A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action
Identifieur interne : 000B05 ( Pmc/Corpus ); précédent : 000B04; suivant : 000B06A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action
Auteurs : Qianying Lin ; Shi Zhao ; Daozhou Gao ; Yijun Lou ; Shu Yang ; Salihu S. Musa ; Maggie H. Wang ; Yongli Cai ; Weiming Wang ; Lin Yang ; Daihai HeSource :
- International Journal of Infectious Diseases [ 1201-9712 ] ; 2020.
Abstract
For the ongoing novel coronavirus disease (CODID-19) outbreak in Wuhan, China, the Chinese government has implemented control measures such as city lockdown to mitigate the impact of the epidemic. We model the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalisation and quarantine) based on some parameters of the 1918 influenza pandemic in London, United Kingdom. We show the different effects of individual reaction and governmental action and preliminarily estimate the magnitude of these effects. We also preliminarily estimate the time-varying reporting ratio.
Url:
DOI: 10.1016/j.ijid.2020.02.058
PubMed: 32145465
PubMed Central: 7102659
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PMC:7102659Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action</title><author><name sortKey="Lin, Qianying" sort="Lin, Qianying" uniqKey="Lin Q" first="Qianying" last="Lin">Qianying Lin</name><affiliation><nlm:aff id="aff0005">Michigan Institute for Data Science, University of Michigan, Ann Arbor, Michigan, USA</nlm:aff></affiliation></author><author><name sortKey="Zhao, Shi" sort="Zhao, Shi" uniqKey="Zhao S" first="Shi" last="Zhao">Shi Zhao</name><affiliation><nlm:aff id="aff0010">JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China</nlm:aff></affiliation><affiliation><nlm:aff id="aff0015">Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China</nlm:aff></affiliation></author><author><name sortKey="Gao, Daozhou" sort="Gao, Daozhou" uniqKey="Gao D" first="Daozhou" last="Gao">Daozhou Gao</name><affiliation><nlm:aff id="aff0020">Mathematics and Science College, Shanghai Normal University, Shanghai, China</nlm:aff></affiliation></author><author><name sortKey="Lou, Yijun" sort="Lou, Yijun" uniqKey="Lou Y" first="Yijun" last="Lou">Yijun Lou</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="Yang, Shu" sort="Yang, Shu" uniqKey="Yang S" first="Shu" last="Yang">Shu Yang</name><affiliation><nlm:aff id="aff0030">College of Medical Information Engineering, Chengdu University of Traditional Chinese Medicine, Chengdu, China</nlm:aff></affiliation></author><author><name sortKey="Musa, Salihu S" sort="Musa, Salihu S" uniqKey="Musa S" first="Salihu S" last="Musa">Salihu S. Musa</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Maggie H" sort="Wang, Maggie H" uniqKey="Wang M" first="Maggie H" last="Wang">Maggie H. Wang</name><affiliation><nlm:aff id="aff0010">JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China</nlm:aff></affiliation><affiliation><nlm:aff id="aff0015">Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China</nlm:aff></affiliation></author><author><name sortKey="Cai, Yongli" sort="Cai, Yongli" uniqKey="Cai Y" first="Yongli" last="Cai">Yongli Cai</name><affiliation><nlm:aff id="aff0035">School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Weiming" sort="Wang, Weiming" uniqKey="Wang W" first="Weiming" last="Wang">Weiming Wang</name><affiliation><nlm:aff id="aff0035">School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, China</nlm:aff></affiliation></author><author><name sortKey="Yang, Lin" sort="Yang, Lin" uniqKey="Yang L" first="Lin" last="Yang">Lin Yang</name><affiliation><nlm:aff id="aff0040">School of Nursing, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="He, Daihai" sort="He, Daihai" uniqKey="He D" first="Daihai" last="He">Daihai He</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">32145465</idno><idno type="pmc">7102659</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7102659</idno><idno type="RBID">PMC:7102659</idno><idno type="doi">10.1016/j.ijid.2020.02.058</idno><date when="2020">2020</date><idno type="wicri:Area/Pmc/Corpus">000B05</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000B05</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action</title><author><name sortKey="Lin, Qianying" sort="Lin, Qianying" uniqKey="Lin Q" first="Qianying" last="Lin">Qianying Lin</name><affiliation><nlm:aff id="aff0005">Michigan Institute for Data Science, University of Michigan, Ann Arbor, Michigan, USA</nlm:aff></affiliation></author><author><name sortKey="Zhao, Shi" sort="Zhao, Shi" uniqKey="Zhao S" first="Shi" last="Zhao">Shi Zhao</name><affiliation><nlm:aff id="aff0010">JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China</nlm:aff></affiliation><affiliation><nlm:aff id="aff0015">Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China</nlm:aff></affiliation></author><author><name sortKey="Gao, Daozhou" sort="Gao, Daozhou" uniqKey="Gao D" first="Daozhou" last="Gao">Daozhou Gao</name><affiliation><nlm:aff id="aff0020">Mathematics and Science College, Shanghai Normal University, Shanghai, China</nlm:aff></affiliation></author><author><name sortKey="Lou, Yijun" sort="Lou, Yijun" uniqKey="Lou Y" first="Yijun" last="Lou">Yijun Lou</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="Yang, Shu" sort="Yang, Shu" uniqKey="Yang S" first="Shu" last="Yang">Shu Yang</name><affiliation><nlm:aff id="aff0030">College of Medical Information Engineering, Chengdu University of Traditional Chinese Medicine, Chengdu, China</nlm:aff></affiliation></author><author><name sortKey="Musa, Salihu S" sort="Musa, Salihu S" uniqKey="Musa S" first="Salihu S" last="Musa">Salihu S. Musa</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Maggie H" sort="Wang, Maggie H" uniqKey="Wang M" first="Maggie H" last="Wang">Maggie H. Wang</name><affiliation><nlm:aff