Data-driven exploration of ‘spatial pattern-time process-driving forces’ associations of SARS epidemic in Beijing, China
Identifieur interne : 000C20 ( Pmc/Corpus ); précédent : 000C19; suivant : 000C21Data-driven exploration of ‘spatial pattern-time process-driving forces’ associations of SARS epidemic in Beijing, China
Auteurs : Jin-Feng Wang ; George Christakos ; Wei-Guo Han ; Bin MengSource :
- Journal of Public Health (Oxford, England) [ 1741-3842 ] ; 2008.
Abstract
Severe Acute Respiratory Syndrome (SARS) was first reported in November 2002 in China, and spreads to about 30 countries over the next few months. While the characteristics of epidemic transmission are individually assessed, there are also important implicit associations between them.
A novel methodological framework was developed to overcome barriers among separate epidemic statistics and identify distinctive SARS features. Individual statistics were pair-wise linked in terms of their common features, and an integrative epidemic network was formulated.
The study of associations between important SARS characteristics considerably enhanced the mainstream epidemic analysis and improved the understanding of the relationships between the observed epidemic determinants. The response of SARS transmission to various epidemic control factors was simulated, target areas were detected, critical time and relevant factors were determined.
It was shown that by properly accounting for links between different SARS statistics, a data-based analysis can efficiently reveal systematic associations between epidemic determinants. The analysis can predict the temporal trend of the epidemic given its spatial pattern, to estimate spatial exposure given temporal evolution, and to infer the driving forces of SARS transmission given the spatial exposure distribution.
Url:
DOI: 10.1093/pubmed/fdn023
PubMed: 18441347
PubMed Central: 2518065
Links to Exploration step
PMC:2518065Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Data-driven exploration of ‘spatial pattern-time process-driving forces’ associations of SARS epidemic in Beijing, China</title><author><name sortKey="Wang, Jin Feng" sort="Wang, Jin Feng" uniqKey="Wang J" first="Jin-Feng" last="Wang">Jin-Feng Wang</name><affiliation><nlm:aff id="af1"><institution>Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences</institution>,<addr-line>A11, Datun Rd, Anwai, Beijing 100101</addr-line>,<country>China</country></nlm:aff></affiliation></author><author><name sortKey="Christakos, George" sort="Christakos, George" uniqKey="Christakos G" first="George" last="Christakos">George Christakos</name><affiliation><nlm:aff id="af2"><addr-line>Department of Geography</addr-line>,<institution>San Diego State University</institution>,<addr-line>San Diego, CA 92182-4493</addr-line>,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Han, Wei Guo" sort="Han, Wei Guo" uniqKey="Han W" first="Wei-Guo" last="Han">Wei-Guo Han</name><affiliation><nlm:aff id="af3"><institution>Center for Spatial Information Science and System, George Mason University</institution>,<addr-line>6301 Ivy Lane, Greenbelt, MD 20770</addr-line>,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Meng, Bin" sort="Meng, Bin" uniqKey="Meng B" first="Bin" last="Meng">Bin Meng</name><affiliation><nlm:aff id="af4"><institution>College of Applied Sciences and Humanities, Beijing Union University</institution>,<addr-line>Beijing 100083</addr-line>,<country>China</country></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">18441347</idno><idno type="pmc">2518065</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2518065</idno><idno type="RBID">PMC:2518065</idno><idno type="doi">10.1093/pubmed/fdn023</idno><date when="2008">2008</date><idno type="wicri:Area/Pmc/Corpus">000C20</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000C20</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Data-driven exploration of ‘spatial pattern-time process-driving forces’ associations of SARS epidemic in Beijing, China</title><author><name sortKey="Wang, Jin Feng" sort="Wang, Jin Feng" uniqKey="Wang J" first="Jin-Feng" last="Wang">Jin-Feng Wang</name><affiliation><nlm:aff id="af1"><institution>Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences</institution>,<addr-line>A11, Datun Rd, Anwai, Beijing 100101</addr-line>,<country>China</country></nlm:aff></affiliation></author><author><name sortKey="Christakos, George" sort="Christakos, George" uniqKey="Christakos G" first="George" last="Christakos">George