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The final size of a SARS epidemic model without quarantine

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The final size of a SARS epidemic model without quarantine

Auteurs : Sze-Bi Hsu [Taïwan] ; Lih-Ing W. Roeger [États-Unis]

Source :

Journal of Mathematical Analysis and Applications [ 0022-247X ] ; 2006.

RBID : PMC:7111549

Abstract

In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, Ying-Hen Hsieh, Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak, SIAM J. Appl. Math. 66 (2006) 627–647]. An “acting basic reproductive number” ψ is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size S∞, the initial susceptible population S0, and ψ. If ψ>1, the disease will prevail and the final size of the susceptible, S∞, becomes zero; therefore, everyone in the population will be infected eventually. If ψ<1, the disease dies out, and then S∞>0 which means part of the population will never be infected. Also, when S∞>0, S∞ is increasing with respect to the initial susceptible population S0, and decreasing with respect to the acting basic reproductive number ψ.

Url:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7111549

DOI: 10.1016/j.jmaa.2006.11.026
PubMed: NONE
PubMed Central: 7111549

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<author><name sortKey="Hsu, Sze Bi" sort="Hsu, Sze Bi" uniqKey="Hsu S" first="Sze-Bi" last="Hsu">Sze-Bi Hsu</name>
<affiliation wicri:level="1"><nlm:aff id="aff001">Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan</nlm:aff>
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<wicri:regionArea>Department of Mathematics, National Tsing Hua University, Hsinchu</wicri:regionArea>
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<author><name sortKey="Roeger, Lih Ing W" sort="Roeger, Lih Ing W" uniqKey="Roeger L" first="Lih-Ing W." last="Roeger">Lih-Ing W. Roeger</name>
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<author><name sortKey="Hsu, Sze Bi" sort="Hsu, Sze Bi" uniqKey="Hsu S" first="Sze-Bi" last="Hsu">Sze-Bi Hsu</name>
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<author><name sortKey="Roeger, Lih Ing W" sort="Roeger, Lih Ing W" uniqKey="Roeger L" first="Lih-Ing W." last="Roeger">Lih-Ing W. Roeger</name>
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<front><div type="abstract" xml:lang="en"><p>In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, Ying-Hen Hsieh, Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak, SIAM J. Appl. Math. 66 (2006) 627–647]. An “acting basic reproductive number” <italic>ψ</italic>
 is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size <inline-formula><mml:math id="M1" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
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, and <italic>ψ</italic>
. If <inline-formula><mml:math id="M3" altimg="si3.gif" overflow="scroll"><mml:mi>ψ</mml:mi>
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<mml:mn>1</mml:mn>
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, the disease will prevail and the final size of the susceptible, <inline-formula><mml:math id="M4" altimg="si4.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
</mml:math>
</inline-formula>
, becomes zero; therefore, everyone in the population will be infected eventually. If <inline-formula><mml:math id="M5" altimg="si5.gif" overflow="scroll"><mml:mi>ψ</mml:mi>
<mml:mo><</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</inline-formula>
, the disease dies out, and then <inline-formula><mml:math id="M6" altimg="si6.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</inline-formula>
 which means part of the population will never be infected. Also, when <inline-formula><mml:math id="M7" altimg="si7.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
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<mml:mn>0</mml:mn>
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 is increasing with respect to the initial susceptible population <inline-formula><mml:math id="M9" altimg="si9.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:math>
</inline-formula>
, and decreasing with respect to the acting basic reproductive number <italic>ψ</italic>
.</p>
</div>
</front>
<back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Diekmann, O" uniqKey="Diekmann O">O. Diekmann</name>
</author>
<author><name sortKey="De Koeijer, A A" uniqKey="De Koeijer A">A.A. de Koeijer</name>
</author>
<author><name sortKey="Metz, J A J" uniqKey="Metz J">J.A.J. Metz</name>
</author>
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</biblStruct>
<biblStruct><analytic><author><name sortKey="Diekmann, O" uniqKey="Diekmann O">O. Diekmann</name>
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<author><name sortKey="Heesterbeek, J A P" uniqKey="Heesterbeek J">J.A.P. Heesterbeek</name>
</author>
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<biblStruct><analytic><author><name sortKey="Hsu, Sze Bi" uniqKey="Hsu S">Sze-Bi Hsu</name>
</author>
<author><name sortKey="Hsieh, Ying Hen" uniqKey="Hsieh Y">Ying-Hen Hsieh</name>
</author>
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<biblStruct><analytic><author><name sortKey="Murray, J D" uniqKey="Murray J">J.D. Murray</name>
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<biblStruct><analytic><author><name sortKey="Waltman, P E" uniqKey="Waltman P">P.E. Waltman</name>
