## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer
 \tag{96}\]  
Let \(\hat{I}_{E}^{\hat{x}_{i}}(t)\) be the area delimited by \(\hat{x}_{i,E}^{+}(t)\) and \(\hat{x}_{i,E}^{-}(t)\),  
\[\hat{I}_{E}^{\hat{x}_{i}}(t)=\{x\in\mathbb{R}\ :\ \hat{x}_{i,E}^{-}(t)\leq x \leq\hat{x}_{i,E}^{+}(t)\}\quad\text{for}\quad i=1,2,3. \tag{97}\]  
**Remark 10**. : _Given information related to the unknown variables \(\alpha x_{3}(t)\), \(q(t)\), \(\theta(t)\), and \(w_{i}(t)\) for \(i=1,2,3\), they might be approximated so estimation errors \(\hat{\theta}_{e}(t)\) and \(\hat{z}_{e}(t)\) smaller than \(\hat{\theta}_{E}(t)\) and \(\hat{z}_{E}(t)\), respectively, could be obtained. For example, let us assume that the total number of deaths, the average transmission rate \(\overline{\overline{\theta}}\) and its average change \(\overline{\overline{q}}\) during the period \([t_{0},t_{1}]\) are given, then \(\alpha x_{3}(t)\) and \(q(t)\) can be approximated as follows:_  
\[\alpha x_{3}(t)\approx\overline{\overline{\alpha}}=\frac{\alpha}{t_{1}-t_{0} }\int_{t_{0}}^{t_{1}}x_{3}(\tau)d\tau, \tag{98}\]  
_where \(\int_{t_{0}}^{t_{1}}x_{3}(\tau)d\tau\) is the total number of deaths during the period \([t_{0},t_{1}]\) (assumed to be known), so \(\overline{\overline{\alpha}}\) is the average number of deaths per day, and_  
\[\theta(t)\approx\overline{\overline{\theta}},\quad q(t)\approx\overline{ \overline{q}}. \tag{99}\]  
_From the approximation \(\alpha x_{3}(t)\approx\overline{\overline{\alpha}}\), one obtains that_  
\[\|w\|\approx\overline{\overline{b}}=\overline{\overline{\alpha}}\|\Big{[}\ CA\quad-(A+C)\quad 1\ \Big{]}\|, \tag{100}\]  
_and_  
\[\|\phi_{w}\|\lesssim\overline{\overline{\phi}}_{w}=\overline{\overline{ \alpha}}\,\|\Big{[}\ -(C+\rho)\quad 1\quad 0\ \Big{]}\|. \tag{101}\]  
_Thus, with Equations (_80_), (_81_) and (_85_), an approximation of the estimation errors can be calculated, that is,_  
\[\|\eta(t)\|\lesssim\eta_{e}(t)=\overline{\Phi}_{\eta}(t,t_{0})\bigg{(}\beta_{ M}\sqrt{(|C|+\rho)^{2}+1}+\|\text{Y}(t)\|\overline{\overline{\theta}}\bigg{)}+ \int_{t_{0}}^{t}\overline{\Phi}_{\eta}(t,t_{0})\Big{(}\Big{(}\|\text{Y}(\tau) \|+\overline{\overline{\phi}}_{w}\Big{)}|\overline{\overline{q}}|+\overline{ \overline{\phi}}_{w}\overline{\overline{\theta}}+\overline{\overline{w}}\Big{]} d\tau, \tag{102}\]  
_and_  
\[|\widetilde{\theta}(t)|\lesssim\hat{\theta}_{e}(t)=\Phi_{\widetilde{\theta}}(t,t_{0})\overline{\overline{\theta}}+\int_{t_{0}}^{t}\Phi_{\widetilde{\theta}}( t,\tau)\big{(}\Gamma\Sigma|\text{Y}_{e}(\tau)|\eta_{e}(\tau)+|\overline{ \overline{q}}|\big{)}d\tau \tag{103}\]  
_from which it follows that_  
\[\|\widetilde{z}(t)\|\lesssim\hat{z}_{e}(t)=\eta_{e}(t)+\|\text{Y}(t)\|\hat{ \theta}_{e}(t). \tag{104}\]  
_All the equations involving the maximum error (e.g., \(\hat{\beta}_{E}(t)\)) are also valid for these errors by substituting the subscript \(E\) with \(e\) (e.g., replacing the subscript \(E\) with \(e\) in (92) yields \(\hat{\beta}_{e}^{\pm}(t)=\text{sat}_{\widehat{\beta}_{m}}^{BM}\left(\hat{\beta} (t)\pm\hat{\beta}_{e}(t)\right)\))._  
**Remark 11**. : _By considering both the maximum and approximated errors, it is possible to obtain an overview of the worst and average scenarios, respectively, which can be beneficial in control applications. Specifically, the approximated error helps to optimize performance under constraints, while the maximum error ensures robustness and safety. For example, if control inputs (e.g.,vaccination) have a constraint (e.g., limited vaccines) and the controller is designed to minimize the number of hospitalized people, the control outputs obtained by considering \(\mathfrak{x}_{3,E}^{+}(t)\) rather than \(\mathfrak{x}_{3,E}^{+}(t)\) will be less strict and beneficial (the control input constraints will be more respected). If the aim is to avoid critical situations, like the number of hospitalized people going beyond the number of available beds in hospitals, \(\mathfrak{x}_{3,E}^{+}(t)\) should be considered instead._