Christakos</name><affiliation><nlm:aff id="af2"><addr-line>Department of Geography</addr-line>,<institution>San Diego State University</institution>,<addr-line>San Diego, CA 92182-4493</addr-line>,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Han, Wei Guo" sort="Han, Wei Guo" uniqKey="Han W" first="Wei-Guo" last="Han">Wei-Guo Han</name><affiliation><nlm:aff id="af3"><institution>Center for Spatial Information Science and System, George Mason University</institution>,<addr-line>6301 Ivy Lane, Greenbelt, MD 20770</addr-line>,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Meng, Bin" sort="Meng, Bin" uniqKey="Meng B" first="Bin" last="Meng">Bin Meng</name><affiliation><nlm:aff id="af4"><institution>College of Applied Sciences and Humanities, Beijing Union University</institution>,<addr-line>Beijing 100083</addr-line>,<country>China</country></nlm:aff></affiliation></author></analytic><series><title level="j">Journal of Public Health (Oxford, England)</title><idno type="ISSN">1741-3842</idno><idno type="eISSN">1741-3850</idno><imprint><date when="2008">2008</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><sec><title>Background</title><p>Severe Acute Respiratory Syndrome (SARS) was first reported in November 2002 in China, and spreads to about 30 countries over the next few months. While the characteristics of epidemic transmission are individually assessed, there are also important implicit associations between them.</p></sec><sec sec-type="methods"><title>Methods</title><p>A novel methodological framework was developed to overcome barriers among separate epidemic statistics and identify distinctive SARS features. Individual statistics were pair-wise linked in terms of their common features, and an integrative epidemic network was formulated.</p></sec><sec><title>Results</title><p>The study of associations between important SARS characteristics considerably enhanced the mainstream epidemic analysis and improved the understanding of the relationships between the observed epidemic determinants. The response of SARS transmission to various epidemic control factors was simulated, target areas were detected, critical time and relevant factors were determined.</p></sec><sec><title>Conclusion</title><p>It was shown that by properly accounting for links between different SARS statistics, a data-based analysis can efficiently reveal systematic associations between epidemic determinants. The analysis can predict the temporal trend of the epidemic given its spatial pattern, to estimate spatial exposure given temporal evolution, and to infer the driving forces of SARS transmission given the spatial exposure distribution.</p></sec></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article" xml:lang="EN"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">J Public Health (Oxf)</journal-id><journal-id journal-id-type="publisher-id">pubmed</journal-id><journal-id journal-id-type="hwp">jphm</journal-id><journal-title>Journal of Public Health (Oxford, England)</journal-title><issn pub-type="ppub">1741-3842</issn><issn pub-type="epub">1741-3850</issn><publisher><publisher-name>Oxford University Press</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">18441347</article-id><article-id pub-id-type="pmc">2518065</article-id><article-id pub-id-type="doi">10.1093/pubmed/fdn023</article-id><article-id pub-id-type="publisher-id">fdn023</article-id><article-categories><subj-group subj-group-type="heading"><subject>Health Protection</subject></subj-group></article-categories><title-group><article-title>Data-driven exploration of ‘spatial pattern-time process-driving forces’ associations of SARS epidemic in Beijing, China</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Wang</surname><given-names>Jin-Feng</given-names></name><role>Professor of Geoinformatics</role><xref ref-type="aff" rid="af1">1</xref><xref ref-type="corresp" rid="cor1"></xref></contrib><contrib contrib-type="author"><name><surname>Christakos</surname><given-names>George</given-names></name><role>Professor of Geography</role><xref ref-type="aff" rid="af2">2</xref></contrib><contrib contrib-type="author"><name><surname>Han</surname><given-names>Wei-Guo</given-names></name><role>Scientist of Computation</role><xref ref-type="aff" rid="af3">3</xref></contrib><contrib contrib-type="author"><name><surname>Meng</surname><given-names>Bin</given-names></name><role>Lecturer of Human Geography</role><xref ref-type="aff" rid="af4">4</xref></contrib></contrib-group><aff