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<biblStruct><analytic><author><name sortKey="World Health Organization" uniqKey="World Health Organization">World Health Organization</name>
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<pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
  <front><journal-meta><journal-id journal-id-type="nlm-ta">J Math Anal Appl</journal-id>
<journal-id journal-id-type="iso-abbrev">J Math Anal Appl</journal-id>
<journal-title-group><journal-title>Journal of Mathematical Analysis and Applications</journal-title>
</journal-title-group>
<issn pub-type="ppub">0022-247X</issn>
<issn pub-type="epub">0022-247X</issn>
<publisher><publisher-name>Elsevier Inc.</publisher-name>
</publisher>
</journal-meta>
<article-meta><article-id pub-id-type="pmc">7111549</article-id>
<article-id pub-id-type="publisher-id">S0022-247X(06)01308-4</article-id>
<article-id pub-id-type="doi">10.1016/j.jmaa.2006.11.026</article-id>
<article-categories><subj-group subj-group-type="heading"><subject>Article</subject>
</subj-group>
</article-categories>
<title-group><article-title>The final size of a SARS epidemic model without quarantine</article-title>
</title-group>
<contrib-group><contrib contrib-type="author"><name><surname>Hsu</surname>
<given-names>Sze-Bi</given-names>
</name>
<email>sbhsu@math.nthu.edu.tw</email>
<xref rid="aff001" ref-type="aff">a</xref>
</contrib>
<contrib contrib-type="author"><name><surname>Roeger</surname>
<given-names>Lih-Ing W.</given-names>
</name>
<email>lih-ing.roeger@ttu.edu</email>
<xref rid="aff002" ref-type="aff">b</xref>
<xref rid="cor001" ref-type="corresp">⁎</xref>
</contrib>
</contrib-group>
<aff id="aff001"><label>a</label>
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan</aff>
<aff id="aff002"><label>b</label>
Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 79409, USA</aff>
<author-notes><corresp id="cor001"><label>⁎</label>
Corresponding author. Fax: +1 806 742 1112. <email>lih-ing.roeger@ttu.edu</email>
</corresp>
</author-notes>
<pub-date pub-type="pmc-release"><day>13</day>
<month>12</month>
<year>2006</year>
</pub-date>
<pmc-comment> PMC Release delay is 0 months and 0 days and was based on .</pmc-comment>
      <pub-date pub-type="ppub"><day>15</day>
<month>9</month>
<year>2007</year>
</pub-date>
<pub-date pub-type="epub"><day>13</day>
<month>12</month>
<year>2006</year>
</pub-date>
<volume>333</volume>
<issue>2</issue>
<fpage>557</fpage>
<lpage>566</lpage>
<history><date date-type="received"><day>10</day>
<month>11</month>
<year>2006</year>
</date>
</history>
<permissions><copyright-statement>Copyright © 2006 Elsevier Inc. All rights reserved.</copyright-statement>
<copyright-year>2006</copyright-year>
<copyright-holder>Elsevier Inc.</copyright-holder>
<license><license-p>Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.</license-p>
</license>
</permissions>
<abstract><p>In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, Ying-Hen Hsieh, Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak, SIAM J. Appl. Math. 66 (2006) 627–647]. An “acting basic reproductive number” <italic>ψ</italic>
 is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size <inline-formula><mml:math id="M1" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
</mml:math>
</inline-formula>
, the initial susceptible population <inline-formula><mml:math id="M2" altimg="si2.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:math>
</inline-formula>
, and <italic>ψ</italic>
. If <inline-formula><mml:math id="M3" altimg="si3.gif" overflow="scroll"><mml:mi>ψ</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</inline-formula>
, the disease will prevail and the final size of the susceptible, <inline-formula><mml:math id="M4" altimg="si4.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
</mml:math>
</inline-formula>
, becomes zero; therefore, everyone in the population will be infected eventually. If <inline-formula><mml:math id="M5" altimg="si5.gif" overflow="scroll"><mml:mi>ψ</mml:mi>
<mml:mo><</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</inline-formula>
, the disease dies out, and then <inline-formula><mml:math id="M6" altimg="si6.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</inline-formula>
 which means part of the population will never be infected. Also, when <inline-formula><mml:math id="M7" altimg="si7.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</inline-formula>
, <inline-formula><mml:math id="M8" altimg="si8.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mo>∞</mml:mo>
</mml:msub>
</mml:math>
</inline-formula>
 is increasing with respect to the initial susceptible population <inline-formula><mml:math id="M9" altimg="si9.gif" overflow="scroll"><mml:msub><mml:mi>S</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:math>
</inline-formula>
, and decreasing with respect to the acting basic reproductive number <italic>ψ</italic>
.</p>
</abstract>
<kwd-group><title>Keywords</title>
<kwd>SARS</kwd>
<kwd>Final size</kwd>
<kwd>Epidemic models</kwd>
</kwd-group>
</article-meta>
<notes><p>Submitted by G. Chen</p>
</notes>
</front>
</pmc>
</record>

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The final size of a SARS epidemic model without quarantine

The final size of a SARS epidemic model without quarantine

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