Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model
Identifieur interne : 000490 ( Pmc/Checkpoint ); précédent : 000489; suivant : 000491Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model
Auteurs : David Yaro ; Wilson Osafo Apeanti ; Saviour Worlanyo Akuamoah ; Dianchen LuSource :
- International Journal of Applied and Computational Mathematics [ 2349-5103 ] ; 2019.
Abstract
The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number
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DOI: 10.1007/s40819-019-0699-7
PubMed: NONE
PubMed Central: 7134539
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model</title><author><name sortKey="Yaro, David" sort="Yaro, David" uniqKey="Yaro D" first="David" last="Yaro">David Yaro</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Apeanti, Wilson Osafo" sort="Apeanti, Wilson Osafo" uniqKey="Apeanti W" first="Wilson Osafo" last="Apeanti">Wilson Osafo Apeanti</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Akuamoah, Saviour Worlanyo" sort="Akuamoah, Saviour Worlanyo" uniqKey="Akuamoah S" first="Saviour Worlanyo" last="Akuamoah">Saviour Worlanyo Akuamoah</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Lu, Dianchen" sort="Lu, Dianchen" uniqKey="Lu D" first="Dianchen" last="Lu">Dianchen Lu</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmc">7134539</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7134539</idno><idno type="RBID">PMC:7134539</idno><idno type="doi">10.1007/s40819-019-0699-7</idno><idno type="pmid">NONE</idno><date when="2019">2019</date><idno type="wicri:Area/Pmc/Corpus">000395</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000395</idno><idno type="wicri:Area/Pmc/Curation">000395</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Curation">000395</idno><idno type="wicri:Area/Pmc/Checkpoint">000490</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Checkpoint">000490</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model</title><author><name sortKey="Yaro, David" sort="Yaro, David" uniqKey="Yaro D" first="David" last="Yaro">David Yaro</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Apeanti, Wilson Osafo" sort="Apeanti, Wilson Osafo" uniqKey="Apeanti W" first="Wilson Osafo" last="Apeanti">Wilson Osafo Apeanti</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Akuamoah, Saviour Worlanyo" sort="Akuamoah, Saviour Worlanyo" uniqKey="Akuamoah S" first="Saviour Worlanyo" last="Akuamoah">Saviour Worlanyo Akuamoah</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author><author><name sortKey="Lu, Dianchen" sort="Lu, Dianchen" uniqKey="Lu D" first="Dianchen" last="Lu">Dianchen Lu</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author></analytic><series><title level="j">International Journal of Applied and Computational Mathematics</title><idno type="ISSN">2349-5103</idno><idno type="eISSN">2199-5796</idno><imprint><date when="2019">2019</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p id="Par1">The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number <inline-formula id="IEq1"><alternatives><tex-math id="M1">\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {R}_{0} $$\end{document}</tex-math><mml:math id="M2"><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula>. We discuss the stability of the disease-free equilibrium when <inline-formula id="IEq2"><alternatives><tex-math id="M3">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} < 1$$\end{document}</tex-math><mml:math id="M4"><mml:mrow><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, and the necessary conditions for the stability of the endemic equilibrium when <inline-formula id="IEq3"><alternatives><tex-math id="M5">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} > 1 $$\end{document}</tex-math><mml:math id="M6"><mml:mrow><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>. A sensitivity analysis shows that <inline-formula id="IEq4"><alternatives><tex-math id="M7">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} $$\end{document}</tex-math><mml:math id="M8"><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula> is most sensitive to the probability of the disease transmission rate. The results from the numerical simulations of optimal control strategies disclose that the utmost way of controlling or probably eradicating the transmission of respiratory diseases should be quarantining the exposed individuals, monitoring and treating infected people for a substantial period.