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Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach

Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transf...

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Detalles Bibliográficos
Autores principales: Prathom, Kiattisak, Jampeepan, Asama
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10313017/
https://www.ncbi.nlm.nih.gov/pubmed/37390099
http://dx.doi.org/10.1371/journal.pone.0287556
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author Prathom, Kiattisak
Jampeepan, Asama
author_facet Prathom, Kiattisak
Jampeepan, Asama
author_sort Prathom, Kiattisak
collection PubMed
description Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transforming the SIR model to an equivalent differential equation. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model’s solutions. The derived Dirichlet solution not only matches the numerical solution obtained by the fourth-order Runge-Kutta method (RK-4), but it also carries the long-run behavior of the system. The SIR solutions obtained by the RK-4 method, an approximated analytical solution, and the Dirichlet series approximants are graphically compared. The Dirichlet series approximants order 15 and the RK-4 method are almost perfectly matched with the mean square error less than 2 × 10(−5). A specific Dirichlet series is considered in the case of the SEIR model. The process to obtain a numerical solution is done in the similar way. The graphical comparisons of the solutions achieved by the Dirichlet series approximants order 20 and the RK-4 method show that both methods produce almost the same solution. The mean square errors of the Dirichlet series approximants order 20 in this case are less than 1.2 × 10(−4).
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spelling pubmed-103130172023-07-01 Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach Prathom, Kiattisak Jampeepan, Asama PLoS One Research Article Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transforming the SIR model to an equivalent differential equation. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model’s solutions. The derived Dirichlet solution not only matches the numerical solution obtained by the fourth-order Runge-Kutta method (RK-4), but it also carries the long-run behavior of the system. The SIR solutions obtained by the RK-4 method, an approximated analytical solution, and the Dirichlet series approximants are graphically compared. The Dirichlet series approximants order 15 and the RK-4 method are almost perfectly matched with the mean square error less than 2 × 10(−5). A specific Dirichlet series is considered in the case of the SEIR model. The process to obtain a numerical solution is done in the similar way. The graphical comparisons of the solutions achieved by the Dirichlet series approximants order 20 and the RK-4 method show that both methods produce almost the same solution. The mean square errors of the Dirichlet series approximants order 20 in this case are less than 1.2 × 10(−4). Public Library of Science 2023-06-30 /pmc/articles/PMC10313017/ /pubmed/37390099 http://dx.doi.org/10.1371/journal.pone.0287556 Text en © 2023 Prathom, Jampeepan https://creativecommons.org/licenses/by/4.0/This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
spellingShingle Research Article
Prathom, Kiattisak
Jampeepan, Asama
Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title_full Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title_fullStr Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title_full_unstemmed Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title_short Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach
title_sort direct numerical solutions of the sir and seir models via the dirichlet series approach
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10313017/
https://www.ncbi.nlm.nih.gov/pubmed/37390099
http://dx.doi.org/10.1371/journal.pone.0287556
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