violated in such a situation. Note that as isolation measures were taken after the early phase in Lombardy, Italy, the real situation was better than those shown in Figure \[fig\_results0\]. Now we investigate the hypothetical scenario where isolation measures are replaced by vaccination.

  parameter          value           
  ------------------ ------------ -- --
  $\lambda$          1/30295         
  $\mu$              1/30295         
  $\alpha$           0.006/day       
  $\beta$            0.75/day        
  $\epsilon$         (1/3)/day       
  $\gamma$           (1/8)/day       
  $N_0$              10 million      
  $T_{\textrm{s}}$   1 day           

  : Parameters of COVID-19 SEIR model estimated from data from Lombardy, Italy in early phase corresponding to no isolation conditions [@Carcione_model].[]{data-label="parameter_SEIR"}

[ll&gt;p[20mm]{}]{}      MTL specification & control effort & computation time\

  -----------------------------------------------------------
   $\varphi_{\textrm{V}}^1=\Box_{[0,100]} (\Delta D\le0.01)$
            $~~~~~~\wedge\Box_{[0,100]} (D\le0.2)$
             $~~~\wedge\Diamond_{[40, 60]}(R\ge6)$
  -----------------------------------------------------------

  : MTL specifications and simulation results for vaccination control.[]{data-label="result_vaccination"}

&      1.104 &      1.128 s\

  ------------------------------------------------------------
   $\varphi_{\textrm{V}}^2=\Box_{[0,100]} (\Delta D\le0.005)$
             $~~~~~\wedge\Box_{[0,100]} (D\le0.2)$
              $~~\wedge\Diamond_{[40, 60]}(R\ge6)$
  ------------------------------------------------------------

  : MTL specifications and simulation results for vaccination control.[]{data-label="result_vaccination"}

&      2.033 &      2.286 s\

  ------------------------------------------------------------
   $\varphi_{\textrm{V}}^3=\Box_{[0,100]} (\Delta D\le0.005)$
             $~~~~~\wedge\Box_{[0,100]} (D\le0.05)$
               $\wedge\Diamond_{[40, 60]}(R\ge6)$
  ------------------------------------------------------------

  : MTL specifications and simulation results for vaccination control.[]{data-label="result_vaccination"}

&      4.022 &      4.609 s\

We use the solver GEKKO [@beal2018gekko] to solve the optimization problems. Figures \[fig\_results1\], \[fig\_results2\] and \[fig\_results3\] show the simulation results for vaccination control of COVID-19 SEIR model with MTL specifications $\varphi_{\textrm{V}}^1$, $\varphi_{\textrm{V}}^2$ and $\varphi_{\textrm{V}}^3$, 