On the Dynamics of COVID-19 Propagation with Vaccination and Optimal Control Strategies

In this paper, a mathematical model is proposed to describe the spread dynamics of COVID-19, considering the factors of self-protection and vaccination. The basic reproduction number of the model, which is a critical signal of the dynamics of COVID-19 transmission, is calculated using the next-generation matrix method. A study of the local stability of steady states has been conducted. Moreover, global stability is demonstrated using Lyapunov's second method and LaSalle's invariance principle. In addition, the effect of vaccination on the evolution of the disease spread has been studied. Further, an optimal control problem is formulated and solved to reduce the number of infected individuals and the cost of the controls by considering self-protection and vaccination as intervention options. Specifically, our results indicate that implementing optimal control strategies, such as time-dependent interventions, reduces disease transmission and overall infection burden. The model demonstrates that strategically timed and intensity-optimized control measures can flatten the epidemic curve, delay peak infection, and minimize the outbreak's cost and duration. Finally, comprehensive simulations were performed across various initial conditions and parameter settings to verify the theoretical findings.