Dynamically consistent nonstandard finite difference schemes for epidemiological models

dc.contributor.authorAnguelov, Roumen
dc.contributor.authorDumont, Yves
dc.contributor.authorLubuma, Jean M.-S.
dc.contributor.authorShillor, M.
dc.contributor.emailjean.lubuma@up.ac.zaen_US
dc.date.accessioned2013-07-08T07:52:12Z
dc.date.available2013-07-08T07:52:12Z
dc.date.issued2014-01
dc.description.abstractThis work is the numerical analysis and computational companion of the paper by Kamgang and Sallet [J.C. Kamgang, G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium (DFE), Mathematical Biosciences 213 (2008) 1–12] where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens’ rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.en_US
dc.description.librarianhb2013en_US
dc.description.sponsorshipSouth African National Research Foundation.Y. D. has been partially supported by the SIT feasibility programme in Reunion, jointly funded by the French Ministry of Health and the European Regional Development Fund (ERDF).en_US
dc.description.urihttp://www.elsevier.com/locate/camen_US
dc.identifier.citationAnguelov, R, Dumont, Y, Lubuma, JM-S & Shillor, M 2014 , 'Dynamically consistent nonstandard finite difference schemes for epidemiological models', Journal of Computational and Applied Mathematics, vol. 255, no. 1, pp. 161-182.en_US
dc.identifier.issn0377-0427 (print)
dc.identifier.issn1879-1778 (online)
dc.identifier.other10.1016/j.cam.2013.04.042
dc.identifier.urihttp://hdl.handle.net/2263/21861
dc.language.isoenen_US
dc.publisherElsevieren_US
dc.rights© 2013 Elsevier. All rights reserved. Notice : this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics, vol. 255, no. 1, 2014. DOI : 10.1016/j.cam.2013.04.042en_US
dc.subjectEpidemiological modelingen_US
dc.subjectDynamical systemsen_US
dc.subjectAsymptotic stabilityen_US
dc.subjectNonstandard finite difference methoden_US
dc.subjectSimulationsen_US
dc.titleDynamically consistent nonstandard finite difference schemes for epidemiological modelsen_US
dc.typePostprint Articleen_US

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