## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer  
With the nonlinear dynamical system (11) and (12) transformed into the observer canonical form (34) and (35), the state variables and the parameter estimation problem will be solved with the adaptive observer proposed in [28], which is given in Theorem 3 of this subsection. Assuming that the terms \(\alpha x_{i}x_{3}\), where \(i=1,2,3\), and \(\dot{\theta}\) are small enough, this observer is characterized by global exponential convergence. Since we are interested in identifying the parameter \(\theta\), let us rewrite the dynamical system (34) and (35) in a more appropriate form, that is,  
\[\dot{z}(t)=A_{c}z(t)+B_{c}u(t)+\phi(y_{1}(t),u_{1}(t),u_{2})\theta,\quad y(t) =C_{c}z(t), \tag{47}\]where  
\[B_{c}=\left[ \begin{array}{cccc}0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array} \right],\quad u(t)=\left[ \begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\\ u_{4}\\ u_{5}\end{array} \right]=\left[ \begin{array}{c}-\frac{v(t)}{N(t)}\\ \rho+\lambda\\ ABC\\ -B(C+A)-AC\ln\frac{v_{1}(t)}{\sigma}\\ B+(A+C)\ln\frac{v_{1}(t)}{\sigma}\end{array} \right], \tag{48}\]  
Regarding \(\phi\), its dependence with respect to the known signals can simply be considered as a dependence of the time \(t\). So, for the sake of simplicity, we will use \(\phi(t)\) instead of \(\phi(y_{1}(t),u_{1}(t),u_{2})\). Now, we will recall some important points for the application of the adaptive observer in [28]. Note that those assumptions and corollaries in [28] have been adapted for our particular case. Assumption 6: Assume a matrix pair \((A_{c},C_{c})\) in system (47) is such that there exists a gain vector \(K\in\mathbb{R}^{3}\) so that the system  
\[\dot{\eta}(t)=[A_{c}-KC_{c}]\eta(t) \tag{49}\]  
is exponentially stable. Considering that \(\lambda=0\) for all \(\lambda\in\sigma(A_{c})\) (i.e., critically stable), where \(\sigma(A_{c})=\{\lambda\in\mathbb{C}\ :\ \det(A_{c}-\lambda\mathbb{I})=0\}\) and \(\mathbb{I}\) stands for the identity matrix, it follows that the condition  
\[\text{rank}\left[ \begin{array}{c}\lambda\mathbb{I}-A_{c}\\ C_{c}\end{array} \right]=3,\quad\forall\lambda\in\sigma(A_{c}), \tag{50}\]  
is satisfied (note that this condition is usually referred to as the Belewith-Popov-Haulus requirement for detectability). Therefore, according to Lemma 7.1.2 in [38], the pair \((A_{c},C_{c})\) is detectable, that is, all the unstable and/or critically stable modes of \(A_{c}\), if any, are observable, and they can be asymptotically stabilized by output feedback through some constant gain matrix \(K\); see the subsequent result. Corollary 2: Given the matrix pair \((A_{c},C_{c})\) satisfying the detectability condition, the gain vector \(K\in\mathbb{R}^{3}\) can be chosen as the solution of the stationary Riccati equation  
\[K=P\mathbb{C}_{c}^{T} \tag{51}\]  
where  
\[G=-B_{c}B_{c}^{T}\quad\text{and}\quad Q=\mathbb{C}_{c}^{T}C_{c}. \tag{52}\]  
The pole placement method is an alternative solution to the stability condition problem. Assumption 7: Let \(\mathrm{Y}(t)\in\mathbb{R}^{3}\) be a vector of signals generated by the ODE system  
\[\dot{\mathrm{Y}}(t)=[A_{c}-KC_{c}]\mathrm{Y}(t)+\phi(t). \tag{53}\]  
Assume that \(\phi(t)\) is persistently exciting so that there exist three positive constants \(\delta\), \(\
u\), \(T\) and some positive \(\Sigma(t)\in\mathbb{R}\) such that for all \(t\), the following inequality holds\[\delta\leq\int_{t}^{t+T}Y^{T}(\tau)C_{c}^{T}\Sigma(\tau)C_{c}Y(\tau)d\tau\leq\
u. \tag{54}\]  
Now that all the necessary assumptions have been stated, the proposed adaptive observer in [28] is given in the following Theorem 3. **Theorem 3** (Adaptive observer [28]). : _Let \(\Gamma\in\mathbb{R}\) be any positive scalar. Under Assumptions 6 and 7 for constant \(\theta\), the ODE system_  
\[ \begin{array}{l}\dot{\hat{z}}(t)=A_{c}\hat{z}(t)+B_{c}u(t)+\phi(t)\hat{\theta }+\big{[}K+Y(t)\Gamma Y^{T}(t)C_{c}^{T}\Sigma(t)\big{]}\big{[}y(t)-C_{c}\hat{z }(t)\big{]}\\ \dot{\hat{\theta}}(t)=\Gamma Y^{T}(t)C_{c}^{T}\Sigma(t)\big{[}y(t)-C_{c}\hat{z }(t)\big{]}\end{array}