Motivated by the challenge of analyzing the dynamics of weekly sea border crossings in the Mediterranean (2015-2025) and the English Channel (2018-2025), we develop a Bayesian dynamic framework for modeling heteroskedastic count time series. Building on theoretical considerations and empirical stylized facts, our approach utilizes a Poisson random walk model that allows for heavy-tailed innovations or stochastic volatility dynamics, while incorporating an explicit mechanism to separate structural from sampling zeros. Posterior inference is carried out via a straightforward Markov chain Monte Carlo algorithm. Applying this methodology to Mediterranean and English Channel data, we compare alternative model specifications through a comprehensive out-of-sample forecasting exercise. Using log predictive scores and empirical coverage at predictive quantiles to evaluate each model, we find strong evidence for stochastic volatility in migration innovations. These models deliver the strongest out-of-sample forecasts with empirical coverage close to nominal levels up to the 99th percentile. Our framework can be used to develop risk indicators with direct policy implications for improving governance and preparedness for migration surges. More broadly, the methodology extends to other zero-inflated non-stationary count time series applications, including epidemiological surveillance and public safety incident monitoring.

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