## 6 Simulation Results  
The results of the estimation methods developed with respect to the output availability will be shown via different theoretical simulations. Version R2022a of MATLAB has been used for the numerical methods, and all the ODEs have been solved with the solver _ode89()_, where the selected time step is 1 day (i.e., the solutions of the ODEs are sampled at equi-spaced intervals of 1 day). Every time a dynamical system was solved, if any of the inputs were discrete time series, they were previously interpolated with the function _interp1()_ to transform them into continuous time inputs (a necessary step since the function _ode89()_ needs continuous time inputs). To show the simulated results, the function _plot()_ has been used, which for all points connects with line segments at each side-by-side point. Since the aim is to show a reliable model and a parameter estimation method, the parameters chosen for this purpose coincide with the COVID-19 characteristics (e.g., the death rate \(\alpha\) corresponds to the COVID-19 death rate); see Table 2 for more details. Then, with a predefined time-varying vaccination function and transmission rate, and the parameters and initial state values shown in Tables 2 and 3, respectively, the differential Equations (11) and (12) were solved from \(t_{0}=0\) to \(t_{f}=1825\) days, from which the outputs new hospital admission (\(y_{1}(t)=\sigma x_{2}(t)\)) and new deaths (\(y_{2}(t)=\alpha x_{3}(t)\)) were gathered. Additionally, the upper bounds and the average values of \(\theta(t)\), \(q(t)\) and \(\alpha x_{3}(t)\) were computed from the obtained solutions; see the resulting values in Table 4. Then, the methods seen in Sections 3-5 were applied. More details about the steps followed are given in the subsequent enumeration:  
1. **Epidemic model simulation and data extraction:** With the vaccination function \[v(t)\equiv\left\{ \begin{array}{ll}0,&\text{if }0\leq t<1459\\ -8.21*10^{-4}\big{(}1+\sin\big{(}\frac{2\pi}{365}t\big{)}\big{)},&\text{if }1459\leq t<t_{f}, \end{array} \right.\] (106) where \(t_{f}=1825\) days, and the parameters and the initial values shown in Tables 2 and 3, respectively, the dynamical system (11) and (12) has been solved for a constant, and a time-varying \(\beta(t)\), where \[\beta(t)\equiv\left\{ \begin{array}{ll}0.36+0.056\sin\bigl{(}\frac{2\pi}{865}t+\pi \bigr{)},&\text{ if }0\leq t<1102\\ 6.77*10^{-5}t+0.289,&\text{ if }1102\leq t<1459\\ 0.382,&\text{ if }1459\leq t<t_{f}\end{array} \right. \tag{107}\]
2. **Parameter estimation with two outputs:** From the epidemic model simulation carried out in step 1, the outputs \(y_{1}(t)\) and \(y_{2}(t)\) were extracted. After verifying that the condition \(A+y_{2}(t)<0\) for all \(t\in[t_{0},t_{f}]\) was satisfied, with the initial value \(\hat{z}(t_{0})\) shown in Table 3, \(\hat{z}\) was computed from the dynamical system (16), so the state estimate \(\hat{x}_{1}(t)\) was calculated from Equation (17); see the left-side subfigures in Figures 1 and 2. After obtaining the time derivative \(\dot{y}_{1}(t)\) with the [PERSON]'s forward method, that is, \[\dot{y}_{1}(t)=\frac{y_{1}(t+h)-y_{1}(t)}{h},\] (108) where \(h=1\,\text{day}\) is the time step, the estimation \(\dot{\beta}(t)\) was computed with Equation (25); see the right-side subfigures in Figures 1 and 2. 3. **Parameter estimation with one output (new hospitalizations)**: From the epidemic model simulation carried out in step 1, the output \(y_{1}(t)\) was gathered. As the change of variables \(z(t)=T(x(t))\) is a diffeomorphism on \(U=\{[x_{1}\,x_{2}\,x_{3}]^{T}\in\mathbb{R}^{3}:x_{2}>0\}\), before proceeding with the solution of the adaptive observer, the condition \(y_{1}(t)>0\) for all \(t\in[t_{0},t_{f}]\) was verified to be true. (a) Given Equations (53) and (55), with the initial values shown in Table 3, and \[K=\left[ \begin{array}{ll}0.145\\ 0.5\\ 0.854\end{array} \right],\quad\Sigma=10,\quad\Gamma=10,\] (109) the estimations \(\hat{z}(t)\) and \(\hat{\theta}(t)\) were obtained.