## 2 Problem Statement  
In this section, the nonlinear dynamical system for the SIHR epidemic model with vaccination is proposed. With some basic assumptions regarding parameters and initial conditions (non-negativeness, input bounds, etc. ), the system solution's well-posedness is ensured. Then, the dynamical system is normalized and the vector notation is given. Note that the normalized system in vector notation will be used throughout the following sections. \begin{table}
\begin{tabular}{l|c c c c c|c} \hline \hline  & \multicolumn{5}{c}{**Data Availability**} \\ \hline \multirow{2}{*}{**County**} & **Positive** & **New Hospital** & **Hospitalized** & **New** & \multirow{2}{*}{**Vaccination**} & \multirow{2}{*}{**Source**} \\  & **Tests** & **Admissions** & **Patients** & **Deaths** & & \\ \hline UK & ✓ & ✓ & ✓ & ✓ & [29; 30] \\ Netherlands & \(\times\) & ✓ & \(\times\) & ✓ & ✓ & [31; 32; 33] \\ France & ✓ & ✓ & ✓ & ✓ & ✓ & [34] \\ Belgium & ✓ & ✓ & ✓ & ✓ & ✓ & [35] \\ Spain & ✓ & \(\times\) & ✓ & ✓ & ✓ & [33; 36] \\ Italy & ✓ & \(\times\) & ✓ & ✓ & ✓ & [33; 37] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some countries’ data availability. A tick (✓) denotes available data, whereas a cross (\(\times\)) denotes unavailable data. The cross mark in some cells does not imply that those datasets do not exist; rather, it indicates that they do not appear in the cited sources. Consider the following SIHR epidemic model:  
\[ \begin{array}{rcl}\dot{S}(t)&=&\Lambda(t)+\rho R(t)-\frac{\beta(t)I(t)S(t)}{N(t)} -\mu S(t)-v(t),\\ \dot{l}(t)&=&\frac{\beta(t)I(t)S(t)}{N(t)}-(\gamma_{1}+\sigma+\mu)I(t),\\ \dot{H}(t)&=&\sigma I(t)-(\gamma_{2}+\alpha+\mu)H(t),\\ \dot{R}(t)&=&\gamma_{1}I(t)+\gamma_{2}H(t)+v(t)-(\rho+\mu)R(t),\end{array}  \tag{3}\]  
where the state variables \(S(t)\), \(I(t)\), \(H(t)\), and \(R(t)\) stand for susceptible, infected, hospitalized, and immunized subpopulations, respectively. From the sum of the equations in (3), the dynamical equation for the total population \(N(t)\) is obtained:  
\[\dot{N}(t)=\Lambda(t)-\mu N(t)-\alpha H(t). \tag{4}\]  
The parameters involved in (3) are the following ones: \(\rho\) (immunity rate), \(\mu\) (natural death rate), \(\gamma_{1}\) (infected people recovery rate), \(\gamma_{2}\) (hospitalized people recovery rate), \(\sigma\) (hospitalization rate), and \(\alpha\) (disease death rate). These parameters determine the amount of people being introduced and removed from each subpopulation. Additionally, the variable \(\Lambda(t)\) stands for the number of newborns per day that are introduced in the susceptible subpopulation (i.e., it is assumed that newborns are healthy, but they do not hold any immunity against the disease), and the known input \(v(t)\) (vaccinated people per day), determines the amount of people moving from the susceptible subpopulation to the immunized subpopulation. Vaccines are assumed to be administered to the susceptible subpopulation for two main reasons: _(i)_ during the COVID-19 epidemic, vaccines were not provided to infected or hospitalized individuals; and _(ii)_ those who had previously been infected or hospitalized generally had to wait until their immunity waned, at which point they transitioned from the immunized to the susceptible subpopulation. The time-varying transmission rate \(\beta(t)\), which is equal to the contact rate multiplied by the probability of being infected after close contact, plays a fundamental role in the spread of any disease, that is, given the epidemic model (3), an increase (decrease) of \(\beta(t)\) implies an increment (diminution) in the appearance of new infected people. Hence, estimating \(\beta(t)\) is important. Note that estimating \(\beta(t)\) implies estimating the contact rate as a consequence of the proportionality between both variables. **Assumption 1** (known parameter positivity).