Abstract
This paper presents a SEIAR-type model considering quarantined individuals (Q), called SQEIAR model. The dynamic of SQEIAR model is defined by six ordinary differential equations that describe the numbers of Susceptible, Quarantined, Exposed, Infected, Asymptomatic, and Recovered individuals. The goal of this paper is to reduce the size of susceptible, infected, exposed and asymptomatic groups to consequently eradicate the infection by using two actions: the quarantine and the treatment of infected people. To reach this purpose, optimal control theory is presented to control the epidemic model over free terminal optimal time control with an optimal cost. Pontryagin's maximum principle is used to characterize the optimal controls and the optimal final time. Also, an impulsive epidemic model of SQEIAR is considered to deal with the potential suddenly increased in population caused by immigration or travel. Since this model is suitable to describe the COVID-19 pandemic, especial attention is devoted to this case. Thus, numerical simulations are given to prove the accuracy of the theoretical claims and applied to the particular data of this infection. Moreover, numerical computations of the COVID-19 are compared with diseases like Ebola and Influenza. In addition, the controller is evaluated with system parameters identified by using actual data of China. Finally, the controller tuned with the estimated parameters of the Chinese data is applied to the actual data of Spain to compare the quarantine and treatment policies in both countries.