id="af1"><label>1</label><institution>Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences</institution>,<addr-line>A11, Datun Rd, Anwai, Beijing 100101</addr-line>,<country>China</country></aff><aff id="af2"><label>2</label><addr-line>Department of Geography</addr-line>,<institution>San Diego State University</institution>,<addr-line>San Diego, CA 92182-4493</addr-line>,<country>USA</country></aff><aff id="af3"><label>3</label><institution>Center for Spatial Information Science and System, George Mason University</institution>,<addr-line>6301 Ivy Lane, Greenbelt, MD 20770</addr-line>,<country>USA</country></aff><aff id="af4"><label>4</label><institution>College of Applied Sciences and Humanities, Beijing Union University</institution>,<addr-line>Beijing 100083</addr-line>,<country>China</country></aff><author-notes><corresp id="cor1">Address correspondence to Jin-Feng Wang, E-mail: <email>wangjf@igsnrr.ac.cn</email></corresp></author-notes><pub-date pub-type="ppub"><month>9</month><year>2008</year></pub-date><pub-date pub-type="epub"><day>26</day><month>4</month><year>2008</year></pub-date><pub-date pub-type="pmc-release"><day>26</day><month>4</month><year>2008</year></pub-date><volume>30</volume><issue>3</issue><fpage>234</fpage><lpage>244</lpage><copyright-statement>© The Author 2008. Published by Oxford University Press on behalf of the Faculty of Public Health. All rights reserved</copyright-statement><copyright-year>2008</copyright-year><license license-type="creative-commons" xlink:href="http://creativecommons.org/licenses/by-nc/2.0/uk/"><p>The online version of this article has been published under an open access model. Users are entitled to use, reproduce, disseminate, or display the open access version of this article for non-commercial purposes provided that: the original authorship is properly and fully attributed; the Journal and Oxford University Press are attributed as the original place of publication with the correct citation details given; if an article is subsequently reproduced or disseminated not in its entirety but only in part or as a derivative work this must be clearly indicated. For commercial re-use, please contact journals.permissions@oupjournals.org</p></license><abstract><sec><title>Background</title><p>Severe Acute Respiratory Syndrome (SARS) was first reported in November 2002 in China, and spreads to about 30 countries over the next few months. While the characteristics of epidemic transmission are individually assessed, there are also important implicit associations between them.</p></sec><sec sec-type="methods"><title>Methods</title><p>A novel methodological framework was developed to overcome barriers among separate epidemic statistics and identify distinctive SARS features. Individual statistics were pair-wise linked in terms of their common features, and an integrative epidemic network was formulated.</p></sec><sec><title>Results</title><p>The study of associations between important SARS characteristics considerably enhanced the mainstream epidemic analysis and improved the understanding of the relationships between the observed epidemic determinants. The response of SARS transmission to various epidemic control factors was simulated, target areas were detected, critical time and relevant factors were determined.</p></sec><sec><title>Conclusion</title><p>It was shown that by properly accounting for links between different SARS statistics, a data-based analysis can efficiently reveal systematic associations between epidemic determinants. The analysis can predict the temporal trend of the epidemic given its spatial pattern, to estimate spatial exposure given temporal evolution, and to infer the driving forces of SARS transmission given the spatial exposure distribution.</p></sec></abstract><kwd-group><title>Keywords</title><kwd>associations</kwd><kwd>determinants</kwd><kwd>epidemic</kwd><kwd>SARS</kwd><kwd>spatial pattern</kwd><kwd>statistics</kwd><kwd>time evolution</kwd></kwd-group></article-meta></front><body><sec id="s1"><title>Introduction</title><p>Severe Acute respiratory Syndrome (SARS) is a viral respiratory illness caused by a coronavirus known as SARS-associated coronavirus, SARS-CoV.<sup><xref ref-type="bibr" rid="FDN023C1">1</xref>,<xref ref-type="bibr" rid="FDN023C2">2</xref></sup> According to the World Health Organization (WHO), a total of 8098 people worldwide became sick with SARS during the 2003 outbreak; of these, 774 people died. SARS quickly obtained the status of the first pandemic of the twenty-first century.