\dot{S} = \lambda N - \mu S - \beta SI/(N+\chi R);\\
\dot{R} = \gamma I - \mu R + V;\\
\dot{D} = - \dot{I}-\dot{E}-\dot{S}-\dot{R},
\end{cases}                 \end{aligned}$$ where the states and parameters are the same as in (\[vaccination\_model\]), while $\chi(\cdot)$ is the shielding strength as time-varying input to be synthesized for the recovered population to substitute the contact for the susceptible population.

![Block diagram of the COVID-19 SEIR model with shield immunity control.[]{data-label="diagram2"}](diagram_covid_shield.pdf)

COVID-19 SUQC model with quarantine control
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The susceptible, un-quarantined infected, quarantined infected, confirmed infected (SUQC) model was recently proposed in [@Zhao2020] based on the COVID-19 data in Wuhan, China. As shown in Figure \[diagram3\], we consider four subgroups in the population:

-   The susceptible population $S$: everyone is susceptible to the disease by birth since immunity is not hereditary;

-   The un-quarantined infected population $U$: the individuals who are infected and un-quarantined, and they can be either asymptomatic or symptomatic;

-   The quarantined infected population $Q$: the individuals who are infectious and quarantined (the un-quarantined infected become quarantined infected by isolation or hospitalization, and the quarantined infected lose the ability of infecting the susceptible individuals);

-   The confirmed infected population $C$: the individuals who are confirmed to 