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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Comparison of methods to Estimate Basic Reproduction Number (<italic>R</italic><sub>0</sub>) of influenza, Using Canada 2009 and 2017-18 A (H1N1) Data</title><author><name sortKey="Nikbakht, Roya" sort="Nikbakht, Roya" uniqKey="Nikbakht R" first="Roya" last="Nikbakht">Roya Nikbakht</name><affiliation><nlm:aff id="aff1">HIV/STI Surveillance Research Center, and WHO Collaborating Center for HIV Surveillance, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Department of Biostatistics and Epidemiology, Faculty of Health Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Baneshi, Mohammad Reza" sort="Baneshi, Mohammad Reza" uniqKey="Baneshi M" first="Mohammad Reza" last="Baneshi">Mohammad Reza Baneshi</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Bahrampour, Abbas" sort="Bahrampour, Abbas" uniqKey="Bahrampour A" first="Abbas" last="Bahrampour">Abbas Bahrampour</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Hosseinnataj, Abolfazl" sort="Hosseinnataj, Abolfazl" uniqKey="Hosseinnataj A" first="Abolfazl" last="Hosseinnataj">Abolfazl Hosseinnataj</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">31523253</idno><idno type="pmc">6670001</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6670001</idno><idno type="RBID">PMC:6670001</idno><idno type="doi">10.4103/jrms.JRMS_888_18</idno><date when="2019">2019</date><idno type="wicri:Area/Pmc/Corpus">000A14</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000A14</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Comparison of methods to Estimate Basic Reproduction Number (<italic>R</italic><sub>0</sub>) of influenza, Using Canada 2009 and 2017-18 A (H1N1) Data</title><author><name sortKey="Nikbakht, Roya" sort="Nikbakht, Roya" uniqKey="Nikbakht R" first="Roya" last="Nikbakht">Roya Nikbakht</name><affiliation><nlm:aff id="aff1">HIV/STI Surveillance Research Center, and WHO Collaborating Center for HIV Surveillance, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Department of Biostatistics and Epidemiology, Faculty of Health Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Baneshi, Mohammad Reza" sort="Baneshi, Mohammad Reza" uniqKey="Baneshi M" first="Mohammad Reza" last="Baneshi">Mohammad Reza Baneshi</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Bahrampour, Abbas" sort="Bahrampour, Abbas" uniqKey="Bahrampour A" first="Abbas" last="Bahrampour">Abbas Bahrampour</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author><author><name sortKey="Hosseinnataj, Abolfazl" sort="Hosseinnataj, Abolfazl" uniqKey="Hosseinnataj A" first="Abolfazl" last="Hosseinnataj">Abolfazl Hosseinnataj</name><affiliation><nlm:aff id="aff2">Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</nlm:aff></affiliation></author></analytic><series><title level="j">Journal of Research in Medical Sciences : The Official Journal of Isfahan University of Medical Sciences</title><idno type="ISSN">1735-1995</idno><idno type="eISSN">1735-7136</idno><imprint><date when="2019">2019</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><sec id="st1"><title>Background:</title><p>The basic reproduction number (<italic>R</italic><sub>0</sub>) has a key role in epidemics and can be utilized for preventing epidemics. In this study, different methods are used for estimating <italic>R</italic><sub>0</sub>'s and their vaccination coverage to find the formula with the best performance.</p></sec><sec id="st2"><title>Materials and Methods:</title><p>We estimated <italic>R</italic><sub>0</sub> for cumulative cases count data from April 18 to July 6, 2009 and 35-2017 to 34-2018 weeks in Canada: maximum likelihood (ML), exponential growth rate (EG), time-dependent reproduction numbers (TD), attack rate (AR), gamma-distributed generation time (GT), and the final size of the epidemic. Gamma distribution with mean and standard deviation 3.6 ± 1.4 is used as GT.</p></sec><sec id="st3"><title>Results:</title><p>The AR method obtained a <italic>R</italic><sub>0 (</sub>95% confidence interval [CI]) value of 1.116 (1.1163, 1.1165) and an EG (95%CI) value of 1.46 (1.41, 1.52). The <italic>R</italic><sub>0</sub> (95%CI) estimate was 1.42 (1.27, 1.57) for the obtained ML, 1.71 (1.12, 2.03) for the obtained TD, 1.49 (1.0, 1.97) for the gamma-distributed GT, and 1.00 (0.91, 1.09) for the final size of the epidemic. The minimum and maximum vaccination coverage were related to AR and TD methods, respectively, where the TD method has minimum mean squared error (MSE). Finally, the <italic>R</italic><sub>0</sub> (95%CI) for 2018 data was 1.52 (1.11, 1.94) by TD method, and vaccination coverage was estimated as 34.2%.</p></sec><sec id="st4"><title>Conclusion:</title><p>For the purposes of our study, the estimation of TD was the most useful tool for computing the <italic>R</italic><sub>0</sub>, because it has the minimum MSE. The estimation <italic>R</italic><sub>0</sub> > 1 indicating that the epidemic has occurred. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">J Res Med Sci</journal-id><journal-id journal-id-type="iso-abbrev">J Res Med Sci</journal-id><journal-id journal-id-type="publisher-id">JRMS</journal-id><journal-title-group><journal-title>Journal of Research in Medical Sciences : The Official Journal of Isfahan University of Medical Sciences</journal-title></journal-title-group><issn pub-type="ppub">1735-1995</issn><issn pub-type="epub">1735-7136</issn><publisher><publisher-name>Wolters Kluwer - Medknow</publisher-name><publisher-loc>India</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">31523253</article-id><article-id pub-id-type="pmc">6670001</article-id><article-id pub-id-type="publisher-id">JRMS-24-67</article-id><article-id pub-id-type="doi">10.4103/jrms.JRMS_888_18</article-id><article-categories><subj-group subj-group-type="heading"><subject>Original Article</subject></subj-group></article-categories><title-group><article-title>Comparison of methods to Estimate Basic Reproduction Number (<italic>R</italic><sub>0</sub>) of influenza, Using Canada 2009 and 2017-18 A (H1N1) Data</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Nikbakht</surname><given-names>Roya</given-names></name><xref ref-type="aff" rid="aff1">1</xref></contrib><contrib contrib-type="author"><name><surname>Baneshi</surname><given-names>Mohammad Reza</given-names></name><xref ref-type="aff" rid="aff2">2</xref></contrib><contrib contrib-type="author"><name><surname>Bahrampour</surname><given-names>Abbas</given-names></name><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="corresp" rid="cor1"></xref></contrib><contrib contrib-type="author"><name><surname>Hosseinnataj</surname><given-names>Abolfazl</given-names></name><xref ref-type="aff" rid="aff2">2</xref></contrib></contrib-group><aff id="aff1"><label>1</label>HIV/STI Surveillance Research Center, and WHO Collaborating Center for HIV Surveillance, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Department of Biostatistics and Epidemiology, Faculty of Health Kerman, Iran</aff><aff id="aff2"><label>2</label>Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran</aff><author-notes><corresp id="cor1"><bold>Address for correspondence:</bold> Prof. Abbas Bahrampour, Department of Biostatistics and Epidemiology, Faculty of Health, Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran. E-mail: <email xlink:href="abahrampour@yahoo.com">abahrampour@yahoo.com</email></corresp></author-notes><pub-date pub-type="collection"><year>2019</year></pub-date><pub-date pub-type="epub"><day>24</day><month>7</month><year>2019</year></pub-date><volume>24</volume><elocation-id>67</elocation-id><history><date date-type="received"><day>18</day><month>11</month><year>2018</year></date><date date-type="rev-recd"><day>13</day><month>3</month><year>2019</year></date><date date-type="accepted"><day>17</day><month>5</month><year>2019</year></date></history><permissions><copyright-statement>Copyright: © 2019 Journal of Research in Medical Sciences</copyright-statement><copyright-year>2019</copyright-year><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by-nc-sa/4.0"><license-p>This is an open access journal, and articles are distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License, which allows others to remix, tweak, and build upon the work non-commercially, as long as appropriate credit is given and the new creations are licensed under the identical terms.</license-p></license></permissions><abstract><sec id="st1"><title>Background:</title><p>The basic reproduction number (<italic>R</italic><sub>0</sub>) has a key role in epidemics and can be utilized for preventing epidemics. In this study, different methods are used for estimating <italic>R</italic><sub>0</sub>'s and their vaccination coverage to find the formula with the best performance.</p></sec><sec id="st2"><title>Materials and Methods:</title><p>We estimated <italic>R</italic><sub>0</sub> for cumulative cases count data from April 18 to July 6, 2009 and 35-2017 to 34-2018 weeks in Canada: maximum likelihood (ML), exponential growth rate (EG), time-dependent reproduction numbers (TD), attack rate (AR), gamma-distributed generation time (GT), and the final size of the epidemic. Gamma distribution with mean and standard deviation 3.6 ± 1.4 is used as GT.</p></sec><sec id="st3"><title>Results:</title><p>The AR method obtained a <italic>R</italic><sub>0 (</sub>95% confidence interval [CI]) value of 1.116 (1.1163, 1.1165) and an EG (95%CI) value of 1.46 (1.41, 1.52). The <italic>R</italic><sub>0</sub> (95%CI) estimate was 1.42 (1.27, 1.57) for the obtained ML, 1.71 (1.12, 2.03) for the obtained TD, 1.49 (1.0, 1.97) for the gamma-distributed GT, and 1.00 (0.91, 1.09) for the final size of the epidemic. The minimum and maximum vaccination coverage were related to AR and TD methods, respectively, where the TD method has minimum mean squared error (MSE). Finally, the <italic>R</italic><sub>0</sub> (95%CI) for 2018 data was 1.52 (1.11, 1.94) by TD method, and vaccination coverage was estimated as 34.2%.