using detailed survival data from epidemiological studies without requiring sampling, avoiding the need for calibration, reducing computational time, and allowing for more robust probabilistic sensitivity analyses. Highlights Time-inhomogeneous semi-Markov models and other time-to-event–based modeling approaches can capture risks that evolve over time spent with a disease. We describe an approach to computing these models that represents them as partial differential equations representing the evolution of the sojourn time probability density. This sojourn time density framework incorporates complex data sources on competing risks and serial events while minimizing computational complexity.