(\[vaccination\_model\]) as follows. $$\begin{aligned}
\begin{cases}
I[k+1] = I[k] + T_{\textrm{s}}\epsilon E[k] - T_{\textrm{s}}(\gamma+\mu+\alpha)I[k];\\
E[k+1] = E[k] + T_{\textrm{s}}\beta S[k]I[k]/N[k] - T_{\textrm{s}}(\mu+\epsilon)E[k];\\
S[k+1] = S[k] + T_{\textrm{s}}\lambda N[k] - T_{\textrm{s}}\mu S[k] - T_{\textrm{s}}\beta S[k]I[k]/N[k] - T_{\textrm{s}}V[k];\\
R[k+1] = R[k] + T_{\textrm{s}}\gamma I[k] - T_{\textrm{s}}\mu R[k] + T_{\textrm{s}}V[k];\\
D[k+1] = D[k] - T_{\textrm{s}}\dot{I}[k]-T_{\textrm{s}}\dot{E}[k]-T_{\textrm{s}}\dot{S}[k]-T_{\textrm{s}}\dot{R}[k],
\end{cases}     
\label{discrete_vaccination_model}\end{aligned}$$ where $T_{\textrm{s}}$ is the sampling period. We also use $\Delta D[k]=D[k+1]-D[k]$ to denote the number of deaths from the infection at day $k$.

Following the notations in Section \[sec\_MTL\], we use $x_{\textrm{V}}=[I, E, S, R, D]$ to denote the state of (\[discrete\_vaccination\_model\]) and $\xi_{\bm\cdot;x^{init}_{\textrm{V}},V}$ to denote the trajectory of (\[discrete\_vaccination\_model\]) starting from $x^{init}_{\textrm{V}}=[I[1], E[1], S[1], R[1], D[1]]$ and vaccination control input $V(\bm\cdot)$. We formulate the problem of vaccination control as follows.

\[Vaccination control\] Given the SEIR model in (\[discrete\_vaccination\_model\]) and an MTL specification $\varphi_{\textrm{V}}$, compute the input signal $V(\bm\cdot)$ that minimize the vaccination control efforts ${\left\lVertV(\bm\cdot)\right\rVert}$ while satisfying $\left[\left[\varphi_{\textrm{V}}\right]\right](\xi_{\bm\cdot;x^{init}_{\textrm{V}},V}, 0)\ge0$, i.e., the trajectory $\xi_{\bm\cdot;x^{init}_{\textrm{V}},V}$ satisfies the MTL specification $\varphi_{\textrm{V}}$. \[problem1\]

The vaccination control synthesis problem can be formulated as a constrained optimization problem as follows. $$\begin{aligned}
\begin{split}
\underset{V(\cdot)}{{\mathrm{arg}\min}} ~ & {\left\lVertV[\cdot]\right\rVert}   \\
\text{subject to:} ~ 
& I[k+1] = I[k] + T_{\textrm{s}}\epsilon E[k] - T_{\textrm{s}}(\gamma+\mu+\alpha)I, \forall k=1,\dots,T-1, \\