id="aff0010">JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China</nlm:aff></affiliation><affiliation><nlm:aff id="aff0015">Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China</nlm:aff></affiliation></author><author><name sortKey="Cai, Yongli" sort="Cai, Yongli" uniqKey="Cai Y" first="Yongli" last="Cai">Yongli Cai</name><affiliation><nlm:aff id="aff0035">School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Weiming" sort="Wang, Weiming" uniqKey="Wang W" first="Weiming" last="Wang">Weiming Wang</name><affiliation><nlm:aff id="aff0035">School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, China</nlm:aff></affiliation></author><author><name sortKey="Yang, Lin" sort="Yang, Lin" uniqKey="Yang L" first="Lin" last="Yang">Lin Yang</name><affiliation><nlm:aff id="aff0040">School of Nursing, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author><author><name sortKey="He, Daihai" sort="He, Daihai" uniqKey="He D" first="Daihai" last="He">Daihai He</name><affiliation><nlm:aff id="aff0025">Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</nlm:aff></affiliation></author></analytic><series><title level="j">International Journal of Infectious Diseases</title><idno type="ISSN">1201-9712</idno><idno type="eISSN">1878-3511</idno><imprint><date when="2020">2020</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><title>Highlights</title><p><list list-type="simple" id="list0005"><list-item id="listitem0005"><label>•</label><p id="p0005">For the ongoing novel coronavirus disease (CODID-19) outbreak in Wuhan, China, the Chinese government has implemented control measures such as city lockdown to mitigate the impact of the epidemic.</p></list-item><list-item id="listitem0010"><label>•</label><p id="p0010">We model the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalisation and quarantine) based on some parameters of the 1918 influenza pandemic in London, United Kingdom.</p></list-item><list-item id="listitem0015"><label>•</label><p id="p0015">We show the different effects of individual reaction and governmental action and preliminarily estimate the magnitude of these effects.</p></list-item><list-item id="listitem0020"><label>•</label><p id="p0020">We also preliminarily estimate the time-varying reporting ratio.</p></list-item></list></p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="National Health Commission Of The People S Republic Of China" uniqKey="National Health Commission Of The People S Republic Of China">National Health Commission of the People's Republic of China</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Wu, P" uniqKey="Wu P">P. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Int J Infect Dis</journal-id><journal-id journal-id-type="iso-abbrev">Int. J. Infect. Dis</journal-id><journal-title-group><journal-title>International Journal of Infectious Diseases</journal-title></journal-title-group><issn pub-type="ppub">1201-9712</issn><issn pub-type="epub">1878-3511</issn><publisher><publisher-name>The Authors. Published by Elsevier Ltd on behalf of International Society for Infectious Diseases.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">32145465</article-id><article-id pub-id-type="pmc">7102659</article-id><article-id pub-id-type="publisher-id">S1201-9712(20)30117-X</article-id><article-id pub-id-type="doi">10.1016/j.ijid.2020.02.058</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action</article-title></title-group><contrib-group><contrib contrib-type="author" id="aut0005"><name><surname>Lin</surname><given-names>Qianying</given-names></name><email>qianying@umich.edu</email><xref rid="aff0005" ref-type="aff">a</xref><xref rid="fn0005" ref-type="fn">1</xref></contrib><contrib contrib-type="author" id="aut0010"><name><surname>Zhao</surname><given-names>Shi</given-names></name><email>zhaoshi.cmsa@gmail.com</email><xref rid="aff0010" ref-type="aff">b</xref><xref rid="aff0015" ref-type="aff">c</xref><xref rid="fn0005" ref-type="fn">1</xref></contrib><contrib contrib-type="author" id="aut0015"><name><surname>Gao</surname><given-names>Daozhou</given-names></name><email>dzgao@shnu.edu.cn</email><xref rid="aff0020" ref-type="aff">d</xref></contrib><contrib contrib-type="author" id="aut0020"><name><surname>Lou</surname><given-names>Yijun</given-names></name><email>yijun.lou@polyu.edu.hk</email><xref rid="aff0025" ref-type="aff">e</xref></contrib><contrib contrib-type="author" id="aut0025"><name><surname>Yang</surname><given-names>Shu</given-names></name><email>sishiyu1978@qq.com</email><xref rid="aff0030" ref-type="aff">f</xref></contrib><contrib contrib-type="author" id="aut0030"><name><surname>Musa</surname><given-names>Salihu S</given-names></name><email>salihu-sabiu.musa@connect.polyu.hk</email><xref rid="aff0025" ref-type="aff">e</xref></contrib><contrib contrib-type="author" id="aut0035"><name><surname>Wang</surname><given-names>Maggie H</given-names></name><email>haitian.wang@gmail.com</email><xref rid="aff0010" ref-type="aff">b</xref><xref rid="aff0015" ref-type="aff">c</xref></contrib><contrib contrib-type="author" id="aut0040"><name><surname>Cai</surname><given-names>Yongli</given-names></name><email>yonglicai@hytc.edu.cn</email><xref rid="aff0035" ref-type="aff">g</xref></contrib><contrib contrib-type="author" id="aut0045"><name><surname>Wang</surname><given-names>Weiming</given-names></name><email>weimingwang2003@163.com</email><xref rid="aff0035" ref-type="aff">g</xref><xref rid="cor0005" ref-type="corresp">⁎</xref></contrib><contrib contrib-type="author" id="aut0050"><name><surname>Yang</surname><given-names>Lin</given-names></name><email>l.yang@polyu.edu.hk</email><xref rid="aff0040" ref-type="aff">h</xref><xref rid="cor0005" ref-type="corresp">⁎</xref></contrib><contrib contrib-type="author" id="aut0055"><name><surname>He</surname><given-names>Daihai</given-names></name><email>daihai.he@polyu.edu.hk</email><xref