Compartmental Disease Models with Time-Varying Transmission Rates: A Bayesian Generalized Hamiltonian Monte Carlo framework with Applications to COVID-19

Epidemic transmission is shaped by changing behaviour, interventions, and unobserved factors, while incidence data are noisy and incomplete.

We develop a Bayesian statistical-mechanistic framework that combines staged SIR/SEIR-like compartmental models with a time-varying transmission rate represented by Bayesian P-splines.

A Negative Binomial observation model links the latent epidemic dynamics to incidence data, and posterior inference is performed using Generalised Hamiltonian Monte Carlo with gradients obtained from ODE sensitivity equations.

Synthetic experiments and an application to COVID-19 incidence in the Basque Country show that the framework can recover temporal variation in transmission and quantify uncertainty.

However, different compartmental structures may fit the observed incidence similarly while producing distinct transmission-rate and reproduction-number estimates.

These findings highlight the importance of convergence diagnostics, sensitivity analyses, and epidemiological knowledge when selecting and interpreting flexible compartmental models.