respectively. The results show that the MTL specifications $\varphi_{\textrm{V}}^1$, $\varphi_{\textrm{V}}^2$ and $\varphi_{\textrm{V}}^3$ are satisfied with the synthesized vaccination control inputs respectively. It can be seen that vaccination within the first 40 days after the outbreak can mitigate the spread of SARS-CoV-2 in the most efficient manner. The results also show that the control effort for satisfying $\varphi_{\textrm{V}}^1$ is less than that for satisfying $\varphi_{\textrm{V}}^2$, which is still less than that for satisfying $\varphi_{\textrm{V}}^3$. This is consistent with the fact that $\varphi_{\textrm{V}}^2$ implies $\varphi_{\textrm{V}}^1$, and $\varphi_{\textrm{V}}^3$ implies both $\varphi_{\textrm{V}}^1$ and $\varphi_{\textrm{V}}^2$. For all three specifications, the computations for control synthesis are finished within 5 seconds.

[0.46]{} ![Simulation results for COVID-19 SEIR model estimated from data from Lombardy, Italy in early phase corresponding to no isolation conditions.[]{data-label="fig_results0"}](case0_a.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for COVID-19 SEIR model estimated from data from Lombardy, Italy in early phase corresponding to no isolation conditions.[]{data-label="fig_results0"}](case0_c.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for COVID-19 SEIR model estimated from data from Lombardy, Italy in early phase corresponding to no isolation conditions.[]{data-label="fig_results0"}](case0_d.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_a.png "fig:"){width="\textwidth"}

[0.46]{} ![Simulation results for the COVID-19 SEIR model with vaccination control and MTL specification 