rid="aff0025" ref-type="aff">e</xref><xref rid="cor0005" ref-type="corresp">⁎</xref></contrib></contrib-group><aff id="aff0005"><label>a</label>Michigan Institute for Data Science, University of Michigan, Ann Arbor, Michigan, USA</aff><aff id="aff0010"><label>b</label>JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China</aff><aff id="aff0015"><label>c</label>Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China</aff><aff id="aff0020"><label>d</label>Mathematics and Science College, Shanghai Normal University, Shanghai, China</aff><aff id="aff0025"><label>e</label>Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China</aff><aff id="aff0030"><label>f</label>College of Medical Information Engineering, Chengdu University of Traditional Chinese Medicine, Chengdu, China</aff><aff id="aff0035"><label>g</label>School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, China</aff><aff id="aff0040"><label>h</label>School of Nursing, Hong Kong Polytechnic University, Hong Kong, China</aff><author-notes><corresp id="cor0005"><label>⁎</label>Corresponding authors. <email>weimingwang2003@163.com</email><email>l.yang@polyu.edu.hk</email><email>daihai.he@polyu.edu.hk</email></corresp><fn id="fn0005"><label>1</label><p id="npar0010">These authors equally contributed..</p></fn></author-notes><pub-date pub-type="pmc-release"><day>4</day><month>3</month><year>2020</year></pub-date><pmc-comment> PMC Release delay is 0 months and 0 days and was based on .</pmc-comment>
<pub-date pub-type="epub"><day>4</day><month>3</month><year>2020</year></pub-date><elocation-id></elocation-id><history><date date-type="received"><day>31</day><month>1</month><year>2020</year></date><date date-type="rev-recd"><day>26</day><month>2</month><year>2020</year></date><date date-type="accepted"><day>27</day><month>2</month><year>2020</year></date></history><permissions><copyright-statement>© 2020 The Authors. Published by Elsevier Ltd on behalf of International Society for Infectious Diseases.</copyright-statement><copyright-year>2020</copyright-year><copyright-holder></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 abstract-type="author-highlights" id="abs0005"><title>Highlights</title><p><list list-type="simple" id="list0005"><list-item id="listitem0005"><label>•</label><p id="p0005">For the ongoing novel coronavirus disease (CODID-19) outbreak in Wuhan, China, the Chinese government has implemented control measures such as city lockdown to mitigate the impact of the epidemic.</p></list-item><list-item id="listitem0010"><label>•</label><p id="p0010">We model the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalisation and quarantine) based on some parameters of the 1918 influenza pandemic in London, United Kingdom.</p></list-item><list-item id="listitem0015"><label>•</label><p id="p0015">We show the different effects of individual reaction and governmental action and preliminarily estimate the magnitude of these effects.</p></list-item><list-item id="listitem0020"><label>•</label><p id="p0020">We also preliminarily estimate the time-varying reporting ratio.</p></list-item></list></p></abstract><abstract id="abs0010"><p>The ongoing Coronavirus Disease 2019 (COVID-19) outbreak, originated in the end of 2019 in Wuhan, China, has claimed more than 2200 lives and posed a huge threat to global public health. The Chinese government has implemented control measures including setting up special hospitals and travel restriction to mitigate the spread. We propose conceptual models for the outbreak in Wuhan with the consideration of individual behavioural reaction and governmental actions, e.g., holiday extension, travel restriction, hospitalisation and quarantine. We employed the estimates of these two key components from the 1918 influenza pandemic in London, United Kingdom, incorporated zoonotic introductions and the emigration, then computed future trends and the reporting ratio. The model is concise in structure, and it successfully captures the course of the COVID-19 outbreak, and thus sheds light on understanding the trends of the outbreak.</p></abstract><kwd-group id="kwd0005"><title>Keywords</title><kwd>COVID-19</kwd><kwd>Epidemic</kwd><kwd>mathematical modelling</kwd><kwd>individual reaction</kwd><kwd>governmental action</kwd><kwd>city lock-down</kwd></kwd-group></article-meta></front><body><sec id="sec0005"><label>1</label><title>Introduction</title><p id="p0025">The ongoing outbreak of Coronavirus Disease 2019, COVID-19, has claimed 2663 lives, along with 77,658 confirmed cases and 2824 suspected cases in China, as of 24 February 2020 (24:00 GMT+8), according to the National Health Commission of the People's Republic of China (<xref rid="bib0005" ref-type="bibr">NHCPRC, 2020</xref>). This number of deaths greatly exceeds the other two coronaviruses (Severe Acure Respiratory Syndrome Coronavirus and Middle East Respiratory Syndrome Coronavirus), and it is still increasing, which posed a huge threat to the global public health and economics (<xref rid="bib0010" ref-type="bibr">[Bogoch et al., in press]</xref>, <xref rid="bib0015" ref-type="bibr">[Wu et al., 2020]</xref>).</p><p id="p0030">The emergence of COVID-19 coincided with the largest annual human migration in the world, i.e., the Spring Festival travel season, which resulted in a rapid national and global spread of the virus. At early stage of the outbreak, most cases were scattered, and some linked to the Huanan Seafood Wholesale Market (<xref rid="bib0015" ref-type="bibr">Wu et al., 2020</xref>). The Chinese government has adopted extreme measures to mitigate outbreak. On 23 January 2020, the local government of Wuhan suspended all public traffics within the city, and closed all inbound and outbound transportation. Other cities in Hubei province announced similar traffic control measures following Wuhan shortly, see Fig. <xref rid="fig0005" ref-type="fig">1</xref>. The resumption date in Wuhan remains unclear as of the submission date of this study on 25 February 2020.