## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer
\tag{76}\]  
The bounds \(\hat{\theta}(t_{0})\in[\beta_{m},\beta_{M}]\), \(0<\hat{x}_{1}(t_{0})\leq 1\), and \(0\leq\hat{x}_{3}(t_{0})\leq 1\), and the condition \(\hat{x}_{1}(t_{0})+\hat{x}_{2}(t_{0})+\hat{x}_{3}(t_{0})\leq 1\) lead to  
\[\|\widetilde{z}(t_{0})\|\leq\beta_{M}\sqrt{\left(|C|+\rho\right)^{2}+1}, \tag{77}\]  
so the inequality (74) holds. **Proposition 4** (Adaptive Observer Maximum Estimation Error). : _Consider the dynamical system (61). Then, given the upper bounds_  
\[\overline{q}=\sup_{t_{0}\leq\tau\leq\infty}|q(\tau)|,\quad\overline{\phi}_{w}= \sup_{t_{0}\leq\tau\leq\infty}\|\phi_{w}(\tau)\|,\quad\beta_{M}=\sup_{t_{0}\leq \tau\leq\infty}|\theta(\tau)|,\quad\overline{w}=\sup_{t_{0}\leq\tau\leq\infty} \|w(\tau)\|, \tag{78}\]  
_the estimated \(\hat{\theta}\) maximum error \(\hat{\theta}_{E}(t)\) is given by_  
\[\hat{\theta}_{E}(t)=\Phi_{\widehat{\theta}}(t,t_{0})\overline{\theta}+\int_{ t_{0}}^{t}\Phi_{\widehat{\theta}}(t,\tau)(\Gamma\Sigma|{\rm Y}_{c}(\tau)|\eta_{E}( \tau)+\overline{q})d\tau, \tag{79}\]  
\[\text{where }{\rm Y}_{c}(t)=C_{c}{\rm Y}(t),\,\[PERSON]_{\widehat{\theta}}(t,\tau)=e^{- \int_{\tau}^{t}\Gamma\Sigma{\rm Y}_{c}^{2}(\tau^{\prime})d\tau^{\prime}},\text { and }\]  
\[\eta_{E}(t)=\overline{\[PERSON]}_{\eta}(t,t_{0})\Big{(}\beta_{M}\sqrt{(|C|+\rho)^{ 2}+1}+\|{\rm Y}(t)\|\overline{\theta}\Big{)}+\int_{t_{0}}^{t}\overline{\[PERSON]}_ {\eta}(t,\tau)\big{[}\big{(}\|{\rm Y}(\tau)\|+\overline{\phi}_{w}\big{)} \overline{q}+\overline{\phi}_{w}\beta_{M}+\overline{w}\big{]}d\tau, \tag{80}\]  
_where \(\overline{\[PERSON]}_{\eta}(t,\tau)=e^{\
u((A_{c}-KC_{c}))(t-\tau)}\sum_{k=0}^{2} \frac{(\|T_{k}\|(t-\tau))^{k}}{k!}\), and \(\overline{\theta}\) is defined as in Lemma 2. The maximum error \(\hat{z}_{E}(t)\) is given by_  
\[\hat{z}_{E}(t)=\eta_{E}(t)+\|{\rm Y}(t)\|\hat{\theta}_{E}(t). \tag{81}\]  
Proof of Proposition 4.: The norm of the solution of the differential Equation (67) satisfies the inequality  
\[\|\eta(t)\|\leq\|\Phi_{\eta}(t,t_{0})\|\|\eta(t_{0})\|+\int_{t_{0}}^{t}\|\Phi_ {\eta}(t,\tau)\|[(\|{\rm Y}(\tau)\|+\|\phi_{w}(\tau)\|)|q(\tau)]+\|\phi_{w}( \tau)\|\theta(t)\|+\|w(\tau)\|]d\tau, \tag{82}\]  
where \(\Phi_{\eta}(t,\tau)\) is the transition matrix of the differential Equation (67). The condition for the norm of exponential matrices given in Lemma 1 leads to  
\[\|\Phi_{\eta}(t,\tau)\|\leq\overline{\Phi}_{\eta}(t,\tau)=e^{\
u((A_{c}-KC_{c }))(t-\tau)}\sum_{k=0}^{2}\frac{(\|T_{k}\|(t-\tau))^{k}}{k!}. \tag{83}\]  
By substituting the upper bound for \(\|\eta(t_{0})\|\) given in (74), the upper bounds in (78), and \(\overline{\Phi}_{\eta}(t,\tau)\) on the right-hand side of the inequality (82), the inequality  
\[\|\eta(t)\|\leq\eta_{E}(t) \tag{84}\]  
is obtained, where \(\eta_{E}(t)\) is defined as in (80). From Equation (68), it follows that  
\[|\widetilde{\theta}(t)|\leq\Phi_{\widehat{\theta}}(t,t_{0})|\widehat{\theta}(t _{0})|+\int_{t_{0}}^{t}\Phi_{\widehat{\theta}}(t,\tau)(\Gamma\Sigma|{\rm Y}_ {c}(\tau)|\|\eta(\tau)\|+|q(\tau)|)d\tau, \tag{85}\]  
where \({\rm Y}_{c}(t)=C_{c}{\rm Y}(t)\) is a scalar function, and \(\Phi_{\widehat{\theta}}\) is the transition matrix of the ODE (68) which fulfills  
\[\overline{q}=\sup_{t_{0}\leq\tau\leq\infty}|q(\tau)|,\quad\overline{\phi}_{w}= \sup_{t_{0}\leq\tau\leq\infty}\|\phi_{w}(\tau)\|,\quad\beta_{M}=\sup_{t_{0}\leq \tau\leq\infty}|\theta(\tau)|,\quad\overline{w}=\sup_{t_{0}\leq\tau\leq\infty} \|w(\tau)\|, \tag{78}\]0  
Considering the upper bounds given in (78), the upper bound for \(|\widetilde{\theta}(t_{0})|\) defined in (72), and the inequality (84), the maximum error of \(\hat{\theta}\) is obtained;  
\[\overline{q}=\sup_{t_{0}\leq\tau\leq\infty}|q(\tau)|,\quad\overline{\phi}_{w}= \sup_{t_{0}\leq\tau\leq\infty}\|\phi_{w}(\tau)\|,\quad\beta_{M}=\sup_{t_{0}\leq \tau\leq\infty}|\theta(\tau)|,\quad\overline{w}=\sup_{t_{0}\leq\tau\leq\infty} \|w(\tau)\|, \tag{78}\]1  
so the inequality \(|\widetilde{\theta}(t)|\leq\hat{\theta}_{E}(t)\) is satisfied.