formulate the problem of quarantine control as follows.

\[Quarantine control\] Given the SUQC with quarantine control model in (\[discrete\_quarantine\_model\]) and an MTL specification $\varphi_{\textrm{Q}}$, compute the input signal $q(\bm\cdot)$ that minimize the quarantine control efforts ${\left\lVertq(\bm\cdot)\right\rVert}$ while satisfying $\left[\left[\varphi_{\textrm{Q}}\right]\right](\xi_{\bm\cdot;x^{init}_{\textrm{Q}},q}, 0)\ge0$, i.e., the trajectory $\xi_{\bm\cdot;x^{init}_{\textrm{Q}},q}$ satisfies the MTL specification $\varphi_{\textrm{Q}}$. \[problem1\]

The control synthesis problem can be formulated as a constrained optimization problem as follows. $$\begin{aligned}
\begin{split}
\underset{q(\cdot)}{{\mathrm{arg}\min}} ~ & {\left\lVertq[\cdot]\right\rVert}  \\
\text{subject to:} ~ 
& S[k+1] = S[k] - T_{\textrm{s}}\beta_0 U[k]S[k]/N[k], \forall k=1,\dots,T-1, \\
& U[k+1] = U[k] + T_{\textrm{s}}\beta_0 U[k]S[k]/N[k] - q[k]U[k], \forall k=1,\dots,T-1,  \\
& Q[k+1] = Q[k] + T_{\textrm{s}}q[k]U[k] - T_{\textrm{s}}[\gamma_2 + (1-\gamma_2)\sigma]Q[k], \forall k=1,\dots,T-1, \\
& C[k+1] = C[k] + T_{\textrm{s}}[\gamma_2 + (1-\gamma_2)\sigma]Q[k], \forall k=1,\dots,T-1,   \\
& 0\le q[k]\le q_{\textrm{max}}, \forall k=1,\dots,T, \nonumber \\ 
& \left[\left[\varphi_{\textrm{Q}}\right]\right](\xi_{\bm\cdot;x^{init}_{\textrm{Q}},q}, 0)\ge 0. 
\end{split}\end{aligned}$$

The above optimization problem is generally a mixed-integer non-linear programming problem. As the change of total population is relatively small compared to the multiplication of the susceptible population and the un-quarantined infectious population, we approximate the term $T_{\textrm{s}}\beta_0 U[k]S[k]/N[k]$ with $T_{\textrm{s}}\beta_0 U[k]S[k]/N_0$. With such approximation, the optimization problem becomes a mixed-integer bi-linear programming problem, which can be more efficiently solved using techniques such as McCormick’s relaxation [@McCormick1976; 