</p></sec><sec id="st4"><title>Conclusion:</title><p>For the purposes of our study, the estimation of TD was the most useful tool for computing the <italic>R</italic><sub>0</sub>, because it has the minimum MSE. The estimation <italic>R</italic><sub>0</sub> > 1 indicating that the epidemic has occurred. Thus, it is required to vaccinate at least 41.5% to prevent and control the next epidemic.</p></sec></abstract><kwd-group><kwd>Basic reproduction number</kwd><kwd>influenza A virus</kwd><kwd>vaccination coverage</kwd></kwd-group></article-meta></front><body><sec sec-type="intro" id="sec1-1"><title>INTRODUCTION</title><p>Pandemic influenza, a global outbreak, defines as spreading influenza virus between peoples (with little or lack of immunity) over a wide geographic field.[<xref rid="ref1" ref-type="bibr">1</xref>] In the 20<sup>th</sup> century, three pandemics of influenza happened which were “Spanish flu,” “Asian flu,” and “Hong Kong flu” in the years “1918–1919,” “1957–1958,” and “1968–1969,” respectively.[<xref rid="ref2" ref-type="bibr">2</xref>] In early 2009, H1N1 influenza at first occurred in Mexico and the United States and speared rapidly worldwide (>200 countries involved).[<xref rid="ref3" ref-type="bibr">3</xref><xref rid="ref4" ref-type="bibr">4</xref>] The influenza virus can spread among people by direct contact (a cough, sneeze or talk), inhalation of virus-laden aerosols, and touch fomites (contaminated objects) that has the flu virus.[<xref rid="ref5" ref-type="bibr">5</xref><xref rid="ref6" ref-type="bibr">6</xref>] The most affected groups for developing flu-related complications are children, pregnant women, elders (adults older than 64-year-old), and persons with a specific disease (chronic pulmonary disease, chronic heart disease, diabetes, etc.,).[<xref rid="ref7" ref-type="bibr">7</xref><xref rid="ref8" ref-type="bibr">8</xref>] The mortality and morbidity related to the annual influenza in the worldwide estimated approximately one million people, a considerable number.[<xref rid="ref9" ref-type="bibr">9</xref>] For example, the number of deaths for “United States flu (2009)” reported 12,469 and for “Asian flu” was 1–4 million.[<xref rid="ref10" ref-type="bibr">10</xref><xref rid="ref11" ref-type="bibr">11</xref>] Therefore, controlling and preventing the epidemic of influenza is an important issue. The basic reproduction number (<italic>R</italic><sub>0</sub>) is an important metric that used for measuring the vaccination coverage (to prevent epidemic), eradicating an infectious disease, controlling and immunizing the disease which is defined the mean number of secondary infections generated by a single infectious individual in a fully susceptible population without immunity and interventions.[<xref rid="ref12" ref-type="bibr">12</xref>] In particular, the <italic>R</italic><sub>0</sub> determines whether an infection spreads through a population.[<xref rid="ref13" ref-type="bibr">13</xref>] The basic reproduction number or threshold parameter applied for determining the critical immunity coverage can be a real number greater than, less than, or equal to one. The disease will fade out when <italic>R</italic><sub>0</sub> < 1 and an epidemic will occur (the infection will grow) if <italic>R</italic><sub>0</sub> ≥ 1, showing an endemic in the population.[<xref rid="ref13" ref-type="bibr">13</xref><xref rid="ref14" ref-type="bibr">14</xref>]</p><p>Since the <italic>R</italic><sub>0</sub> has a key role in measuring the transmission of diseases and is crucial in preventing epidemics, thus it is important to know which methods and formulas to apply to estimate <italic>R</italic><sub>0</sub> and have better performance. We estimate the <italic>R</italic><sub>0</sub> and its related vaccination coverage for Canadian influenza data during 2009 and 2017–2018.</p></sec><sec sec-type="materials|methods" id="sec1-2"><title>MATERIALS AND METHODS</title><sec id="sec2-1"><title>Objectives</title><p>In this study, we reviewed the investigated methods and formulas used for estimating the <italic>R</italic><sub>0</sub> of influenza in various published research papers from 1954 to 2017. After a scientific systematic review on <italic>R</italic><sub>0</sub>, we found out that there are many basic reproduction formula which are applied for determining the vaccination coverage so it is necessary to characterize a formula which gives more accurate result to use in vaccination strategies which leads to optimize the costs. We extracted more commonly-utilized formulas [<xref rid="T5" ref-type="table">Appendix Table 1</xref>]. We considered six common formulas and applied them to real data to determine which formula most closely approximates the real epidemic threshold parameter with high efficacy.</p><p>Then, <italic>R</italic><sub>0</sub>s and related vaccination coverage of these methods was estimated for a secondary real data of Canadian influenza (2009). The calculated <italic>R</italic><sub>0</sub> was compared with <italic>R</italic><sub>0</sub> of the Canadian paper[<xref rid="ref15" ref-type="bibr">15</xref>] and also simulations were performed. Finally, the best method was chosen based on mean squared error (MSE), then <italic>R</italic><sub>0</sub> calculated by selected method for the H1N1 Canadian data in the 35<sup>th</sup> week in 2017–34<sup>th</sup> week in 2018.</p></sec><sec id="sec2-2"><title>Data</title><p>In Canada, circulating of influenza A virus is very common. The data sets in this study were obtained from the Public Health Agency of Canada (PHAC) website[<xref rid="ref16" ref-type="bibr">16</xref>] and the last FluWatch weekly report of the 2017–2018 influenza surveillance season achieved from the Respiratory Virus Detections in Canada Report website.[<xref rid="ref17" ref-type="bibr">17</xref>]</p><p>The total number of patients was 927 during the 2009 influenza season which were based on month/day and the number of new cases was 1280 for Canada 2017–2018 H1N1 data which report every Thursday in Canada. We fitted all the six models to Canadian 2009 pH 1N1 cumulative cases data.[<xref rid="ref16" ref-type="bibr">16</xref>] Then, the best model was applied to the data of Canada (34<sup>th</sup> week in 2017 to 34<sup>th</sup> week in 2018).[<xref rid="ref17" ref-type="bibr">17</xref>]</p></sec><sec id="sec2-3"><title>Statistical analysis</title><p>The models used in this article included the Richard model, attack rate (AR), exponential growth rate (EG), maximum likelihood (ML), time-dependent reproduction numbers (TD), gamma-distributed generation time (GT), and <italic>R</italic><sub>0</sub> using the final size of the epidemic. The above mentioned methods were applied for estimating <italic>R</italic><sub>0</sub> using R software (<italic>R</italic><sub>0</sub> package and programming). R software was created by Ross Ihaka and Robert Gentleman at the University of Auckland, New Zealand, and is currently developed by the R Development Core Team (of which Chambers is a member).</p><sec id="sec3-1"><title>Generation time</title><p>The time-gap between infection of a primary case and infection of a secondary case that is generated by the primary case.[<xref rid="ref18" ref-type="bibr">18</xref>]</p></sec><sec id="sec3-2"><title>The attack rate</title><p>The <italic>R</italic><sub>0</sub> can be described by the AR with the following formula:</p><p><inline-graphic xlink:href="JRMS-24-67-g001.jpg"></inline-graphic></p><p>where AR defines the ratio of the people generating an infection disease and <italic>S</italic><sub>0</sub> show the initial susceptible ratio.[<xref rid="ref19" ref-type="bibr">19</xref>]</p></sec><sec id="sec3-3"><title>The exponential growth rate</title><p>The following formula was applied for computing the R:</p><p><inline-graphic xlink:href="JRMS-24-67-g002.jpg"></inline-graphic></p><p>In this formula, <italic>M</italic> is the moment-generating function of the GT.[<xref rid="ref20" ref-type="bibr">20</xref>] The parameter <italic>r</italic> is determined by the Poisson regression. Furthermore, the parameter <italic>w</italic> is GT.</p></sec><sec id="sec3-4"><title>The maximum likelihood</title><p>Let <italic>N</italic><sub>0</sub>, <italic>N</italic><sub>1</sub>,...., <italic>N</italic><sub>T</sub> identify incident cases over sequential time. The log-likelihood function is:</p><p><inline-graphic xlink:href="JRMS-24-67-g003.jpg"></inline-graphic></p><p>where</p><p><inline-graphic xlink:href="JRMS-24-67-g004.jpg"></inline-graphic></p><p>and <italic>R</italic> is the maximum value of the log-likelihood function.[<xref rid="ref21" ref-type="bibr">21</xref>] Furthermore, the parameter w is estimated by maximizing log-likelihood is GT.</p></sec><sec id="sec3-5"><title>Time-dependent reproduction numbers</title><p>In this method, <italic>R</italic><sub>t</sub> is computed by averaging <italic>R</italic><sub>j</sub>, which is the mean of all transmission networks corresponding to the cases observed.[<xref rid="ref22" ref-type="bibr">22</xref>]</p><p><inline-graphic xlink:href="JRMS-24-67-g005.jpg"></inline-graphic></p><p>where</p><p><inline-graphic xlink:href="JRMS-24-67-g006.jpg"></inline-graphic></p><p>And</p><p><inline-graphic xlink:href="JRMS-24-67-g007.jpg"></inline-graphic></p><p>Consider that person <italic>i</italic> and person <italic>j</italic> are in times <italic>t</italic><sub>i</sub> and <italic>t</italic><sub>j</sub>, respectively, then displays the probability of infection transmission from person <italic>j</italic> to person i so <italic>R</italic><sub>t</sub> compute by averaging all <italic>R</italic><sub>j</sub> which is the mean of all transmission networks correspondent with the cases that observed.