<sup><xref ref-type="bibr" rid="FDN023C3">3</xref>,<xref ref-type="bibr" rid="FDN023C4">4</xref></sup> In this work we focus on the SARS epidemic in the city of Beijing (China) during the year 2003.</p><p>SARS usually spreads by person-to-person transmission. The virus transmits most readily by respiratory droplets produced when an infected person coughs or sneezes. The virus can also spread when a person touches a surface or object contaminated with infectious droplets and then touches his/her mouth, nose or eyes. In addition, it is possible that the SARS virus might spread more broadly through the air (airborne spread) or by other ways that are not now known.<sup><xref ref-type="bibr" rid="FDN023C5">5</xref></sup> Naturally, composite space-time variations, disease parameters and determinants (proximity to infected persons, population density, number of doctors, etc.) are essential features of the SARS epidemic.<sup><xref ref-type="bibr" rid="FDN023C6">6</xref></sup> Understanding these features is critical for the scientific representation of the epidemic and the efficient control of its transmission.</p><p>In general, the evolution of an epidemic is represented in terms of some kind of a mechanistic model.<sup><xref ref-type="bibr" rid="FDN023C7">7</xref>–<xref ref-type="bibr" rid="FDN023C14">14</xref></sup> However, in the case of SARS (as well as in many other epidemic situations) only datasets of various kinds are available, whereas little is known about the underlying disease mechanisms. Mainstream statistics, time series, meta-analysis, randomized control trials and spatial techniques have been used to detect disease characteristics in a given dataset,<sup><xref ref-type="bibr" rid="FDN023C15">15</xref>–<xref ref-type="bibr" rid="FDN023C18">18</xref></sup> but the substantive relationships between the disease characteristics across space-time often remain unknown. Similarly, the procedure generating epidemiologic curves<sup><xref ref-type="bibr" rid="FDN023C19">19</xref></sup> (i.e. curves showing the day-to-day rate of SARS growth) does not account for important associations between epidemic characteristics and dominant space-time disease patterns and dependencies. Despite the aforementioned limitations of the mainstream approaches, it is highly desirable to develop methods that can incorporate the scattered and incomplete datasets generated by various surveillance systems<sup><xref ref-type="bibr" rid="FDN023C20">20</xref>–<xref ref-type="bibr" rid="FDN023C22">22</xref></sup> and by different programmatic and sectoral actions<sup><xref ref-type="bibr" rid="FDN023C23">23</xref></sup> in order to evaluate salient associations between epidemic determinants and quantify consequential space-time variations and patterns.</p><p>In view of similar considerations, the aim of the present work is to develop a mathematically rigorous and scientifically meaningful SARS modelling framework that accounts for the crucial epidemic associations mentioned above by combining information about space, time, disease parameters and epidemic determinants from diverse datasets. The notation used in this paper is described in Appendix 1; the reader should peruse it before proceeding.</p></sec><sec id="s2"><title>Data and methods</title><p>SARS is a typical case of a communicable disease about which a set of observed data has been obtained across space and time, and its potential determinants are available in a geographical information system (GIS) format. Below, several tools are considered for exploring epidemic determinants, spatial patterns and temporal processes on the basis of the available datasets. Certain statistics pairs share common elements that can be used to link these determinants, patterns and processes, and to formulate informative associations between them.</p><p>In this study, it is assumed that the pair-wise linking of the various SARS determinants is mathematically represented by a system of the general form <italic>f</italic>(<italic>R</italic>, <italic>C</italic>, <italic>F</italic>, Θ, <italic>s</italic>, <italic>t</italic>) = 0, which is constructed mainly on the basis of the available datasets. The underlying epidemic mechanisms (usually represented in terms of differential equations) are not considered in this formulation, but they will be the topic of forthcoming publications.