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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Estimation of basic reproduction number of the Middle East respiratory syndrome coronavirus (MERS-CoV) during the outbreak in South Korea, 2015</title><author><name sortKey="Chang, Hyuk Jun" sort="Chang, Hyuk Jun" uniqKey="Chang H" first="Hyuk-Jun" last="Chang">Hyuk-Jun Chang</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">28610609</idno><idno type="pmc">5470331</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5470331</idno><idno type="RBID">PMC:5470331</idno><idno type="doi">10.1186/s12938-017-0370-7</idno><date when="2017">2017</date><idno type="wicri:Area/Pmc/Corpus">000340</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000340</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Estimation of basic reproduction number of the Middle East respiratory syndrome coronavirus (MERS-CoV) during the outbreak in South Korea, 2015</title><author><name sortKey="Chang, Hyuk Jun" sort="Chang, Hyuk Jun" uniqKey="Chang H" first="Hyuk-Jun" last="Chang">Hyuk-Jun Chang</name><affiliation><nlm:aff id="Aff1"></nlm:aff></affiliation></author></analytic><series><title level="j">BioMedical Engineering OnLine</title><idno type="eISSN">1475-925X</idno><imprint><date when="2017">2017</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><sec><title>Background</title><p>In South Korea, an outbreak of Middle East respiratory syndrome (MERS) occurred in 2015. It was the second largest MERS outbreak. As a result of the outbreak in South Korea, 186 infections were reported, and 36 patients died. At least 16,693 people were isolated with suspicious symptoms. This paper estimates the basic reproduction number of the MERS coronavirus (CoV), using data on the spread of MERS in South Korea.</p></sec><sec><title>Methods</title><p>The basic reproduction number of an epidemic is defined as the average number of secondary cases that an infected subject produces over its infectious period in a susceptible and uninfected population. To estimate the basic reproduction number of the MERS-CoV, we employ data from the 2015 South Korea MERS outbreak and the susceptible-infected-removed (SIR) model, a mathematical model that uses a set of ordinary differential equations (ODEs).</p></sec><sec><title>Results</title><p>We fit the model to the epidemic data of the South Korea outbreak minimizing the sum of the squared errors to identify model parameters. Also we derive the basic reproductive number as the terms of the parameters of the SIR model. Then we determine the basic reproduction number of the MERS-CoV in South Korea in 2015 as 8.0977. It is worth comparing with the basic reproductive number of the 2014 Ebola outbreak in West Africa including Guinea, Sierra Leone, and Liberia, which had values of 1.5–2.5.</p></sec><sec><title>Conclusions</title><p>There was no intervention to control the infection in the early phase of the outbreak, thus the data used here provide the best conditions to evaluate the epidemic characteristics of MERS, such as the basic reproduction number. An evaluation of basic reproduction number using epidemic data could be problematic if there are stochastic fluctuations in the early phase of the outbreak, or if the report is not accurate and there is bias in the data. Such problems are not relevant to this study because the data used here were precisely reported and verified by Korea Hospital Association.</p></sec></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Su, S" uniqKey="Su S">S Su</name></author><author><name sortKey="Wong, G" uniqKey="Wong G">G Wong</name></author><author><name sortKey="Liu, Y" uniqKey="Liu Y">Y Liu</name></author><author><name sortKey="Gao, Gf" uniqKey="Gao G">GF Gao</name></author><author><name sortKey="Li, S" uniqKey="Li S">S Li</name></author><author><name sortKey="Bi, Y" uniqKey="Bi Y">Y Bi</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hyun, Mh" uniqKey="Hyun M">MH Hyun</name></author><author><name sortKey="Park, Y" uniqKey="Park Y">Y Park</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Anderson, Rm" uniqKey="Anderson R">RM Anderson</name></author><author><name sortKey="May, Rm" uniqKey="May R">RM May</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Diekmann, O" uniqKey="Diekmann O">O Diekmann</name></author><author><name sortKey="Heesterbeek, Jap" uniqKey="Heesterbeek J">JAP Heesterbeek</name></author><author><name sortKey="Metz, Jaj" uniqKey="Metz J">JAJ Metz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Heesterbeek, Jap" uniqKey="Heesterbeek J">JAP Heesterbeek</name></author><author><name sortKey="Dietz, K" uniqKey="Dietz K">K Dietz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Diekmann, O" uniqKey="Diekmann O">O Diekmann</name></author><author><name sortKey="Heesterbeek, Jap" uniqKey="Heesterbeek J">JAP Heesterbeek</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Van Den Driessche, P" uniqKey="Van Den Driessche P">P van den Driessche</name></author><author><name sortKey="Watmough, J" uniqKey="Watmough J">J Watmough</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Heesterbeek, Jap" uniqKey="Heesterbeek J">JAP Heesterbeek</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Heffernan, Jm" uniqKey="Heffernan J">JM Heffernan</name></author><author><name sortKey="Smith, Rj" uniqKey="Smith R">RJ Smith</name></author><author><name sortKey="Wahl, Lm" uniqKey="Wahl L">LM Wahl</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chowell, G" uniqKey="Chowell G">G Chowell</name></author><author><name sortKey="Blumberg, S" uniqKey="Blumberg S">S Blumberg</name></author><author><name sortKey="Simonsen, L" uniqKey="Simonsen L">L Simonsen</name></author><author><name sortKey="Miller, Ma" uniqKey="Miller M">MA Miller</name></author><author><name sortKey="Viboud, C" uniqKey="Viboud C">C Viboud</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Distefano, J" uniqKey="Distefano J">J DiStefano</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Yashima, K" uniqKey="Yashima K">K Yashima</name></author><author><name sortKey="Sasaki, A" uniqKey="Sasaki A">A Sasaki</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Anderson, Rm" uniqKey="Anderson R">RM Anderson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Weiss, H" uniqKey="Weiss H">H Weiss</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cowling, Bj" uniqKey="Cowling B">BJ Cowling</name></author><author><name sortKey="Park, M" uniqKey="Park M">M Park</name></author><author><name sortKey="Fang, Vj" uniqKey="Fang V">VJ Fang</name></author><author><name sortKey="Wu, P" uniqKey="Wu P">P Wu</name></author><author><name sortKey="Leung, Gm" uniqKey="Leung G">GM Leung</name></author><author><name sortKey="Wu, Jt" uniqKey="Wu J">JT Wu</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Wallinga, J" uniqKey="Wallinga J">J Wallinga</name></author><author><name sortKey="Teunis, P" uniqKey="Teunis P">P Teunis</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Mills, Ce" uniqKey="Mills C">CE Mills</name></author><author><name sortKey="Robins, Jm" uniqKey="Robins J">JM Robins</name></author><author><name sortKey="Lipsitch, M" uniqKey="Lipsitch M">M Lipsitch</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Althaus, Cl" uniqKey="Althaus C">CL Althaus</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chowell, G" uniqKey="Chowell G">G Chowell</name></author><author><name sortKey="Hengartner, Nw" uniqKey="Hengartner N">NW Hengartner</name></author><author><name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C Castillo-Chavez</name></author><author><name sortKey="Fenimore, Pw" uniqKey="Fenimore P">PW Fenimore</name></author><author><name sortKey="Hyman, Jm" uniqKey="Hyman J">JM Hyman</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Smith, Dl" uniqKey="Smith D">DL Smith</name></author><author><name sortKey="Mckenzie, Fe" uniqKey="Mckenzie F">FE McKenzie</name></author><author><name sortKey="Snow, Rw" uniqKey="Snow R">RW Snow</name></author><author><name sortKey="Hay, Si" uniqKey="Hay S">SI Hay</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ejima, K" uniqKey="Ejima K">K Ejima</name></author><author><name sortKey="Aihara, K" uniqKey="Aihara K">K Aihara</name></author><author><name sortKey="Nishiura, H" uniqKey="Nishiura H">H Nishiura</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ha, K M" uniqKey="Ha K">K-M Ha</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Assiri, A" uniqKey="Assiri A">A Assiri</name></author><author><name sortKey="Mcgeer, A" uniqKey="Mcgeer A">A McGeer</name></author><author><name sortKey="Perl, Tm" uniqKey="Perl T">TM Perl</name></author><author><name sortKey="Price, Cs" uniqKey="Price C">CS Price</name></author><author><name sortKey="Al Rabeeah, Aa" uniqKey="Al Rabeeah A">AA Al Rabeeah</name></author><author><name sortKey="Cummings, Dat" uniqKey="Cummings D">DAT Cummings</name></author><author><name sortKey="Alabdullatif, Zn" uniqKey="Alabdullatif Z">ZN Alabdullatif</name></author><author><name sortKey="Assad, M" uniqKey="Assad M">M Assad</name></author><author><name sortKey="Almulhim, A" uniqKey="Almulhim A">A Almulhim</name></author><author><name sortKey="Makhdoom, H" uniqKey="Makhdoom H">H Makhdoom</name></author><author><name sortKey="Madani, H" uniqKey="Madani H">H Madani</name></author><author><name sortKey="Alhakeem, R" uniqKey="Alhakeem R">R Alhakeem</name></author><author><name sortKey="Al Tawfiq, Ja" uniqKey="Al Tawfiq J">JA Al-Tawfiq</name></author><author><name sortKey="Cotten, M" uniqKey="Cotten M">M Cotten</name></author><author><name sortKey="Watson, Sj" uniqKey="Watson S">SJ Watson</name></author><author><name sortKey="Kellam, P" uniqKey="Kellam P">P Kellam</name></author><author><name sortKey="Zumla, Ai" uniqKey="Zumla A">AI Zumla</name></author><author><name sortKey="Memish, Za" uniqKey="Memish Z">ZA Memish</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Area, I" uniqKey="Area I">I Area</name></author><author><name sortKey="Batarfi, H" uniqKey="Batarfi H">H Batarfi</name></author><author><name sortKey="Losada, J" uniqKey="Losada J">J Losada</name></author><author><name sortKey="Nieto, Jj" uniqKey="Nieto J">JJ Nieto</name></author><author><name sortKey="Shammakh, W" uniqKey="Shammakh W">W Shammakh</name></author><author><name sortKey="Torres, A" uniqKey="Torres A">Á Torres</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Nishiura, H" uniqKey="Nishiura H">H Nishiura</name></author><author><name sortKey="Ejima, K" uniqKey="Ejima K">K Ejima</name></author><author><name sortKey="Mizumoto, K" uniqKey="Mizumoto K">K Mizumoto</name></author><author><name sortKey="Nakaoka, S" uniqKey="Nakaoka S">S Nakaoka</name></author><author><name sortKey="Inaba, H" uniqKey="Inaba H">H Inaba</name></author><author><name sortKey="Imoto, S" uniqKey="Imoto S">S Imoto</name></author><author><name sortKey="Yamaguchi, R" uniqKey="Yamaguchi R">R Yamaguchi</name></author><author><name sortKey="Saito, M" uniqKey="Saito M">M Saito</name></author></analytic></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Biomed Eng Online</journal-id><journal-id journal-id-type="iso-abbrev">Biomed Eng Online</journal-id><journal-title-group><journal-title>BioMedical Engineering OnLine</journal-title></journal-title-group><issn pub-type="epub">1475-925X</issn><publisher><publisher-name>BioMed Central</publisher-name><publisher-loc>London</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">28610609</article-id><article-id pub-id-type="pmc">5470331</article-id><article-id pub-id-type="publisher-id">370</article-id><article-id pub-id-type="doi">10.1186/s12938-017-0370-7</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research</subject></subj-group></article-categories><title-group><article-title>Estimation of basic reproduction number of the Middle East respiratory syndrome coronavirus (MERS-CoV) during the outbreak in South Korea, 2015</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name><surname>Chang</surname><given-names>Hyuk-Jun</given-names></name><address><email>hchang@kookmin.ac.kr</email></address><xref ref-type="aff" rid="Aff1"></xref></contrib><aff id="Aff1"><institution-wrap><institution-id institution-id-type="ISNI">0000 0001 0788 9816</institution-id><institution-id institution-id-type="GRID">grid.91443.3b</institution-id><institution>School of Electrical Engineering,</institution><institution>Kookmin University,</institution></institution-wrap>77 Jeongneung-ro, Seongbuk-gu, 136-702 Seoul, Korea</aff></contrib-group><pub-date pub-type="epub"><day>13</day><month>6</month><year>2017</year></pub-date><pub-date pub-type="pmc-release"><day>13</day><month>6</month><year>2017</year></pub-date><pub-date pub-type="collection"><year>2017</year></pub-date><volume>16</volume><elocation-id>79</elocation-id><history><date date-type="received"><day>31</day><month>1</month><year>2017</year></date><date date-type="accepted"><day>5</day><month>6</month><year>2017</year></date></history><permissions><copyright-statement>© The Author(s) 2017</copyright-statement><license license-type="OpenAccess"><license-p><bold>Open Access</bold>This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link>), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/publicdomain/zero/1.0/">http://creativecommons.org/publicdomain/zero/1.0/</ext-link>) applies to the data made available in this article, unless otherwise stated.