the population we are investigating.

We consider an SEIR epidemic disease model [@Carcione_model; @Elie2020] with vaccination control [@Alonso2012] as follows. $$\begin{aligned}
\begin{cases}
\dot{I} = \epsilon E - (\gamma+\mu+\alpha)I;\\
\dot{E} = \beta SI/N - (\mu+\epsilon)E;\\
\dot{S} = \lambda N - \mu S - \beta SI/N - V;\\
\dot{R} = \gamma I - \mu R + V;\\
\dot{D} = - \dot{I}-\dot{E}-\dot{S}-\dot{R},
\end{cases}            
\label{vaccination_model}\end{aligned}$$ where the control input $V$ is the number of vaccinated individuals per day, $N = S + E + I + R\le N_0$ is the total population in the region ($N_0$ is the initial total population in the region), $S$, $E$, $I$, $R$ and $D$ are the number of susceptible, exposed, infectious and recovered population in the region, respectively, and $D$ is the number of deaths from SARS-CoV-2 in the region. For the parameters, $\lambda$ denotes the per-capita birth rate, $\mu$ is the per-capita natural death rate (death rate from causes unrelated to SARS-CoV-2), $\alpha$ is the SARS-CoV-2 virus-induced average fatality rate, $\beta$ is the probability of disease transmission per contact (dimensionless) times the number of contacts per unit time, $\epsilon$ is the rate of progression from exposed to infectious (the reciprocal is the incubation period), and $\gamma$ is 