Frequency-Domain Evaluation and Estimation of Semi-Markov Disease Natural History Models

Continuous-time semi-Markov disease natural history models can be calibrated without time discretization or Monte Carlo simulation by solving Laplace-domain convolution equations, enabling exact gradients and Hessians.

• Why it matters: Natural history data often do not directly observe disease transitions, and Markov approximations require extra states while discretization and simulation add approximation and Monte Carlo error.
• What they did: The authors derive convolution equations for model-implied prevalence, incidence, and cause-specific mortality, solve a small linear system on complex-frequency grids, and invert via an accelerated Fourier series.
• The result: In a urinary bladder cancer model calibrated to SEER incidence and mortality by age and stage, targets and derivatives matched an independent implementation (max relative difference 8x10-6) with runtimes of 1.3, 2.5, and 9.1 ms, and Newton-type maximum likelihood and Bayesian estimates agreed.