$\varphi_{\textrm{V}}^1$.[]{data-label="fig_results1"}](case1_b.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^1$.[]{data-label="fig_results1"}](case1_c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^1$.[]{data-label="fig_results1"}](case1_d.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^2$.[]{data-label="fig_results2"}](case2_a.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^2$.[]{data-label="fig_results2"}](case2_b.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^2$.[]{data-label="fig_results2"}](case2_c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^2$.[]{data-label="fig_results2"}](case2_d.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^3$.[]{data-label="fig_results3"}](case3_a.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^3$.[]{data-label="fig_results3"}](case3_b.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^3$.[]{data-label="fig_results3"}](case3_c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification $\varphi_{\textrm{V}}^3$.[]{data-label="fig_results3"}](case3_d.png "fig:"){width="\textwidth"}

COVID-19 SEIR Model with Shield Immunity Control
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We use the same parameters of the COVID-19 SEIR model as shown in Table \[parameter\_SEIR\]. We also choose the same initial values of the states as $I[1]=1$ (person), $E[1]=0.02$ million, 