Some notes on the dynamic behavior of COVID-19 epidemic disease governed by modified SEIR epidemic model with saturated treatment function

In this work, based on theory of differential systems and stability theory, we are in the first stage to study some applications of differential systems in real-world modelling problems. Here, we propose a mathematical epidemic model, namely SEIR epidemic model with saturated treatment to describe the COVID-19 infection. Based on the next–generation matrix method and the theory of Lyapunov stability, we evaluate the basic reproduction number  and investigate the asymptotic behavior of disease-free equilibrium. Additionally, in the case , we also prove the existence of a unique endemic equilibrium. Finally, in order to dicuss the effect of isolation control strategies, we study the optimal control problem for this epidemic model.