Optimal Control Problem and Its Application in COVID-19 Transmission Dynamics

Optimal control techniques are among the most significant technical concepts that can be applied to medical concerns. Establishing a control law for a particular system mathematical model under specified restrictions to meet a specific optimality criterion is known as optimal control. To evaluate the effect of vaccination on the spread of infectious disease and try to decrease the infected group, an optimal control strategy known as Pontryagin’s principle applied in this project. This work develops and thoroughly analyzes and assesses two crucial COVID-19 therapeutic interventions: immunizing susceptible individuals and treating infected individuals under quarantine. A relevant set of conditions is met by proving the presence, distinctiveness, positivity, and unchanging domain of response, among other essential aspects of the model system. The model shows two equilibrium points: one is disease-free and the other that, under some circumstances, is endemic. The fundamental reproduction number R0 is obtained using the next-generation matrix technique, and the model’s dynamic behavior is thoroughly examined. When the corresponding primary reproduction number is smaller than unity, the analytical study shows that the unaffected-by-illness balance approach is both globally and locally stable over time, indicating that COVID-19 has died out in the population. Additionally, anytime the corresponding primary reproduction number surpasses a unity, suggesting that COVID-19 becomes established among the population. The primary reproduction number is subjected to a sensitivity analysis to determine the most critical factors that influence the spread of infection and its control. These are factors that intervention efforts should focus on.