</license-p></license></permissions><abstract id="Abs1"><sec><title>Background</title><p>In South Korea, an outbreak of Middle East respiratory syndrome (MERS) occurred in 2015. It was the second largest MERS outbreak. As a result of the outbreak in South Korea, 186 infections were reported, and 36 patients died. At least 16,693 people were isolated with suspicious symptoms. This paper estimates the basic reproduction number of the MERS coronavirus (CoV), using data on the spread of MERS in South Korea.</p></sec><sec><title>Methods</title><p>The basic reproduction number of an epidemic is defined as the average number of secondary cases that an infected subject produces over its infectious period in a susceptible and uninfected population. To estimate the basic reproduction number of the MERS-CoV, we employ data from the 2015 South Korea MERS outbreak and the susceptible-infected-removed (SIR) model, a mathematical model that uses a set of ordinary differential equations (ODEs).</p></sec><sec><title>Results</title><p>We fit the model to the epidemic data of the South Korea outbreak minimizing the sum of the squared errors to identify model parameters. Also we derive the basic reproductive number as the terms of the parameters of the SIR model. Then we determine the basic reproduction number of the MERS-CoV in South Korea in 2015 as 8.0977. It is worth comparing with the basic reproductive number of the 2014 Ebola outbreak in West Africa including Guinea, Sierra Leone, and Liberia, which had values of 1.5–2.5.</p></sec><sec><title>Conclusions</title><p>There was no intervention to control the infection in the early phase of the outbreak, thus the data used here provide the best conditions to evaluate the epidemic characteristics of MERS, such as the basic reproduction number. An evaluation of basic reproduction number using epidemic data could be problematic if there are stochastic fluctuations in the early phase of the outbreak, or if the report is not accurate and there is bias in the data. Such problems are not relevant to this study because the data used here were precisely reported and verified by Korea Hospital Association.</p></sec></abstract><kwd-group xml:lang="en"><title>Keywords</title><kwd>Basic reproduction number</kwd><kwd>Susceptible-infected-removed (SIR) model</kwd><kwd>Middle East respiratory syndrome (MERS)</kwd><kwd>MERS coronavirus (MERS-CoV)</kwd></kwd-group><funding-group><award-group><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100003725</institution-id><institution>National Research Foundation of Korea</institution></institution-wrap></funding-source><award-id>NRF-2014R1A1A1003056</award-id></award-group></funding-group><custom-meta-group><custom-meta><meta-name>issue-copyright-statement</meta-name><meta-value>© The Author(s) 2017</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="Sec1"><title>Background</title><p id="Par2">The Middle East respiratory syndrome (MERS) is caused by a coronavirus (CoV), the MERS-CoV. In Saudi Arabia, the first case of the disease was reported in 2012 [<xref ref-type="bibr" rid="CR1">1</xref>]. The first case of MERS in the Republic of Korea was identified on 20 May 2015 [<xref ref-type="bibr" rid="CR2">2</xref>]. A significant outbreak of MERS occurred in South Korea and lasted for almost three months, from May to July 2015.</p><p id="Par3">The 2015 MERS spread in South Korea is the second largest outbreak recorded to date [<xref ref-type="bibr" rid="CR3">3</xref>]. As a result of the outbreak in South Korea, 186 infections were reported, and 36 patients died. At least 16,693 people were isolated with suspicious symptoms [<xref ref-type="bibr" rid="CR4">4</xref>]. This paper evaluates the basic reproduction number of MERS-CoV, using data from the 2015 South Korea outbreak.</p><p id="Par4">The basic reproduction number (generally denoted as <inline-formula id="IEq2"><alternatives><tex-math id="M1">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M2"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>) of an epidemic is defined as the average number of secondary cases that an infected subject produces over its infectious period in a susceptible and uninfected population [<xref ref-type="bibr" rid="CR5">5</xref>–<xref ref-type="bibr" rid="CR11">11</xref>]. It can estimate the growth rate of an infectious disease at the early stage of the outbreak, when most individuals are susceptible [<xref ref-type="bibr" rid="CR12">12</xref>]. The basic reproduction number of an epidemic is useful for determining whether an outbreak of the disease will occur or not [<xref ref-type="bibr" rid="CR13">13</xref>], and for analyzing epidemic properties of the disease further [<xref ref-type="bibr" rid="CR14">14</xref>].</p><p id="Par5">Note that <inline-formula id="IEq3"><alternatives><tex-math id="M3">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M4"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula> is referred to as the basic reproductive (or reproduction) number (or ratio). The basic reproductive (or reproduction) rate is incorrect nomenclature because <inline-formula id="IEq4"><alternatives><tex-math id="M5">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M6"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula> is a dimensionless number that is not related to any physical quantity corresponding to rate.</p><p id="Par6">Based on [<xref ref-type="bibr" rid="CR11">11</xref>], the reasons to estimate the basic reproduction number of an epidemic are summarized as follows: First, we can relatively evaluate the risk of the corresponding epidemic using <inline-formula id="IEq5"><alternatives><tex-math id="M7">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M8"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula>. In other words, we can compare the infectivity of the epidemic with others, already familiar to us. Second, the reproduction number can be evaluated multiple times, e.g., before (and after) an infection control measure intervention. To this end, it is needed to distinguish the reproduction number after control intervention from the basic reproductive number <inline-formula id="IEq6"><alternatives><tex-math id="M9">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M10"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq6.gif"></inline-graphic></alternatives></inline-formula>, which is estimated before the intervention. Now we refer to the reproduction number after control intervention as the effective reproduction number, denoted by <inline-formula id="IEq7"><alternatives><tex-math id="M11">\documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}$$\end{document}</tex-math><mml:math id="M12"><mml:msub><mml:mi>R</mml:mi><mml:mtext>eff</mml:mtext></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq7.gif"></inline-graphic></alternatives></inline-formula>. Then we can compare <inline-formula id="IEq8"><alternatives><tex-math id="M13">\documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}$$\end{document}</tex-math><mml:math id="M14"><mml:msub><mml:mi>R</mml:mi><mml:mtext>eff</mml:mtext></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq8.gif"></inline-graphic></alternatives></inline-formula><bold>with</bold><inline-formula id="IEq9"><alternatives><tex-math id="M15">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M16"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq9.gif"></inline-graphic></alternatives></inline-formula>, and we can evaluate the efficacy of a control measure quantitatively based on <inline-formula id="IEq10"><alternatives><tex-math id="M17">\documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}$$\end{document}</tex-math><mml:math id="M18"><mml:msub><mml:mi>R</mml:mi><mml:mtext>eff</mml:mtext></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq10.gif"></inline-graphic></alternatives></inline-formula>. By doing so, eventually we can determine how to apply control intervention to reduce <inline-formula id="IEq11"><alternatives><tex-math id="M19">\documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}$$\end{document}</tex-math><mml:math id="M20"><mml:msub><mml:mi>R</mml:mi><mml:mtext>eff</mml:mtext></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq11.gif"></inline-graphic></alternatives></inline-formula> to less than one. If <inline-formula id="IEq12"><alternatives><tex-math id="M21">\documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}} < 1$$\end{document}</tex-math><mml:math id="M22"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>eff</mml:mtext></mml:msub><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq12.gif"></inline-graphic></alternatives></inline-formula>, it is concluded that the control intervention works effectively, and that the outbreak will eventually be controlled by reducing the reproduction number to less than the threshold level, 1.</p><p id="Par7">In this paper, to estimate the basic reproduction number of the MERS-CoV, we employ data from the 2015 South Korea MERS outbreak and the susceptible-infected-removed (SIR) model [<xref ref-type="bibr" rid="CR13">13</xref>, <xref ref-type="bibr" rid="CR15">15</xref>, <xref ref-type="bibr" rid="CR16">16</xref>], a mathematical model that uses a set of ordinary differential equations (ODEs). Because the availability of epidemic data is limited, we usually employ non-structured deterministic models to evaluate <inline-formula id="IEq13"><alternatives><tex-math id="M23">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M24"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq13.gif"></inline-graphic></alternatives></inline-formula> [<xref ref-type="bibr" rid="CR11">11</xref>].</p><p id="Par8">Based on the data reported from the 2015 South Korea outbreak of MERS, we evaluate the basic reproductive number of the virus, MERS-CoV. We fit the model to epidemic data from the South Korea outbreak, and identify model parameters and the basic reproduction number, <inline-formula id="IEq14"><alternatives><tex-math id="M25">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M26"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq14.gif"></inline-graphic></alternatives></inline-formula>. Note that other epidemiological parameters, such as incubation period and serial interval, have been discussed in [<xref ref-type="bibr" rid="CR17">17</xref>] for the outbreak.</p><p id="Par9">Preliminary work relating to this paper is presented in [<xref ref-type="bibr" rid="CR18">18</xref>]. This paper includes an analysis of the derivation of the basic reproduction number, careful screening of the reported data, and a sophisticated approach using the sum of the squared errors to evaluate the basic reproduction number precisely.