COVID-19 transmission, incorporating the nonlinear effects of population caution on infection rates. The model`s validity is confirmed by demonstrating the existence of positive bounded solutions. The basic reproduction number is calculated, and the local stability of both the disease-free equilibrium (DFE) and endemic equilibrium (EE) is assessed, revealing that the EE only exists when the basic reproduction number exceeds a critical threshold. Global stability for both equilibria is established using Lyapunov functions. Additionally, an optimal control strategy for vaccination is proposed, proving its existence and uniqueness, with simulations showing that the strategy effectively minimizes infection rates and associated costs. The impact of integrating public education into the model is also explored, emphasizing its critical role in enhancing vaccination coverage and reducing overall transmission.