<fig id="fig0005"><label>Fig. 1</label><caption><p>The timeline of the facts of COVID-19 and control measures implemented in Wuhan, China from December 2019 to February 2020. The red dots are the events in the COVID-19 outbreak, and the blue dots are the control measures.</p></caption><graphic xlink:href="gr1_lrg"></graphic></fig></p><p id="p0035">The public panic in face of the ongoing COVID-19 outbreak reminds us the history of the 1918 influenza pandemic in London, United Kingdom. Furthermore, its characteristics of mild symptoms in most cases and short serial interval (i.e., 4-5 days) (<xref rid="bib0020" ref-type="bibr">[Nishiura et al., 2020]</xref>, <xref rid="bib0025" ref-type="bibr">[You et al., 2002]</xref>) are similar to pandemic influenza, rather than other two coronaviruses (i.e., SARS-CoV and MERS-CoV). In 1918, a significant proportion of the deaths were from pneumonia followed influenza infection. Thus, it might be reasonable to revisit the modelling framework of 1918 influenza pandemic, and in particular, to capture the effects of the individual reaction (to the risk of infection) and government action. In (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>), we proposed a model incorporating individual reaction, holiday effects as well as weather conditions (temperature in London, United Kingdom) which successfully captured the multiple-wave feature in the influenza-associated mortality in London.</p><p id="p0040">In this study, we followed the form of individual reaction and governmental action effects in (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>), except for the effects of weather condition due to limited knowledge on weather effects on the transmission of coronaviruses. We note that the governmental action, in both 1918 and current time, summarized all measures including holiday extension, city lock-down, hospitalisation and quarantine of patients. We presume it will last for the next few months for the moment, and will update later if things change. The parameter values may be improved when more information is available. We argue that all prevention and control measures may be categorised into two large groups, which are described by either a step function or a response function, respectively. We also consider zoonotic transmission period of one month and a huge emigration from Wuhan (35.7% of the population). Nevertheless, our model is a preliminary conceptual model, intending to lay a foundation for further modelling studies, but we can easily tune our model so that the outcomes of our model are in line with the observation and previous studies (<xref rid="bib0015" ref-type="bibr">[Wu et al., 2020]</xref>, <xref rid="bib0035" ref-type="bibr">[Mahase, 2020]</xref>).</p></sec><sec id="sec0010"><label>2</label><title>A conceptual model</title><p id="p0045">We adopt the ‘Susceptible-Exposed-Infectious-Removed’ (SEIR) framework with the total population size <inline-formula><mml:math id="M1" altimg="si19.svg"><mml:mi>N</mml:mi></mml:math></inline-formula> with two extra classes (1) “<inline-formula><mml:math id="M2" altimg="si20.svg"><mml:mi>D</mml:mi></mml:math></inline-formula>” mimicking the public perception of risk regarding the number of severe and critical cases and deaths; and (2) “<inline-formula><mml:math id="M3" altimg="si21.svg"><mml:mi>C</mml:mi></mml:math></inline-formula>” representing the number of cumulative cases (both reported and not reported). Let <inline-formula><mml:math id="M4" altimg="si22.svg"><mml:mi>S</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M5" altimg="si23.svg"><mml:mi>E</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M6" altimg="si24.svg"><mml:mi>I</mml:mi></mml:math></inline-formula> represent the susceptible, exposed and infectious populations and <inline-formula><mml:math id="M7" altimg="si25.svg"><mml:mi>R</mml:mi></mml:math></inline-formula> represent the removed population (i.e., recovered or dead). In a recent study (<xref rid="bib0040" ref-type="bibr">Wu and McGoogan, 2020</xref>), Wu and McGoogan found that 81% of cases were of mild symptom (without pneumonia or only mild pneumonia), 14% were severe case with difficulty breathing, and 5% were critical with respiratory failure, septic shock, and/or multiple organ dysfunction or failure.</p><p id="p0050">We adopt the transmission rate function from (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>). We rename the school term effect as the governmental action effect, since the former belongs to the latter. We also assume a period of zoonotic transmission during December 2019. We model the zoonotic transmission (denoted as <inline-formula><mml:math id="M8" altimg="si7.svg"><mml:mi>F</mml:mi></mml:math></inline-formula>) as a stepwise function, which takes zero after the shutdown of Huanan seafood market (presumably). We then only model the sustained human-to-human transmission of COVID-19 after this date, along with the emigration of 5 million population before Wuhan was officially locked down (<xref rid="bib0045" ref-type="bibr">South China Morning Post, 2020</xref>). Thus, a compartmental model is formulated as follows:<disp-formula id="eq0005"><label>(1)</label><mml:math id="M9" altimg="si26.svg"><mml:mfenced open="{"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>S</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi mathvariant="italic">SF</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="italic">SI</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mi>S</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>E</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi mathvariant="italic">SF</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="italic">SI</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>I</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mi>E</mml:mi><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>R</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mi>R</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>N</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>D</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mtext> </mml:mtext><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:mi>λ</mml:mi><mml:mi>D</mml:mi><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi mathvariant="normal">and</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:msup><mml:mi>C</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd columnalign="left"><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd><mml:mtd columnalign="left"></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></disp-formula>where<disp-formula id="eq0010"><label>(2)</label><mml:math id="M10" altimg="si27.svg"><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mi>D</mml:mi><mml:mi>N</mml:mi></mml:mfrac></mml:mrow></mml:mfenced></mml:mrow><mml:mi>κ</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula></p><p id="p0055">The efficient transmission rate, <inline-formula><mml:math id="M11" altimg="si28.svg"><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> in Equation <xref rid="eq0010" ref-type="disp-formula">(2)</xref>, incorporates the impact of governmental action (all actions which can be modelled as a step function), and the decreasing contacts among individuals responding to the proportion of deaths (i.e., the severity of the epidemic). We also incorporate the individuals leaving Wuhan before the lock-down in the model. We assume (i) the zoonotic cases only make impacts during December 2019 (<xref rid="bib0050" ref-type="bibr">Huang et al., 2020</xref>); (ii) the effect of governmental action starts on 23 January 2020 (in particular, <inline-formula><mml:math id="M12" altimg="si29.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.4249</mml:mn></mml:math></inline-formula> during 23-29 January 2020 and <inline-formula><mml:math id="M13" altimg="si6.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.8478</mml:mn></mml:math></inline-formula> after that); (iii) the emigration from Wuhan starts on 31 December 2019 and ends on 22 January 2020.</p><p id="p0060">In this outbreak it seems children are spared. Only 0.9% cases are from age 15 or less (<xref rid="bib0055" ref-type="bibr">Guan et al., 2020</xref>), while In China, 0-14 years are 17.2%. To take this effect into account, we assume 10% of the population are ‘protected’. Recently studies showed the serial interval of COVID-19 could be as short as 5 days (<xref rid="bib0060" ref-type="bibr">[Nishiura et al., 2020]</xref>, <xref rid="bib0065" ref-type="bibr">[You et al., 2020]</xref>), and the median incubation period could be as short as 3 days (<xref rid="bib0055" ref-type="bibr">Guan et al., 2020</xref>). These characteristics imply short latent period and infectious period. Thus, we adopt a relatively shorter mean latent period (3 days) and mean infectious period (4 days). Different from (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>), we use the severe cases and death in the individual reaction function, instead of death only. We also increase the intensity of the governmental action such that the model outcomes (increments in cases) largely match the observed, with a reporting ratio. Namely only a proportion of the model generated cases will be reported in reality. Many evidences and studies, e.g., (<xref rid="bib0070" ref-type="bibr">[Tuite and Fisman, 2020]</xref>, <xref rid="bib0075" ref-type="bibr">[Zhao et al., 2020]</xref>, <xref rid="bib0080" ref-type="bibr">[Zhao et al., 2020]</xref>), suggest the reporting ratio is time-varying. We summarise our parameters in Table <xref rid="tbl0005" ref-type="table">1</xref>.<table-wrap position="float" id="tbl0005"><label>Table 1</label><caption><p>Summary table of the parameters in model <xref rid="eq0005" ref-type="disp-formula">(1)</xref>.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">Parameter</th><th align="left">Notation</th><th align="left">Value or Range</th><th align="left">Remark</th><th align="left">Reference</th></tr></thead><tbody><tr><td align="left">number of zoonotic cases</td><td align="left"><inline-formula><mml:math id="M14" altimg="si7.svg"><mml:mi>F</mml:mi></mml:math></inline-formula></td><td align="left">{0, 10}</td><td align="left">a stepwise function</td><td align="left">(<xref rid="bib0015" ref-type="bibr">Wu et al., 2020</xref>)</td></tr><tr><td align="left">initial population size</td><td align="left"><inline-formula><mml:math id="M15" altimg="si8.svg"><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula></td><td align="left">14 million</td><td align="left">constant</td><td align="left">(<xref rid="bib0045" ref-type="bibr">South China Morning Post, 2020</xref>)</td></tr><tr><td align="left">initial susceptible population</td><td align="left"><inline-formula><mml:math id="M16" altimg="si9.svg"><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula></td><td align="left">0.9<inline-formula><mml:math id="M17" altimg="si8.svg"><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula></td><td align="left">constant</td><td align="left">Assumed</td></tr><tr><td align="left">transmission rate</td><td align="left"><inline-formula><mml:math id="M18" altimg="si10.svg"><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula></td><td align="left">{0.5944, 1.68}<xref rid="tblfn0005" ref-type="table-fn">*</xref> (day<inline-formula><mml:math id="M19" altimg="si11.svg"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>)</td><td