</p><p id="Par10">A number of papers and books have been dedicated to the study of <inline-formula id="IEq15"><alternatives><tex-math id="M27">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M28"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq15.gif"></inline-graphic></alternatives></inline-formula> for other infectious diseases. A few of them are as follows: see [<xref ref-type="bibr" rid="CR19">19</xref>] for the <inline-formula id="IEq16"><alternatives><tex-math id="M29">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M30"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq16.gif"></inline-graphic></alternatives></inline-formula> of severe acute respiratory syndrome(SARS), [<xref ref-type="bibr" rid="CR20">20</xref>] for the <inline-formula id="IEq17"><alternatives><tex-math id="M31">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M32"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq17.gif"></inline-graphic></alternatives></inline-formula> of influenza, [<xref ref-type="bibr" rid="CR21">21</xref>, <xref ref-type="bibr" rid="CR22">22</xref>] for the <inline-formula id="IEq18"><alternatives><tex-math id="M33">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M34"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq18.gif"></inline-graphic></alternatives></inline-formula> of Ebola, and [<xref ref-type="bibr" rid="CR23">23</xref>] for the <inline-formula id="IEq19"><alternatives><tex-math id="M35">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M36"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq19.gif"></inline-graphic></alternatives></inline-formula> of malaria.</p><p id="Par11">Although some literature on the study of <inline-formula id="IEq20"><alternatives><tex-math id="M37">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M38"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq20.gif"></inline-graphic></alternatives></inline-formula> of the MERS-CoV has been reported such as [<xref ref-type="bibr" rid="CR12">12</xref>, <xref ref-type="bibr" rid="CR24">24</xref>], the MERS outbreak in South Korea is unique [<xref ref-type="bibr" rid="CR25">25</xref>]. MERS spread almost naturally without any intervention in the early stage, and the Korean government did not respond appropriately [<xref ref-type="bibr" rid="CR3">3</xref>, <xref ref-type="bibr" rid="CR25">25</xref>]. The list of medical facilities involved was not even announced to public. Ironically, it is the ideal condition to fit a mathematical model to the clinical epidemic data and to evaluate epidemic properties of the MERS-CoV, including the basic reproduction number.</p><p id="Par12">The paper is organized as follows: In “<xref rid="Sec2" ref-type="sec">Methods</xref>” section, a mathematical model, which comprises a set of ordinary differential equations, is introduced, and an estimation method is discussed for the basic reproductive number. “<xref rid="Sec5" ref-type="sec">Results</xref>” section describes the evaluation result, providing further discussion. “<xref rid="Sec6" ref-type="sec">Conclusions</xref>” section concludes the paper by suggesting additional research.</p></sec><sec id="Sec2"><title>Methods</title><p id="Par13">In this section, we briefly discuss the method employed in this paper, including a definition of the basic reproduction number, and we introduce the SIR model.</p><sec id="Sec3"><title>Basic reproduction number</title><p id="Par14">The basic reproduction number is defined as the number of secondary cases that one infected primary subject causes on average in an uninfected and totally susceptible population, over the infectious period [<xref ref-type="bibr" rid="CR8">8</xref>–<xref ref-type="bibr" rid="CR10">10</xref>]. Based on this definition, we obtain a mathematical description of <inline-formula id="IEq21"><alternatives><tex-math id="M39">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M40"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq21.gif"></inline-graphic></alternatives></inline-formula> via the so-called ‘survival function’ [<xref ref-type="bibr" rid="CR7">7</xref>, <xref ref-type="bibr" rid="CR11">11</xref>].</p><p id="Par15">Considering a large population, this description is given by<disp-formula id="Equ1"><label>1</label><alternatives><tex-math id="M41">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_0 = \int _0^\infty {b(a)F(a)}da \end{aligned}$$\end{document}</tex-math><mml:math id="M42" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>∞</mml:mi></mml:msubsup><mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ1.gif" position="anchor"></graphic></alternatives></disp-formula>where <italic>F</italic>(<italic>a</italic>) is the survival function describing the probability of a newly infected patient will be infectious at least to time <italic>a</italic>, and <italic>b</italic>(<italic>a</italic>) is the average number of infected subjects whom one infected patient will produce per unit time at time <italic>a</italic> [<xref ref-type="bibr" rid="CR7">7</xref>, <xref ref-type="bibr" rid="CR11">11</xref>]. The notation of (<xref rid="Equ1" ref-type="">1</xref>) follows the usage therein.</p><p id="Par16">Formula (<xref rid="Equ1" ref-type="">1</xref>) is derived from the <inline-formula id="IEq22"><alternatives><tex-math id="M43">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M44"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq22.gif"></inline-graphic></alternatives></inline-formula> definition and can thus be used for any mathematical model, not just models given by ODEs. However, it requires explicit expressions for <inline-formula id="IEq23"><alternatives><tex-math id="M45">\documentclass[12pt]{minimal}
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\begin{document}$$F(\cdot )$$\end{document}</tex-math><mml:math id="M46"><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq23.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq24"><alternatives><tex-math id="M47">\documentclass[12pt]{minimal}
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\begin{document}$$b(\cdot )$$\end{document}</tex-math><mml:math id="M48"><mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq24.gif"></inline-graphic></alternatives></inline-formula>, which are functions of time. This paper employs the SIR model described by ODEs that is introduced in the next section. Because the model does not provide explicit descriptions for <inline-formula id="IEq25"><alternatives><tex-math id="M49">\documentclass[12pt]{minimal}
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\begin{document}$$F(\cdot )$$\end{document}</tex-math><mml:math id="M50"><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq25.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq26"><alternatives><tex-math id="M51">\documentclass[12pt]{minimal}
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\begin{document}$$b(\cdot )$$\end{document}</tex-math><mml:math id="M52"><mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq26.gif"></inline-graphic></alternatives></inline-formula>, we use an alternative expression for <inline-formula id="IEq27"><alternatives><tex-math id="M53">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M54"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq27.gif"></inline-graphic></alternatives></inline-formula>, derived from the SIR model.</p></sec><sec id="Sec4"><title>SIR model</title><p id="Par17">To investigate the spread dynamics of epidemics, several nonlinear mathematical models have been studied. We employ one of the models, the SIR model that is described by<disp-formula id="Equ2"><label>2</label><alternatives><tex-math id="M55">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dot{S}&= - \beta S I, \nonumber \\ \dot{I}&= \beta S I - \nu I, \nonumber \\ \dot{R}&= \nu I, \end{aligned}$$\end{document}</tex-math><mml:math id="M56" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mover accent="true"><mml:mi>S</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="italic">β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mrow></mml:mrow><mml:mover accent="true"><mml:mi>I</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant="italic">β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mrow></mml:mrow><mml:mover accent="true"><mml:mi>R</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ2.gif" position="anchor"></graphic></alternatives></disp-formula>where the states <italic>S</italic>, <italic>I</italic>, and <italic>R</italic> correspond to the number of susceptible people, the number of infected, and the number of removed, respectively. Note that state <italic>R</italic> includes both deceased and recovered patients. For each subject group of <italic>S</italic>, <italic>I</italic>, and <italic>R</italic>, we assume that the properties of the subjects (for example, infectiveness, susceptibility, and so forth) are homogeneous. The parameter <inline-formula id="IEq28"><alternatives><tex-math id="M57">\documentclass[12pt]{minimal}
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\begin{document}$$\beta$$\end{document}</tex-math><mml:math id="M58"><mml:mi mathvariant="italic">β</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq28.gif"></inline-graphic></alternatives></inline-formula> is the disease transmission rate and the parameter <inline-formula id="IEq29"><alternatives><tex-math id="M59">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M60"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq29.gif"></inline-graphic></alternatives></inline-formula> is the removed rate. Note that both parameters are positive real.</p><p id="Par18">The notation of (<xref rid="Equ2" ref-type="">2</xref>) follows that used in [<xref ref-type="bibr" rid="CR16">16</xref>]. Table <xref rid="Tab1" ref-type="table">1</xref> presents descriptions of system parameters and state variables for model (<xref rid="Equ2" ref-type="">2</xref>). Parameters <inline-formula id="IEq30"><alternatives><tex-math id="M61">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa$$\end{document}</tex-math><mml:math id="M62"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq30.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq31"><alternatives><tex-math id="M63">\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}</tex-math><mml:math id="M64"><mml:mi mathvariant="italic">τ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq31.gif"></inline-graphic></alternatives></inline-formula> are discussed further in the “<xref rid="Sec5" ref-type="sec">Results</xref>” section. For a further explanation of the SIR model, see [<xref ref-type="bibr" rid="CR13">13</xref>, <xref ref-type="bibr" rid="CR15">15</xref>, <xref ref-type="bibr" rid="CR16">16</xref>].<table-wrap id="Tab1"><label>Table 1</label><caption><p>Descriptions of system parameters and state variables of model (<xref rid="Equ2" ref-type="">2</xref>) and the equation (<xref rid="Equ3" ref-type="">3</xref>)</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">State variables, parameters</th><th align="left">Descriptions</th></tr></thead><tbody><tr><td align="left"><italic>S</italic></td><td align="left">Number of susceptible subjects</td></tr><tr><td align="left"><italic>I</italic></td><td align="left">Number of infected subjects</td></tr><tr><td align="left"><italic>R</italic></td><td align="left">Number of removed subjects</td></tr><tr><td align="left"><inline-formula id="IEq32"><alternatives><tex-math id="M65">\documentclass[12pt]{minimal}
