Theoretical State and Time-Varying Parameter Estimation for a Susceptible, Infected, Hospitalized, and Immunized Epidemic Model Based on New Hospital Admission and Death Data

The transmission rate (β(t)) plays a crucial role in disease spread, making its measurement essential for effective control. However, existing techniques for its estimation are impractical as they rely on typically unavailable data. To address this, a prior analysis of the most frequently reported data during the Coronavirus Disease 2019 (COVID-19) pandemic has been carried out, namely the number of new hospital admissions (y1(t)) and deaths (y2(t)). Based on this analysis, an SIHR epidemic model is presented, where S, I, H, and R represent susceptible, infected, hospitalized, and immunized subpopulations, respectively, and various observers tailored to the available data are proposed. Assuming both y1(t) and y2(t) are available, an exponential observer has been designed for the state estimation, from which β(t) is obtained. If only y1(t) is available, a combination of output injection and output diffeomorphism is employed to transform the nonlinear system into its observer canonical form, enabling the design of an adaptive observer for estimating both the states and β(t). By considering only y2(t), an explicit equation has been obtained to estimate y1(t), enabling the application of the previously developed methods. These approaches have been validated through a theoretical framework and numerical simulations; the first and third methods successfully estimate the unknown states and β(t) with high accuracy. The second method yields a bounded region where the true value is expected to lie.