An improved method of global dynamics: Analyzing the COVID-19 model with time delays and exposed infection.

Considering the transmission characteristics of the coronavirus disease 2019 (COVID-19), there are certain time delays in the transition from susceptible individuals to exposed individuals after contact with exposed, symptomatically infected, and asymptomatically infected individuals. A COVID-19 model with time delays and exposed infection is developed and then the global dynamics of this model is investigated by an improved method; moreover, the numerical simulations are carried out. It is shown that the COVID-19-free equilibrium T0 is globally asymptotically stable (GAS) if and only if the control reproduction number Rc≤1, while T0 is unstable and the COVID-19 equilibrium T∗ is GAS if and only if Rc>1. The numerical results reveal that strengthening quarantine measures is helpful to control the COVID-19 epidemic in India. Furthermore, when Rc<1, the numbers of symptomatically infected, asymptomatically infected, and quarantined individuals eventually tend to the zero equilibrium state, and with the increase in the time delay, the three kinds of variables change faster and their peaks become larger; when Rc>1, the three kinds of variables eventually tend to the positive equilibrium state, which are oscillatory and the amplitudes of the oscillation enlarge as the value of time delay increases. The numerical results show that when Rc<1, the smaller the value of time delay, the smaller the final epidemic size. In short, the longer it takes time for susceptible individuals to transform exposed individuals, the harder COVID-19 will be controlled.