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\begin{document}$$\beta$$\end{document}</tex-math><mml:math id="M66"><mml:mi mathvariant="italic">β</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq32.gif"></inline-graphic></alternatives></inline-formula></td><td align="left">Disease transmission rate</td></tr><tr><td align="left"><inline-formula id="IEq33"><alternatives><tex-math id="M67">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M68"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq33.gif"></inline-graphic></alternatives></inline-formula></td><td align="left">Removed rate</td></tr><tr><td align="left"><inline-formula id="IEq34"><alternatives><tex-math id="M69">\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}</tex-math><mml:math id="M70"><mml:mi mathvariant="italic">τ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq34.gif"></inline-graphic></alternatives></inline-formula></td><td align="left">Transmissibility of the infection</td></tr><tr><td align="left"><inline-formula id="IEq35"><alternatives><tex-math id="M71">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa$$\end{document}</tex-math><mml:math id="M72"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq35.gif"></inline-graphic></alternatives></inline-formula></td><td align="left">Number of transmittable contacts by infected patient per unit time</td></tr></tbody></table></table-wrap></p><p id="Par19">Any term related to birth and death in the population that is not caused by MERS is not included in the model (<xref rid="Equ2" ref-type="">2</xref>). The dynamics of the disease (e.g., infection or recovery) is assumed to be significantly faster than that of birth and death in the population. Generally, epidemic models such as SIR do not include birth and death because zero net change of the population is assumed. If we model an infectious disease with comparatively slow dynamics (e.g., an endemic disease), we must consider dynamic terms describing birth and death.</p><p id="Par20">The system parameters, which are all rates, are positive and real in the model (<xref rid="Equ2" ref-type="">2</xref>). Because of the definition of <inline-formula id="IEq36"><alternatives><tex-math id="M73">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M74"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq36.gif"></inline-graphic></alternatives></inline-formula>, an alternative description for <inline-formula id="IEq37"><alternatives><tex-math id="M75">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M76"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq37.gif"></inline-graphic></alternatives></inline-formula> can be derived from model (<xref rid="Equ2" ref-type="">2</xref>):<disp-formula id="Equ3"><label>3</label><alternatives><tex-math id="M77">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_0 = \kappa \tau d, \end{aligned}$$\end{document}</tex-math><mml:math id="M78" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mi>d</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ3.gif" position="anchor"></graphic></alternatives></disp-formula>where <italic>d</italic> is the infection duration, <inline-formula id="IEq38"><alternatives><tex-math id="M79">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa$$\end{document}</tex-math><mml:math id="M80"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq38.gif"></inline-graphic></alternatives></inline-formula> is the transmittable contact number by one infected subject per one unit time, and <inline-formula id="IEq39"><alternatives><tex-math id="M81">\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}</tex-math><mml:math id="M82"><mml:mi mathvariant="italic">τ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq39.gif"></inline-graphic></alternatives></inline-formula> is the <italic>transmissibility</italic> of the infectious disease, which corresponds to the probability of infection per one contact between an infected patient and a susceptible individual.</p><p id="Par21">It is notable that the removed rate is reciprocal to the infection duration by the assumption of constant rates for the SIR model (<xref rid="Equ2" ref-type="">2</xref>):<disp-formula id="Equ9"><alternatives><tex-math id="M83">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d = \frac{1}{\nu }. \end{aligned}$$\end{document}</tex-math><mml:math id="M84" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi mathvariant="italic">ν</mml:mi></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ9.gif" position="anchor"></graphic></alternatives></disp-formula></p></sec></sec><sec id="Sec5"><title>Results</title><p id="Par22">In Table <xref rid="Tab2" ref-type="table">2</xref>, the history of the MERS-CoV spread status is presented. The Ministry of Health and Welfare, Korea officially announced the data. “Infected” represents the accumulated number of infected patients. “Deceased” is the number of dead subjects. “Recovered” is the number of individuals returning to healthy status. All entries in the table are as of the “Date”. The number of infected patients includes both removed and recovered patients.</p><p id="Par23">Table <xref rid="Tab2" ref-type="table">2</xref> shows the MERS-CoV spread data for only the initial phase of the outbreak, i.e., from 20 May 2015 to 12 June 2015.</p><p id="Par24">On 7 June 2015, the South Korean government disclosed to the public the list of all hospitals exposed to MERS-CoV, with the dates and duration of exposure [<xref ref-type="bibr" rid="CR4">4</xref>]. This is the first intervention of the government to control the spread. Before this date, there was no control action that could affect estimation of the basic reproduction number of MERS-CoV.</p><p id="Par25">The incubation period of MERS-CoV that can range from 2 to 14 days, is 5 days on average [<xref ref-type="bibr" rid="CR26">26</xref>]. Thus, we use reported data up to 12 June 2015.<table-wrap id="Tab2"><label>Table 2</label><caption><p>Accumulated MERS-CoV patients in Korea, 2015 [<xref ref-type="bibr" rid="CR4">4</xref>]</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">Date</th><th align="left">Infected</th><th align="left">Deceased</th><th align="left">Recovered</th></tr></thead><tbody><tr><td align="left">20 May</td><td char="." align="char">2</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">21 May</td><td char="." align="char">3</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">22 May</td><td char="." align="char">3</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">23 May</td><td char="." align="char">3</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">24 May</td><td char="." align="char">3</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">25 May</td><td char="." align="char">3</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">26 May</td><td char="." align="char">5</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">27 May</td><td char="." align="char">5</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">28 May</td><td char="." align="char">7</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">29 May</td><td char="." align="char">13</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">30 May</td><td char="." align="char">15</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">31 May</td><td char="." align="char">18</td><td char="." align="char">0</td><td char="." align="char">0</td></tr><tr><td align="left">1 June</td><td char="." align="char">25</td><td char="." align="char">1</td><td char="." align="char">0</td></tr><tr><td align="left">2 June</td><td char="." align="char">30</td><td char="." align="char">1</td><td char="." align="char">0</td></tr><tr><td align="left">3 June</td><td char="." align="char">30</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">4 June</td><td char="." align="char">36</td><td char="." align="char">4</td><td char="." align="char">0</td></tr><tr><td align="left">5 June</td><td char="." align="char">42</td><td char="." align="char">5</td><td char="." align="char">1</td></tr><tr><td align="left">6 June</td><td char="." align="char">64</td><td char="." align="char">5</td><td char="." align="char">1</td></tr><tr><td align="left">7 June</td><td char="." align="char">87</td><td char="." align="char">5</td><td char="." align="char">1</td></tr><tr><td align="left">8 June</td><td char="." align="char">95</td><td char="." align="char">7</td><td char="." align="char">2</td></tr><tr><td align="left">9 June</td><td char="." align="char">108</td><td char="." align="char">7</td><td char="." align="char">3</td></tr><tr><td align="left">10 June</td><td char="." align="char">122</td><td char="." align="char">9</td><td char="." align="char">4</td></tr><tr><td align="left">11 June</td><td char="." align="char">126</td><td char="." align="char">10</td><td char="." align="char">7</td></tr><tr><td align="left">12 June</td><td char="." align="char">138</td><td char="." align="char">13</td><td char="." align="char">9</td></tr></tbody></table></table-wrap></p><p id="Par26">The total population size is denoted by <italic>N</italic>(<italic>t</italic>):<disp-formula id="Equ10"><alternatives><tex-math id="M85">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} N(t):=S(t)+I(t)+R(t). \end{aligned}$$\end{document}</tex-math><mml:math id="M86" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ10.gif" position="anchor"></graphic></alternatives></disp-formula>It can be seen that<disp-formula id="Equ11"><alternatives><tex-math id="M87">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dot{N}(t)&=\dot{S}(t) + \dot{I}(t) + \dot{R}(t)\\&= 0 \end{aligned}$$\end{document}</tex-math><mml:math id="M88" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mover accent="true"><mml:mi>N</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>S</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mi>I</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mi>R</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ11.gif" position="anchor"></graphic></alternatives></disp-formula>from the SIR model (<xref rid="Equ2" ref-type="">2</xref>) (see the left-hand sides of the SIR model equations and their sum). This implies that <italic>N</italic>(<italic>t</italic>) can be assumed to be constant for the SIR model (<xref rid="Equ2" ref-type="">2</xref>), <italic>i.e.,</italic><inline-formula id="IEq40"><alternatives><tex-math id="M89">\documentclass[12pt]{minimal}
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\begin{document}$$N(t)=N(0):=N$$\end{document}</tex-math><mml:math id="M90"><mml:mrow><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq40.gif"></inline-graphic></alternatives></inline-formula>. With this assumption, we consider the state <italic>S</italic>(<italic>t</italic>) as <inline-formula id="IEq41"><alternatives><tex-math id="M91">\documentclass[12pt]{minimal}
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\begin{document}$$N - I(t) - R(t)$$\end{document}</tex-math><mml:math id="M92"><mml:mrow><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq41.gif"></inline-graphic></alternatives></inline-formula>. Then, we can describe the SIR model (<xref rid="Equ2" ref-type="">2</xref>) as a two-dimensional model.</p><p id="Par27">Considering the magnitude of the numbers in Table <xref rid="Tab2" ref-type="table">2</xref>, it is assumed that<disp-formula id="Equ4"><label>4</label><alternatives><tex-math id="M93">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} N \cong S. \end{aligned}$$\end{document}</tex-math><mml:math id="M94" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≅</mml:mo><mml:mi>S</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ4.gif" position="anchor"></graphic></alternatives></disp-formula>Compared to the total size of the population, the number of outbreak cases is small. The <italic>N</italic> of South Korea is now known to be over 51 million. If the number of outbreak cases is much smaller than the total size of the population, the number <italic>N</italic> does not need to be exact to estimate system parameters of the SIR model (<xref rid="Equ2" ref-type="">2</xref>) [<xref ref-type="bibr" rid="CR21">21</xref>, <xref ref-type="bibr" rid="CR22">22</xref>].</p><p id="Par28">From [<xref ref-type="bibr" rid="CR16">16</xref>],<disp-formula id="Equ12"><alternatives><tex-math id="M95">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \beta = \frac{\kappa \tau }{N}, \end{aligned}$$\end{document}</tex-math><mml:math id="M96" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi mathvariant="italic">β</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ12.gif" position="anchor"></graphic></alternatives></disp-formula>where <inline-formula id="IEq42"><alternatives><tex-math id="M97">\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}</tex-math><mml:math id="M98"><mml:mi mathvariant="italic">τ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq42.gif"></inline-graphic></alternatives></inline-formula> is the transmissibility of the infectious disease and <inline-formula id="IEq43"><alternatives><tex-math id="M99">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa$$\end{document}</tex-math><mml:math id="M100"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq43.gif"></inline-graphic></alternatives></inline-formula> is the transmittable contact number by one infected subject per one unit time.