</p><sec id="s2a"><title>Datasets</title><p>The first SARS case in Beijing was recorded on 1 March 2003. The city consists of 16 districts and 2 counties, with a total population of 12.5 million people occupying an area of 17 800 km<sup>2</sup>. Information about SARS cases was available daily from authorized reports, beginning on 20 April 2003 and continuing till the end of the epidemic on 24 June 2003. Since April 27, the data in each of the 18 Beijing districts were obtained in GIS format. GIS data also provided the residential locations of all 11 108 people who had been in close contact with infected persons. This information was collected by exhaustively tracing SARS cases from 20 April 2003 onwards and backwards. Other data that were considered relevant to the epidemic were also obtained, including population counts in 246 census units, hospital locations, number of doctors in each hospital and the topography of the main traffic lines.</p><p>By way of a summary, the following datasets were systematically used in the present SARS study:
<list list-type="roman-lower"><list-item><p>New SARS cases, <italic>R</italic>(<italic>s</italic>, <italic>t</italic>), reported daily in all 18 districts; <italic>s</italic> refers to the geographical district and <italic>t</italic> to the day considered.</p></list-item><list-item><p>Epidemic-relevant determinants, <italic>F</italic>(<italic>s</italic>), i.e. population (<italic>P</italic>), number of doctors (<italic>D</italic>), number of hospitals (Hosp) and proximity to traffic (<italic>T</italic>) for each of the 18 districts.</p></list-item><list-item><p>Residential locations, <italic>H</italic>(s), of each one of the 11 108 individual close contacts to the infected persons.</p></list-item></list></p></sec><sec id="s2b"><title>Temporal evolution of infected cases</title><p>A SEIR (susceptible-exposed-infectious-recovered) model was developed to represent SARS epidemic transmission in a dynamic context. The model basically involves the following successive stages of the individuals involved:
<disp-formula><graphic xlink:href="fdn02305.jpg" position="float"></graphic></disp-formula>see Appendix 2. The SEIR model for the SARS epidemic is based on the theoretical considerations discussed in Hamer<sup><xref ref-type="bibr" rid="FDN023C24">24</xref></sup> and Anderson.<sup><xref ref-type="bibr" rid="FDN023C25">25</xref></sup> Similar dynamic models have been used in early SARS studies by Lipsitch <italic>et al.</italic><sup><xref ref-type="bibr" rid="FDN023C3">3</xref></sup> and Riley <italic>et al.</italic>,<sup><xref ref-type="bibr" rid="FDN023C5">5</xref></sup> which concluded that the SARS coronavirus, if uncontrolled, would infect the majority of people wherever it was introduced. Epidemic parameters (as described in Chowell <italic>et al.</italic><sup><xref ref-type="bibr" rid="FDN023C26">26</xref></sup>), temporal process and total size can be easily simulated using a time-dependent SEIR model and a small amount of data.</p></sec><sec id="s2c"><title>Spatial distribution of identified contacts</title><p>In general, the determination of spatial dependence contributes significantly to adequate risk exposure assessment.<sup><xref ref-type="bibr" rid="FDN023C27">27</xref>–<xref ref-type="bibr" rid="FDN023C31">31</xref></sup> In the present study, the nearest-neighbour hierarchical clustering (NHC) model (Appendix 3) was used to identify spatial patterns of SARS distribution. Points were randomly distributed in space and circled in a spatial cluster, if their inter-distance was significantly smaller than the mean distance. Note that the primary spatial clusters may be further clustered using the same algorithm.</p></sec><sec id="s2d"><title>Temporal changes in spatial clustering of cases</title><p>We measured the degree of global spatial data clustering using Moran's <italic>I</italic><sub><italic>M</italic></sub> coefficient (Appendix 4). A positive value of the coefficient meant that adjacent districts had similar values, whereas a negative value indicated that adjacent districts were dissimilar. A temporal measure of spatial clustering was obtained by calculating this coefficient on a daily basis. This time series was then filtered using a discrete wavelet transform implemented by the MatLab computer library (<uri xlink:type="simple" xlink:href="http://www.mathworks.com/">http://www.mathworks.com/</uri>). In this way, the time series was decomposed into low-frequency components (reflecting the fundamental trend of a time series) and high-frequency components (reflecting noise caused by random factors).