## 1 Introduction
In the second case, following the methodologies in [23; 24; 25; 26], the epidemic model is transformed into an observer canonical form using output injection and output diffeomorphism. Then, the adaptive observer described in [27; 28] is applied. In the third case, an explicit equation is derived to estimate new hospital admissions, enabling the use of the previously developed methods. The proposed approaches are validated through a theoretical framework and numerical simulations, demonstrating their performance across different scenarios. The solutions developed for the first and third cases successfully estimate both the states and the time-varying transmission rate. In the second case, the proposed observer provides a bounded region where the true values lie. Although these bounds are generally informative, the method fails to accurately estimate the bounds for one of the state variables. A key advantage of these methods over the previously discussed approaches is that they rely on commonly available data, ensuring their applicability, and they do not require prior knowledge of initial values for the unknown states and parameters. Althoughnumerical simulations generally yield good results, these methods require a convergence period (i.e., time to produce estimates that closely match real values), which can be a drawback in situations with short time-period data, such as at the onset of a pandemic. Due to the big impact of COVID-19 and its durability over time, many countries have gathered large datasets regarding the epidemic (e.g., the UK collected data from 2020 to 2023; see Table 1 for details on data sources). The availability of such datasets ensures the convergence of estimated values, making these methods well suited for estimating the transmission rate during the COVID-19 epidemic. In the Section 8, transmission rate estimation methods are proposed for application to data regarding COVID-19. Furthermore, a hypothetical insight into their implications for control is given. The rest of this work is organized as follows: Section 2 presents the epidemic model. In Section 3, two outputs are considered, namely new hospital admissions and new deaths, and an exponential observer and an explicit equation, from which the unknown states and the transmission rate are estimated, are presented. In Section 4, with the output new hospital admissions, a diffeomorphism is applied to the dynamical system so the observer canonical form is obtained, and the joint parameter and state estimation problem is solved. In Section 5, where only data regarding new deaths are assumed to be accessible, the number of new hospitalized people is inferred by an explicit equation, allowing the usage of any of the methods developed in Sections 3 and 4. Section 6 illustrates the numerical simulations carried out. Section 7 presents an overview of theoretical and simulation results. Finally, concluding remarks and a discussion of future directions are given in Section 8. Notation  
1. Let the function \(\mathrm{sat}_{a}^{b}(x)\) be defined as \[\mathrm{sat}_{a}^{b}(x)=\min{(\max{(x,a)},b)}.\] (1)
2. Given a vector \(v\in\mathbb{C}^{n}\), the notation \([v]_{i}\) refers to the \(i\)-th component of the vector, that is, if \(v\) is defined as \[v=[v_{1},\ldots,v_{n}]^{T},\] (2) then \([v]_{i}=v_{i}\) for all \(i=1,\ldots,n\).