A hybrid integer-Caputo fractional order dengue transmission model: Parameter optimization and empirical study with real-world data

Dengue fever is a major viral disease that spreads through mosquitoes and is a public health concern, especially in some tropical and subtropical regions. Traditional integer-order compartmental models often do not work well at modeling how disease spreads over time, which is often affected by past infection rates and environmental factors. We propose a hybrid SEISRD-SI model that combines integer-order and fractional-order dynamics with the Caputo derivative. It also includes compartments for severe dengue and dengue-induced mortality to better represent how the disease spreads and what happens as a result. The existence and uniqueness of the fractional model are proved using the Banach Fixed Point Theorem. The basic reproduction number R₀ is derived using the next generation matrix method, which provides key insights into disease spread thresholds. The hybrid model is calibrated using weekly dengue incidence data from Brazil, and parameters are optimized through Particle Swarm Optimization. The optimized hybrid model lowers the mean relative error (MRE) by up to 6.12% compared to the Caputo fractional-order model (MRE: 20.90%) and the integer-order model (MRE: 24.15%). These findings highlight the ability of hybrid modeling to capture both peak and non-peak epidemic dynamics and underscore the value of fractional calculus in advancing epidemiological modeling frameworks.