## 4 Parameter Estimation with One Output (New Hospitalizations)
### Adaptive Observer
 \tag{55}\]  
_is globally exponential adaptive observer for system (47), i.e., for any initial conditions \(\hat{z}(t_{0})\), \(\hat{\theta}(t_{0})\), the errors \(\hat{z}(t)-z(t)\) and \(\hat{\theta}(t)-\theta\) tend to zero exponentially fast when \(t\to\infty\)._  
**Remark 5**. : _The diffeomorphism \(\Gamma(x(t))\) expressed by (33) in Theorem 2 considers \(\theta\) constant, while its estimated value (\(\hat{\theta}(t)\)) is time-varying. However, \(\Gamma(x(t))\) remains valid as a diffeomorphism even when \(\theta\) is time-varying because the mapping (the transformation from the \(x\)-coordinates to the \(z\)-coordinates and vice versa) is still smooth, invertible, and well defined for any \(\theta(t)\). In cases where \(\theta\) varies with respect to time, we will refer to the diffeomorphism as \(T(x(t),\theta(t))\)._  
**Remark 6** (Initial conditions \(\hat{z}(t_{0})\) and \(\hat{\theta}(t_{0})\)). : _Given Assumption 2, the values of \(\hat{\theta}(t_{0})\) must be chosen so their values lie on the region delimited by the upper and lower bounds \(\beta_{M}\) and \(\beta_{m}\), respectively (i.e., \(\hat{\theta}(t_{0})\in[\beta_{m},\beta_{M}]\)). Note that considering \(\hat{\theta}(t_{0})\in[\beta_{m},\beta_{M}]\) implies \(\hat{\beta}(t_{0})=\hat{\theta}(t_{0})\); see expression (9). Although bounds delimiting the value of \(\hat{z}(t_{0})  
Considering the term \(\alpha x_{i}(t)x_{3}(t)\) as an unknown error \(w_{i}(t)\) for each \(i=1,2,3\), the dynamical system (11) can be rewritten as  
\[\dot{x}(t)=f(x(t))+\sum_{i}^{2}g_{i}u_{i}(t)+\left[ \begin{array}{c}w_{1}(t)\\ w_{2}(t)\\ w_{3}(t)\end{array} \right] \tag{57}\]  
where \(f(x(t))\) is defined as in (32), and the expressions for \(g_{i}\) and \(u_{i}\) are given in (12) and (13), respectively. Then, the resulting dynamical system after applying the diffeomorphism \(T(x(t),\theta(t))\) is given in the subsequent Corollary 3. Corollary 3: _Let us consider \(w_{i}(t)=\alpha x_{i}(t)x_{3}(t)\) for \(i=1,2,3\), and a time-varying \(\theta\), so the change of variables \(z(t)=T(x(t),\theta(t))\) given in (33) transforms the dynamical system (57) into_  
\[\dot{z}(t)=A_{c}z(t)+B_{c}u(t)+\phi(t)\theta(t)+\phi_{w}(t)\big{[}\dot{\theta }(t)+\theta(t)\big{]}+w(t), \tag{58}\]  
_where_  
\[\phi_{w}(t)=\left[ \begin{array}{c}-C(w_{1}(t)+w_{2}(t))-\rho w_{3}(t)\\ w_{1}(t)+w_{2}(t)\\ 0\end{array} \right]\quad\text{and}\quad w(t)=\left[ \begin{array}{c}CA\sigma \frac{w_{2}(t)}{y_{1}(t)}\\ -(A+C)\sigma\frac{w_{2}(t)}{y_{1}(t)}\\ \sigma\frac{w_{2}(t)}{y_{1}(t)}\end{array} \right], \tag{59}\]  
_and \(\phi_{w}(t)\in\mathbb{R}^{3}\), and \(w(t)\in\mathbb{R}^{3}\) are the regression vector function error and the state error, respectively._  
Remark 7: _The expressions for \(w_{2}(t)\) and \(y_{1}(t)\) given in the statement of Corollary 3 and in the Equation (15), respectively, lead to_  
\[\sigma\frac{w_{2}(t)}{y_{1}(t)}=\alpha x_{3}(t), \tag{60}\]  
_such that for a bounded \(w_{2}(t)\), the term \(\sigma\frac{w_{2}(t)}{y_{1}(t)}\to\infty\) as \(y_{1}(t)\to 0\) (i.e., there is not a singularity). Since one of the objectives in this subsection is to analyze the error on the adaptive observer, results in terms of errors derived from some assumptions (e.g., the error \(w_{2}(t)\) comes from Assumption 5), we prefer to maintain \(\sigma\frac{w_{2}(t)}{y_{1}}\) instead of \(\alpha x_{3}(t)\) for clarity purposes._  
Remark 8: _Considering that all the normalized subpopulations are bounded (i.e., \(x_{i}(t)\leq 1\)), then \(w_{i}(t)=\alpha x_{i}(t)x_{3}(t)\) is also bounded for all \(i=1,2,3\) and for all \(t\in[t_{0},\infty)\). Given the expression for \(\phi_{w}(t)\) in (59), since its terms are linear in \(w_{i}(t)\), it is straightforward that each component of \(\phi_{w}(t)\) is also bounded.