Introduction: Influenza remains a major public health concern due to rapid transmission and seasonal recurrence, straining healthcare systems. Mathematical models help understand disease dynamics and evaluate interventions. Vaccination is highly effective, yet resource constraints require optimally designed policies. This study develops a controlled SEIR model to investigate efficient vaccination strategies for influenza.

Methods: A time-dependent vaccination rate is introduced, representing the proportion of susceptible individuals immunized over time. The model incorporates a vaccine efficacy parameter, accounting for imperfect protection. Fundamental properties, including positivity and boundedness of solutions, are established. The basic reproduction number R₀ was derived using the next generation matrix approach. An optimal control problem was formulated to minimize infectious individuals and vaccination costs over a fixed horizon. The objective functional uses a quadratic control cost, reflecting nonlinear costs and diminishing returns in public health interventions. Pontryagin's Maximum Principle provides necessary optimality conditions, solved numerically via forward-backward sweep with fourth order Runge–Kutta discretization.

Results: Using influenza-calibrated parameters with R₀ approximately 1.8, optimal vaccination significantly suppresses epidemic spread. Simulations show peak infection reduced by 62% and total cumulative cases by 47% compared to uncontrolled scenarios. The optimal strategy also achieves 23% lower cumulative vaccination costs than constant policies while maintaining strong control.

Conclusions: Dynamically adjusted vaccination strategies substantially outperform static interventions for influenza. This study demonstrates the value of optimal control frameworks, incorporating vaccine efficacy and nonlinear costs, to guide efficient public health policies under resource constraints.