Hybrid Euler method and Pontryagin Principle in Fractional Dengue Model with Sex Classification and Optimal Controls

This study develops a fractional epidemiological model to investigate the dynamics of dengue transmission, incorporating biological and behavioral differences between male and female human populations. The model utilizes fractional calculus to capture memory effects, which are essential for understanding the long-term behavior of infectious diseases. Control variables representing fumigation and preventive measures are introduced to evaluate intervention strategies, formulating a fractional optimal control problem. To solve the model, Euler’s method is employed for numerical approximation of the fractional differential equations, while Pontryagin’s Minimum Principle and a forward-backward numerical approach are applied to determine optimal strategies. Parameter estimation is conducted using the least-squares fitting method based on cumulative dengue hemorrhagic fever (DHF) case data from West Java Province, Indonesia from 2014 to 2023. The estimation yields a Mean Absolute Percentage Error (MAPE) of 4.32% for males and 4.50% for females, with an overall fit of 4.41%. As one of the key parameters affecting basic reproduction number, proportion of additional immunity levels among females is simulated to examine its influence on infection outcomes. The results show that increasing this parameter reduces female infections, which subsequently lowers vector transmission and indirectly decreases male infections. In the control simulation, lower fractional orders enhance the efficiency of system dynamics, leading to faster convergence towards the desired infection reduction. Additionally, the cost-effectiveness analysis indicates that the implementation of a combined strategy, incorporating both fumigation and preventive measures, provides the most efficient intervention.