The global crisis of the COVID-19 pandemic has highlighted the need for mathematical models to inform public health strategies. The present study introduces a novel six-compartment epidemiological model that uniquely incorporates a higher isolation rate for unreported symptomatic cases of COVID-19 compared to reported cases, aiming to enhance prediction accuracy and address the challenge of initial underreporting. Additionally, we employ optimal control theory to assess the cost-effectiveness of interventions and adapt these strategies to specific epidemiological scenarios, such as varying transmission rates and the presence of asymptomatic carriers. By applying this model to COVID-19 data from India (30 January 2020 to 24 November 2020), chosen to capture the initial outbreak and subsequent waves, we calculate a basic reproduction number of 2.147, indicating the high transmissibility of the virus during this period in India. A sensitivity analysis reveals the critical impact of detection rates and isolation measures on disease progression, showing the robustness of our model in estimating the basic reproduction number. Through optimal control simulations, we demonstrate that increasing isolation rates for unreported cases and enhancing detection reduces the spread of COVID-19. Furthermore, our cost-effectiveness analysis establishes that a combined strategy of isolation and treatment is both more effective and economically viable. This research offers novel insights into the efficacy of non-pharmaceutical interventions, providing a tool for strategizing public health interventions and advancing our understanding of infectious disease dynamics.