</p></sec><sec id="sec3-6"><title>The gamma-distributed generation time</title><p>The number of cases on the day “<italic>t</italic>,” denoted by <italic>n</italic><sub>t</sub> in (<italic>t</italic><sub>1</sub>, <italic>t</italic><sub>2</sub>) grows exponentially where</p><p><italic>n</italic><sub>t</sub> = <italic>n</italic><sub>t1</sub> exp (<italic>r</italic>[<italic>t</italic> – <italic>t</italic>1]) (9)</p><p><inline-graphic xlink:href="JRMS-24-67-g008.jpg"></inline-graphic></p><p>And</p><p><inline-graphic xlink:href="JRMS-24-67-g009.jpg"></inline-graphic></p><p>The EG denotes by <italic>r</italic>. The mean and standard deviation of the GT are μ and σ, respectively, where <italic>a</italic> = μ<sup>2</sup>/σ<sup>2</sup> and <italic>b</italic> = μ/σ<sup>2</sup>.[<xref rid="ref23" ref-type="bibr">23</xref>]</p></sec><sec id="sec3-7"><title><italic>R</italic><sub>0</sub> using the final size of the epidemic</title><p>The <italic>R</italic><sub>0</sub> can be estimated with the below formula:</p><p><inline-graphic xlink:href="JRMS-24-67-g010.jpg"></inline-graphic></p><p>where the total population at risk and total number of infections are denoted by <italic>N</italic> and <italic>C</italic>, respectively.[<xref rid="ref24" ref-type="bibr">24</xref>]</p></sec></sec><sec id="sec2-4"><title>Vaccination coverage</title><p>The vaccination coverage is computed by the basic reproduction number with formula:</p><p><inline-graphic xlink:href="JRMS-24-67-g011.jpg"></inline-graphic></p><p>which shows the proportion of peoples who should be received the vaccine.[<xref rid="ref7" ref-type="bibr">7</xref>]</p></sec><sec id="sec2-5"><title>Comparison of methods</title><p>For exploring the closeness of the estimation of the mentioned methods to the actual <italic>R</italic><sub>0</sub>s and comparing them with each other, we applied 10000 times simulation for each formula based on the Canada data. The epidemics were simulated with the following properties. The distribution of the GT was considered gamma with the mean of 3.6 and standard deviation of 1.4. According to real data (the Canada data), the length of the epidemic was 80 days. Moreover, the peak value (the threshold value for the incidence before epidemics begin decreasing) for the Canada data occurred in the day 54. Therefore, we applied the value equal to 54 for the peak value in the simulation command [For details, see the simulation command under <xref rid="T1" ref-type="table">Table 1</xref> in the results section]. Simulation of the basic reproduction number was made with above characteristics and the MSE was calculated for evaluating the performance of models with below formula. The lowest MSE value corresponds to the method which fitted the data best.</p><table-wrap id="T1" position="float"><label>Table 1</label><caption><p>The simulated R<sub>0</sub>s and their 95% confidence interval for each method</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="3" colspan="1">Actual R<sub>0</sub></th><th align="center" colspan="6" rowspan="1">R<sub>0</sub> (95% CI)</th></tr><tr><th align="center" colspan="6" rowspan="1"><hr></hr></th></tr><tr><th align="center" rowspan="1" colspan="1">ML</th><th align="center" rowspan="1" colspan="1">EG</th><th align="center" rowspan="1" colspan="1">TD</th><th align="center" rowspan="1" colspan="1">AR</th><th align="center" rowspan="1" colspan="1">Gamma-distributed generation time</th><th align="center" rowspan="1" colspan="1">R<sub>0</sub> using the final size of the epidemic</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">1.23 (1.03, 1.47)</td><td align="center" rowspan="1" colspan="1">1.26 (1.19, 1.34)</td><td align="center" rowspan="1" colspan="1">1.17 (0.93, 1.42)</td><td align="center" rowspan="1" colspan="1">1.000003 (1.000003, 1.000004)</td><td align="center" rowspan="1" colspan="1">1.25 (0.98, 1.52)</td><td align="center" rowspan="1" colspan="1">0.91 (0.23, 1.58)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.116</td><td align="center" rowspan="1" colspan="1">1.27 (1.08, 1.49)</td><td align="center" rowspan="1" colspan="1">1.33 (1.26, 1.40)</td><td align="center" rowspan="1" colspan="1">1.24 (1.0, 1.47)</td><td align="center" rowspan="1" colspan="1">1.000004 (1.000004, 1.000005)</td><td align="center" rowspan="1" colspan="1">1.30 (1.03, 1.57)</td><td align="center" rowspan="1" colspan="1">0.93 (0.39, 1.46)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.42</td><td align="center" rowspan="1" colspan="1">1.43 (1.28, 1.61)</td><td align="center" rowspan="1" colspan="1">1.54 (1.48, 1.61)</td><td align="center" rowspan="1" colspan="1">1.47 (1.29, 1.65)</td><td align="center" rowspan="1" colspan="1">1.000009 (1.000008, 1.000009)</td><td align="center" rowspan="1" colspan="1">1.48 (1.21, 1.75)</td><td align="center" rowspan="1" colspan="1">0.97 (0.71, 1.23)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.46</td><td align="center" rowspan="1" colspan="1">1.47 (1.32, 1.63)</td><td align="center" rowspan="1" colspan="1">1.59 (1.53, 1.66)</td><td align="center" rowspan="1" colspan="1">1.51 (1.34, 1.69)</td><td align="center" rowspan="1" colspan="1">1.000007 (1.000007, 1.000009)</td><td align="center" rowspan="1" colspan="1">1.52 (1.26, 1.79)</td><td align="center" rowspan="1" colspan="1">0.98 (0.75, 1.20)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.49</td><td align="center" rowspan="1" colspan="1">1.48 (1.33, 1.64)</td><td align="center" rowspan="1" colspan="1">1.59 (1.53, 1.65)</td><td align="center" rowspan="1" colspan="1">1.51 (1.33, 1.69)</td><td align="center" rowspan="1" colspan="1">1.000008 (1.000008, 1.000009)</td><td align="center" rowspan="1" colspan="1">1.54 (1.27, 1.81)</td><td align="center" rowspan="1" colspan="1">0.98 (0.75, 1.21)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.68</td><td align="center" rowspan="1" colspan="1">1.60 (1.47, 1.73)</td><td align="center" rowspan="1" colspan="1">1.75 (1.69, 1.81)</td><td align="center" rowspan="1" colspan="1">1.64 (1.44, 1.84)</td><td align="center" rowspan="1" colspan="1">1.000006 (1.000006, 1.000007)</td><td align="center" rowspan="1" colspan="1">1.64 (1.37, 1.91)</td><td align="center" rowspan="1" colspan="1">0.99 (0.80, 1.73)</td></tr><tr><td align="left" rowspan="1" colspan="1">1.71</td><td align="center" rowspan="1" colspan="1">1.60 (1.48, 1.73)</td><td align="center" rowspan="1" colspan="1">1.76 (1.71, 1.83)</td><td align="center" rowspan="1" colspan="1">1.66 (1.44, 1.88)</td><td align="center" rowspan="1" colspan="1">1.000006 (1.000005, 1.000006)</td><td align="center" rowspan="1" colspan="1">1.64 (1.38, 1.91)</td><td align="center" rowspan="1" colspan="1">0.99 (0.81, 1.17)</td></tr><tr><td align="left" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">1.56 (1.47, 1.66)</td><td align="center" rowspan="1" colspan="1">1.80 (1.76, 1.85)</td><td align="center" rowspan="1" colspan="1">1.83 (1.53, 2.13)</td><td align="center" rowspan="1" colspan="1">1.000005 (1.000005, 1.000006)</td><td align="center" rowspan="1" colspan="1">1.67 (1.41, 1.94)</td><td align="center" rowspan="1" colspan="1">0.99 (0.83, 1.16)</td></tr><tr><td align="left" rowspan="1" colspan="1">2.5</td><td align="center" rowspan="1" colspan="1">1.36 (1.29, 1.42)</td><td align="center" rowspan="1" colspan="1">1.6 (1.57, 1.63)</td><td align="center" rowspan="1" colspan="1">2.16 (1.71, 2.60)</td><td align="center" rowspan="1" colspan="1">1.000004 (1.000003, 1.000004)</td><td align="center" rowspan="1" colspan="1">1.62 (1.35, 1.89)</td><td align="center" rowspan="1" colspan="1">1 (0.82, 1.17)</td></tr><tr><td align="left" rowspan="1" colspan="1">3</td><td align="center" rowspan="1" colspan="1">1.26 (1.21, 1.33)</td><td align="center" rowspan="1" colspan="1">1.46 (1.43, 1.48)</td><td align="center" rowspan="1" colspan="1">2.47 (1.87, 3.06)</td><td align="center" rowspan="1" colspan="1">1.000003 (1.000003, 1.000004)</td><td align="center" rowspan="1" colspan="1">1.56 (1.29, 1.82)</td><td align="center" rowspan="1" colspan="1">1 (0.81, 1.18)</td></tr></tbody></table><table-wrap-foot><fn><p>Sim.epid (epid.n b=10000, GT=Generation.time (“gamma”, c [3, 1.4]), R0 =r0, epid.length=80, family=“poisson”, peak.value=54). AR=Attack rate; R0 =Reproduction number; CI=Confidence interval; EG=Exponential growth rate; TD=Time dependent reproduction numbers; ML=Maximum likelihood; This is simulation command in R<sub>0</sub> package of R software</p></fn></table-wrap-foot></table-wrap><p><inline-graphic xlink:href="JRMS-24-67-g012.jpg"></inline-graphic></p></sec></sec><sec sec-type="results" id="sec1-3"><title>RESULTS</title><sec id="sec2-6"><title>Canadian 2009 H1N1 influenza data</title><p>We fitted the six models to the daily dataset of Canada, throughout the 80-day period of the studies. All dates of the Canada data were based on month/day form 18 April, 2009 to 6 July, 2009. Moreover, the number of infected people was plotted as frequency [<xref ref-type="fig" rid="F1">Figure 1</xref>].</p><fig id="F1" position="float"><label>Figure 1</label><caption><p>The incidence case counts influenza data of Canada during 18 April, 2009–6 July, 2009</p></caption><graphic xlink:href="JRMS-24-67-g013"></graphic></fig><p>In order to demonstrate the difference in modeling with various formulas, the result of the Richard model (presented in Hsieh's study)[<xref rid="ref15" ref-type="bibr">15</xref>] as well as the results of the other six models are presented in <xref rid="T2" ref-type="table">Table 2</xref>. The reported <italic>R</italic><sub>0</sub> (95% confidence interval [CI]) (vaccination coverage%) using the Richard model was 1.68 (1.45, 1.91) (40.47) that means every person infected 1.68 other people on average during the infection period. Note that, <italic>R</italic><sub>0</sub> (95%CI) (vaccination coverage%) for the estimation of TD (1.71 [1.12, 2.03] [41.52]) was clearly close to <italic>R</italic><sub>0</sub> for the Richard model. The second method with the closest <italic>R</italic><sub>0</sub> (95%CI) to that of the Richard model was the gamma-distributed GT (1.49 [1.0, 1.97] [32.88]). On the other hand, the computed <italic>R</italic><sub>0</sub> (95%CI) using the EG was 1.46 (1.41, 1.52) (31.51). The ML method revealed that the calculated <italic>R</italic><sub>0</sub> (95% CI) for this model was different from that for the Richard model (1.42 [1.27, 1.57] [29.58]). In addition, the estimated <italic>R</italic><sub>0</sub> (95% CI) (vaccination coverage%) by the AR with two approaches was 1.000388 (1.000383, 1.000392) (0.04) and 1.1164 (1.1163, 1.1165) (10.43). The minimum computed <italic>R</italic><sub>0</sub> (95% CI) was related to the estimation of the final size of the epidemic obtained as 1.0 (0.91, 1.09). The estimates of vaccination coverage for the six methods were vary. The lowest and highest vaccination coverage values in this setting were associated with AR and TD methods, respectively.