Optimal Control Strategies for COVID-19 Epidemic Management: A Mathematical Modeling Approach Using the SEIQR Framework

The COVID-19 pandemic has necessitated the development of robust mathematical models to understand and mitigate its impact. This study presents a compartmental model for the Indian pandemic COVID-19 dynamics, incorporating key compartments such as susceptible, exposed, infected, quarantined, and recovered populations. The positivity boundedness of solutions are rigorously analyzed to ensure that the model remains biologically meaningful over time. A detailed exploration of basic reproduction number R-0 is conducted using the next-generation matrix approach, identifying it as a pivotal threshold parameter dictating disease dynamics. The equilibria of the system, including the Disease-Free Equilibrium (DFE) and the Endemic Equilibrium (EE), are derived analyzed for their stability properties. The local stability of the DFE established for R-0 < 1, while conditions for the existence and stability of the EE are explored for R-0 > 1. Additionally, the study employs Lyapunov functions to assess the global stability of equilibria, ensuring the robustness of the proposed model under varying initial conditions. The Pontryagin's Maximum Principle is utilized to derive optimal trol strategies, focusing on minimizing the number of infections optimizing interventions such as vaccination, treatment, and quarantine measures like wearing a face mask and hand washing. Numerical simulations validate the theoretical findings, providing critical insights into effectiveness of various control measures. This comprehensive framework contributes to the mathematical understanding of COVID-19 dynamics and offers valuable guidance for public health decision-making.