:=&-\left[\left[ \varphi\right]\right](\xi, k),\\
\left[\left[\varphi_1\vee\varphi_2\right]\right](\xi, k)  :=&\max\big(\left[\left[ \varphi_1\right]\right](\xi, k),\left[\left[ \varphi_2\right]\right](\xi, k)\big),\\
\left[\left[\varphi_1\mathcal{U}_{\mathcal{I}}\varphi_{2}\right]\right](\xi, k)  :=&\max_{k'\in (k+\mathcal{I})}\Big(\min\big(\left[\left[ \varphi_2\right]\right](\xi, k'),\\& \min_{k\leq k''<k'}\left[\left[\varphi_1\right]\right]
(\xi,k'')\big)\Big).  
\end{split}$$ As defined, $\left[\left[\varphi\right]\right](\xi, k)\ge0$ if $\xi$ satisfies $\varphi$ at time index $k$.

COVID-19 Models with Control Strategies {#sec_models}
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In this section, we study three models for COVID-19 epidemic [@Carcione_model; @Weitz_nature; @Zhao2020] and introduce the corresponding models with vaccination control, shield immunity control and quarantine control.

COVID-19 SEIR Model with Vaccination Control
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The susceptible, exposed, infectious, recovered (SEIR) model has been frequently used in epidemic analyses. As shown in Figure \[diagram1\], the total population is divided into five parts:

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

-   The exposed population $E$: the individuals who have been exposed to the disease, but are still not infectious;

-   The infectious population $I$: the individuals who are infectious;

-   The immune (recovered) population $R$: the individuals who are vaccinated or recovered from the disease, i.e., the population who are immune to the disease;

-   The dead population $D$: the dead individuals from the disease.

For simplicity, we assume that the birth rate and the natural death rate are the same for 