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]+\chi[k]R[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]+\chi[k]R[k]);\\
R[k+1] = R[k] + T_{\textrm{s}}\gamma I[k] - T_{\textrm{s}}\mu R[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_shield_model}\end{aligned}$$ where $T_{\textrm{s}}$ is the sampling period.\
Following the notations in Section \[sec\_MTL\], we use $x_{\textrm{S}}=[I, E, S, R, D]$ to denote the state of (\[discrete\_shield\_model\]) and $\xi_{\bm\cdot;x^{init}_{\textrm{S}},\chi}$ to denote the trajectory of (\[discrete\_shield\_model\]) starting from $x^{init}_{\textrm{S}}=[I[1], E[1], S[1], R[1], D[1]]$ and shield immunity control input $\chi(\bm\cdot)$. We formulate the problem of shield immunity control as follows.

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

The control synthesis problem can be formulated as a constrained optimization problem as follows. $$\begin{aligned}
\begin{split}
\underset{\chi(\cdot)}{{\mathrm{arg}\min}} ~ & {\left\lVert\chi[\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, \\
& E[k+1] = E[k] + T_{\textrm{s}}\beta S[k]I[k]/(N[k]+\chi[k]R[k]) - T_{\textrm{s}}(\mu+\epsilon)E[k], \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\forall k=1,\dots,T-1,  \\
& 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]+\chi[k]R[k]), \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\forall 