Bayesian parameter estimation of SIR and SIRD models for COVID-19 in Indonesia using Metropolis–Hastings and Hamiltonian Monte Carlo

Abstract

Accurate parameter estimation is essential for reliable epidemic modeling and forecasting. This study presents a Bayesian framework for estimating parameters of the SIR and SIRD epidemiological models using two Markov Chain Monte Carlo (MCMC) methods: Metropolis–Hastings (MH) and Hamiltonian Monte Carlo (HMC). Daily COVID-19 surveillance data from Indonesia spanning January to April 2022 are used to estimate the transmission rate \((\beta)\), recovery rate \((\gamma)\), and death rate \((\kappa)\). Model performance is evaluated through convergence diagnostics (Gelman–Rubin statistic), sampling efficiency (effective sample size, ESS), and predictive accuracy (root mean squared error, RMSE). Results show that the MH algorithm exhibits severe convergence failure, with Gelman–Rubin statistics reaching up to \(66.63\) and ESS values as low as \(7.75\), indicating highly autocorrelated and inefficient sampling. In contrast, HMC achieves perfect convergence \((\widehat{R} = \ 1)\) across all parameters with ESS values ranging from about 790 to more than \(1,600\), reflecting substantially more efficient posterior exploration. Although both algorithms yield high RMSE under the beta prior for \(\beta\), adopting a lognormal prior reduces the RMSE dramatically—from over one million to approximately \(67,000\) for both the SIR and SIRD models when combined with HMC. Among all configurations tested, the SIRD model estimated via HMC with a lognormal prior for \(\beta\) achieves the lowest RMSE (\(67,255.57\)), demonstrating superior predictive performance. These findings suggest that the choice of both the sampling algorithm and the prior distribution critically determines estimation quality in compartmental epidemic models, with HMC and lognormal priors offering a robust and efficient framework for Bayesian epidemiological inference.