## 8 Conclusions  
When data on new hospital admissions and new deaths, or only new deaths, are available, the proposed observers and numerical differentiation enable an accurate estimation of both state variables and the transmission rate. However, since numerical differentiation is highly sensitive to noise, smoothing the data is recommended to prevent error amplification. Additionally, these observers require a fixed convergence time to produce reliable estimates, which can be a limitation for short time-series datasets. When only new hospital admissions data are available, the proposed method does not yield exact estimates but instead provides bounds within which the true values are expected to lie. Determining these bounds requires prior knowledge of certain parameters, such as upper limits for the transmission rate and the number of deaths. While this approach produces reasonably accurate bounds for the susceptible subpopulation and transmission rate, it struggles to define a suitable bound for hospitalized cases. Furthermore, as this method relies on an observer, its estimates are restrained to a convergence time. Compared to previous methods, its numerical implementation is also more complex, requiring the solution of dynamical equations, application of error propagation formulas, and other computational steps. For large datasets, such as those related to the COVID-19 epidemic, the proposed observer's convergence is ensured. Consequently, these methods enable the estimation of the transmission rate and provide valuable insights into its dynamics, aiding the development of more effective control policies. For instance, if the estimated transmission rate consistently increases during the winter seasons, strengthening vaccination efforts in the preceding months could help to prevent hospitalizations and deaths during winter. While controlling disease spread is the primary goal, it cannot be achieved without a validated model. Validating the model and the proposed estimation methods is not as straightforward as simply collecting real data and estimating the time-varying transmission rate and state variables. Commonly, a model is said to be validated if the model can closely reproduce real data. Therefore, a challenge arises: if the susceptible subpopulation and transmission rate are computed using the proposed model and methods, how can these outputs be compared to the real values when no direct data exist for either the susceptible subpopulation or the transmission rate? To address this, future work will focus on transmission rate characterization by establishing a mathematical relationship between the transmission rate and correlated and measurable variables, such as mobility and seasonal changes. This relationship will enable the estimation of the transmission rate from available data, which will then be used to simulate the SIHR epidemic model. Finally, validation will be performed by comparing the model's predictions (e.g., new hospital admissions and deaths) with real data to assess its accuracy. A successfully validated model will not only enable accurate disease spread predictions but also provide a framework for designing optimal control strategies. Conceptualization, C.L., M.D.I.S., and S.A.-Q. ; methodology, C.L. and S.A.-Q. ; software, C.L. ; validation, S.A.-Q. ; formal analysis, C.L., M.D.I.S., and S.A.-Q. ; investigation, C.L., M.D.I.S., and S.A.-Q. ; resources, C.L. ; data curation, C.L. ; writing--original draft preparation, C.L. ; writing--review and editing, S.A.-Q. ; visualization, C.L. ; supervision, M.D.I.S. ; project administration, M.D.I.S. ; funding acquisition, M.D.I.S. All authors have read and agreed to the published version of the manuscript. This research received support from the Basque Government through grant IT1555-22. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.