</p><table-wrap id="T2" position="float"><label>Table 2</label><caption><p>The Reproduction number estimation by the different methods for the Canada data (2009)</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="1" colspan="1">Method</th><th align="center" rowspan="1" colspan="1">R<sub>0</sub> (95% CI for R<sub>0</sub>)</th><th align="center" rowspan="1" colspan="1">Vaccination coverage (%)</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">Richard model</td><td align="center" rowspan="1" colspan="1">1.68 (1.45, 1.91)</td><td align="center" rowspan="1" colspan="1">40.47</td></tr><tr><td align="left" rowspan="2" colspan="1">AR</td><td align="center" rowspan="1" colspan="1">1.000388 (1.000383, 1.000392)<sup>a</sup></td><td align="center" rowspan="1" colspan="1">0.04</td></tr><tr><td align="center" rowspan="1" colspan="1">1.1164 (1.1163, 1.1165)<sup>b</sup></td><td align="center" rowspan="1" colspan="1">10.43</td></tr><tr><td align="left" rowspan="1" colspan="1">EG</td><td align="center" rowspan="1" colspan="1">1.46 (1.41, 1.52)</td><td align="center" rowspan="1" colspan="1">31.51</td></tr><tr><td align="left" rowspan="1" colspan="1">ML</td><td align="center" rowspan="1" colspan="1">1.42 (1.27, 1.57)</td><td align="center" rowspan="1" colspan="1">29.58</td></tr><tr><td align="left" rowspan="1" colspan="1">TD</td><td align="center" rowspan="1" colspan="1">1.71 (1.12, 2.03)</td><td align="center" rowspan="1" colspan="1">41.52</td></tr><tr><td align="left" rowspan="1" colspan="1">Gamma-distributed generation time</td><td align="center" rowspan="1" colspan="1">1.49 (1.0, 1.97)</td><td align="center" rowspan="1" colspan="1">32.88</td></tr><tr><td align="left" rowspan="1" colspan="1">R<sub>0</sub> using the final size of the epidemic</td><td align="center" rowspan="1" colspan="1">1.0 (0.91, 1.09)</td><td align="center" rowspan="1" colspan="1">0</td></tr></tbody></table><table-wrap-foot><fn><p><sup>a</sup>AR based on incidence (<italic>n</italic>=33,630,000), <sup>b</sup>AR based on reported AR=0.201. R<sub>0</sub>: Reproduction number; TD=Time-dependent reproduction numbers; ML=Maximum likelihood; EG=Exponential growth rate; AR=Attack rate, CI=Confidence interval</p></fn></table-wrap-foot></table-wrap><p>In order to compare the mentioned models to find the formula with better fit to the actual values, we conducted a simulation with R software and calculated <italic>R</italic><sub>0</sub> based on the six models reported in <xref rid="T2" ref-type="table">Table 2</xref>. We used gamma distribution for the GT with the mean of 3.6 and standard deviation of 1.4. The peak value determined right over the original data were equal to 54. Then, using the above parameters, the simulation was implemented and <italic>R</italic><sub>0</sub> was computed for each method. The simulation results for comparing the quality of the six methods are represented in <xref rid="T1" ref-type="table">Table 1</xref> and <xref ref-type="fig" rid="F2">Figure 2</xref>. In order to carry out the simulation, the number of runs to achieve the <italic>R</italic><sub>0</sub> was 10000.</p><fig id="F2" position="float"><label>Figure 2</label><caption><p>The plots of the actual and simulated <italic>R</italic><sub>0</sub> compared for each method</p></caption><graphic xlink:href="JRMS-24-67-g014"></graphic></fig><p>The results, given in <xref rid="T1" ref-type="table">Table 1</xref>, indicated that there were differences between the actual and simulated <italic>R</italic><sub>0</sub>; however, the TD method had the closest value to the <italic>R</italic><sub>0</sub> calculated from the simulation compared to the other methods. Surprisingly, some variation was considered for the ML estimations when the actual values were equal to one, between one and two and greater than two. In the ML method, we found that the simulated <italic>R</italic><sub>0</sub> for small values was very close to that for the actual values when the actual values were between 1.42 and 1.71; while the simulated <italic>R</italic><sub>0</sub> for large values was very different from that for the actual values. For the gamma-distributed GT approach, the simulated <italic>R</italic><sub>0</sub> grew out of the actual values for values close to one. In contrast, the results showed that the computed values for <italic>R</italic><sub>0</sub> in the simulated system were slightly greater than the actual values when we applied <italic>R</italic><sub>0</sub> between 1.42 and 2. By following the same interpretation, we can infer that the EG method had a small variation for small <italic>R</italic><sub>0</sub> values (1.4 < <italic>R</italic><sub>0</sub> < 2). On the other hand, the <italic>R</italic><sub>0</sub> estimations using the EG diverged from the actual <italic>R</italic><sub>0</sub> but was not significant. Finally, the computed <italic>R</italic><sub>0</sub> by the AR and final size of the epidemic methods seemed likely to reflect stability for all <italic>R</italic><sub>0</sub>s. In particular, for the latest assumed <italic>R</italic><sub>0</sub>s, the estimated <italic>R</italic><sub>0</sub> was equal to one.</p><p>We also plotted [<xref ref-type="fig" rid="F2">Figure 2</xref>] the actual <italic>R</italic><sub>0</sub> and simulated <italic>R</italic><sub>0</sub> based on six methods with the parameters described in <xref rid="T1" ref-type="table">Table 1</xref>. For evaluating the performance of models, we computed MSE for all methods [<xref rid="T3" ref-type="table">Table 3</xref>]. The TD method had the lowest MSE value in comparison to other methods. The MSE of AR and final size of the epidemic methods was very varied. In addition, MSE of ML, EG, and gamma-distributed GT methods were also calculated. For ML, EG, and gamma-distributed GT, the mean of MSE of all points were 4.85, 3.81, and 3.31, respectively. As noted above, the TD introduced the approach with the nearest estimation to the actual <italic>R</italic><sub>0</sub> based on MSE criterion.</p><table-wrap id="T3" position="float"><label>Table 3</label><caption><p>Mean squared error of reproduction number estimation for each method</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="3" colspan="1">R<sub>0</sub></th><th align="center" colspan="6" rowspan="1">Method</th></tr><tr><th align="center" colspan="6" rowspan="1"><hr></hr></th></tr><tr><th align="center" rowspan="1" colspan="1">ML</th><th align="center" rowspan="1" colspan="1">EG</th><th align="center" rowspan="1" colspan="1">TD</th><th align="center" rowspan="1" colspan="1">AR</th><th align="center" rowspan="1" colspan="1">Gamma-distributed generation time</th><th align="center" rowspan="1" colspan="1">The final size of the epidemic</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">0.061</td><td align="center" rowspan="1" colspan="1">0.090</td><td align="center" rowspan="1" colspan="1">0.042</td><td align="center" rowspan="1" colspan="1">1.036e-11</td><td align="center" rowspan="1" colspan="1">0.080</td><td align="center" rowspan="1" colspan="1">0.015</td></tr><tr><td align="left" rowspan="1" colspan="1">1.116</td><td align="center" rowspan="1" colspan="1">0.038</td><td align="center" rowspan="1" colspan="1">0.072</td><td align="center" rowspan="1" colspan="1">0.030</td><td align="center" rowspan="1" colspan="1">0.014</td><td align="center" rowspan="1" colspan="1">0.055</td><td align="center" rowspan="1" colspan="1">0.043</td></tr><tr><td align="left" rowspan="1" colspan="1">1.42</td><td align="center" rowspan="1" colspan="1">0.027</td><td align="center" rowspan="1" colspan="1">0.064</td><td align="center" rowspan="1" colspan="1">0.025</td><td align="center" rowspan="1" colspan="1">0.178</td><td align="center" rowspan="1" colspan="1">0.043</td><td align="center" rowspan="1" colspan="1">0.207</td></tr><tr><td align="left" rowspan="1" colspan="1">1.46</td><td align="center" rowspan="1" colspan="1">0.027</td><td align="center" rowspan="1" colspan="1">0.065</td><td align="center" rowspan="1" colspan="1">0.022</td><td align="center" rowspan="1" colspan="1">0.212</td><td align="center" rowspan="1" colspan="1">0.041</td><td align="center" rowspan="1" colspan="1">0.236</td></tr><tr><td align="left" rowspan="1" colspan="1">1.49</td><td align="center" rowspan="1" colspan="1">0.027</td><td align="center" rowspan="1" colspan="1">0.059</td><td align="center" rowspan="1" colspan="1">0.022</td><td align="center" rowspan="1" colspan="1">0.240</td><td align="center" rowspan="1" colspan="1">0.040</td><td align="center" rowspan="1" colspan="1">0.266</td></tr><tr><td align="left" rowspan="1" colspan="1">1.68</td><td align="center" rowspan="1" colspan="1">0.035</td><td align="center" rowspan="1" colspan="1">0.050</td><td align="center" rowspan="1" colspan="1">0.014</td><td align="center" rowspan="1" colspan="1">0.046</td><td align="center" rowspan="1" colspan="1">0.026</td><td align="center" rowspan="1" colspan="1">0.482</td></tr><tr><td align="left" rowspan="1" colspan="1">1.71</td><td align="center" rowspan="1" colspan="1">0.042</td><td align="center" rowspan="1" colspan="1">0.050</td><td align="center" rowspan="1" colspan="1">0.016</td><td align="center" rowspan="1" colspan="1">0.505</td><td align="center" rowspan="1" colspan="1">0.028</td><td align="center" rowspan="1" colspan="1">0.524</td></tr><tr><td align="left" rowspan="1" colspan="1">2.0</td><td