Research article
Special Issues
Stability and estimation problems related to a stagestructured epidemic model

1.
UMIIRD209 UMMISCO, and LANI, Université Gaston Berger, SaintLouis, 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, 24210201, Brazil

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




In this work, we consider a class of stagestructured SusceptibleInfectious (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 stateestimators: the first one being the classical highgain 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.
Citation: Mamadou L. Diouf, Abderrahman Iggidr, Max O. Souza. Stability and estimation problems related to a stagestructured epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 44154432. doi: 10.3934/mbe.2019220

Abstract
In this work, we consider a class of stagestructured SusceptibleInfectious (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 stateestimators: the first one being the classical highgain 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
[1]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 15131532.

[2]

A. Iggidr, J. Mbang, G. Sallet, et al., Multicompartment models. Discrete Contin. Dyn. Syst. Supplements, suppl. volume(Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference), 2007, 506–519.

[3]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513–525.

[4]

P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equi libria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.

[5]

J.P. LaSalle, Stability theory for ordinary differential equations. J. Differ. Equations, 4 (1968), 57–65.

[6]

J.P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Autom. Control, 37 (1992), 875–880.

[7]

G. Bornard and H. Hammouri, A high gain observer for a class of uniformly observable systems. In Proceedings of the 30th IEEE Conference on Decision and Control, 1991, 1494–1496.

[8]

F. Deza, E. Busvelle, J.P. Gauthier, et al., High gain estimation for nonlinear systems, Syst. Control Lett., 18 (1992), 295–299.

[9]

A. Guiro, A. Iggidr, D. Ngom, et al., On the stock estimation for some fishery systems, Rev. Fish Biol. Fisher., 19 (2009), 313–327.


