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

Introduction: Natural history models must often be estimated from data that do not directly capture disease transitions. Semi-Markov processes let the risk of progression depend on the time spent in a health state, which Markov models need extra states to represent. Their limitation is computational: discretizing time introduces approximation error, and simulating individuals introduces Monte Carlo error and a non-differentiable calibration objective. Methods: We show that the model-implied prevalence, incidence and cause-specific mortality of a continuous-time semi-Markov model satisfy convolution equations that become a small linear system when Laplace-transformed. Solving it on a grid of complex frequencies and inverting with an accelerated Fourier series returns the targets without a time grid or Monte Carlo error, and yields exact first- and second-order derivatives for one additional linear solve each. These support Newton-type maximum likelihood estimation, gradient-informed Bayesian sampling, and local identifiability diagnostics. We apply our open-source implementation to a urinary bladder cancer model calibrated to SEER incidence and mortality by age and stage. Results: The method reproduced an independent implementation to within its accuracy (maximum relative difference 8x10 -6 ) and evaluated all targets, their gradient, and their Hessian in 1.3, 2.5, and 9.1 milliseconds; a one-year-cycle cohort approximation deviated by up to 42% and a two-million-person microsimulation by up to 811%. Maximum likelihood and Bayesian estimates agreed, and second-order information showed that the aggregate targets constrain where individuals go from preclinical states far better than how long they stay. Early-stage incidence and mortality showed systematic misfit across age. Conclusions: The framework supplies existing calibration algorithms with fast, exact model evaluations and derivatives for models whose transition hazards depend on time in state rather than on age.