align="center" rowspan="1" colspan="1">0.242</td><td align="center" rowspan="1" colspan="1">0.089</td><td align="center" rowspan="1" colspan="1">0.043</td><td align="center" rowspan="1" colspan="1">1.001</td><td align="center" rowspan="1" colspan="1">0.118</td><td align="center" rowspan="1" colspan="1">1.014</td></tr><tr><td align="left" rowspan="1" colspan="1">2.5</td><td align="center" rowspan="1" colspan="1">1.345</td><td align="center" rowspan="1" colspan="1">0.862</td><td align="center" rowspan="1" colspan="1">0.141</td><td align="center" rowspan="1" colspan="1">2.252</td><td align="center" rowspan="1" colspan="1">0.784</td><td align="center" rowspan="1" colspan="1">2.267</td></tr><tr><td align="left" rowspan="1" colspan="1">3.0</td><td align="center" rowspan="1" colspan="1">3.011</td><td align="center" rowspan="1" colspan="1">2.405</td><td align="center" rowspan="1" colspan="1">0.321</td><td align="center" rowspan="1" colspan="1">4.004</td><td align="center" rowspan="1" colspan="1">2.097</td><td align="center" rowspan="1" colspan="1">4.022</td></tr><tr><td align="left" rowspan="1" colspan="1">Total mean</td><td align="center" rowspan="1" colspan="1">4.855</td><td align="center" rowspan="1" colspan="1">3.806</td><td align="center" rowspan="1" colspan="1">0.676</td><td align="center" rowspan="1" colspan="1">8.452</td><td align="center" rowspan="1" colspan="1">3.312</td><td align="center" rowspan="1" colspan="1">9.076</td></tr></tbody></table><table-wrap-foot><fn><p>R<sub>0</sub>=Reproduction number; TD=Time-dependent reproduction numbers; ML=Maximum likelihood; EG=Exponential growth rate; AR=Attack rate</p></fn></table-wrap-foot></table-wrap><p>We also performed a sensitivity analysis with the incidence data of Canada on the GT with the gamma distribution [<xref ref-type="fig" rid="F3">Figure 3</xref>]. The sensitivity analysis demonstrated that <italic>R</italic><sub>0</sub> (95% CI) for the mean GT (days) of 3.6 and 4.9 was estimated as 1.47 (1.41, 1.53) and 1.67 (1.58, 1.76). Thus, the computed <italic>R</italic><sub>0</sub> was approximately near that of the Richard and TD methods when the mean GT was equal to 4.9.</p><fig id="F3" position="float"><label>Figure 3</label><caption><p>Sensitivity of <italic>R</italic><sub>0</sub> to mean generation time to select the generation time</p></caption><graphic xlink:href="JRMS-24-67-g015"></graphic></fig></sec><sec id="sec2-7"><title>Canadian 2017–2018 H1N1 influenza data</title><p>The incidence data are reported based on week/year from the 35<sup>th</sup> week in 2017 to the 34<sup>th</sup> week in 2018. Peak value for this data has occurred in the 12<sup>th</sup> week in 2018 after starting the epidemics. The number of infected cases is plotted in <xref ref-type="fig" rid="F4">Figure 4</xref>.</p><fig id="F4" position="float"><label>Figure 4</label><caption><p>The incidence case counts influenza data of Canada from the 35<sup>th</sup> week in 2017 to the 34<sup>th</sup> week in 2018</p></caption><graphic xlink:href="JRMS-24-67-g016"></graphic></fig><p>For the given data, <italic>R</italic><sub>0</sub> (95% CI) and vaccination coverage based on TD method was computed. Indeed, we found that the estimated <italic>R</italic><sub>0</sub> by TD method was (1.52 95% CI: 1.11, 1.94). In addition, the estimates of vaccination coverage were 34.2% for 2017–2018.</p></sec></sec><sec sec-type="discussion" id="sec1-4"><title>DISCUSSION</title><p>We implemented six methods (the ML, EG, TD, AR, gamma-distributed and final size of the epidemic), which permitted the estimation of the <italic>R</italic><sub>0</sub> as key parameters of the epidemic based on the A/H1N1 Influenza cumulative case counts data in Canada (2009). The <italic>R</italic><sub>0</sub> for the ML, EG, TD, AR, gamma-distributed and final size of the epidemic methods were estimated 1.42, 1.46, 1.71, 1.116, 1.49, and 1.0, respectively. In most cases, the <italic>R</italic><sub>0</sub> was greater than unity; hence, the epidemic outbreak was observed. In addition, the computed <italic>R</italic><sub>0</sub> for Canadian data (2018) by TD method was greater than one indicating that an epidemic occurred in Canada (<italic>R</italic><sub>0</sub> > 1). Thus, it seems necessary to consider appropriate solutions in order to control, decrease and prevent the epidemic or pandemic of influenza. One of the most effective methods to protect people against influenza is vaccination that can be determined by using <italic>R</italic><sub>0</sub> (vaccination coverage = 1 − 1/<italic>R</italic><sub>0</sub>). On the other hand, annual influenza vaccination in the high-risk groups such as elderly people, ill person, pregnant woman, and children can reduce mortality rate. In addition, vaccination can also reduce the incidence of disease, cost, exacerbations of the disease, and hospitalizations. The vaccination coverage for Canada (2009) ranged between 10.43 and 41.52 using various methods and this value was 34.2% for 2017–2018 influenza Canada data.</p><p>Moreover, we performed a simulation using R software for several <italic>R</italic><sub>0</sub> and obtained their estimates based on the epidemic data of Canada (2009) for the six methods. The computed <italic>R</italic><sub>0</sub> in the TD method was nearly the same as the actual <italic>R</italic><sub>0</sub> based on MSE criterion. Comparing the simulation results from the ML, gamma-distributed GT and EG methods showed variation for different values of the actual <italic>R</italic><sub>0</sub>; however, some of the calculated <italic>R</italic><sub>0</sub>s applying the simulation were close to the actual values. For the most actual <italic>R</italic><sub>0</sub>, the simulated <italic>R</italic><sub>0</sub> by the AR and final size of the epidemic methods was equal to one. Whereas these type of modeling approaches are not able to differentiate between various <italic>R</italic><sub>0</sub>. We believe that this may correspond to the small number of the infected cases compared to the susceptible cases.</p><p>Note that, our basic reproduction number estimated using the TD method was consistent with that derived from the Richard model in the Canadian papers.[<xref rid="ref15" ref-type="bibr">15</xref>] Not only the simulated <italic>R</italic><sub>0</sub> for the value 1.68 almost agreed with that of the TD approach but also the other simulated <italic>R</italic><sub>0</sub> by the TD method was nearly consistent with the actual <italic>R</italic><sub>0</sub>. In other words, the lowest MSE values were obtained for TD method.</p><p>From the methods reviewed in <xref rid="T6" ref-type="table">Appendix Table 2</xref>, which can be applied to estimate the <italic>R</italic><sub>0</sub>, the approaches presented in <xref rid="T1" ref-type="table">Table 1</xref> fitted to the cumulative cases data. All the methods reviewed in this paper, as any modeling techniques, had advantageous and disadvantageous. One of the strengths of this study is to review all studies done related to influenza and then selected some of the frequently used model and determine their strengths and weaknesses; seven of them used for the <italic>R</italic><sub>0</sub> estimation in the Canada data, as shown in <xref rid="T1" ref-type="table">Table 1</xref>, are explained in details in <xref rid="T4" ref-type="table">Table 4</xref>.</p><table-wrap id="T4" position="float"><label>Table 4</label><caption><p>Limitation and power of the methods used for the cumulative case counts data</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="1" colspan="1">Models</th><th align="left" rowspan="1" colspan="1">Advantageous</th><th align="left" rowspan="1" colspan="1">Disadvantageous</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">The Richard model</td><td align="left" rowspan="1" colspan="1">For cumulative case count, it gives simple means of fitting <break></break>For modeling, it only needs cumulative case counts Initial estimation of R<sub>0</sub> is fairly stable and credible</td><td align="left" rowspan="1" colspan="1">Missing data provide problems (which may be nonrandom) <break></break>Data quality (real-time modeling) is important</td></tr><tr><td align="left" rowspan="1" colspan="1">ML</td><td align="left" rowspan="1" colspan="1">Serial interval estimates by this formulation and then details of the disease dynamics can be characterized <break></break>The MLE and posterior mode (with uninformative gamma prior distribution) are equal when the serial interval is known[<xref rid="ref25" ref-type="bibr">25</xref>] <break></break>The MLE approach is the least biased <break></break>The approach used for missing data in the ML method is similar to McBryde in Bayesian[<xref rid="ref26" ref-type="bibr">26</xref>]</td><td align="left" rowspan="1" colspan="1">Some of the assumptions of the models are: no imported cases, no missing data and uniformly-mixed population. <break></break>Violation of any of these assumptions changes the results[<xref rid="ref27" ref-type="bibr">27</xref>] <break></break>In the long period for the aggregated data, the estimation of the reproduction number tends to be increasingly underestimated</td></tr><tr><td align="left" rowspan="1" colspan="1">EG</td><td align="left" rowspan="1" colspan="1">Aggregated data and dispersion are least impressed on the estimation of reproduction</td><td align="left" rowspan="1" colspan="1">For the initial phase of the epidemic, this simple method may not be always powerful <break></break>The assumptions should be checked and the method should be used with caution[<xref rid="ref28" ref-type="bibr">28</xref>]</td></tr><tr><td align="left" rowspan="1" colspan="1">TD</td><td align="left" rowspan="1" colspan="1">It is the least biased <break></break>Importation of the cases can be accounted within the epidemic</td><td align="left" rowspan="1" colspan="1">In the long period for the aggregated data, the estimation of the reproduction number tends to be increasingly underestimated <break></break>In the TD approach, the R<sub>0</sub> depends on time and changes with it and no solution exists for correcting this method</td></tr><tr><td align="left" rowspan="1" colspan="1">AR</td><td align="left" rowspan="1" colspan="1">The least information is needed for this approach[<xref rid="ref28" ref-type="bibr">28</xref>] <break></break>The AR method, unlike the other models, does not require the GT distribution (there may be no prior knowledge about the GT distribution)</td><td align="left" rowspan="1" colspan="1">It is useful when the epidemic ends <break></break>No intervention is required to set up during outbreak <break></break>This method is applied in particular limited settings such as army and schools[<xref rid="ref29" ref-type="bibr">29</xref>] It does not require the GT distribution</td></tr><tr><td align="left" rowspan="1" colspan="1">The gamma-distributed generation time</td><td align="left" rowspan="1" colspan="1">Only the number of cases on each day and generation time distribution are needed for modeling</td><td align="left" rowspan="1" colspan="1">The growth in case number over time should be specified; the violation of this condition can be problematic</td></tr><tr><td align="left" rowspan="1" colspan="1">R<sub>0</sub> using the final size of the epidemic</td><td align="left" rowspan="1" colspan="1">For modeling, the total population at risk and total number of infections for a fully susceptible population are only required</td><td align="left" rowspan="1" colspan="1">It is useful when the epidemic ends <break></break>It does not require the generation time distribution</td></tr></tbody></table><table-wrap-foot><fn><p>GT=Generation time; R<sub>0</sub>=Reproduction number; TD=Time-dependent reproduction numbers; ML=Maximum likelihood; MLE=ML estimation; EG=Exponential growth rate; AR=Attack rate</p></fn></table-wrap-foot></table-wrap><p>Regarding <xref rid="T4" ref-type="table">Table 4</xref>, it seemed that the TD, ML and EG methods had superiority compared to the other methods. These models were used by researchers to estimate <italic>R</italic><sub>0</sub> of influenza.</p><p>Some studies estimated the <italic>R</italic><sub>0</sub> from influenza data using different models and compared the results. Obadia <italic>et al.</italic> obtained estimates of <italic>R</italic><sub>0</sub> from the “Germany 1918” epidemic data based on five approaches which including the AR, ML, sequential Bayesian and TD methods. In addition, comparing results from different methods showed that the biased ML and TD methods were least.[<xref rid="ref30" ref-type="bibr">30</xref>] Another study applied four different methods (the EG, simple susceptible-exposed-infectious-recovered [SEIR], more complex SEIR-type model, and ML model) in order to compare these estimation approaches. The EG had large uncertainty while ML had a consistent estimate with the estimate of the autumn wave.[<xref rid="ref20" ref-type="bibr">20</xref>] In general, the TD had a good fit on the data as confirmed with the Richard model and MSE criterion.</p><p>A weakness of this study is that the 2009 Canada data have been used for comparing methods, which looks old. The reason for this, is comparing <italic>R</italic><sub>0</sub> with pervious article[<xref rid="ref15" ref-type="bibr">15</xref>] and comparing the methods with the actual values which are exist on this data in the mentioned paper. Finally, a more comprehensive study for influenza as an annual national disaster using new method such as Bayesian is needed that we are going to do in the future research.</p></sec><sec sec-type="conclusion" id="sec1-5"><title>CONCLUSION</title><p>Awareness of the basic reproduction number of influenza is useful for calculating vaccination coverage and then applying vaccine strategy. Therefore, it is necessary to know the method which has better performance for influenza data that our results showed the TD method is preferred. One advantage of the TD method in compared to the other methods was that it was useful for computing the <italic>R</italic><sub>0</sub> regarding the real cumulative case count data. Another advantage of the mentioned modeling was that it did not require extensive, detailed data as well as more parameters to calculate the basic reproduction number. Therefore, we recommend using this method in order to estimate the basic reproduction number.</p><sec id="sec2-8"><title>Financial support and sponsorship</title><p>Nil.</p></sec><sec id="sec2-9" sec-type="COI-statement"><title>Conflicts of interest</title><p>There are no conflicts of interest.</p></sec></sec></body><back><ack><title>Acknowledgments</title><p>This research is part of Roya Nikbakht's PhD dissertation. We are very grateful to the Public Health Agency of Canada (PHAC) website and the Respiratory Virus Detections in Canada Report website to provide online database and access to use their data.</p></ack><app-group><app id="App1"><title>APPENDIX</title><sec id="sec1-6"><title>SEARCH STRATEGY</title><p>In order to review the literature on basic reproduction number of influenza, we searched in the electronic databases such as the web of knowledge, PubMed, EMBASE, and Google Scholar to find published papers between 1954 and 2017. The medical subject heading was applied to find a wide range of keywords that had a maximum sensitivity. The following keywords were searched: influenza, human, and reproduction number. In detail searched keywords were (“influenza, human”[MeSH Terms] OR (“influenza”[All Fields] AND “human”[All Fields]) OR “human influenza”[All Fields] OR “influenza”[All Fields]) AND ((“reproduction”[MeSH Terms] OR “reproduction”[All Fields]) AND number [All Fields]).</p></sec><sec id="sec1-7"><title>STUDY SELECTION</title><p>Two reviewers independently extracted relevant studies from the keywords search. All types of original articles were investigated. The studies which included “influenza reproduction number” in their titles or abstracts were included. The irrelevant articles, based on the title and abstract evaluation, were excluded. Moreover, we eliminated the duplicated articles to determine unique studies. Animal studies and human studies that included special populations such as pregnant women and schizophrenia were excluded. We then extracted data and formulas from the full text of the included studies.</p><p><xref ref-type="fig" rid="F5">Figure 1</xref> shows the search strategy, through which 1213 papers were obtained in the initial round. The number of the retained papers was 910, which estimated <italic>R</italic><sub>0</sub> for epidemic or pandemic influenza with A/H1N1, A/H1N5, H1N2, H1N3, H5N1, pH 1N1, A/H3N2, influenza B, A (H7N9), Spanish flu, H2N2, H3N2, AH1, AH3, A (H5N1), and Asian flu. The number of papers identified through other sources was 5. Overall, 89 papers presented the basic reproduction number estimation and its formula, as summarized in <xref rid="T1" ref-type="table">Table 1</xref>.</p><fig id="F5" position="anchor"><label>Figure 1</label><caption><p>PRISMA flowchart of the article selection for the reproduction number and influenza literature review</p></caption><graphic xlink:href="JRMS-24-67-g017"></graphic></fig><p>In addition, detailed information of the study characteristics provided in the systematic review is given in <xref rid="T2" ref-type="table">Table 2</xref>, of which 10 studies were taken into consideration. In some of the studies, p-H1N1, A (H1N1), A (H3N2), type B, and A (H7N9) were reported as types of influenza. The models used for estimating <italic>R</italic><sub>0</sub> in these 8 studies were the multi-control measure, growth rate of exponential, and multi-phase Richards. In several of the studies, laboratory-confirmed cases were investigated for determining the reproduction number of influenza. Maximum, minimum and median of the reproduction number were 10.03 (in Mainland China), 0.08 (in China) and 1.39, respectively. The reproduction number of the influenza type A (H1N1) in Taiwan (2013) and Mexico was reported 1.54 (95% confidence interval [CI]: 0.22–8.88) and 1.69 (95% CI: 1.65–1.73), respectively. For A (H7N9), the reproduction number and its 95% CI in China for the first wave was estimated 0.27 (0.14, 0.44).</p><table-wrap id="T5" position="anchor"><label>Appendix Table 1</label><caption><p>The formula which applied for calculating reproduction number in different studies</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="1" colspan="1">Id</th><th align="center" rowspan="1" colspan="1">Formula</th><th align="left" rowspan="1" colspan="1">Reference</th><th align="center" rowspan="1" colspan="1">Id</th><th align="center" rowspan="1" colspan="1">Formula</th><th align="left" rowspan="1" colspan="1">Reference</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g018.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Ajelli and Merler, 2012</td><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g019.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Ajelli and Merler, 2012, Chowell <italic>et al</italic>., 2008, Chowell <italic>et al</italic>., 2012, Chowell <italic>et al</italic>., 2011, Pamaran <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">3</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g020.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Ajelli <italic>et al</italic>., 2014</td><td align="center" rowspan="1" colspan="1">4</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g021.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chowell <italic>et al</italic>., 2007b</td></tr><tr><td align="center" rowspan="1" colspan="1">5</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g022.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chen and Liao, 2010, Chen and Liao, 2013, Chong <italic>et al</italic>., 2016</td><td align="center" rowspan="1" colspan="1">6</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g023.