</p><p id="Par29">Then, the model where the dimension is reduced by assumption (<xref rid="Equ4" ref-type="">4</xref>) is<disp-formula id="Equ5"><label>5</label><alternatives><tex-math id="M101">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dot{I}&= \kappa \tau I - \nu I, \nonumber \\ \dot{R}&= \nu I. \end{aligned}$$\end{document}</tex-math><mml:math id="M102" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mover accent="true"><mml:mi>I</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mrow></mml:mrow><mml:mover accent="true"><mml:mi>R</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>I</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ5.gif" position="anchor"></graphic></alternatives></disp-formula>The initial state for the SIR model is provided by Table <xref rid="Tab2" ref-type="table">2</xref>, given by<disp-formula id="Equ6"><label>6</label><alternatives><tex-math id="M103">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{}[I(0), R(0)]=[2, 0]. \end{aligned}$$\end{document}</tex-math><mml:math id="M104" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mrow></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ6.gif" position="anchor"></graphic></alternatives></disp-formula>Now, we only need to know the parameter pair, (<inline-formula id="IEq44"><alternatives><tex-math id="M105">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M106"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq44.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq45"><alternatives><tex-math id="M107">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M108"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq45.gif"></inline-graphic></alternatives></inline-formula>), to solve model (<xref rid="Equ5" ref-type="">5</xref>) numerically. To solve the mathematical model and to evaluate the basic reproduction number of the model, we do not need to know each <inline-formula id="IEq46"><alternatives><tex-math id="M109">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa$$\end{document}</tex-math><mml:math id="M110"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq46.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq47"><alternatives><tex-math id="M111">\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}</tex-math><mml:math id="M112"><mml:mi mathvariant="italic">τ</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq47.gif"></inline-graphic></alternatives></inline-formula> necessarily, but we can use the value of the product, i.e., <inline-formula id="IEq48"><alternatives><tex-math id="M113">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M114"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq48.gif"></inline-graphic></alternatives></inline-formula>.</p><p id="Par30">We search for the parameter pair (<inline-formula id="IEq49"><alternatives><tex-math id="M115">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M116"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq49.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq50"><alternatives><tex-math id="M117">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M118"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq50.gif"></inline-graphic></alternatives></inline-formula>) such that can respond appropriately with the data in Table <xref rid="Tab2" ref-type="table">2</xref>. To evaluate how closely system response is fitted to the data in Table <xref rid="Tab2" ref-type="table">2</xref>, we employ a quantitative measure, <italic>the sum of the squared errors</italic>. Once we obtain the optimal values for the parameters with respect to this measure, we can estimate the basic reproduction number as described in the “<xref rid="Sec2" ref-type="sec">Methods</xref>” section.</p><p id="Par31">We define the measure <inline-formula id="IEq51"><alternatives><tex-math id="M119">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}(\cdot , \cdot )$$\end{document}</tex-math><mml:math id="M120"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>,</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq51.gif"></inline-graphic></alternatives></inline-formula> as<disp-formula id="Equ7"><label>7</label><alternatives><tex-math id="M121">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f_{E}(\kappa _i \tau _i, \nu _i) = \sqrt{\sum _{k=0}^{n} \left( \left( I_i(k) - I_T(k) \right) ^2 + \left( R_i(k) - R_T(k) \right) ^2 \right) } \end{aligned}$$\end{document}</tex-math><mml:math id="M122" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="italic">κ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">τ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>I</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mfenced></mml:mrow></mml:msqrt></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ7.gif" position="anchor"></graphic></alternatives></disp-formula>where <inline-formula id="IEq52"><alternatives><tex-math id="M123">\documentclass[12pt]{minimal}
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\begin{document}$$n = 23$$\end{document}</tex-math><mml:math id="M124"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>23</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq52.gif"></inline-graphic></alternatives></inline-formula> and <italic>k</italic> corresponds to the date. For example, <inline-formula id="IEq53"><alternatives><tex-math id="M125">\documentclass[12pt]{minimal}
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\begin{document}$$k = 0$$\end{document}</tex-math><mml:math id="M126"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq53.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq54"><alternatives><tex-math id="M127">\documentclass[12pt]{minimal}
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\begin{document}$$k = 23$$\end{document}</tex-math><mml:math id="M128"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>23</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq54.gif"></inline-graphic></alternatives></inline-formula> indicate the dates 20 May 2015 and 12 June 2015, respectively. <inline-formula id="IEq55"><alternatives><tex-math id="M129">\documentclass[12pt]{minimal}
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\begin{document}$$I_i(\cdot )$$\end{document}</tex-math><mml:math id="M130"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq55.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq56"><alternatives><tex-math id="M131">\documentclass[12pt]{minimal}
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\begin{document}$$R_i(\cdot )$$\end{document}</tex-math><mml:math id="M132"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq56.gif"></inline-graphic></alternatives></inline-formula> are from the simulation result of model (<xref rid="Equ5" ref-type="">5</xref>) with the initial condition (<xref rid="Equ6" ref-type="">6</xref>) and the corresponding parameters, i.e., the function arguments (<inline-formula id="IEq57"><alternatives><tex-math id="M133">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _i \tau _i$$\end{document}</tex-math><mml:math id="M134"><mml:mrow><mml:msub><mml:mi mathvariant="italic">κ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">τ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq57.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq58"><alternatives><tex-math id="M135">\documentclass[12pt]{minimal}
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\begin{document}$$\nu _i$$\end{document}</tex-math><mml:math id="M136"><mml:msub><mml:mi mathvariant="italic">ν</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq58.gif"></inline-graphic></alternatives></inline-formula>). <inline-formula id="IEq59"><alternatives><tex-math id="M137">\documentclass[12pt]{minimal}
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\begin{document}$$R_T(k)$$\end{document}</tex-math><mml:math id="M138"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq59.gif"></inline-graphic></alternatives></inline-formula> is the sum of the values of the “Deceased” and “Recovered” at <italic>k</italic> in Table <xref rid="Tab2" ref-type="table">2</xref>. <inline-formula id="IEq60"><alternatives><tex-math id="M139">\documentclass[12pt]{minimal}
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\begin{document}$$I_T(k)$$\end{document}</tex-math><mml:math id="M140"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq60.gif"></inline-graphic></alternatives></inline-formula> is the value, <inline-formula id="IEq61"><alternatives><tex-math id="M141">\documentclass[12pt]{minimal}
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\begin{document}$$R_T(k)$$\end{document}</tex-math><mml:math id="M142"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq61.gif"></inline-graphic></alternatives></inline-formula> subtracted from the value of the “Infected” at <italic>k</italic>. See Table <xref rid="Tab3" ref-type="table">3</xref>. Based on the data of Table <xref rid="Tab2" ref-type="table">2</xref>, <inline-formula id="IEq62"><alternatives><tex-math id="M143">\documentclass[12pt]{minimal}
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\begin{document}$$I_T(k)$$\end{document}</tex-math><mml:math id="M144"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq62.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq63"><alternatives><tex-math id="M145">\documentclass[12pt]{minimal}
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\begin{document}$$R_T(k)$$\end{document}</tex-math><mml:math id="M146"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq63.gif"></inline-graphic></alternatives></inline-formula> along with <italic>k</italic> are shown in Table <xref rid="Tab3" ref-type="table">3</xref>.<table-wrap id="Tab3"><label>Table 3</label><caption><p><inline-formula id="IEq64"><alternatives><tex-math id="M147">\documentclass[12pt]{minimal}
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\begin{document}$$I_T(k)$$\end{document}</tex-math><mml:math id="M148"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq64.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq65"><alternatives><tex-math id="M149">\documentclass[12pt]{minimal}
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\begin{document}$$R_T(k)$$\end{document}</tex-math><mml:math id="M150"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq65.gif"></inline-graphic></alternatives></inline-formula> of function (<xref rid="Equ7" ref-type="">7</xref>), derived from Table <xref rid="Tab2" ref-type="table">2</xref></p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left"><italic>k</italic></th><th align="left"><inline-formula id="IEq66"><alternatives><tex-math id="M151">\documentclass[12pt]{minimal}
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\begin{document}$$I_T(k)$$\end{document}</tex-math><mml:math id="M152"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq66.gif"></inline-graphic></alternatives></inline-formula></th><th align="left"><inline-formula id="IEq67"><alternatives><tex-math id="M153">\documentclass[12pt]{minimal}