</p></sec><sec id="s2e"><title>Factors associated with dispersion of infection</title><p>The Black-White (BW) join-count test (Appendix 5) was used to measure the extent to which districts of a disease network shared the same infection pattern. We started with an initial disease dispersal network, which consisted of connections (or joints) between districts. Each day a district was coded black (B) if a SARS case was reported on that day; otherwise it was coded white (W). Every network joint connected two B districts (BB), two W districts (WW), or a B and a W district (BW). The observed number of BW joints was compared with the expected number, and a standard normal deviation (z-score) was used to test the significance. High negative values of the statistics indicated clustering of infected cases on the network, whereas high positive values provided evidence of spacing. Seven networks were considered as follows:
<list list-type="simple"><list-item><p>N1. Local transmission: Two districts were considered connected if they shared a common geographical boundary.</p></list-item><list-item><p>N2. Nearest district: Each district was connected to its nearest neighbour as measured by the distance between the centroids of the districts.</p></list-item><list-item><p>N3. Population size: Districts were ranked according to population size, and consecutive districts in the corresponding hierarchy were appropriately connected.</p></list-item><list-item><p>N4. Population density: The same as in N3, but ranked by population density, instead.</p></list-item><list-item><p>N5. Number of doctors: The same as in N3, but ranked by number of doctors in the district.</p></list-item><list-item><p>N6. Number of hospitals: The same as for N3, but ranked by number of hospitals in the district.</p></list-item><list-item><p>N7. Urban-rural: Eight districts were designated urban and the rest rural; a rural-urban pair was considered connected if (i) the districts shared a boundary, or (ii) the urban district could be reached from the rural district by passing through just one other rural district, or (iii) the rural district could be reached from the urban district by passing through just one other urban district.</p></list-item></list>For each network, the BW join-count statistics was calculated for each day, and statistics changes were plotted over time.</p></sec><sec id="s2f"><title>Combining space, time, parameters and factors of epidemic transmission</title><p>The above separate SARS statistics were combined by means of pair-wise linking of common items to form a network connecting (Table <xref ref-type="table" rid="FDN023TB1">1</xref>):
<list list-type="bullet"><list-item><p>–The time series of infectives <italic>R</italic>(<italic>t</italic>) and epidemic parameters Θ(<italic>t</italic>); time <italic>t</italic> is here the common item.</p></list-item><list-item><p>–The spatial pattern of risk exposure <italic>C</italic>(<italic>s</italic>) and the city traffic loop <italic>T</italic>(<italic>s</italic>); space <italic>s</italic> is the common item.</p></list-item><list-item><p>–The time process of spatial clusters <italic>C</italic>(<italic>t</italic>) and the transmission determinants <italic>F</italic>(<italic>t</italic>), including spatial proximity <italic>T</italic>(<italic>t</italic>); a link is established by the consistency between the spatial patterns of <italic>C</italic> and <italic>T</italic>.</p></list-item></list></p><table-wrap id="FDN023TB1" position="float"><label>Table 1</label><caption><p>Common items of different statistics</p></caption><table frame="hsides" rules="groups"><colgroup span="1"><col align="left" span="1"></col><col align="left" span="1"></col><col align="left" span="1"></col><col align="left" span="1"></col><col align="left" span="1"></col></colgroup><thead><tr><th align="left" rowspan="1" colspan="1"><italic>Common item</italic></th><th align="left" rowspan="1" colspan="1">R(t) <italic>and Θ</italic>(t)</th><th align="left" rowspan="1" colspan="1">C(s) ∼ T(s)</th><th align="left" rowspan="1" colspan="1">C(t)</th><th align="left" rowspan="1" colspan="1">F(t): T, P, D, …</th></tr></thead><tbody><tr><td rowspan="1" colspan="1"><italic>R</italic><italic>(t)</italic> and Θ<italic>(t)</italic></td><td rowspan="1" colspan="1"></td><td rowspan="1" colspan="1"></td><td rowspan="1" colspan="1"><italic>T</italic></td><td rowspan="1" colspan="1"><italic>t</italic></td></tr><tr><td rowspan="1" colspan="1"><italic>C</italic>(<italic>s</italic>) ∼ <italic>T</italic>(s)</td><td rowspan="1" colspan="1"></td><td rowspan="1" colspan="1"></td><td rowspan="1" colspan="1"><italic>C</italic></td><td rowspan="1" colspan="1"><italic>T</italic></td></tr><tr><td rowspan="1" colspan="1"><italic>C</italic>(<italic>t</italic>)</td><td rowspan="1" colspan="1"><italic>t</italic></td><td rowspan="1" colspan="1"><italic>C</italic></td><td rowspan="1" colspan="1"></td><td rowspan="1" colspan="1"><italic>t</italic></td></tr><tr><td rowspan="1" colspan="1"><italic>F</italic>(<italic>t</italic>): <italic>T</italic>, <italic>P</italic>, <italic>D</italic>, …</td><td rowspan="1" colspan="1"><italic>t</italic></td><td rowspan="1" colspan="1"><italic>T</italic></td><td rowspan="1" colspan="1"><italic>T</italic></td><td rowspan="1" colspan="1"></td></tr></tbody></table><table-wrap-foot><fn><p><italic>Note</italic>: ‘<italic>C</italic> ∼ <italic>T’</italic> denotes that <italic>C</italic> and <italic>T</italic> are associated (connected).</p></fn></table-wrap-foot></table-wrap><p>Careful considerations of the combinations described in Table <xref ref-type="table" rid="FDN023TB1">1</xref> play a central role in the evaluation of determinant associations in a space-time context.</p></sec></sec><sec id="s3"><title>Results</title><p>The statistics described above were applied to the datasets, in which case a systematic connection between the separate statistics was established and described below.</p><sec id="s3a"><title>Separate statistics</title><p>The time process of SARS transmission, <italic>R</italic>(<italic>t</italic>), and the transmission rate in Θ(<italic>t</italic>) were simulated using the SEIR model. The risk of spatial exposure to the epidemic, <italic>C</italic>(<italic>s</italic>), was disclosed by mapping data on the residences of the 11 108 identified contacts of the Beijing SARS cases using the NHC model.<sup><xref ref-type="bibr" rid="FDN023C32">32</xref></sup> The first-order clusters indicated spatial clustering of high-risk susceptibles, whereas second-order clusters indicated regions with a high concentration of primary clusters. The <italic>C</italic>(<italic>s</italic>) exhibited a strong visual association with traffic infrastructure, <italic>T</italic>(<italic>s</italic>) (Fig. <xref ref-type="fig" rid="FDN023F1">1</xref>).</p><fig id="FDN023F1" position="float"><label>Fig. 1</label><caption><p>Identified contacts (11 108) of Beijing SARS cases and their spatial patterns (modified from Wang <italic>et al.</italic>, 2006).</p></caption><graphic xlink:href="fdn02301"></graphic></fig><p>Temporal changes in the spatial clustering of cases, <italic>C</italic>(<italic>t</italic>), were determined by the time-variation of Moran's <italic>I</italic><sub><italic>M</italic></sub> coefficient. The <italic>I</italic><sub><italic>M</italic></sub>-curve was then filtered using a discrete wavelet transform implemented by MatLab (with Daubechies db3 as mother wave and four decomposition levels). In this way, the curve was decomposed into a low-frequency component (<italic>a</italic><sub>4</sub>) and high-frequency components (<italic>d</italic><sub>1</sub>, <italic>d</italic><sub>2</sub>, <italic>d</italic><sub>3</sub>, <italic>d</italic><sub>4</sub>) (Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>). Daily data on new SARS cases were grouped according to the district of residence. The time-variation of <italic>a</italic><sub>4</sub> indicated that SARS cases in Beijing became increasingly clustered towards the end of April, 2003, followed by a decrease in clustering during 1–8 May 2003. After 9 May 2003, the time-variation of <italic>a</italic><sub>4</sub> and <italic>d</italic><sub>4</sub> indicated that spatial clustering was sharply declining.</p><fig id="FDN023F2" position="float"><label>Fig. 2</label><caption><p>Wavelet decomposition of the time series of Moran's coefficient <italic>I</italic><sub><italic>M</italic></sub> of spatial clustering (<italic>a</italic><sub><italic>i</italic></sub> and <italic>d</italic><sub><italic>i</italic></sub>, <italic>i</italic> = 1, …, 4, correspond to low and high frequency signals respectively; grey figures were further decomposed in the arrows' directions).</p></caption><graphic xlink:href="fdn02302"></graphic></fig><p>Seven suspect determinants of the space-time SARS transmission, jointly denoted by <italic>F</italic>(<italic>t</italic>), were investigated by means of the BW join-count test.