SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^2$.[]{data-label="fig_results_shield2"}](case_shield_phi2c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^2$.[]{data-label="fig_results_shield2"}](case_shield_phi2d.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^3$.[]{data-label="fig_results_shield3"}](case_shield_phi3a.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^3$.[]{data-label="fig_results_shield3"}](case_shield_phi3b.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^3$.[]{data-label="fig_results_shield3"}](case_shield_phi3c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with shield immunity control and MTL specification $\varphi_{\textrm{S}}^3$.[]{data-label="fig_results_shield3"}](case_shield_phi3d.png "fig:"){width="\textwidth"}

COVID-19 SUQC model with quarantine control
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The parameters of the COVID-19 SUQC model are shown in Table \[parameter\_SUQC\]. They were estimated in [@Zhao2020] from the data in Wuhan, China. We choose the initial values of the states as $S[1]=8.9$ million, $U[1]=0.001$ million, $Q[1]=0$ and $C[1]=0$. We set $q_{\textrm{max}}=1$. Figure \[fig\_results\_quarantine0\] shows the simulation results for the COVID-19 SUQC model estimated from data in Stage I of Wuhan, China. It can be seen that the three MTL specifications $\varphi_{\textrm{Q}}^1$, $\varphi_{\textrm{Q}}^2$ and $\varphi_{\textrm{Q}}^3$ are all violated in such a situation (with quarantine rate being always 0.063). Now we investigate the scenario where the quarantine rate 