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\begin{document}$$R_T(k)$$\end{document}</tex-math><mml:math id="M154"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq67.gif"></inline-graphic></alternatives></inline-formula></th></tr></thead><tbody><tr><td align="left">0</td><td char="." align="char">2</td><td char="." align="char">0</td></tr><tr><td align="left">1</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">2</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">3</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">4</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">5</td><td char="." align="char">3</td><td char="." align="char">0</td></tr><tr><td align="left">6</td><td char="." align="char">5</td><td char="." align="char">0</td></tr><tr><td align="left">7</td><td char="." align="char">5</td><td char="." align="char">0</td></tr><tr><td align="left">8</td><td char="." align="char">7</td><td char="." align="char">0</td></tr><tr><td align="left">9</td><td char="." align="char">13</td><td char="." align="char">0</td></tr><tr><td align="left">10</td><td char="." align="char">15</td><td char="." align="char">0</td></tr><tr><td align="left">11</td><td char="." align="char">18</td><td char="." align="char">0</td></tr><tr><td align="left">12</td><td char="." align="char">24</td><td char="." align="char">1</td></tr><tr><td align="left">13</td><td char="." align="char">29</td><td char="." align="char">1</td></tr><tr><td align="left">14</td><td char="." align="char">27</td><td char="." align="char">3</td></tr><tr><td align="left">15</td><td char="." align="char">32</td><td char="." align="char">4</td></tr><tr><td align="left">16</td><td char="." align="char">36</td><td char="." align="char">6</td></tr><tr><td align="left">17</td><td char="." align="char">58</td><td char="." align="char">6</td></tr><tr><td align="left">18</td><td char="." align="char">81</td><td char="." align="char">6</td></tr><tr><td align="left">19</td><td char="." align="char">86</td><td char="." align="char">9</td></tr><tr><td align="left">20</td><td char="." align="char">98</td><td char="." align="char">10</td></tr><tr><td align="left">21</td><td char="." align="char">109</td><td char="." align="char">13</td></tr><tr><td align="left">22</td><td char="." align="char">109</td><td char="." align="char">17</td></tr><tr><td align="left">23</td><td char="." align="char">116</td><td char="." align="char">22</td></tr></tbody></table></table-wrap></p><p id="Par32">To compare the quantitative measures for each pair of parameters, we consider the plane, i.e., 2-dimensional space, of the parameters, (<inline-formula id="IEq68"><alternatives><tex-math id="M155">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M156"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq68.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq69"><alternatives><tex-math id="M157">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M158"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq69.gif"></inline-graphic></alternatives></inline-formula>). We can obtain a surface in 3-dimensional space by plotting the corresponding measure as the value along the third axis.</p><p id="Par33">We explore the plane for wide ranges of parameters, and one of the results is shown in Fig. <xref rid="Fig1" ref-type="fig">1</xref>. This figure can help show the relation between <inline-formula id="IEq70"><alternatives><tex-math id="M159">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}$$\end{document}</tex-math><mml:math id="M160"><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq70.gif"></inline-graphic></alternatives></inline-formula> and the system parameters. To effectively capture the characteristics of the measure, the ranges of <inline-formula id="IEq71"><alternatives><tex-math id="M161">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M162"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq71.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq72"><alternatives><tex-math id="M163">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M164"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq72.gif"></inline-graphic></alternatives></inline-formula> in the figure are [0.00, 0.26] and [0.00, 0.12], respectively.<fig id="Fig1"><label>Fig. 1</label><caption><p>Plot of <inline-formula id="IEq73"><alternatives><tex-math id="M165">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}(\kappa \tau , \nu )$$\end{document}</tex-math><mml:math id="M166"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq73.gif"></inline-graphic></alternatives></inline-formula> of (<xref rid="Equ7" ref-type="">7</xref>) on the plane of (<inline-formula id="IEq74"><alternatives><tex-math id="M167">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M168"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq74.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq75"><alternatives><tex-math id="M169">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M170"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq75.gif"></inline-graphic></alternatives></inline-formula>). <inline-formula id="IEq76"><alternatives><tex-math id="M171">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}(\cdot , \cdot )$$\end{document}</tex-math><mml:math id="M172"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>,</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq76.gif"></inline-graphic></alternatives></inline-formula> is a quantitative measure, the sum of the squared errors, that describes how the system response of model (<xref rid="Equ5" ref-type="">5</xref>) with corresponding parameters is close to the data in Table <xref rid="Tab2" ref-type="table">2</xref>. The ranges of <inline-formula id="IEq77"><alternatives><tex-math id="M173">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M174"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq77.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq78"><alternatives><tex-math id="M175">\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M176"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq78.gif"></inline-graphic></alternatives></inline-formula> in this figure are [0.00, 0.26] and [0.00, 0.12], respectively</p></caption><graphic xlink:href="12938_2017_370_Fig1_HTML" id="MO12"></graphic></fig></p><p id="Par34">Based on characteristics shown in Fig. <xref rid="Fig1" ref-type="fig">1</xref>, we search for precise values for <inline-formula id="IEq79"><alternatives><tex-math id="M177">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M178"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq79.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq80"><alternatives><tex-math id="M179">\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M180"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq80.gif"></inline-graphic></alternatives></inline-formula> to minimize the measure, <inline-formula id="IEq81"><alternatives><tex-math id="M181">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}$$\end{document}</tex-math><mml:math id="M182"><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq81.gif"></inline-graphic></alternatives></inline-formula>. Consider the parameter plane with the range [0.2050, 0.2150] of <inline-formula id="IEq82"><alternatives><tex-math id="M183">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M184"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq82.gif"></inline-graphic></alternatives></inline-formula> and with the range [0.0200, 0.0300] of <inline-formula id="IEq83"><alternatives><tex-math id="M185">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M186"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq83.gif"></inline-graphic></alternatives></inline-formula> in Fig. <xref rid="Fig2" ref-type="fig">2</xref>. Then we can identify the precise values for <inline-formula id="IEq84"><alternatives><tex-math id="M187">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M188"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq84.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq85"><alternatives><tex-math id="M189">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M190"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq85.gif"></inline-graphic></alternatives></inline-formula> using Fig. <xref rid="Fig2" ref-type="fig">2</xref>. The system parameters are <inline-formula id="IEq86"><alternatives><tex-math id="M191">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau = 0.21153$$\end{document}</tex-math><mml:math id="M192"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.21153</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq86.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq87"><alternatives><tex-math id="M193">\documentclass[12pt]{minimal}
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\begin{document}$$\nu = 0.026122$$\end{document}</tex-math><mml:math id="M194"><mml:mrow><mml:mi mathvariant="italic">ν</mml:mi><mml:mo>=</mml:mo><mml:mn>0.026122</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq87.gif"></inline-graphic></alternatives></inline-formula> evaluated by Fig. <xref rid="Fig2" ref-type="fig">2</xref> to minimise the quantitative measure <inline-formula id="IEq88"><alternatives><tex-math id="M195">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}$$\end{document}</tex-math><mml:math id="M196"><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq88.gif"></inline-graphic></alternatives></inline-formula><fig id="Fig2"><label>Fig. 2</label><caption><p>Plot of <inline-formula id="IEq89"><alternatives><tex-math id="M197">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}(\kappa \tau , \nu )$$\end{document}</tex-math><mml:math id="M198"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="italic">ν</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq89.gif"></inline-graphic></alternatives></inline-formula> of (<xref rid="Equ7" ref-type="">7</xref>) on the plane of (<inline-formula id="IEq90"><alternatives><tex-math id="M199">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M200"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq90.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq91"><alternatives><tex-math id="M201">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M202"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq91.gif"></inline-graphic></alternatives></inline-formula>). Compared to Fig. <xref rid="Fig1" ref-type="fig">1</xref>, the ranges of <inline-formula id="IEq92"><alternatives><tex-math id="M203">\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \tau$$\end{document}</tex-math><mml:math id="M204"><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq92.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq93"><alternatives><tex-math id="M205">\documentclass[12pt]{minimal}
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\begin{document}$$\nu$$\end{document}</tex-math><mml:math id="M206"><mml:mi mathvariant="italic">ν</mml:mi></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq93.gif"></inline-graphic></alternatives></inline-formula> in this figure are [0.2050, 0.2150] and [0.0200, 0.0300], respectively, and they can be used to obtain precise values of system parameters to minimize the quantitative measure <inline-formula id="IEq94"><alternatives><tex-math id="M207">\documentclass[12pt]{minimal}
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\begin{document}$$f_{E}$$\end{document}</tex-math><mml:math id="M208"><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq94.gif"></inline-graphic></alternatives></inline-formula></p></caption><graphic xlink:href="12938_2017_370_Fig2_HTML" id="MO13"></graphic></fig></p><p id="Par35">The function <inline-formula id="IEq95"><alternatives><tex-math id="M209">\documentclass[12pt]{minimal}
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\begin{document}$$f_E(\cdot , \cdot )$$\end{document}</tex-math><mml:math id="M210"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>,</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq95.gif"></inline-graphic></alternatives></inline-formula> describes the sum of squared error between the outbreak data of Table <xref rid="Tab2" ref-type="table">2</xref> and the simulation result of model (<xref rid="Equ5" ref-type="">5</xref>) with the initial condition (<xref rid="Equ6" ref-type="">6</xref>) and the function arguments. Thus, if we find the parameters minimizing the function (<xref rid="Equ7" ref-type="">7</xref>), then these parameters can be considered to correspond to the case of Table <xref rid="Tab2" ref-type="table">2</xref>.</p><p id="Par36">Figure <xref rid="Fig3" ref-type="fig">3</xref> shows the data presented in Table <xref rid="Tab2" ref-type="table">2</xref> and the state trajectories of model (<xref rid="Equ5" ref-type="">5</xref>) with the parameter values obtained from Fig. <xref rid="Fig2" ref-type="fig">2</xref>. The top panel of Fig. <xref rid="Fig3" ref-type="fig">3</xref> shows infected patients, and the bottom panel shows deceased or recovered patients. In both panels, the circle marks and the solid line display patient numbers based on Table <xref rid="Tab2" ref-type="table">2</xref> and the state transition of model (<xref rid="Equ5" ref-type="">5</xref>), respectively.