Using the Hybrid GARCH–STARIMA Model to Model the Potential Spread of Measles in Diyala Governorate

Abstract

This study aims to model and analyze the spatio-temporal dynamics of measles incidence in Diyala Governorate using a hybrid framework that combines the Space–Time Autoregressive Integrated Moving Average (STARIMA) model with the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, where the parameters of STARIMA are estimated using the Cat Swarm Optimization (CSO) algorithm. Weekly measles case data from eight health sectors in Diyala were used over a period of 130 weeks from January 1, 2023, to July 1, 2025. Preliminary analysis using the Mann–Kendall test revealed the presence of significant temporal trends in all series, indicating non-stationarity, which was corrected by applying first-order differencing. The Moran’s I test confirmed the existence of significant spatial autocorrelation among the sectors, implying that measles transmission follows a spatially dependent pattern. Several STARIMA models with different orders were estimated using CSO and evaluated based on the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Hannan–Quinn Criterion (HQ). The results showed that the STARIMA model with first-order spatial lag (Lag = 1) provided the best fit. The residuals of the selected STARIMA model were further modeled using a GARCH(1,1) specification, which revealed significant conditional heteroskedasticity, reflecting the volatility associated with epidemic outbreaks. The integration of STARIMA and GARCH enabled a more accurate representation of both the conditional mean and the conditional variance of the measles incidence series, leading to improved short- and medium-term forecasting performance. The findings demonstrate that the proposed STARIMA–GARCH hybrid model provides a robust and efficient statistical framework for analyzing and forecasting spatio-temporal epidemic data, offering valuable support for public health planning and disease control strategies.