Dynamics of SCIR Modeling for COVID-19 with Immigration

Abstract

In this study, for COVID-19, we divide people into four categories: susceptible $S\left(t\right)$ , closely contacted $C\left(t\right)$ , infective $I\left(t\right)$ , and removed $R\left(t\right)$ according to the current epidemic situation and then investigate two models: the SCIR models with immigration (Model (2) and without immigration (Model 1). For the former, Model 1, we obtain the condition for global stability of its disease-free equilibrium. For the latter, Model 2, we establish the local asymptotic stability of its endemic equilibrium by constructing Lyapunov function. Afterwards, by the bifurcation theory, we qualitatively analyze the properties of its Hopf bifurcations of the latter. Finally, numerical simulations are given to illustrate the obtained results of two models. The results imply the importance of finding closely contacted and overseas imports on epidemic control. It indicates that not only the incubation delay $\tau$ is crucial for the containment of the COVID-19 but also the scientific and rigorous containment measures are the key factors of the success of the containment.