## 4 Parameter Estimation with One Output (New Hospitalizations)
### Observer Canonical Form with Output Injection and Output Diffeomorphism  
In this subsection, with the diffeomorphism proposed in Theorem 2, the dynamical systems (11) and (12) are transformed into the observer canonical form. Note that this transformation is only viable if one assumes that the term \(\alpha x_{i}(t)x_{3}(t)\), where \(i=1,2,3\), is small enough so its effects are negligible in system (12). **Assumption 5**. : _Let the parameter \(\theta\) be constant and the term \(\alpha x_{i}(t)x_{3}(t)\), \(\forall t\in[t_{0},\infty)\) and for \(i=1,2,3\) be negligible enough such that \(f(x(t),\theta(t))\) in (12) can be approximated as_  
\[f(x(t),\theta)\approx\left[ \begin{array}{c}Ax_{1}(t)-\theta x_{1}(t)x_{2}(t )-\rho(x_{2}(t)+x_{3}(t))\\ Bx_{2}(t)+\theta x_{1}(t)x_{2}(t)\\ Cx_{3}(t)+\sigma x_{2}(t)\end{array} \right]. \tag{32}\]  
In this context, the conditions for the existence of diffeomorphism that transforms the nonlinear dynamical system into the observer canonical form given in [25; 26] are fulfilled, so the steps to obtain such a change of variables have been followed. This leads to the diffeomorphism shown in the subsequent Theorem 2. **Theorem 2**. : _Under Assumption 5, the change of coordinates_  
\[z(t)=T(x(t))=\left[ \begin{array}{c}-C[\theta(x_{1}(t)+x_{2}(t))-A\ln\left(x_ {2}(t)\right)]-\theta\rho x_{3}(t)\\ \theta(x_{1}(t)+x_{2}(t))-(A+C)\ln\left(x_{2}(t)\right)\\ \ln\left(x_{2}(t)\right)\end{array} \right], \tag{33}\]  
_where \(T:U\subset\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\) and \(U=\{[x_{1}(t)\ x_{2}(t)\ x_{3}(t)]^{T}\in\mathbb{R}^{3}:x_{2}(t)>0\), \(t\in[t_{0},\infty)\}\), is a local diffeomorphism which transforms the system (32) into the observer canonical form_  
\[\dot{z}(t)=A_{c}z(t)+\psi(y_{1}(t),u_{1}(t),u_{2},\theta),\quad y(t)=C_{c}z(t), \tag{34}\]  
_with a state vector \(z(t)=[z_{1}(t)\ z_{2}(t)\ z_{3}(t)]^{T}\), and_  
\[A_{c}=\left[ \begin{array}{ccc}0&0&0\\ 1&0&0\\ 0&1&0\end{array} \right],\quad C_{c}=\left[ \begin{array}{ccc}0&0&1\\ \end{array} \right], \tag{35}\] \[\psi(y_{1}(t),u_{1}(t),u_{2},\theta)=\left[ \begin{array}{c}ABC+ \theta(C(\rho-B)-\rho\sigma)\frac{y_{1}(t)}{\sigma}-C\theta(u_{1}(t)+u_{2}) \\ -B(C+A)+\theta(B+C-\rho)\frac{y_{1}(t)}{\sigma}-AC\ln\left(\frac{y_{1}(t)}{ \sigma}\right)+\theta(u_{1}(t)+u_{2})\\ B-\theta\frac{y_{1}(t)}{\sigma}+(A+C)\ln\left(\frac{y_{1}(t)}{\sigma}\right) \end{array} \right].\]  
**Proof of Theorem 2. ** The inverse function of \(T\) is given by the expression  
\[x=T^{-1}(z)=\left[ \begin{array}{c}\frac{1}{\theta}(z_{2}-\theta e^{z_{3}}+(A+C) z_{3})\\ e^{z_{3}}\\ -\frac{1}{\theta\rho}(z_{1}+C(z_{2}+Cz_{3}))\end{array} \right]. \tag{36}\]  
Then, it is straightforward that \(T^{-1}(T(x))=x\) for each \(x\in U\), and \(T(T^{-1}(z))=z\) for each \(z\in\mathbb{R}^{3}\). Thus, the function \(T:U\subset\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\) is bijective. The Jacobian matrix of the function \(T(x)\) leads to  
\[\frac{\partial T}{\partial x}=\left[ \begin{array}{ccc}-C\theta&-C(\theta- \frac{A}{x_{2}})&-\theta\rho\\ \theta&\theta-\frac{(A+C)}{x_{2}}&0\\ 0&\frac{1}{x_{2}}&0\end{array} \right] \tag{37}\]  
whose terms exist and are continuous on \(U\subset\mathbb{R}^{3}\). Regarding the inverse function \(T^{-1}(z)\), it follows that  
\[\frac{\partial T^{-1}}{\partial z}=\left[ \begin{array}{ccc}0&\frac{1}{ \theta}&\frac{1}{\theta}(-\theta e^{z_{3}}+A+C)\\ 0&0&e^{z_{3}}\\ -\frac{1}{\theta\rho}&-\frac{C}{\theta\rho}&-\frac{C^{2}}{\theta\rho}\end{array}  \right], \tag{38}\]  
whose terms also exist and are continuous on \(\mathbb{R}^{3}\). Since the function \(T(x)\) has been proven to be bijective, and \(T(x)\) and \(T^{-1}(z)\) are continuously differentiable on their domain, the map \(T:U\subset\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\) is a diffeomorphism.