Medical treatment, vaccination, and quarantine are the most efficacious controls in preventing the spread of contagious epidemics such as COVID-19. In this paper, we demonstrate the global stability of the endemic and disease-free equilibrium by using the Lyapunov function. Moreover, we apply the three measures to minimize the density of infected people and also reduce the cost of controls. Furthermore, we use the Pontryagin Minimum Principle in order to characterize the optimal controls. Finally, we execute some numerical simulations to approve and verify our theoretical results using the fourth order Runge-Kutta approximation through Matlab.
