## 7 Results  
This work presents an SIHR epidemic model, whose normalized form is characterized by a set of differential equations with a state vector \(x(t)=[x_{1}(t)\ x_{2}(t)\ x_{3}(t)]^{T}\), and based on the most commonly reported data such as the new hospital admissions per day (\(y_{1}(t)\)) or the new deaths per day (\(y_{2}(t)\)), different state and time-varying transmission rate estimation methods have been developed, so any of these will suit users' needs.  
In the first method, both datasets are assumed to be available, and an exponential observer for the state \(x_{1}(t)\) and the transmission rate \(\beta(t)\) estimation is proposed. In the second method, only \(y_{1}(t)\) is considered as the system's output, and the nonlinear dynamical system is transformed into the observer canonical form with the change of variable \(z=T(x(t),\beta(t))\), which is a diffeomorphism obtained from an output injection and an output diffeomorphism. Once the system is transformed to this new form, an adaptive observer is proposed, so the estimations \(\hat{z}(t)\) and \(\hat{\beta}(t)\) are obtained. Finally, the diffeomorphism \(T^{-1}(\hat{z}(t),\hat{\beta}(t))\) is applied to obtain \(\hat{x}(t)\). When the change of variables \(T(x(t),\beta(t))\) was being built up, two assumptions were made: _(i)_ the terms \(\alpha x_{i}x_{3}\) are negligible for \(i=1,2,3\); _(ii)_\(\beta(t)\) varies slowly with respect to time (\(\hat{\beta}(t)\approx 0\)). Due to these approximations, the model in its canonical form contains errors, causing the computed estimations to deviate from real values. Without additional information, these deviations remain uncertain. However, assuming that the bounds of those errors are known, equations to calculate the maximum estimation upper (\(\hat{\beta}_{E}^{+}(t)\)) and lower (\(\hat{\beta}_{E}^{-}(t)\)) bounds of \(\beta(t)\), and the maximum estimations of the upper (\(\hat{x}_{i,E}^{+}(t)\)) and lower (\(\hat{x}_{i,E}^{-}(t)\)) bounds of \(x_{i}(t)\) for \(i=1,2,3\) are obtained, so \(\beta(t)\in[\hat{\beta}_{E}^{-sponding to the COVID-19 epidemic, predefined initial state values, a vaccination function, and a transmission rate function, the SIHR epidemic model has been simulated. For each method, the corresponding output has been provided (e.g., to validate the second method, the output \(y_{1}(t)\) has been gathered from simulated data), and the estimations have been computed. Note that, in the third method, after computing \(y_{1}(t)\) from the output \(y_{2}(t)\), for simplicity reasons, we turn to the first method instead of the second.  
The first and the third methods have the advantage of giving good estimations, despite a fixed convergence. With the second method, in addition to obtaining \(\hat{\beta}(t)\) and \(\hat{x}_{i}(t)\) for \(i=1,2,3\), the maximum estimation bounds were also computed along with the error bounds. Then, the conditions \(\hat{\beta}(t)\in[\hat{\beta}_{E}^{-}(t),\hat{\beta}_{E}^{+}(t)]\) and \(x_{i}(t)\in[\hat{x}_{i,E}^{-}(t),\hat{x}_{i,E}^{+}(t)]\) for \(i=1,2\) were evaluated and found to be true. Given the average values of the errors, the approximated estimation error bounds were calculated, and they were shown to be more informative than the maximum estimation bounds since their values were closer to the real ones and the conditions \(\hat{\beta}(t)\in[\hat{\beta}_{e}^{-}(t),\hat{\beta}_{e}^{+}(t)]\) and \(x_{i}(t)\in[\hat{x}_{i,e}^{-}(t),\hat{x}_{i,e}^{+}(t)]\) for \(i=1,2\) were generally satisfied. However, considering the big difference between \(x_{3}(t)\) and the calculated estimation bounds, it follows that this method is not appropriate for \(x_{3}(t)\).