& E[k+1] = E[k] + T_{\textrm{s}}\beta S[k]I[k]/N[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] - T_{\textrm{s}}V[k], \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\forall k=1,\dots,T-1, \\
& R[k+1] = R[k] + T_{\textrm{s}}\gamma I[k] - T_{\textrm{s}}\mu R[k] + T_{\textrm{s}}V[k], \forall k=1,\dots,T-1, \\
& D[k] = N_0 - I[k] - E[k] - S[k] -R[k], \forall k=1,\dots,T,  \\
& 0\le V[k]\le V_{\textrm{max}}, \forall k=1,\dots,T, \nonumber \\ 
& \left[\left[\varphi_{\textrm{V}}\right]\right](\xi_{\bm\cdot;x^{init}_{\textrm{V}},V}, 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 infectious population, we approximate the term $T_{\textrm{s}}\beta S[k]I[k]/N[k]$ with $T_{\textrm{s}}\beta S[k]I[k]/N_0$. 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; @Gupte2013SolvingMI]. Furthermore, if the MTL specification $\phi$ consists of only conjunctions ($\wedge$) and the always operator ($\Box$), the integers in the optimization problem can be eliminated and the problem becomes a bi-linear programming problem.\
**Shield immunity control**: For the COVID-19 SEIR model with shield immunity control, we discretize the model in (\[vaccination\_model\]) as follows. $$\begin{aligned}
\begin{cases}
I[k+1] = I[k] + T_{\textrm{s}}\epsilon 