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Pongsumpum, P" uniqKey="Pongsumpum P">P Pongsumpum</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Tan, X" uniqKey="Tan X">X Tan</name></author><author><name sortKey="Yuan, L" uniqKey="Yuan L">L Yuan</name></author><author><name sortKey="Zhou, J" uniqKey="Zhou J">J Zhou</name></author><author><name sortKey="Zheng, Y" uniqKey="Zheng Y">Y Zheng</name></author><author><name sortKey="Yang, F" uniqKey="Yang F">F Yang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lee, J" uniqKey="Lee J">J Lee</name></author><author><name 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Int J Appl Comput Math</journal-id><journal-id journal-id-type="iso-abbrev">Int J Appl Comput Math</journal-id><journal-title-group><journal-title>International Journal of Applied and Computational Mathematics</journal-title></journal-title-group><issn pub-type="ppub">2349-5103</issn><issn pub-type="epub">2199-5796</issn><publisher><publisher-name>Springer India</publisher-name><publisher-loc>New Delhi</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmc">7134539</article-id><article-id pub-id-type="publisher-id">699</article-id><article-id pub-id-type="doi">10.1007/s40819-019-0699-7</article-id><article-categories><subj-group subj-group-type="heading"><subject>Original Paper</subject></subj-group></article-categories><title-group><article-title>Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">http://orcid.org/0000-0002-6382-9727</contrib-id><name><surname>Yaro</surname><given-names>David</given-names></name><address><email>ortaega36@yahoo.com</email></address><xref ref-type="aff" rid="Aff1"></xref></contrib><contrib contrib-type="author"><name><surname>Apeanti</surname><given-names>Wilson Osafo</given-names></name><address><email>woapeanti@outlook.com</email></address><xref ref-type="aff" rid="Aff1"></xref></contrib><contrib contrib-type="author"><name><surname>Akuamoah</surname><given-names>Saviour Worlanyo</given-names></name><address><email>awalimemab@yahoo.com</email></address><xref ref-type="aff" rid="Aff1"></xref></contrib><contrib contrib-type="author"><name><surname>Lu</surname><given-names>Dianchen</given-names></name><address><email>dclu@ujs.edu.cn</email></address><xref ref-type="aff" rid="Aff1"></xref></contrib><aff id="Aff1"><institution-wrap><institution-id institution-id-type="ISNI">0000 0001 0743 511X</institution-id><institution-id institution-id-type="GRID">grid.440785.a</institution-id><institution>Faculty of Science,</institution><institution>Jiangsu University,</institution></institution-wrap>ZhenJiang, People’s Republic of China</aff></contrib-group><pub-date pub-type="epub"><day>15</day><month>7</month><year>2019</year></pub-date><pub-date pub-type="ppub"><year>2019</year></pub-date><volume>5</volume><issue>4</issue><elocation-id>116</elocation-id><permissions><copyright-statement>© Springer Nature India Private Limited 2019</copyright-statement><license><license-p>This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.</license-p></license></permissions><abstract id="Abs1"><p id="Par1">The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number <inline-formula id="IEq1"><alternatives><tex-math id="M1">\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {R}_{0} $$\end{document}</tex-math><mml:math id="M2"><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula>. We discuss the stability of the disease-free equilibrium when <inline-formula id="IEq2"><alternatives><tex-math id="M3">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} < 1$$\end{document}</tex-math><mml:math id="M4"><mml:mrow><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, and the necessary conditions for the stability of the endemic equilibrium when <inline-formula id="IEq3"><alternatives><tex-math id="M5">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} > 1 $$\end{document}</tex-math><mml:math id="M6"><mml:mrow><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>. A sensitivity analysis shows that <inline-formula id="IEq4"><alternatives><tex-math id="M7">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{0} $$\end{document}</tex-math><mml:math id="M8"><mml:msub><mml:mi mathvariant="script">R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="40819_2019_699_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula> is most sensitive to the probability of the disease transmission rate. The results from the numerical simulations of optimal control strategies disclose that the utmost way of controlling or probably eradicating the transmission of respiratory diseases should be quarantining the exposed individuals, monitoring and treating infected people for a substantial period.</p></abstract><kwd-group xml:lang="en"><title>Keywords</title><kwd>Fractional calculus</kwd><kwd>Caputo fractional derivative</kwd><kwd>Respiratory epidemic model</kwd><kwd>Stability analysis</kwd><kwd>Optimal control</kwd><kwd>Numerical simulations</kwd></kwd-group><custom-meta-group><custom-meta><meta-name>issue-copyright-statement</meta-name><meta-value>© Springer Nature India Private Limited 2019</meta-value></custom-meta></custom-meta-group></article-meta></front></pmc><affiliations><list></list><tree><noCountry><name sortKey="Akuamoah, Saviour Worlanyo" sort="Akuamoah, Saviour Worlanyo" uniqKey="Akuamoah S" first="Saviour Worlanyo" last="Akuamoah">Saviour Worlanyo Akuamoah</name><name sortKey="Apeanti, Wilson Osafo" sort="Apeanti, Wilson Osafo" uniqKey="Apeanti W" first="Wilson Osafo" last="Apeanti">Wilson Osafo Apeanti</name><name sortKey="Lu, Dianchen" sort="Lu, Dianchen" uniqKey="Lu D" first="Dianchen" last="Lu">Dianchen Lu</name><name sortKey="Yaro, David" sort="Yaro, David" uniqKey="Yaro D" first="David" last="Yaro">David Yaro</name></noCountry></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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