## 4 Parameter Estimation with One Output (New Hospitalizations)
### Observer Canonical Form with Output Injection and Output Diffeomorphism
Differentiating \(z\) with respect to time, where \(z=[z_{1}\;z_{2}\;z_{3}]^{T}\), leads directly to the expression (34), that is,  
\[z_{1}=ABC+\theta(C(\rho-B)-\rho\sigma)x_{2}-C\theta(u_{1}+u_{2}). \tag{39}\]  
Then, the time derivative of the term \(z_{2}\) gives  
\[\dot{z}_{2}=\theta(Ax_{1}+Bx_{2}-\rho(x_{3}))-(B+\theta x_{1})(A+C)+\theta(u_{ 1}+u_{2}). \tag{40}\]  
Since \(x\in U=\{[x_{1}\,x_{2}\,x_{3}]^{T}\in\mathbb{R}^{3}:x_{2}>0\}\), expressions like \(\ln x_{2}\) are well defined through the proof, so the difference between \(z_{2}\) and \(z_{1}\), which leads to  
\[\dot{z}_{2}-z_{1}=-B(C+A)+\theta(B+C-\rho)x_{2}-AC\ln x_{2}+\theta(u_{1}+u_{2}), \tag{41}\]  
is well defined, and the expression for \(\dot{z}_{2}\) in (34) is obtained by isolating \(\dot{z}_{2}\) from the expression (41). Finally, differentiating \(z_{3}\) with respect to time yields  
\[\dot{z}_{3}=\theta x_{1}+B, \tag{42}\]  
and, by calculating \(\dot{z}_{3}-z_{2}\) and isolating \(\dot{z}_{3}\), one obtains that  
\[\dot{z}_{3}=z_{2}+B-\theta x_{2}+(A+C)\ln x_{2}. \tag{43}\]  
Thus, by substituting \(x_{2}\) with \(\frac{v_{1}}{\sigma}\) into the expressions (39), (41) and (43), the observer canonical form (34) is obtained. **Remark 4**. : _Assuming that at \(t_{1}\), where \(t_{0}\leq t_{1}\), the case \(x_{2}(t_{1})=0\) occurs, hospitalized people, if any, will disappear at any \(t\geq t_{1}\), and susceptible people will not be infected anymore since the infected subpopulation does not exist, so \(x_{2}(t)=0\) will be perpetuated for all \(t\in[t_{1},\infty)\). Therefore, this refers to a situation where the disease becomes extinct, and for this reason, the diffeomorphism \(T(x)\) is well defined over the set \(U\), and it does not apply to cases where \(x_{2}=0\)._The requisites for the existence of a diffeomorphism that transforms (11) and (12) into (34) can be used in two different ways depending on whether the inputs are user-defined. For inputs that are not user-defined, such as those in the dynamical system (11) and (12), the next step is to verify whether the conditions for the existence of the diffeomorphism are met. In contrast, in cases where the inputs can be defined, they can be chosen in a way that ensures the existence of the diffeomorphism. Specifically, when the transmission rate \(\theta(t)\) in (11) and (12) is considered as an input, the diffeomorphism remains valid under the assumption \(\theta(t)\approx 0\). However, if \(\theta(t)\) does not satisfy the condition condition (i.e., \(\dot{\theta}(t)\
ot\approx 0\)), a time-varying function for \(\theta(t)\) can be assigned for the existence of the diffeomorphism. These results are formalized in Corollary 1. Under Assumption 5, the change of coordinates (33) transforms the system (32) into the observer canonical form (34) for a time-dependent \(\theta(t)\) whose variations with respect to time are small enough to assume that  
\[\dot{\theta}(t)\approx 0, \tag{44}\]  
where \(\psi(y_{1},u)\) is given in (35), or a time-dependent transmission rate with the form  
\[\theta^{\prime}(t)=\theta+\frac{u_{3}(t)}{x_{1}(t)}, \tag{45}\]  
for any \(x_{1}>0\), where  
\[\psi(y_{1}(t),u_{1}(t),u_{2},u_{3}(t),\theta)=\left[ \begin{array}{c}ABC+ \theta(C(\rho-B)-\sigma\rho)\frac{y_{1}(t)}{\sigma}-C\theta(u_{1}(t)+u_{2})+CAu _{3}(t)\\ -(B+u_{3}(t))(C+A)+\theta(B+C-\rho)\frac{y_{1}(t)}{\sigma}-AC\ln\left(\frac{y_ {1}(t)}{\sigma}\right)+\theta(u_{1}(t)+u_{2})\\ B+u_{3}(t)-\theta\frac{y_{1}(t)}{\sigma}+(A+C)\ln\left(\frac{y_{1}(t)}{\sigma }\right)\end{array} \right]. \tag{46}\]  
The proof of Corollary (1) follows directly from the following steps: _(i)_ substituting \(\theta\) with \(\theta^{\prime}\) in expression (32), and maintaining \(\theta\) in (33); _(ii)_ differentiating with respect to time the change of variables given by the expression (33), as seen in the proof of Theorem 2. Furthermore, from the point _i)_ in Proposition 1, it follows that \(x_{1}>0\), so the function \(\theta^{\prime}\) given in (45) is defined and well defined on the domain \(x_{1}>0\).