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Time-continuous and time-discrete SIR models revisited: theory and applications

Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this...

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Autores principales: Wacker, Benjamin, Schlüter, Jan
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7538854/
https://www.ncbi.nlm.nih.gov/pubmed/33042201
http://dx.doi.org/10.1186/s13662-020-02995-1
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author Wacker, Benjamin
Schlüter, Jan
author_facet Wacker, Benjamin
Schlüter, Jan
author_sort Wacker, Benjamin
collection PubMed
description Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people–infectious people–recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.
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spelling pubmed-75388542020-10-07 Time-continuous and time-discrete SIR models revisited: theory and applications Wacker, Benjamin Schlüter, Jan Adv Differ Equ Research Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people–infectious people–recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran. Springer International Publishing 2020-10-07 2020 /pmc/articles/PMC7538854/ /pubmed/33042201 http://dx.doi.org/10.1186/s13662-020-02995-1 Text en © The Author(s) 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Research
Wacker, Benjamin
Schlüter, Jan
Time-continuous and time-discrete SIR models revisited: theory and applications
title Time-continuous and time-discrete SIR models revisited: theory and applications
title_full Time-continuous and time-discrete SIR models revisited: theory and applications
title_fullStr Time-continuous and time-discrete SIR models revisited: theory and applications
title_full_unstemmed Time-continuous and time-discrete SIR models revisited: theory and applications
title_short Time-continuous and time-discrete SIR models revisited: theory and applications
title_sort time-continuous and time-discrete sir models revisited: theory and applications
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7538854/
https://www.ncbi.nlm.nih.gov/pubmed/33042201
http://dx.doi.org/10.1186/s13662-020-02995-1
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