## 1 Introduction  
When an outbreak occurs and no vaccine is available, non-pharmaceutical interventions such as school closure or mobility reduction are crucial to lessen the transmission rate and prevent the spread of the virus. While reducing the transmission rate, achieved with a reduction in the average number of close contacts, is effective for disease eradication, it might negatively affect the economy and people's mental health, as happened during the Coronavirus Disease 2019 (COVID-19) epidemic [1, 2, 3, 4, 5]. Finding a balance between controlling the disease spread and minimizing these negative impacts requires a precise understanding of the transmission rate dynamics. Therefore, accurate estimation and characterization of the transmission rate are essential for optimizing control inputs. Furthermore, the importance of state estimation cannot be overstated, as it gives a more precise understanding of the system's dynamics, facilitating more effective intervention strategies. With a transmission rate and state estimation, policymakers can implement interventions that not only reduce transmission but also minimize broader societal consequences. The most commonly used epidemic models are based on the framework developed in [6]. This model is described by a nonlinear dynamical system, where the state variables represent different subpopulations: susceptible, infected, and recovered. Hence, it is named the SIR epidemic model. Since then, more complex models have been developed depending on diseases and societal characteristics such as models incorporating age groups [7; 8; 9], geographical divisions [10; 11; 12], and more. In contrast to black-box models, these are grounded in physical principles, so their solutions have clear physical interpretations, aiding in the understanding of epidemic dynamics, disease spread, and control strategies. Moreover, since these models explicitly depend on the transmission rate, they can be used to estimate its value. Various approaches to estimating the transmission rate have been developed. Using a deterministic SIR model and prior knowledge of infected individuals, explicit equations for the time-varying rate can be derived [13; 14]. Additionally, least squares and functional minimization methods have been widely applied for parameter estimation [15; 16; 17; 18]. However, these methods typically require knowledge of the initial state and transmission rate values, which are often unavailable in real-world scenarios. Observer theory offers a promising alternative since it considers available data to reconstruct unknown states and parameters. Recent studies have explored the use of observers [19; 20] or transformed epidemic models into simpler forms to facilitate observer-based estimations [21; 22]. However, these approaches assume that the number of infected individuals or newly infected cases is known, whereas such data are often unavailable, as shown in Table 1, as with the COVID-19 epidemic data. Furthermore, they primarily focus on estimating a constant transmission rate, even though it can vary over time, for instance, due to seasonal effects. This limitation suggests that while observer theory is a valuable tool, its application in current studies may not be fully suitable due to the lack of appropriate data. To address this problem, this paper proposes different methods for estimating the state and time-varying transmission rates based on the most commonly available data. Specifically, we develop a susceptible, infected, hospitalized, and immunized (SIHR) epidemic model and solve the observer problem for three different scenarios:  
1. Both _new hospital admissions_ and _new deaths_ are available. 2. Only _new hospital admissions_ are available. 3. Only _new deaths_ are available. In the first case, an exponential observer is designed to estimate the susceptible subpopulation, while an explicit equation is used to estimate the transmission rate.