align="left">a stepwise function</td><td align="left">Assumed</td></tr><tr><td align="left">governmental action strength</td><td align="left"><inline-formula><mml:math id="M20" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula></td><td align="left">{0,0.4239,0.8478}</td><td align="left">a stepwise function</td><td align="left">(<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>)</td></tr><tr><td align="left">intensity of responds</td><td align="left"><inline-formula><mml:math id="M21" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula></td><td align="left">1117.3</td><td align="left">constant</td><td align="left">(<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>)</td></tr><tr><td align="left">emigration rate</td><td align="left"><inline-formula><mml:math id="M22" altimg="si12.svg"><mml:mi>μ</mml:mi></mml:math></inline-formula></td><td align="left">{0, 0.0205} (day<inline-formula><mml:math id="M23" altimg="si11.svg"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>)</td><td align="left">a stepwise function</td><td align="left">(<xref rid="bib0045" ref-type="bibr">South China Morning Post, 2020</xref>)</td></tr><tr><td align="left">mean latent period</td><td align="left"><inline-formula><mml:math id="M24" altimg="si13.svg"><mml:msup><mml:mi>σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></td><td align="left">3 (days)</td><td align="left">constant</td><td align="left">(<xref rid="bib0015" ref-type="bibr">Wu et al., 2020</xref>)</td></tr><tr><td align="left">mean infectious period</td><td align="left"><inline-formula><mml:math id="M25" altimg="si14.svg"><mml:msup><mml:mi>γ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></td><td align="left">5 (days)</td><td align="left">constant</td><td align="left">(<xref rid="bib0015" ref-type="bibr">Wu et al., 2020</xref>)</td></tr><tr><td align="left">proportion of severe cases</td><td align="left"><inline-formula><mml:math id="M26" altimg="si15.svg"><mml:mi>d</mml:mi></mml:math></inline-formula></td><td align="left">0.2</td><td align="left">constant</td><td align="left">(<xref rid="bib0085" ref-type="bibr">Worldometers., 2020</xref>)</td></tr><tr><td align="left">mean duration of public reaction</td><td align="left"><inline-formula><mml:math id="M27" altimg="si16.svg"><mml:msup><mml:mi>λ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></td><td align="left">11.2 (days)</td><td align="left">constant</td><td align="left">(<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>)</td></tr></tbody></table><table-wrap-foot><fn id="tblfn0005"><label>*</label><p id="npar0005">It is derived by assuming that the basic reproduction number, <inline-formula><mml:math id="M28" altimg="si17.svg"><mml:mtext>R0 </mml:mtext><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mi>γ</mml:mi></mml:mfrac><mml:mo>·</mml:mo><mml:mfrac><mml:mi>σ</mml:mi><mml:mrow><mml:mi>σ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>2.8</mml:mn></mml:math></inline-formula> (referring to (<xref rid="bib0090" ref-type="bibr">[Wu et al., 2020]</xref>, <xref rid="bib0080" ref-type="bibr">[Zhao et al., 2020]</xref>, <xref rid="bib0075" ref-type="bibr">[Zhao et al., 2020]</xref>, <xref rid="bib0095" ref-type="bibr">[Natsuko, 2020]</xref>, <xref rid="bib0100" ref-type="bibr">[Riou and Althaus, 2020]</xref>, <xref rid="bib0105" ref-type="bibr">[Li et al., 2002]</xref>)) when <inline-formula><mml:math id="M29" altimg="si18.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, by using the next generation matrix approach (<xref rid="bib0110" ref-type="bibr">van den Driessche and Watmough, 2002</xref>). The time unit is in year if not mentioned.</p></fn></table-wrap-foot></table-wrap></p></sec><sec id="sec0015"><label>3</label><title>Data analyses</title><p id="p0065">We summarise the officially reported data from Wuhan, China in <xref rid="fig0010" ref-type="fig">Fig. 2</xref>. There is an increasing trend of daily new confirmations and deaths. We argue that these data were heavily impacted by availability of medical supplies and health care workers.<fig id="fig0010"><label>Fig. 2</label><caption><p>The daily number of (a) cases or (b) deaths, cumulative number of (c) cases or (d) deaths, and the percentage of (e) cases or (f) deaths, of COVID-19 in Wuhan, China. In panel (f), the <inline-formula><mml:math id="M30" altimg="si1.svg"><mml:mn>100</mml:mn><mml:mo>%</mml:mo></mml:math></inline-formula> represents the count of deaths or cured cases.</p></caption><graphic xlink:href="gr2_lrg"></graphic></fig></p><p id="p0070">Fig. <xref rid="fig0015" ref-type="fig">3</xref><fig id="fig0015"><label>Fig. 3</label><caption><p>Comparison between different sources of reported cases: official released data (<xref rid="bib0005" ref-type="bibr">NHCPRC, 2020</xref>) in red, data from Li et al (denoted as NEJM) (<xref rid="bib0115" ref-type="bibr">Li et al., 2020</xref>) in green, from Liu et al (denoted as GDCDC) (<xref rid="bib0120" ref-type="bibr">Liu et al., 2020</xref>) in blue, and from Wu et al (denoted as Eurosurv) (<xref rid="bib0015" ref-type="bibr">Wu et al., 2020</xref>) in purple.</p></caption><graphic xlink:href="gr3_lrg"></graphic></fig></p><p id="p0075">The official data report was not available before January 15, 2020. We fill the missing data before that from several retrospective studies. Among them data in Li et al (<xref rid="bib0115" ref-type="bibr">Li et al., 2020</xref>) are daily symptom onset records, while those in Liu et al (<xref rid="bib0120" ref-type="bibr">Liu et al., 2020</xref>) are daily confirmations. We notice that there is a delay of 14 days between symptom onset and laboratory confirmation of COVID-19 between the two datasets which are largely the same group of patients. Namely if we put back data in Li et al (<xref rid="bib0115" ref-type="bibr">Li et al., 2020</xref>) by 14 days, it largely matches data in (<xref rid="bib0120" ref-type="bibr">Liu et al., 2020</xref>). Thus, we assume a proportion of daily cases (reporting rate) will be reported after 14 days since their infectiousness onset (which is generally no later than their symptom onset).