## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer
By substituting \(\eta_{E}(t)\) and \(\hat{\theta}_{E}(t)\) into (64), the expression (81) is obtained, which stands for the maximum error of \(\hat{z}\), and fulfills the inequality \(\|\widetilde{z}(t)\|\leq\hat{z}_{E}(t)\). **Remark 9**. : _The term \(\sigma w_{2}(t)/y_{1}(t)\) is equal to the number of deaths at a time \(t\) (see expression (60) in Remark 7), so let the maximum number of deaths during the epidemic be defined as \(\overline{\alpha}\), such that \(\overline{\alpha}\geq\sigma\frac{w_{2}(t)}{y_{1}(t)}\) for all \(t\in[t_{0},\infty)\); then, an upper bound for \(w(t)\) and \(\phi_{w}(t)\) can be obtained from (59), that is,_  
\[\overline{w}=\overline{\alpha}\|\Big{[} \begin{array}{cc}CA&-(A+C)&1\end{array}  \Big{]}\|, \tag{88}\]  
_and_  
\[\overline{\phi}_{w}=\overline{\alpha}\|\Big{[} \begin{array}{cc}-(C+\rho)&1 &0\end{array} \Big{]}\|. \tag{89}\]  
_Thus, if the values for \(\overline{q}\), \(\beta_{M}\), and \(\overline{\alpha}\) are given, the maximum estimation error can be calculated with Equations (79)-(81)._  
To obtain \(\overline{\phi}_{w}\) from \(\phi_{w}(t)\), first, the equality \(w_{i}(t)=\alpha x_{i}(t)x_{3}(t)\) for \(i=1,2,3\) is taken into account, so  
\[\phi_{w}(t)=\alpha x_{3}(t)\|\Big{[} \begin{array}{cc}-C(x_{1}(t)+x_{2}(t))& x_{1}(t)+x_{2}(t)&0\end{array} \Big{]}^{T}\|\leq\overline{\alpha}\|\Big{[} \begin{array}{cc}-C(x_{1}(t)+x_{2}(t))&x_{1}(t)+x_{2}(t)&0\end{array} \Big{]} ^{T}\|. \tag{90}\]  
The sum of normalized subpopulations cannot exceed \(1\) for all \(t\in[t_{0},\infty)\), hence the expression (89). Since \(\beta(t)\) is bounded, the transmission rate maximum estimation error \(\hat{\beta}_{E}(t)\) can be determined as  
\[\hat{\beta}_{E}(t)=\min\big{(}\beta_{M}-\beta_{m},\hat{\theta}_{E}(t)\big{)}, \tag{91}\]  
so  
\[\hat{\beta}_{E}^{\pm}(t)=\text{sat}_{\hat{P}_{m}}^{\beta_{M}}(\hat{\beta}(t) \pm\hat{\beta}_{E}(t)) \tag{92}\]  
where \(\hat{\beta}_{E}^{\pm}(t)\) and \(\hat{\beta}_{E}^{-}(t)\) stand for \(\beta\)'s maximum upper and lower estimation bounds, respectively. Let \(I_{E}^{\beta}(t)\) be the area delimited by \(\hat{\beta}_{E}^{+}(t)\) and \(\hat{\beta}_{E}^{-}(t)\), that is,  
\[I_{E}^{\beta}(t)=\{x\in\mathbb{R}\,:\,\hat{\beta}_{E}^{-}(t)\leq x\leq\hat{ \beta}_{E}^{+}(t)\}. \tag{93}\]  
Once the estimations \(\hat{z}(t)\) and \(\hat{\beta}(t)\), and their respective maximum errors \(\hat{\beta}_{E}(t)\) and \(\hat{z}_{E}(t)\) are calculated, with the diffeomorphism given in (36), the maximum error propagation formula defined as  
\[\sum_{i=1}^{n}\Big{|}\frac{\partial f}{\partial x_{i}}\Big{|}\Delta x_{i} \tag{94}\]  
where \(f(x_{1},\ldots,x_{n})\) stands for any function with variables \(x_{1},\ldots,x_{n}\), can be applied so the maximum error corresponding to the states \(x_{i}(t)\) for all \(i=1,2,3\) is obtained. The result of this process is given in the next corollary. **Corollary 4**. : _(Maximum State Estimation Error) Given the diffeomorphism \(\hat{x}(t)=T^{-1}(\hat{z}(t)\), \(\hat{\beta}(t))\) and the maximum estimation errors \(\hat{\beta}_{E}(t)\) and \(\hat{z}_{E}(t)\), the maximum estimation error of \(\hat{x}_{i}(t)\) is given by_  
\[\begin{array}{l}\hat{x}_{1,E}(t)=\Big{|}\frac{1}{\beta(t)}\Big{|}\hat{z}_{E} (t)+\Big{|}\frac{-\hat{\beta}(t)e^{\hat{\beta}(t)+A+C}}{\hat{\beta}  
Taking into account that the normalized subpopulations fulfill the conditions \(x_{1}(t)\in(0,1]\) and \(x_{2,3}(t)\in[0,1]\) (see Proposition 1), the maximum upper and lower bounds of the state estimations, \(\hat{x}_{i,E}^{+}(t)\) and \(\hat{x}_{i,E}^{-}(t)\) for \(i=1,2,3\), respectively, are given by the following expressions:  
\[ \begin{array}{l}\hat{x}_{1,E}^{\pm}(t)=\text{sat}_{e}^{1}(\hat{x}_{1}(t)\pm \hat{x}_{1,E}(t))\\ \hat{x}_{i,E}^{\pm}(t)=\text{sat}_{0}^{1}(\hat{x}_{i}(t)\pm\hat{x}_{i,E}(t)) \ :\quad i=2,3.\end{array}