Due to the current COVID-19 pandemic, much effort has been put on studying the spread of infectious diseases to propose more adequate health politics. The most effective surveillance system consists of doing massive tests. Nonetheless, many countries cannot afford this class of health campaigns due to limited resources. Thus, a transmission model is a viable alternative to study the dynamics of the pandemic. The most used are the Susceptible, Infected and Removed type models (SIR). In this study, we tackle the population estimation problem of the A-SIR model, which takes into account asymptomatic or undetected individuals. By means of an algebraic differential approach, we design a model-free (no copy system) reduced-order estimation algorithm (observer) to determine the different non-measured population groups. We study two types of estimation algorithms: Proportional and Proportional-Integral. Both shown fast convergence speed, as well as a minimal estimation error. Additionally, we introduce random fluctuations in our analysis to represent changes in the external conditions and which result in poor measurements. The numerical results reveal that both model-free estimators are robust despite the presence of these fluctuations. As a point of reference, we apply the classical Luenberger type observer to our estimation problem and compare the results. Finally, we consider real data of infected individuals in Mexico City, reported from February 2020 to March 2021, and estimate the non-measured populations. Our work’s main goal is to proportionate a simple and therefore, an accessible methodology to estimate the behavior of the COVID-19 pandemic from the available data, such that the competent authorities can propose more adequate health politics.
This paper presents a methodology and design of a model-free-based proportional-integral reduced-order observer for a class of nondifferentially flat systems. The problem is tackled from a differential algebra point of view, that is, the state observer for nondifferentially flat systems is based on algebraic differential polynomials of the output. The observation problem is treated together with that of a synchronization between a chaotic system and the designed observer. Some basic notions of differential algebra and concepts related to chaotic synchronization are introduced. The PI observer design methodology is given and it is proven that the estimation error is uniformly ultimately bounded. To exemplify the effectiveness of the PI observer, some cases and their respective numerical simulation results are presented.
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