</p></sec><sec id="sec0020"><label>4</label><title>Model simulation</title><p id="p0080">We show our simulations in Fig. <xref rid="fig0020" ref-type="fig">4</xref>. Under the naive scenario, we assume governmental action strength <inline-formula><mml:math id="M31" altimg="si18.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and intensity of individual reaction <inline-formula><mml:math id="M32" altimg="si30.svg"><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, which is unlikely. The second scenario is when we only consider “individual reaction”, both the peak value and the number of cumulative cases are substantially reduced. The third scenario is considering both “individual reaction” and “governmental action”, and the reduction becomes even further. We highlight the third scenario, as we know the individual reaction and governmental action existed and played important role in previous epidemic and pandemic (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>). Our third scenario implies that<list list-type="simple" id="list0010"><list-item id="listitem0025"><label>•</label><p id="p0085">The total number of zoonotic infections was 145 which corresponds to the reported 41 zoonotic cases with a reporting rate of <inline-formula><mml:math id="M33" altimg="si31.svg"><mml:mo>≈</mml:mo></mml:math></inline-formula> 28%. This level is largely in line with estimates of (<xref rid="bib0100" ref-type="bibr">[Riou and Althaus, 2020]</xref>, <xref rid="bib0125" ref-type="bibr">[Nishiura et al., 2020]</xref>, <xref rid="bib0105" ref-type="bibr">[Li et al., 2002]</xref>).</p></list-item><list-item id="listitem0030"><label>•</label><p id="p0090">The cumulative number of cases in Wuhan was 4,648 by January 18, 2020, which is in line with estimates of other teams (<xref rid="bib0010" ref-type="bibr">[Bogoch et al., in press]</xref>, <xref rid="bib0015" ref-type="bibr">[Wu et al., 2020]</xref>, <xref rid="bib0130" ref-type="bibr">[NCPERET, 2020]</xref>).</p></list-item><list-item id="listitem0035"><label>•</label><p id="p0095">The cumulative number of cases in Wuhan was 16,589 by January 27, 2020. Compared with estimates 25,630 (95%CI: 12,260–44,440), announced by University of Hong Kong team on January 27, 2020, our estimate is low but in their the 95% CI.</p></list-item><list-item id="listitem0040"><label>•</label><p id="p0100">The cumulative infections could be 84,116 in Wuhan by the end of April 2020.</p></list-item><list-item id="listitem0045"><label>•</label><p id="p0105">We compare simulated and reported numbers, and reconstruct the daily reporting ratio, which an improvement from a level of below 10% to around 50% from January 2020 to February 2020 which reflects the reality.</p></list-item><list-item id="listitem0050"><label>•</label><p id="p0110">Due to adjustment of the reporting policy, i.e., an effort to report all clinical cases accumulated in the past few days/weeks, there are a few days where the number of reported cases are artificially high than simulated cases. The reason is that the reported cases in these few days included clinical cases but not laboratory confirmed that are accumulated in the past few days, also weeks.</p></list-item></list><fig id="fig0020"><label>Fig. 4</label><caption><p>(a) Daily new cases with a reporting delay of 14 days under three scenarios: naive (i.e., no action taken) as grey dotted curve, individual reaction regarding to the outbreak as red dashed curve, and individual reaction plus governmental action as green solid curve and reported cases (from official release and (<xref rid="bib0115" ref-type="bibr">Li et al., 2020</xref>) as grey curve with dotes. (b) The reporting ratio between reported cases and estimates when individual reaction and governmental action are involved.</p></caption><graphic xlink:href="gr4_lrg"></graphic></fig></p><p id="p0115">The main purpose of this work is to propose a conceptual model to address the individual reaction (controlled by <inline-formula><mml:math id="M34" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula>) and governmental action (controlled by <inline-formula><mml:math id="M35" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula>), as well as time-varying reporting rate. We perform a simple sensitive analyses on <inline-formula><mml:math id="M36" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M37" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula> in Fig. <xref rid="fig0025" ref-type="fig">5</xref>, where we can see that both <inline-formula><mml:math id="M38" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M39" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula> are needed to capture the observed pattern. In particular, when <inline-formula><mml:math id="M40" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula> is around 0.9 and <inline-formula><mml:math id="M41" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula> is greater than 1100, the simulated largely match the observed.<fig id="fig0025"><label>Fig. 5</label><caption><p>Sensitivity analyses on <inline-formula><mml:math id="M42" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M43" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula>. We simulate the base model with both individual reaction and governmental action while varying <inline-formula><mml:math id="M44" altimg="si2.svg"><mml:mi>α</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M45" altimg="si3.svg"><mml:mi>κ</mml:mi></mml:math></inline-formula>. We show model outcome when (a) <inline-formula><mml:math id="M46" altimg="si4.