Stability and estimation problems related to a stage-structured epidemic model

Mamadou L. Diouf; Abderrahman Iggidr; Max O. Souza; Mamadou L. Diouf; Abderrahman Iggidr; Max O. Souza

doi:10.3934/mbe.2019220

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4415-4432. doi: 10.3934/mbe.2019220

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Stability and estimation problems related to a stage-structured epidemic model

1.
UMI-IRD-209 UMMISCO, and LANI, Université Gaston Berger, Saint-Louis, Sénégal
2.
Université de Lorraine, CNRS, Inria, IECL, F57000 Metz, France
3.
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, , Niterói, Rio de Janeiro, 24210-201, Brazil

Received: 14 January 2019 Accepted: 08 May 2019 Published: 20 May 2019

In this work, we consider a class of stage-structured Susceptible-Infectious (SI) epidemic models which includes, as special cases, a number of models already studied in the literature. This class allows for n different stages of infectious individuals, with all of them being able to infect susceptible individuals, and also allowing for different death rates for each stage—this helps to model disease induced mortality at all stages. Models in this class can be considered as a simplified modelling approach to chronic diseases with progressive severity, as is the case with AIDS for instance. In contradistinction to most studies in the literature, we consider not only the questions of local and global stability, but also the observability problem. For models in this class, we are able to construct two different state-estimators: the first one being the classical high-gain observer, and the second one being the extended Kalman filter. Numerical simulations indicate that both estimators converge exponentially fast, but the former can have large overshooting, which is not present in the latter. The Kalman observer turns out to be more robust to noise in measurable data.
- stage-structured models,
- chronic diseases,
- Lyapunov functions,
- Kalman filter,
- Luenberger observer
Citation: Mamadou L. Diouf, Abderrahman Iggidr, Max O. Souza. Stability and estimation problems related to a stage-structured epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4415-4432. doi: 10.3934/mbe.2019220

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Abstract

In this work, we consider a class of stage-structured Susceptible-Infectious (SI) epidemic models which includes, as special cases, a number of models already studied in the literature. This class allows for n different stages of infectious individuals, with all of them being able to infect susceptible individuals, and also allowing for different death rates for each stage—this helps to model disease induced mortality at all stages. Models in this class can be considered as a simplified modelling approach to chronic diseases with progressive severity, as is the case with AIDS for instance. In contradistinction to most studies in the literature, we consider not only the questions of local and global stability, but also the observability problem. For models in this class, we are able to construct two different state-estimators: the first one being the classical high-gain observer, and the second one being the extended Kalman filter. Numerical simulations indicate that both estimators converge exponentially fast, but the former can have large overshooting, which is not present in the latter. The Kalman observer turns out to be more robust to noise in measurable data.

References

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