State Observer for Time Delay Systems Applied to SIRS Compartmental Epidemiological Model for COVID-19

Raúl Villafuerte-Segura (), Jorge A. Hernández-Ávila, Gilberto Ochoa-Ortega () and Mario Ramirez-Neria ()
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Raúl Villafuerte-Segura: Centro de Investigación en Tecnologías de Información y Sistemas, Universidad Autónoma del Estado de Hidalgo, Pachuca 42184, Mexico
Jorge A. Hernández-Ávila: División de Mecatrónica, Universidad Politécnica del Valle de México, Tultitlán 54910, Mexico
Gilberto Ochoa-Ortega: División de Mecatrónica, Universidad Politécnica del Valle de México, Tultitlán 54910, Mexico
Mario Ramirez-Neria: Instituto de Investigación Aplicada y Tecnología, Universidad Iberoamericana, Ciudad de Mexico 01219, Mexico

Mathematics, 2024, vol. 12, issue 24, 1-18

Abstract: This manuscript presents a Luenberger-type state observer for a class of nonlinear systems with multiple delays. Sufficient conditions are provided to ensure practical stability of the error dynamics. The exponential decay of the observation error dynamics is guaranteed through the use of Lyapunov–Krasovskii functionals and the feasibility of linear matrix inequalities (LMIs). Additionally, a time delay SIRS compartmental epidemiological model is introduced, where the time delays correspond to the transition rates between compartments. The model considers that a portion of the recovered population becomes susceptible again after a period that follows its recovery. Three time delays are considered, representing the exchange of individuals between the following compartments: τ 1 , 2 , 3 , the time it takes for an individual to recover from the disease, the time it takes for an individual to lose immunity to the disease, and the incubation period associated to the disease. It is shown that the effective reproduction number of the model depends on the rate at which the susceptible population becomes infected and, after a period of incubation, starts to be infectious, and the fraction of the infectious that recovers after a a certain period of time. An estimation problem is then addressed for the resulting delay model. The observer is capable of estimating the compartmental populations of Susceptible S ( t ) and Recovered R ( t ) based solely on the real data available, which correspond to the Infectious population I r ( t ) . The I r ( t ) data used for the state estimation are from a 55-day period of the pandemic in Mexico, reported by the World Health Organization (WHO), before vaccination.

Keywords: COVID-19; Luenberger-type state observer; SIRS epidemiological model; time delay systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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