On the Properties of a Newly Susceptible, Non-Seriously Infected, Hospitalized, and Recovered Subpopulation Epidemic Model

The COVID-19 outbreak has brought to the forefront the importance of predicting and controlling an epidemic outbreak with policies such as vaccination or reducing social contacts. This paper studies an SIHR epidemic model characterized by susceptible ( S ), non-seriously infected ( I ), hospitalized ( H ), and recovered ( R ) subpopulations, and dynamic vaccination; vaccination itself and H are fed back, and its dynamics are also determined by a free-design time-dependent function and parameters. From a theoretical analysis, the well-posedness of the model is demonstrated; positivity and the disease-free ( P d f ) and endemic ( P e e ) equilibrium points are analyzed. The controlled reproduction number ( R c ) is proved to be a threshold for the local asymptotic stability of P d f and the existence P e e ; when R c < 1 ( R c > 1 ), then P d f is (not) locally asymptotically stable and P e e does not (does) exist. Simulations have been carried out with data concerning COVID-19 where the importance of keeping R c < 1 to prevent the disease spreading and future deaths is highlighted. We design the control input, since it can be easily adapted to match the user specification, to obtain impulsive and regular vaccination and fulfill the condition R c < 1 .