Stability Analysis and Optimal Control of the SEIR-SI Model for the Spread of Dengue Hemorrhagic Fever with Treatment Control

Abstract

Dengue Hemorrhagic Fever (DHF) is an infectious disease caused by the dengue virus and transmitted through the bite of the Aedes aegypti mosquito. This study aims to analyze the stability and optimal control of the SEIR-SI mathematical model for the spread of DHF. The model was developed from the existing SIR-SI model, incorporating the addition of an Exposed class (Eh) to represent the virus incubation period in humans, as well as a control variable u(t) representing medical intervention in the form of accelerated recovery of infected individuals. The model consists of six compartments : Susceptible human (Sh), Exposed human (Eh), Infected human (Ih), Recovered human (Rh), Susceptible vector (Sv), and Infected vector (Iv). Equilibrium analysis yields two equilibrium points: the disease-free equilibrium (E₀) and the endemic equilibrium (E₁). The basic reproduction number (R₀) is determined using the Next Generation Matrix method. Stability analysis based on the Routh-Hurwitz criterion shows that E₀ is locally asymptotically stable when R₀ < 1, and E₁ is locally asymptotically stable when R₀ > 1. Optimal control is formulated using Pontryagin's Maximum Principle, with an objective function that minimizes the number of infected individuals and intervention costs. Numerical simulation is carried out using the Forward-Backward Sweep method with the RK45 solver in Python over a time horizon of T = 12 months. Simulation results show that the application of optimal control reduces the peak number of infected humans by 14.05% and the total cumulative disease burden by 20.93% compared to the no-intervention scenario. The optimal control profile u(t) ≈ 0.9 for nearly the entire control period indicates that consistent and sustained medical intervention is required to effectively suppress the spread of DHF.