Time-continuous and time-discrete SIR models revisited: theory and applications.
Identifieur interne : 000084 ( Main/Exploration ); précédent : 000083; suivant : 000085Time-continuous and time-discrete SIR models revisited: theory and applications.
Auteurs : Benjamin Wacker [Allemagne] ; Jan Schlüter [Allemagne]Source :
- Advances in difference equations [ 1687-1839 ] ; 2020.
Abstract
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.
DOI: 10.1186/s13662-020-02995-1
PubMed: 33042201
PubMed Central: PMC7538854
Affiliations:
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applications.</title><author><name sortKey="Wacker, Benjamin" sort="Wacker, Benjamin" uniqKey="Wacker B" first="Benjamin" last="Wacker">Benjamin Wacker</name><affiliation wicri:level="3"><nlm:affiliation>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany.</nlm:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen</wicri:regionArea><placeName><region type="land" nuts="2">Basse-Saxe</region><settlement type="city">Göttingen</settlement></placeName></affiliation></author><author><name sortKey="Schluter, Jan" sort="Schluter, Jan" uniqKey="Schluter J" first="Jan" last="Schlüter">Jan Schlüter</name><affiliation wicri:level="3"><nlm:affiliation>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany.</nlm:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen</wicri:regionArea><placeName><region type="land" nuts="2">Basse-Saxe</region><settlement type="city">Göttingen</settlement></placeName></affiliation><affiliation wicri:level="3"><nlm:affiliation>Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany.</nlm:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen</wicri:regionArea><placeName><region type="land" nuts="2">Basse-Saxe</region><settlement type="city">Göttingen</settlement></placeName></affiliation></author></analytic><series><title level="j">Advances in difference equations</title><idno type="ISSN">1687-1839</idno><imprint><date when="2020" type="published">2020</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">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.</div></front></TEI><pubmed><MedlineCitation Status="PubMed-not-MEDLINE" Owner="NLM"><PMID Version="1">33042201</PMID><DateRevised><Year>2020</Year><Month>10</Month><Day>13</Day></DateRevised><Article PubModel="Print-Electronic"><Journal><ISSN IssnType="Print">1687-1839</ISSN><JournalIssue CitedMedium="Print"><Volume>2020</Volume><Issue>1</Issue><PubDate><Year>2020</Year></PubDate></JournalIssue><Title>Advances in difference equations</Title><ISOAbbreviation>Adv Differ Equ</ISOAbbreviation></Journal><ArticleTitle>Time-continuous and time-discrete SIR models revisited: theory and applications.</ArticleTitle><Pagination><MedlinePgn>556</MedlinePgn></Pagination><ELocationID EIdType="doi" ValidYN="Y">10.1186/s13662-020-02995-1</ELocationID><Abstract><AbstractText>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.</AbstractText><CopyrightInformation>© The Author(s) 2020.</CopyrightInformation></Abstract><AuthorList CompleteYN="Y"><Author ValidYN="Y"><LastName>Wacker</LastName><ForeName>Benjamin</ForeName><Initials>B</Initials><Identifier Source="ORCID">0000-0002-0071-8819</Identifier><AffiliationInfo><Affiliation>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany.</Affiliation><Identifier Source="GRID">grid.419514.c</Identifier><Identifier Source="ISNI">0000 0004 0491 5187</Identifier></AffiliationInfo></Author><Author ValidYN="Y"><LastName>Schlüter</LastName><ForeName>Jan</ForeName><Initials>J</Initials><Identifier Source="ORCID">0000-0002-3363-8663</Identifier><AffiliationInfo><Affiliation>Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany.</Affiliation><Identifier Source="GRID">grid.419514.c</Identifier><Identifier Source="ISNI">0000 0004 0491 5187</Identifier></AffiliationInfo><AffiliationInfo><Affiliation>Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany.</Affiliation><Identifier Source="GRID">grid.7450.6</Identifier><Identifier Source="ISNI">0000 0001 2364 4210</Identifier></AffiliationInfo></Author></AuthorList><Language>eng</Language><PublicationTypeList><PublicationType UI="D016428">Journal Article</PublicationType></PublicationTypeList><ArticleDate DateType="Electronic"><Year>2020</Year><Month>10</Month><Day>07</Day></ArticleDate></Article><MedlineJournalInfo><Country>Germany</Country><MedlineTA>Adv Differ Equ</MedlineTA><NlmUniqueID>101670234</NlmUniqueID><ISSNLinking>1687-1839</ISSNLinking></MedlineJournalInfo><KeywordList Owner="NOTNLM"><Keyword MajorTopicYN="N">COVID-19</Keyword><Keyword MajorTopicYN="N">Difference equations</Keyword><Keyword MajorTopicYN="N">Existence and uniqueness</Keyword><Keyword MajorTopicYN="N">Mathematical epidemiology</Keyword><Keyword MajorTopicYN="N">Nonlinear ordinary differential equations</Keyword><Keyword MajorTopicYN="N">Numerical analysis</Keyword><Keyword MajorTopicYN="N">SIR model</Keyword><Keyword MajorTopicYN="N">Well-posedness</Keyword></KeywordList><CoiStatement>Competing interestsThe authors declare that they have no competing interests.</CoiStatement></MedlineCitation><PubmedData><History><PubMedPubDate PubStatus="received"><Year>2020</Year><Month>05</Month><Day>12</Day></PubMedPubDate><PubMedPubDate PubStatus="accepted"><Year>2020</Year><Month>09</Month><Day>22</Day></PubMedPubDate><PubMedPubDate PubStatus="entrez"><Year>2020</Year><Month>10</Month><Day>12</Day><Hour>5</Hour><Minute>29</Minute></PubMedPubDate><PubMedPubDate PubStatus="pubmed"><Year>2020</Year><Month>10</Month><Day>13</Day><Hour>6</Hour><Minute>0</Minute></PubMedPubDate><PubMedPubDate PubStatus="medline"><Year>2020</Year><Month>10</Month><Day>13</Day><Hour>6</Hour><Minute>1</Minute></PubMedPubDate></History><PublicationStatus>ppublish</PublicationStatus><ArticleIdList><ArticleId IdType="pubmed">33042201</ArticleId><ArticleId IdType="doi">10.1186/s13662-020-02995-1</ArticleId><ArticleId IdType="pii">2995</ArticleId><ArticleId IdType="pmc">PMC7538854</ArticleId></ArticleIdList><pmc-dir>pmcsd</pmc-dir>
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