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chowell <italic>et al</italic>., 2007c</td></tr><tr><td align="center" rowspan="1" colspan="1">7</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g024.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Andreasen <italic>et al</italic>., 2008, Haghdoost <italic>et al</italic>., 2012, Jackson <italic>et al</italic>., 2009</td><td align="center" rowspan="1" colspan="1">8</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g025.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chowell <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">9</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g026.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Andreasen <italic>et al</italic>., 2008, Barakat <italic>et al</italic>., 2012, Buckley and Bulger, 2011, Jackson <italic>et al</italic>., 2009</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g027.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chowell <italic>et al</italic>., 2010</td></tr><tr><td align="center" rowspan="1" colspan="1">11</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g028.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Caley <italic>et al</italic>., 2008</td><td align="center" rowspan="1" colspan="1">12</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g029.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Cowling <italic>et al</italic>., 2010</td></tr><tr><td align="center" rowspan="1" colspan="1">13</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g030.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chen <italic>et al</italic>., 2009</td><td align="center" rowspan="1" colspan="1">14</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g031.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Dorigatti <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">15</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g032.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chen <italic>et al</italic>., 2015</td><td align="center" rowspan="1" colspan="1">16</td><td align="center" rowspan="1" colspan="1"><italic>f<sub>p</sub></italic>=(1–<italic>p</italic>).1+<italic>p.f<sub>i</sub></italic>=1+<italic>p</italic>(<italic>f<sub>i</sub></italic>–1)<break></break><italic>R</italic>=<italic>S<sub>init</sub></italic>×<italic>R</italic><sub>0</sub></td><td align="left" rowspan="1" colspan="1">Earn <italic>et al</italic>., 2014</td></tr><tr><td align="center" rowspan="1" colspan="1">17</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g033.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Sheikh Taslim <italic>et al</italic>., 2013</td><td align="center" rowspan="1" colspan="1">18</td><td align="center" rowspan="1" colspan="1"><italic>R</italic><sub>0</sub>=<italic>kR</italic><sub>3</sub>+(1–<italic>k</italic>)<italic>R</italic><sub>4</sub><break></break><italic>R</italic><sub>0</sub>=<italic>R</italic><sub>1</sub>+<italic>αR</italic><sub>2</sub></td><td align="left" rowspan="1" colspan="1">Ejima <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">19</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g034.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chen <italic>et al</italic>., 2016</td><td align="center" rowspan="1" colspan="1">20</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g035.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Fielding <italic>et al</italic>., 2015</td></tr><tr><td align="center" rowspan="1" colspan="1">21</td><td align="center" rowspan="1" colspan="1"><italic>R</italic>=1+<italic>rT</italic><sub>c</sub></td><td align="left" rowspan="1" colspan="1">Chen <italic>et al</italic>., 2017</td><td align="center" rowspan="1" colspan="1">22</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g036.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Fierro and Liccardo, 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">23</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g037.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Cheng <italic>et al</italic>., 2016</td><td align="center" rowspan="1" colspan="1">24</td><td align="center" rowspan="1" colspan="1"><italic>R<sub>A</sub></italic>=(<italic>n</italic>–1)<italic>P</italic></td><td align="left" rowspan="1" colspan="1">Furuya, 2007</td></tr><tr><td align="center" rowspan="1" colspan="1">25</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g038.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Fraser <italic>et al</italic>., 2011</td><td align="center" rowspan="1" colspan="1"> </td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g039.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Hsieh, 2010a</td></tr><tr><td align="center" rowspan="1" colspan="1">27</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g040.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chong <italic>et al</italic>., 2016</td><td align="center" rowspan="1" colspan="1">28</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g041.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Nishiura, 2007</td></tr><tr><td align="center" rowspan="1" colspan="1">29</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g042.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chong <italic>et al</italic>., 2017</td><td align="center" rowspan="1" colspan="1">30</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g043.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Glass <italic>et al</italic>., 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">31</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g044.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Chowell <italic>et al</italic>., 2007a</td><td align="center" rowspan="1" colspan="1">32</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g045.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Gran <italic>et al</italic>., 2010</td></tr><tr><td align="center" rowspan="1" colspan="1">33</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g046.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Gurav <italic>et al</italic>., 2017</td><td align="center" rowspan="1" colspan="1">34</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g047.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Meeyai <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">35</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g048.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Hens <italic>et al</italic>., 2012</td><td align="center" rowspan="1" colspan="1">36</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g049.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Modchang <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">37</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g050.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Hens <italic>et al</italic>., 2011</td><td align="center" rowspan="1" colspan="1">38</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g051.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Mostaco-Guidolin <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">39</td><td align="center" rowspan="1" colspan="1"><italic>R</italic><sub>0</sub>=exp(<italic>rT</italic>)</td><td align="left" rowspan="1" colspan="1">Hsieh, 2010b, Hsieh <italic>et al</italic>., 2011a, Hsieh <italic>et al</italic>., 2010, Hsieh <italic>et al</italic>., 2016, Hsieh <italic>et al</italic>., 2011b, Liu <italic>et al</italic>., 2015b, Mostaco-Guidolin <italic>et al</italic>., 2012</td><td align="center" rowspan="1" colspan="1">40</td><td align="center" rowspan="1" colspan="1"><italic>R<sub>ij</sub></italic>=<italic>Rs<sub>i</sub>m<sub>ij</sub></italic><break></break> R is defined as the dominant eigenvalue of the next-generation matrix</td><td align="left" rowspan="1" colspan="1">Nishiura <italic>et al</italic>., 2010</td></tr><tr><td align="center" rowspan="1" colspan="1">41</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g052.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Inaba and Nishiura, 2008</td><td align="center" rowspan="1" colspan="1">42</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g053.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Nishiura <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">43</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g054.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Obadia <italic>et al</italic>., 2012</td><td align="center" rowspan="1" colspan="1">44</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g055.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Nyirenda <italic>et al</italic>., 2016</td></tr><tr><td align="center" rowspan="1" colspan="1">45</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g056.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Kadi and Avaradi, 2015</td><td align="center" rowspan="1" colspan="1">46</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g057.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Kelso <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">47</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g058.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Lin <italic>et al</italic>., 2016</td><td align="center" rowspan="1" colspan="1">48</td><td align="center" rowspan="1" colspan="1"><italic>R</italic><sub>0</sub>=(1+rT<sub>I</sub>)(1+rT<sub>L</sub>)</td><td align="left" rowspan="1" colspan="1">Rizzo <italic>et al</italic>., 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">49</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g059.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Kim <italic>et al</italic>., 2017</td><td align="center" rowspan="1" colspan="1">50</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g060.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Roberts and Nishiura, 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">51</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g061.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Kucharski and Edmunds, 2015</td><td align="center" rowspan="1" colspan="1">52</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g062.