Bayesian Parameter Estimation and Stochastic Extinction Analysis in a Seasonal Bubonic Plague Transmission Model

Mathematical models of plague dynamics provide insight into zoonotic transmission processes and epidemic persistence but often neglect stochastic effects and seasonal variability. In this study, we examine how seasonal variability influences zoonotic disease transmission through a rat-flea-human plague model. Periodic environmental forcing modulates rat abundance, flea survival, and biting activity, generating time-dependent rates that capture realistic ecological seasonality. To represent demographic randomness, we formulate a continuous-time Markov chain (CTMC) model and apply a multi-type branching process approximation near the disease-free equilibrium (DFE) to derive approximate disease extinction probabilities using probability generating functions (PGFs) and the backward Kolmogorov differential equation (BKDE). Our results show that random fluctuations can prevent an epidemic outbreak even when the seasonal reproduction number satisfies $R_0>1$ . We also quantify the spillover probability arising from a single infectious flea, demonstrating that seasonal forcing significantly alters outbreak risk. Along with this, an expression for time to extinction is also derived. To ensure biological understanding, we estimate model parameters from historical data using a Bayesian Markov chain Monte Carlo (MCMC) approach implemented via the Delayed Rejection Adaptive Metropolis (DRAM) algorithm. Through this analysis, we highlight the critical role of seasonality in shaping outbreak risk and demonstrate the limitations of relying solely on deterministic models.