Dynamical Behavior and Optimal Control Analysis of a Stochastic Influenza Model with Quarantine

This study investigates the transmission dynamics of influenza through an SEIQR framework incorporating both deterministic and stochastic effects. The model explicitly accounts for quarantine, vaccination, social distancing, and treatment as time-dependent control strategies. The analytical investigation involves computing the basic reproduction number and establishing both local and global stability properties of the disease-free and endemic equilibrium states through Lyapunov-based techniques. An optimal control problem is formulated to minimize the combined cost of infection prevalence, quarantine burden, and intervention efforts, and the necessary optimality conditions are obtained via Pontryagin’s Maximum Principle. To capture environmental and behavioral uncertainties, stochastic perturbations driven by independent Wiener processes are introduced, and numerical solutions are obtained using the Euler–Maruyama scheme. Simulation results demonstrate that the implementation of optimal combined control strategies significantly suppresses infection levels and reduces stochastic fluctuations compared to uncontrolled scenarios. The findings highlight the effectiveness and robustness of integrated intervention policies for mitigating influenza outbreaks under uncertainty.