<fig id="Fig3"><label>Fig. 3</label><caption><p>Data presented in Table <xref rid="Tab2" ref-type="table">2</xref> and the state trajectories of model (<xref rid="Equ5" ref-type="">5</xref>) with the estimated parameters. In the<italic> top panel</italic>, the<italic> circle marks</italic> plot the number of “Infected” patients from Table <xref rid="Tab2" ref-type="table">2</xref>, and the<italic> solid line</italic> depicts the transition of state <italic>I</italic> of model (<xref rid="Equ5" ref-type="">5</xref>). In the<italic> bottom panel</italic>, the<italic> circle marks</italic> indicate the sum of the numbers of “Deceased” and “Recovered” patients, and the<italic> solid line</italic> depicts the transition of state <italic>R</italic> of the model</p></caption><graphic xlink:href="12938_2017_370_Fig3_HTML" id="MO14"></graphic></fig></p><p id="Par37">We derive the <italic>basic reproductive number</italic> for the SIR model (<xref rid="Equ2" ref-type="">2</xref>) as<disp-formula id="Equ8"><label>8</label><alternatives><tex-math id="M211">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_0=\frac{\kappa \tau }{\nu } \end{aligned}$$\end{document}</tex-math><mml:math id="M212" display="block"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="italic">κ</mml:mi><mml:mi mathvariant="italic">τ</mml:mi></mml:mrow><mml:mi mathvariant="italic">ν</mml:mi></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12938_2017_370_Article_Equ8.gif" position="anchor"></graphic></alternatives></disp-formula>in the “<xref rid="Sec2" ref-type="sec">Methods</xref>” section, based on [<xref ref-type="bibr" rid="CR16">16</xref>]. Equation (<xref rid="Equ8" ref-type="">8</xref>) can be obtained alternatively by using the next generator approach [<xref ref-type="bibr" rid="CR6">6</xref>, <xref ref-type="bibr" rid="CR8">8</xref>, <xref ref-type="bibr" rid="CR11">11</xref>, <xref ref-type="bibr" rid="CR27">27</xref>] to model (<xref rid="Equ5" ref-type="">5</xref>).</p><p id="Par38">We determine the <inline-formula id="IEq96"><alternatives><tex-math id="M213">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M214"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq96.gif"></inline-graphic></alternatives></inline-formula> of the MERS-CoV in South Korea in 2015 as 8.0977 (<italic>i.e.,</italic> 0.21153/0.026122). It is worth comparing with <inline-formula id="IEq97"><alternatives><tex-math id="M215">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M216"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq97.gif"></inline-graphic></alternatives></inline-formula> of the 2014 Ebola outbreak, which had values of 1.5–2.5 [<xref ref-type="bibr" rid="CR21">21</xref>].</p></sec><sec id="Sec6"><title>Conclusions</title><p id="Par39">In this paper, we evaluated the basic reproduction number of the MERS-CoV outbreak that occurred in 2015 in South Korea, using officially reported data. We employed a mathematical dynamic model, the SIR model. We first fit the response of the SIR model to the epidemic curve data reported from the MERS outbreak. Then, we identified the system parameters of the model to estimate the basic reproduction number.</p><p id="Par40">Because there was no intervention to control the infection in the early phase of the outbreak, the data used here provide the best conditions to evaluate the epidemic characteristics of MERS, such as the basic reproduction number. An evaluation of <inline-formula id="IEq99"><alternatives><tex-math id="M217">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M218"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq99.gif"></inline-graphic></alternatives></inline-formula> using epidemic data could be problematic if there are stochastic fluctuations in the early phase of the outbreak, or if the report is not accurate and there is bias in the data [<xref ref-type="bibr" rid="CR11">11</xref>]. Such problems are not relevant to this study because the data used here were precisely reported and verified by [<xref ref-type="bibr" rid="CR4">4</xref>].</p><p id="Par41">We conclude this paper with the following discussion on future work to overcome the limitations of research, derived from assumptions in the paper.</p><sec id="Sec7"><title>Further research direction</title><p id="Par42"><list list-type="bullet"><list-item><p id="Par43">Behind the SIR model (<xref rid="Equ2" ref-type="">2</xref>), there are several strong assumptions, one of which is a zero latent period, i.e., the incubation period is zero. This implies that a patient becomes infectious immediately after infection. However, the incubation phase occurs during the course of the MERS outbreak. To address this weak point, in future work we could consider the 4-dimensional SEIR (i.e., susceptible-exposed-infectious-removed) model, which has been employed in [<xref ref-type="bibr" rid="CR28">28</xref>] to study Ebola epidemic model. The additional state in the SEIR model can help us deal with the latent period.</p></list-item><list-item><p id="Par44">In this paper, we considered the epidemic curve data in [<xref ref-type="bibr" rid="CR4">4</xref>] only from the early stage of the 2015 MERS outbreak in South Korea, where there was no intervention to control the spread. Accordingly, we evaluated <inline-formula id="IEq100"><alternatives><tex-math id="M219">\documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M220"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq100.gif"></inline-graphic></alternatives></inline-formula> based on the data. In future work, we will also consider the epidemic data in [<xref ref-type="bibr" rid="CR4">4</xref>] from the later (or closing) stage of the MERS spread in South Korea in 2015, so we can estimate the effective production number (i.e., <inline-formula id="IEq101"><alternatives><tex-math id="M221">\documentclass[12pt]{minimal}
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\begin{document}$$R_\mathrm{eff}$$\end{document}</tex-math><mml:math id="M222"><mml:msub><mml:mi>R</mml:mi><mml:mi mathvariant="normal">eff</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq101.gif"></inline-graphic></alternatives></inline-formula>), which is the production number resulting from interventions, such as education, quarantine, and the tracing of contacts by infected patients. By doing so, we can evaluate the effectiveness of each control measure on the spread of the infectious disease[<xref ref-type="bibr" rid="CR29">29</xref>]. Eventually, such evaluation could help us improve public health policy.</p></list-item></list></p></sec></sec></body><back><ack><title>Acknowledgements</title><p>A preliminary form of this research was presented at the 16th International Conference on Control, Automation and Systems, ICCAS 2016.</p><sec id="d29e3625" sec-type="COI-statement"><title>Competing interests</title><p>The author declares that no competing interests.</p></sec><sec id="d29e3630" sec-type="data-availability"><title>Availability of data and materials</title><p>All data generated or analysed during this study are included in this published article.</p></sec><sec id="d29e3635"><title>Funding</title><p>This work was supported by National Research Foundation of Korea - Grant funded by the Korean Government (NRF-2014R1A1A1003056).</p></sec><sec id="d29e3640"><title>Publisher's Note</title><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></sec></ack><ref-list id="Bib1"><title>References</title><ref id="CR1"><label>1.</label><mixed-citation publication-type="other">World Health Organization: Fact Sheet 401: Middle East respiratory syndrome coronavirus (MERS-CoV). 2015. <ext-link ext-link-type="uri" xlink:href="http://www.who.int/mediacentre/factsheets/mers-cov/en/">http://www.who.int/mediacentre/factsheets/mers-cov/en/</ext-link>. Accessed 8 May 2017.</mixed-citation></ref><ref id="CR2"><label>2.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Su</surname><given-names>S</given-names></name><name><surname>Wong</surname><given-names>G</given-names></name><name><surname>Liu</surname><given-names>Y</given-names></name><name><surname>Gao</surname><given-names>GF</given-names></name><name><surname>Li</surname><given-names>S</given-names></name><name><surname>Bi</surname><given-names>Y</given-names></name></person-group><article-title>MERS in South Korea and China: a potential outbreak threat?</article-title><source>Lancet</source><year>2015</year><volume>385</volume><issue>9985</issue><fpage>2349</fpage><lpage>2350</lpage><pub-id pub-id-type="doi">10.1016/S0140-6736(15)60859-5</pub-id><pub-id pub-id-type="pmid">26088634</pub-id></element-citation></ref><ref id="CR3"><label>3.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hyun</surname><given-names>MH</given-names></name><name><surname>Park</surname><given-names>Y</given-names></name></person-group><article-title>Re: Why the panic? South Korea’s MERS response questioned</article-title><source>BMJ</source><year>2015</year><volume>350</volume><fpage>12</fpage><pub-id pub-id-type="doi">10.1136/bmj.h12</pub-id></element-citation></ref><ref id="CR4"><label>4.</label><mixed-citation publication-type="other">Korea Hospital Association MERS. White Paper by Korea Hospital Association. Korea Hospital Association, Seoul; 2015.</mixed-citation></ref><ref id="CR5"><label>5.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Anderson</surname><given-names>RM</given-names></name><name><surname>May</surname><given-names>RM</given-names></name></person-group><source>Infectious diseases of humans: dynamics and control</source><year>1992</year><publisher-loc>New York</publisher-loc><publisher-name>Oxfor University Press</publisher-name></element-citation></ref><ref id="CR6"><label>6.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Diekmann</surname><given-names>O</given-names></name><name><surname>Heesterbeek</surname><given-names>JAP</given-names></name><name><surname>Metz</surname><given-names>JAJ</given-names></name></person-group><article-title>On the definition and the computation of the basic reproduction ratio <inline-formula id="IEq106"><alternatives><tex-math id="M223">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${R}_0$$\end{document}</tex-math><mml:math id="M224"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq106.gif"></inline-graphic></alternatives></inline-formula> in models for infectious diseases in heterogeneous populations</article-title><source>J Math Biol</source><year>1990</year><volume>28</volume><issue>4</issue><fpage>365</fpage><lpage>382</lpage><pub-id pub-id-type="doi">10.1007/BF00178324</pub-id><pub-id pub-id-type="pmid">2117040</pub-id></element-citation></ref><ref id="CR7"><label>7.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Heesterbeek</surname><given-names>JAP</given-names></name><name><surname>Dietz</surname><given-names>K</given-names></name></person-group><article-title>The concept of <inline-formula id="IEq108"><alternatives><tex-math id="M225">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M226"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq108.gif"></inline-graphic></alternatives></inline-formula> in epidemic theory</article-title><source>Stat Neerl</source><year>1996</year><volume>50</volume><issue>1</issue><fpage>89</fpage><lpage>110</lpage><pub-id pub-id-type="doi">10.1111/j.1467-9574.1996.tb01482.x</pub-id></element-citation></ref><ref id="CR8"><label>8.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Diekmann</surname><given-names>O</given-names></name><name><surname>Heesterbeek</surname><given-names>JAP</given-names></name></person-group><source>Mathematical epidemiology of infectious diseases: model building. Analysis and interpretation</source><year>2000</year><publisher-loc>New York</publisher-loc><publisher-name>Wiley</publisher-name></element-citation></ref><ref id="CR9"><label>9.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>van den Driessche</surname><given-names>P</given-names></name><name><surname>Watmough</surname><given-names>J</given-names></name></person-group><article-title>Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission</article-title><source>Math Biosci</source><year>2002</year><volume>180</volume><issue>1–2</issue><fpage>29</fpage><lpage>48</lpage><pub-id pub-id-type="doi">10.1016/S0025-5564(02)00108-6</pub-id><pub-id pub-id-type="pmid">12387915</pub-id></element-citation></ref><ref id="CR10"><label>10.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Heesterbeek</surname><given-names>JAP</given-names></name></person-group><article-title>A brief history of <inline-formula id="IEq110"><alternatives><tex-math id="M227">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