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo></mml:math></inline-formula> 0.5 (black solid), 0.6 (red dotted), 0.7 (green dashed), 0.8 (blue dot-dashed) and 0.9 (cyan long dashed curve), while <inline-formula><mml:math id="M47" altimg="si5.svg"><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>1117.3</mml:mn></mml:math></inline-formula>, when (b) <inline-formula><mml:math id="M48" altimg="si4.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo></mml:math></inline-formula> 100 (black solid), 500 (red dotted), 900 (green dashed), 1300 (blue dotted dash) and 1700 (cyan long dashed curve), while <inline-formula><mml:math id="M49" altimg="si6.svg"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.8478</mml:mn></mml:math></inline-formula>. Grey dots show the reported cases.</p></caption><graphic xlink:href="gr5_lrg"></graphic></fig></p></sec><sec id="sec0025"><label>5</label><title>Discussion and conclusions</title><p id="p0120">We use some parameter estimates from (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>). The estimates were obtained via fitting a mechanistic model to the observed weekly influenza and pneumonia mortality in England and Wales during the 1918 influenza pandemic. Recent studies showed that COVID-19 transmitted rapidly. In this regard, it resembles influenza rather than SARS. In our 1918 influenza work (<xref rid="bib0030" ref-type="bibr">He et al., 2013</xref>), we built a similar model as we introduced here, and we fitted that model to weekly influenza and pneumonia mortality in 334 administrative units. Note that 1918 influenza had an infection-fatality-rate of 2%, which is at the same level of the case-fatality-rate of COVID-19 in Wuhan, China.</p><p id="p0125">The merit of our model is that we considered some essential elements, including individual behavioural response, governmental actions, zoonotic transmission and emigration of a large proportion of the population in a short time period. Meanwhile, our model is relatively simple and our estimates are in line with previous studies (<xref rid="bib0015" ref-type="bibr">[Wu et al., 2020]</xref>, <xref rid="bib0090" ref-type="bibr">[Wu et al., 2020]</xref>, <xref rid="bib0095" ref-type="bibr">[Natsuko, 2020]</xref>). Thus, our model should be considered as a baseline model for further improvement.</p><p id="p0130">We did not fit model to data in conventional way. Instead, we use a simple model framework to discuss what elements might be needed. For instance, in order to achieve a good fit, one obviously needs to include a time-varying report rate (as we reconstructed in <xref rid="fig0020" ref-type="fig">Fig. 4</xref>b), which was caused by the availability of medical supplies, hospital capacities and changing testing/reporting policies. Thus it would be challenging given a relatively short time series, and several other unknown parameters to be estimated. We employ some estimated parameter values from the 1918 influenza pandemic, given the similar characteristics of COVID-19 and influenza (most cases are mild) and the similar level of mitigation. Transmission from asymptotically infected cases is reported but the contribution of asymptomatic transmission is unclear (presumably small), which shall be further investigated in future studies.</p><p id="p0135">In this work, we focused on the transmission of COVID-19 in Wuhan, China. Our conceptual framework can be applied to other cities/countries, or be built into one (multiple-patched) model for multiple cities/countries. Our model can be fitted to daily data when more information (e.g., daily number of tests) is available.</p></sec><sec id="sec0030"><title>Declarations</title><sec id="sec0035"><title>Ethics approval and consent to participate</title><p id="p0140">Since no individual patient's data was collected, the ethical approval or individual consent was not applicable.</p></sec><sec sec-type="data-availability" id="sec0040"><title>Availability of data and materials</title><p id="p0145">All data are publicly available.</p></sec><sec id="sec0045"><title>Funding</title><p id="p0150">This research was supported by National Natural Science Foundation of China (Grant number 61672013 and 11601336), Huaian Key Laboratory for Infectious Diseases Control and Prevention (HAP201704), and General Research Fund (Grant Number 15205119) of the Research Grants Council (RGC) of Hong Kong, China.</p></sec></sec><sec id="sec0050"><title>Disclaimer</title><p id="p0155">The funding agencies had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.</p></sec><sec id="sec0055"><title>Conflict of Interests</title><p id="p0160">The authors declare that they have no competing interests.</p></sec><sec id="sec0060"><title>Authors’ Contributions</title><p id="p0165">Conceptualization: Qianying Lin, Shi Zhao, Daozhou Gao, Yijun Lou, Salihu S Musa, Shu Yang, Maggie H Wang, Yongli Cai, Weiming Wang, Lin Yang and Daihai He; Formal analysis: Qianying Lin, Shi Zhao, Daozhou Gao, Yijun Lou, Salihu S Musa, Shu Yang, Maggie H Wang, Weiming Wang, Lin Yang and Daihai He; Visualization: Lin Yang; Writing – original draft: Qianying Lin, Shi Zhao, Daozhou Gao, Yijun Lou, Salihu S Musa, Shu Yang, Maggie H Wang, Yongli Cai, Weiming Wang and Lin Yang; Writing – review & editing, Lin Yang and Daihai He.</p></sec></body><back><ref-list id="bibl0005"><title>References</title><ref id="bib0005"><label>NHCPRC, 2020</label><element-citation publication-type="book" id="sbref0005"><person-group person-group-type="author"><name><surname>National Health Commission of the People's Republic of China</surname></name></person-group><chapter-title>The updated news of COVID-19 as of 24 February 2020</chapter-title><year>2020</year><comment>24: 00;. 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