## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer
Furthermore, from the equivalence provided by Equation (60) and the expression for \(w(t)\) in (59), it follows that each term of \(w(t)\) is bounded too._  
Due to the existence of the regression vector function error \(\phi_{w}\) and the state error \(w\), the estimations given by the adaptive observer (55) will not converge to the exact state and transmission rate values, that is, the adaptive observer (55) is designed for the dynamical system (47) and (48), which is known to have some inherent errors, see expressions (58) and (59) in Corollary 3. Therefore, as these errors are not being considered in (55), the solutions of the adaptive observer are found to not converge to the real values. This idea is formally developed in the subsequent theorem. Theorem 4 (Adaptive Observer for the system (34)-(35) in the Presence of Errors): _Under Assumptions 6 and 7, and considering that \(\phi_{w}(t)\) and \(w(t)\) are bounded, given \(\dot{\theta}(t)=q(t)\) such that_\[ \begin{split} z(t)&=A_{c}z(t)+B_{c}u(t)+\phi(t)\theta(t)+ \phi_{w}(t)\big{[}\dot{\theta}(t)+\theta(t)\big{]}+w(t)\\ \dot{\theta}(t)&=q(t)\\ y(t)&=C_{c}z(t),\end{split}  \tag{61}\]  
_the adaptive observer (55) gives a state estimate \(\hat{x}(t)\) and a parameter estimate \(\hat{\theta}(t)\) with bounded errors._  
Proof of Theorem 4.: In [28], it is assumed that the regression vector function error does not exist. Here, the proof provided in [28] will be extended to the system (61), where \(\phi_{w}\) is considered. Additionally, to prove that Theorem 4 can be generalized to any pair matrix and not only for those with the form expressed in (35), during the proof, the matrices \(A_{c}\) and \(C_{c}\) will be considered as any constant matrices. Let us calculate the term \(\widetilde{z}=\hat{z}-z\),  
\[\dot{\widetilde{z}}=(A_{c}-KC_{c})\widetilde{z}(t)+\phi(t)\widetilde{\theta}+ \mathrm{Y}(t)\dot{\hat{\theta}}(t)-\phi_{w}(t)\big{[}\dot{\theta}(t)+\theta(t) \big{]}-w(t) \tag{62}\]  
and taking into account that \(\widetilde{\theta}=\hat{\theta}-\theta\), it follows that  
\[\dot{\widetilde{z}}(t)=(A_{c}-KC_{c})\widetilde{z}(t)+\phi(t)\widetilde{ \theta}(t)+\mathrm{Y}(t)\dot{\widetilde{\theta}}(t)+\mathrm{Y}(t)\dot{\theta }(t)-\phi_{w}(t)\big{[}\dot{\theta}(t)+\theta(t)\big{]}-w(t) \tag{63}\]  
Let us define \(\eta(t)\) as  
\[\eta(t)=\widetilde{z}(t)-\mathrm{Y}(t)\widetilde{\theta}(t), \tag{64}\]  
whose time derivative  
\[\dot{\eta}(t)=\dot{\widetilde{z}}(t)-\dot{\mathrm{Y}}(t)\widetilde{\theta}(t) -\mathrm{Y}(t)\dot{\widetilde{\theta}}(t) \tag{65}\]  
leads to  
\[\dot{\eta}(t)=(A_{c}-KC_{c})\eta+\big{[}(A_{c}-KC_{c})\mathrm{Y}(t)+\phi(t)- \dot{\mathrm{Y}}(t)\big{]}\widetilde{\theta}(t)+\mathrm{Y}(t)\dot{\theta}(t) -\phi_{w}(t)\big{[}\dot{\theta}(t)+\theta(t)\big{]}-w(t) \tag{66}\]  
By taking into account Equation (53) (see Assumption 7), then  
\[\dot{\eta}(t)=(A_{c}-KC_{c})\eta(t)+\big{[}\mathrm{Y}(t)-\phi_{w}(t)|q(t)- \phi_{w}(t)\theta(t)-w(t). \tag{67}\]  
Note that, from Equation (53), \(\mathrm{Y}(t)\) has a bounded solution for bounded \(\phi(t)\). Thus, for bounded \(q(t)\), \(\phi_{w}(t)\) and \(w(t)\), since the homogeneous part is exponentially stable, the solution for the ODE (67) will also be bounded and will be characterized by an exponential convergence. Regarding \(\dot{\widetilde{\theta}}(t)\),  
\[\widetilde{\theta}(t)=-\Gamma\mathrm{Y}^{T}(t)C_{c}^{T}\Sigma C_{c}\mathrm{Y} (t)\widetilde{\theta}(t)-\Gamma\mathrm{Y}^{T}(t)C_{c}^{T}\Sigma C_{c}\eta(t)- q(t). \tag{68}\]  
Thus, as the homogeneous part of the equation above is globally exponentially stable (see the proof of Theorem 1 in [28]), then the solution for the forced system will exponentially converge towards a bounded value. As a consequence of the results stated in Theorem 4 and its proof, the existence of an estimation error is proved, and its values are defined by the solutions of Equations (64)-(68).