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\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M228"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq110.gif"></inline-graphic></alternatives></inline-formula> and a recipe for its calculation</article-title><source>Acta Biotheor</source><year>2002</year><volume>50</volume><issue>3</issue><fpage>189</fpage><lpage>204</lpage><pub-id pub-id-type="doi">10.1023/A:1016599411804</pub-id><pub-id pub-id-type="pmid">12211331</pub-id></element-citation></ref><ref id="CR11"><label>11.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Heffernan</surname><given-names>JM</given-names></name><name><surname>Smith</surname><given-names>RJ</given-names></name><name><surname>Wahl</surname><given-names>LM</given-names></name></person-group><article-title>Perspectives on the basic reproductive ratio</article-title><source>J R Soc Interface</source><year>2005</year><volume>2</volume><issue>4</issue><fpage>281</fpage><lpage>293</lpage><pub-id pub-id-type="doi">10.1098/rsif.2005.0042</pub-id><pub-id pub-id-type="pmid">16849186</pub-id></element-citation></ref><ref id="CR12"><label>12.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chowell</surname><given-names>G</given-names></name><name><surname>Blumberg</surname><given-names>S</given-names></name><name><surname>Simonsen</surname><given-names>L</given-names></name><name><surname>Miller</surname><given-names>MA</given-names></name><name><surname>Viboud</surname><given-names>C</given-names></name></person-group><article-title>Synthesizing data and models for the spread of MERS-CoV, 2013: key role of index cases and hospital transmission</article-title><source>Epidemics</source><year>2014</year><volume>9</volume><fpage>40</fpage><lpage>51</lpage><pub-id pub-id-type="doi">10.1016/j.epidem.2014.09.011</pub-id><pub-id pub-id-type="pmid">25480133</pub-id></element-citation></ref><ref id="CR13"><label>13.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>DiStefano</surname><given-names>J</given-names><suffix>III</suffix></name></person-group><source>Dynamic systems biology modeling and simulation</source><year>2015</year><publisher-loc>London</publisher-loc><publisher-name>Academic Press</publisher-name></element-citation></ref><ref id="CR14"><label>14.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Yashima</surname><given-names>K</given-names></name><name><surname>Sasaki</surname><given-names>A</given-names></name></person-group><article-title>Spotting epidemic keystones by <inline-formula id="IEq112"><alternatives><tex-math id="M229">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$R_0$$\end{document}</tex-math><mml:math id="M230"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12938_2017_370_Article_IEq112.gif"></inline-graphic></alternatives></inline-formula> sensitivity analysis: high-risk stations in the Tokyo metropolitan area</article-title><source>PLOS ONE</source><year>2016</year><volume>11</volume><issue>9</issue><fpage>1</fpage><lpage>19</lpage><pub-id pub-id-type="doi">10.1371/journal.pone.0162406</pub-id></element-citation></ref><ref id="CR15"><label>15.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Anderson</surname><given-names>RM</given-names></name></person-group><article-title>Discussion: the Kermack-McKendrick epidemic threshold theorem</article-title><source>Bull Math Biol</source><year>1991</year><volume>53</volume><issue>1–2</issue><fpage>1</fpage><lpage>32</lpage><pub-id pub-id-type="doi">10.1007/BF02464422</pub-id></element-citation></ref><ref id="CR16"><label>16.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Weiss</surname><given-names>H</given-names></name></person-group><article-title>The SIR model and the foundations of public health</article-title><source>MATerials MATemàtics</source><year>2013</year><volume>3</volume><issue>17</issue><fpage>1</fpage><lpage>17</lpage></element-citation></ref><ref id="CR17"><label>17.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Cowling</surname><given-names>BJ</given-names></name><name><surname>Park</surname><given-names>M</given-names></name><name><surname>Fang</surname><given-names>VJ</given-names></name><name><surname>Wu</surname><given-names>P</given-names></name><name><surname>Leung</surname><given-names>GM</given-names></name><name><surname>Wu</surname><given-names>JT</given-names></name></person-group><article-title>Preliminary epidemiologic assessment of MERS-CoV outbreak in South Korea, May to June 2015</article-title><source>Euro Surveill.</source><year>2015</year><volume>20</volume><issue>25</issue><fpage>1</fpage><lpage>7</lpage><pub-id pub-id-type="doi">10.2807/1560-7917.ES2015.20.25.21163</pub-id><pub-id pub-id-type="pmid">26132766</pub-id></element-citation></ref><ref id="CR18"><label>18.</label><mixed-citation publication-type="other">Chang H. Evaluation of the basic reproduction number of MERS-CoV during the 2015 outbreak in South Korea. In: Proceedings of 16th international conference on control, automation and systems. 2016. p. 981–984</mixed-citation></ref><ref id="CR19"><label>19.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Wallinga</surname><given-names>J</given-names></name><name><surname>Teunis</surname><given-names>P</given-names></name></person-group><article-title>Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures</article-title><source>Am J Epidemiol</source><year>2004</year><volume>160</volume><issue>6</issue><fpage>509</fpage><lpage>516</lpage><pub-id pub-id-type="doi">10.1093/aje/kwh255</pub-id><pub-id pub-id-type="pmid">15353409</pub-id></element-citation></ref><ref id="CR20"><label>20.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Mills</surname><given-names>CE</given-names></name><name><surname>Robins</surname><given-names>JM</given-names></name><name><surname>Lipsitch</surname><given-names>M</given-names></name></person-group><article-title>Transmissibility of 1918 pandemic influenza</article-title><source>Nature</source><year>2004</year><volume>432</volume><fpage>904</fpage><lpage>906</lpage><pub-id pub-id-type="doi">10.1038/nature03063</pub-id><pub-id pub-id-type="pmid">15602562</pub-id></element-citation></ref><ref id="CR21"><label>21.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Althaus</surname><given-names>CL</given-names></name></person-group><article-title>Estimating the reproduction number of ebola virus (EBOV) during the 2014 outbreak in west africa</article-title><source>PLOS Curr Outbreaks</source><year>2014</year><volume>6</volume><fpage>1</fpage><lpage>5</lpage></element-citation></ref><ref id="CR22"><label>22.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chowell</surname><given-names>G</given-names></name><name><surname>Hengartner</surname><given-names>NW</given-names></name><name><surname>Castillo-Chavez</surname><given-names>C</given-names></name><name><surname>Fenimore</surname><given-names>PW</given-names></name><name><surname>Hyman</surname><given-names>JM</given-names></name></person-group><article-title>The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda</article-title><source>J Theor Biol</source><year>2004</year><volume>229</volume><issue>1</issue><fpage>119</fpage><lpage>126</lpage><pub-id pub-id-type="doi">10.1016/j.jtbi.2004.03.006</pub-id><pub-id pub-id-type="pmid">15178190</pub-id></element-citation></ref><ref id="CR23"><label>23.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Smith</surname><given-names>DL</given-names></name><name><surname>McKenzie</surname><given-names>FE</given-names></name><name><surname>Snow</surname><given-names>RW</given-names></name><name><surname>Hay</surname><given-names>SI</given-names></name></person-group><article-title>Revisiting the basic reproductive number for malaria and its implications for malaria control</article-title><source>PLoS Biol</source><year>2007</year><volume>5</volume><issue>3</issue><fpage>1</fpage><lpage>12</lpage><pub-id pub-id-type="doi">10.1371/journal.pbio.0050042</pub-id></element-citation></ref><ref id="CR24"><label>24.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ejima</surname><given-names>K</given-names></name><name><surname>Aihara</surname><given-names>K</given-names></name><name><surname>Nishiura</surname><given-names>H</given-names></name></person-group><article-title>Probabilistic differential diagnosis of middle east respiratory syndrome (MERS) using the time from immigration to illness onset among imported cases</article-title><source>J Theor Biol</source><year>2014</year><volume>346</volume><fpage>47</fpage><lpage>53</lpage><pub-id pub-id-type="doi">10.1016/j.jtbi.2013.12.024</pub-id><pub-id pub-id-type="pmid">24406808</pub-id></element-citation></ref><ref id="CR25"><label>25.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ha</surname><given-names>K-M</given-names></name></person-group><article-title>A lesson learned from the MERS outbreak in South Korea in 2015</article-title><source>J Hosp Infect</source><year>2016</year><volume>92</volume><issue>3</issue><fpage>232</fpage><lpage>234</lpage><pub-id pub-id-type="doi">10.1016/j.jhin.2015.10.004</pub-id><pub-id pub-id-type="pmid">26601605</pub-id></element-citation></ref><ref id="CR26"><label>26.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Assiri</surname><given-names>A</given-names></name><name><surname>McGeer</surname><given-names>A</given-names></name><name><surname>Perl</surname><given-names>TM</given-names></name><name><surname>Price</surname><given-names>CS</given-names></name><name><surname>Al Rabeeah</surname><given-names>AA</given-names></name><name><surname>Cummings</surname><given-names>DAT</given-names></name><name><surname>Alabdullatif</surname><given-names>ZN</given-names></name><name><surname>Assad</surname><given-names>M</given-names></name><name><surname>Almulhim</surname><given-names>A</given-names></name><name><surname>Makhdoom</surname><given-names>H</given-names></name><name><surname>Madani</surname><given-names>H</given-names></name><name><surname>Alhakeem</surname><given-names>R</given-names></name><name><surname>Al-Tawfiq</surname><given-names>JA</given-names></name><name><surname>Cotten</surname><given-names>M</given-names></name><name><surname>Watson</surname><given-names>SJ</given-names></name><name><surname>Kellam</surname><given-names>P</given-names></name><name><surname>Zumla</surname><given-names>AI</given-names></name><name><surname>Memish</surname><given-names>ZA</given-names></name></person-group><article-title>Hospital outbreak of middle east respiratory syndrome coronavirus</article-title><source>N Engl J Med</source><year>2013</year><volume>369</volume><issue>5</issue><fpage>407</fpage><lpage>416</lpage><pub-id pub-id-type="doi">10.1056/NEJMoa1306742</pub-id><pub-id pub-id-type="pmid">23782161</pub-id></element-citation></ref><ref id="CR27"><label>27.</label><mixed-citation publication-type="other">Area I, Losada J, Ndaïrou F, Nieto JJ, Tcheutia DD. Mathematical modeling of 2014 Ebola outbreak. Math Methods Appl Sci. 2015. doi:10.1002/mma.3794.</mixed-citation></ref><ref id="CR28"><label>28.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Area</surname><given-names>I</given-names></name><name><surname>Batarfi</surname><given-names>H</given-names></name><name><surname>Losada</surname><given-names>J</given-names></name><name><surname>Nieto</surname><given-names>JJ</given-names></name><name><surname>Shammakh</surname><given-names>W</given-names></name><name><surname>Torres</surname><given-names>Á</given-names></name></person-group><article-title>On a fractional order Ebola epidemic model</article-title><source>Adv Differ Equ</source><year>2015</year><volume>2015</volume><issue>1</issue><fpage>278</fpage><pub-id pub-id-type="doi">10.1186/s13662-015-0613-5</pub-id></element-citation></ref><ref id="CR29"><label>29.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Nishiura</surname><given-names>H</given-names></name><name><surname>Ejima</surname><given-names>K</given-names></name><name><surname>Mizumoto</surname><given-names>K</given-names></name><name><surname>Nakaoka</surname><given-names>S</given-names></name><name><surname>Inaba</surname><given-names>H</given-names></name><name><surname>Imoto</surname><given-names>S</given-names></name><name><surname>Yamaguchi</surname><given-names>R</given-names></name><name><surname>Saito</surname><given-names>M</given-names></name></person-group><article-title>Cost-effective length and timing of school closure during an influenza pandemic depend on the severity</article-title><source>Theor Biol Med Model</source><year>2014</year><volume>11</volume><issue>1</issue><fpage>5</fpage><pub-id pub-id-type="doi">10.1186/1742-4682-11-5</pub-id><pub-id pub-id-type="pmid">24447310</pub-id></element-citation></ref></ref-list></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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