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Roll <italic>et al</italic>., 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">53</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g063.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Liu <italic>et al</italic>., 2015a</td><td align="center" rowspan="1" colspan="1">54</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g064.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Sattenspiel, 2011, Sertsou <italic>et al</italic>., 2006</td></tr><tr><td align="center" rowspan="1" colspan="1">55</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g065.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Marquetoux <italic>et al</italic>., 2012</td><td align="center" rowspan="1" colspan="1">56</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g066.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Shil <italic>et al</italic>., 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">57</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g067.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Massad <italic>et al</italic>., 2007</td><td align="center" rowspan="1" colspan="1">58</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g068.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Tan <italic>et al</italic>., 2013</td></tr><tr><td align="center" rowspan="1" colspan="1">59</td><td align="center" rowspan="1" colspan="1"><italic>R</italic>=<italic>Z</italic>(<italic>t</italic>)<italic>R</italic><sub>0</sub></td><td align="left" rowspan="1" colspan="1">Mathews <italic>et al</italic>., 2010</td><td align="center" rowspan="1" colspan="1">60</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g069.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Tang <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">61</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g070.jpg"></inline-graphic> and <inline-graphic xlink:href="JRMS-24-67-g071.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Truscott <italic>et al</italic>., 2009</td><td align="center" rowspan="1" colspan="1">62</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g072.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Tsukui, 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">63</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g073.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Van Kerckhove <italic>et al</italic>., 2013</td><td align="center" rowspan="1" colspan="1">64</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g074.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Ward <italic>et al</italic>., 2009</td></tr><tr><td align="center" rowspan="1" colspan="1">65</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g075.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">White <italic>et al</italic>., 2009</td><td align="center" rowspan="1" colspan="1">66</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g076.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Xiao <italic>et al</italic>., 2014</td></tr><tr><td align="center" rowspan="1" colspan="1">67</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g077.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Yang <italic>et al</italic>., 2015</td><td align="center" rowspan="1" colspan="1">68</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g078.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Yu <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">69</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g079.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Zhang <italic>et al</italic>., 2010</td><td align="center" rowspan="1" colspan="1">70</td><td align="center" rowspan="1" colspan="1"><italic>P</italic>(z,θ|<italic>y</italic>)∝<italic>P</italic>(y|z)<italic>P</italic>(z|θ)<italic>P</italic>(θ)</td><td align="left" rowspan="1" colspan="1">Cauchemez <italic>et al</italic>., 2011</td></tr><tr><td align="center" rowspan="1" colspan="1">71</td><td align="center" rowspan="1" colspan="1"><italic>b</italic>(<italic>R</italic>)=exp(<italic>τγ</italic>(<italic>R</italic>–1))</td><td align="left" rowspan="1" colspan="1">Kelly <italic>et al</italic>., 2010</td><td align="center" rowspan="1" colspan="1">72</td><td align="center" rowspan="1" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g080.jpg"></inline-graphic></td><td align="left" rowspan="1" colspan="1">Tang <italic>et al</italic>., 2010</td></tr></tbody></table></table-wrap><table-wrap id="T6" position="anchor"><label>Appendix Table 2</label><caption><p>Characteristics of several included studies</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="1" colspan="1">Id</th><th align="left" rowspan="1" colspan="1">Author (published date)</th><th align="left" rowspan="1" colspan="1">Place of study</th><th align="left" rowspan="1" colspan="1">Subject</th><th align="left" rowspan="1" colspan="1">Type of influenza</th><th align="center" rowspan="1" colspan="1">R0 (95% CI)</th><th align="center" rowspan="1" colspan="1">Formula</th><th align="left" rowspan="1" colspan="1">Method</th><th align="left" rowspan="1" colspan="1">Model</th><th align="left" rowspan="1" colspan="1">Refrence</th></tr></thead><tbody><tr><td align="left" rowspan="4" colspan="1">1</td><td align="left" rowspan="4" colspan="1">Y. H. Cheng (2013)</td><td align="left" rowspan="4" colspan="1">Taiwan</td><td align="left" rowspan="4" colspan="1">Elementary school</td><td align="left" rowspan="1" colspan="1">p-H1N1</td><td align="center" rowspan="1" colspan="1">3.30 (0.75, 11.47)</td><td align="center" rowspan="4" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g081.jpg"></inline-graphic></td><td align="left" rowspan="4" colspan="1">Branching process</td><td align="left" rowspan="4" colspan="1">Multi-control measure model</td><td align="left" rowspan="4" colspan="1">Cheng and Liao, 2013</td></tr><tr><td align="left" rowspan="1" colspan="1">A (H1N1)</td><td align="center" rowspan="1" colspan="1">1.54 (0.22, 8.88)</td></tr><tr><td align="left" rowspan="1" colspan="1">A (H3N2)</td><td align="center" rowspan="1" colspan="1">1.11 (0.18, 6.20)</td></tr><tr><td align="left" rowspan="1" colspan="1">Type B</td><td align="center" rowspan="1" colspan="1">1.11 (0.12, 8.52)</td></tr><tr><td align="left" rowspan="3" colspan="1">2</td><td align="left" rowspan="3" colspan="1">K. C. Chong (2016)</td><td align="left" rowspan="3" colspan="1">Zhejiang Province, China</td><td align="left" rowspan="3" colspan="1">Laboratory- confirmed patients</td><td align="left" rowspan="1" colspan="1">A (H7N9) first wave</td><td align="center" rowspan="1" colspan="1">0.27 (0.14, 0.44)</td><td align="center" rowspan="3" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g082.jpg"></inline-graphic></td><td align="left" rowspan="3" colspan="1">MCMC</td><td align="left" rowspan="3" colspan="1">Susceptible (S [t]), infectious (I [t]), or recovered</td><td align="left" rowspan="3" colspan="1">Chong <italic>et al</italic>., 2016</td></tr><tr><td align="left" rowspan="1" colspan="1">A (H7N9) second wave</td><td align="center" rowspan="1" colspan="1">0.15 (0.09, 0.24)</td></tr><tr><td align="left" rowspan="1" colspan="1">A (H7N9) third wave</td><td align="center" rowspan="1" colspan="1">0.15 (0.06, 0.26)</td></tr><tr><td align="left" rowspan="1" colspan="1">3</td><td align="left" rowspan="1" colspan="1">K. C. Chong (2017)</td><td align="left" rowspan="1" colspan="1">Mexico</td><td align="left" rowspan="1" colspan="1">New influenza pandemic</td><td align="left" rowspan="1" colspan="1">A/H1N1</td><td align="center" rowspan="1" colspan="1">1.69 (1.65, 1.73)</td><td align="center" rowspan="1" colspan="1"><italic>I</italic>(<italic>t</italic>)=<italic>I</italic>(0)exp[(<italic>β–γ</italic>)<italic>t</italic>]=exp[(<italic>R</italic><sub>0</sub>–1)<italic>γt</italic>]</td><td align="left" rowspan="1" colspan="1">A likelihood- based method</td><td align="left" rowspan="1" colspan="1">SIR</td><td align="left" rowspan="1" colspan="1">Chong <italic>et al</italic>., 2017</td></tr><tr><td align="left" rowspan="6" colspan="1">5</td><td align="left" rowspan="6" colspan="1">G. Chowell (2012)</td><td align="left" rowspan="6" colspan="1">Chile- Northern area</td><td align="left" rowspan="6" colspan="1">All hospitalizations</td><td align="left" rowspan="6" colspan="1">A/H1N1</td><td align="center" rowspan="1" colspan="1">1.19 (1.13, 1.24)</td><td align="center" rowspan="6" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g083.jpg"></inline-graphic></td><td align="left" rowspan="6" colspan="1">Maximum likelihood</td><td align="left" rowspan="6" colspan="1">Growth rate of the exponential pandemic</td><td align="center" rowspan="6" colspan="1">Chowell <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">1.25 (1.18, 1.32)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.32 (1.27, 1.37)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.43 (1.36, 1.50)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.58 (1.45, 1.72)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.81 (1.62, 2.0)</td></tr><tr><td align="left" rowspan="4" colspan="1">6</td><td align="left" rowspan="4" colspan="1">I. Dorigatti (2012)</td><td align="left" rowspan="4" colspan="1">Italy</td><td align="left" rowspan="4" colspan="1">Surveillance data</td><td align="left" rowspan="4" colspan="1">A/H1N1</td><td align="center" rowspan="1" colspan="1">1.42 (1.41, 1.424)</td><td align="center" rowspan="4" colspan="1"><inline-graphic xlink:href="JRMS-24-67-g084.jpg"></inline-graphic></td><td align="left" rowspan="4" colspan="1">MCMC, Bayesian</td><td align="left" rowspan="4" colspan="1">SEIR</td><td align="left" rowspan="4" colspan="1">Dorigatti <italic>et al</italic>., 2012</td></tr><tr><td align="center" rowspan="1" colspan="1">1.38 (1.37, 1.39)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.32 (1.30, 1.34)</td></tr><tr><td align="center" rowspan="1" colspan="1">1.31 (1.282, 1.35)</td></tr><tr><td align="left" rowspan="2" colspan="1">7</td><td align="left" rowspan="2" colspan="1">Y. H. 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