Hybrid extreme value and deep learning models for tail-risk analysis of COVID-19 dynamics across nine island regions

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This paper investigates two complementary hybrid modelling frameworks for analysing newly confirmed coronavirus disease 2019 (COVID-19) cases across nine island regions: Cabo Verde, Comoros, Madagascar, Maldives, Mauritius, Mayotte, Réunion, Sao Tome and Principe, and Seychelles, from 3 January 2020 to 28 February 2023. The first framework combines autoregressive moving-average (ARMA) models with generalised autoregressive conditional heteroscedasticity (GARCH) or exponential GARCH (EGARCH) models and extreme value mixture models (EVMMs). Alternative ARMA–GARCH and ARMA–EGARCH specifications were first fitted to capture temporal dependence and conditional heteroscedasticity, after which the standardised residuals were modelled using continuity-constrained EVMMs with likelihood-estimated thresholds. The kernel density estimator–Generalised Pareto distribution (KDE–GPD) and GPD–KDE–GPD formulations frequently provided the best fit. The second framework combined recurrent and convolutional deep learning models with a rolling-threshold GPD applied to forecasting residuals, allowing the threshold and, where required, the tail scale parameter to adapt to changing epidemic conditions. The deep learning models were validated using a chronological $$70\%$$ training, $$15\%$$ validation, and $$15\%$$ test split, 40-trial hyperparameter optimisation, early stopping, and held-out evaluation. Model adequacy was assessed using appropriate stationarity, dependence, heteroscedasticity, and goodness-of-fit diagnostics, with declustering applied where necessary. Additional analyses included formal forecast-comparison tests, such as the Diebold–Mariano test, calibration assessment, residual diagnostics, sensitivity analyses, and uncertainty intervals for the estimated tail quantities and return levels. Return levels were transformed to the original case-count scale to assess potential future epidemic surges. For a return period of 100 days under the mean-forecast transformation, both frameworks identified Réunion as the most concerning region, with estimated return levels of 80040 cases and a corresponding $$95\%$$ confidence interval of [43798, 173035] under the ARMA-(E)GARCH-EVMM framework, compared with 180645 cases and a corresponding $$95\%$$ confidence interval of [160, 288616] under the deep learning-GPD framework. Mauritius and Seychelles also exhibited comparatively large extreme case estimates, whereas Comoros and Sao Tome and Principe displayed lower estimated risk. The deep learning-GPD framework generally produced larger estimates at the longer return periods, while the ARMA-(E)GARCH-EVMM framework provided greater interpretability through explicit volatility modelling.

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