<sup><xref ref-type="bibr" rid="FDN023C33">33</xref></sup> The spatial proximity factor is denoted by <italic>T</italic>(<italic>t</italic>). Accordingly, Fig. <xref ref-type="fig" rid="FDN023F3">3</xref> displays the determinants associated with SARS transmission dynamics.</p><fig id="FDN023F3" position="float"><label>Fig. 3</label><caption><p>Relationships between incidence of new SARS cases and spatial spread factors. Each panel tests one suspect determinant (N1, …, N7); dots are BW scores in each date; solid line is simple regression of the BW zigzag; horizontal dashed line denotes a significant threshold (modified from Meng <italic>et al.</italic>, 2005).</p></caption><graphic xlink:href="fdn02303"></graphic></fig></sec><sec id="s3b"><title>Systematic analysis</title><p>The findings obtained on the basis of the different statistics of <italic>R</italic>(<italic>t</italic>), <italic>C</italic>(<italic>t</italic>), <italic>C</italic>(<italic>s</italic>) and <italic>F</italic>(<italic>t</italic>) above were pair-wise combined by means of their common elements and the corresponding network was formed as shown in Fig. <xref ref-type="fig" rid="FDN023F4">4</xref>. Logical inferences concerning the SARS epidemic evolution based on the connected network of Fig. <xref ref-type="fig" rid="FDN023F4">4</xref> are discussed.</p><fig id="FDN023F4" position="float"><label>Fig. 4</label><caption><p>Network of systematic connections between different statistics.</p></caption><graphic xlink:href="fdn02304"></graphic></fig><p>In the following inference analysis, the symbol ‘<italic>A</italic>&B’ means <italic>A</italic> and <italic>B</italic>; ‘<italic>A</italic> ∼ <italic>B</italic>’ denotes that the entities <italic>A</italic> and <italic>B</italic> are connected; the ‘<italic>A</italic> ∼ <italic>m</italic> ∼ <italic>B</italic>’ means that <italic>A</italic> is linked to <italic>B</italic> via <italic>m;</italic> and the ‘<italic>A</italic>→<italic>B</italic>’ denotes that <italic>B</italic> is inferred from <italic>A</italic>.</p><sec id="s3b1"><title>Evolution of spatial clusters</title><p>Inference concerning the evolution of spatial SARS clusters was based on the rule
<disp-formula id="FDN023M1"><label>1</label><graphic xlink:href="fdn02306.jpg" position="float"></graphic></disp-formula>Figure <xref ref-type="fig" rid="FDN023F1">1</xref> illustrates that the larger spatial clusters <italic>C</italic>(<italic>s</italic>) were controlled by the urban traffic loops <italic>T</italic>(<italic>s</italic>), i.e. a proxy of the macro spatial proximity, <italic>T</italic>(<italic>t</italic>), the influence of which to the epidemic is detectable and changes with time (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>b and c). The smaller spatial clusters in Fig. <xref ref-type="fig" rid="FDN023F1">1</xref> are randomly distributed over space, which is consistent with the spatial distributions of the population <italic>P</italic> and the number of doctors <italic>D</italic>, the two factors playing a significant role in the epidemic decline period (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>d–f). We concluded that the larger spatial clusters were dominant during the peak epidemic period and the smaller ones became more active during the decline period.</p><p>A similar conclusion was drawn by means of the inference rule
<disp-formula id="FDN023M2"><label>2</label><graphic xlink:href="fdn02307.jpg" position="float"></graphic></disp-formula></p><p>In other words, Fig. <xref ref-type="fig" rid="FDN023F1">1</xref> clearly shows that there are spatial clusters at two <italic>C</italic>(<italic>s</italic>) scales, whereas Fig. <xref ref-type="fig" rid="FDN023F2">2</xref> depicts the time series of the spatial clustering of the two scales. The large cluster (lower frequency part, left curve in Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>) grew until 30 April 2003, and diffused into smaller clusters thereafter (as shown in the right part of Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>), when there was a slight increase in the building up of higher-frequency components (or, equivalently, of smaller spatial clusters). Both the larger and the smaller clusters lost their strength after 8 May 2003 (Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>).</p></sec><sec id="s3b2"><title>Driving forces of the temporal process</title><p>Inference concerning the driving forces of the temporal process was based on the rule
<disp-formula id="FDN023M3"><label>3</label><graphic xlink:href="fdn02308.jpg" position="float"></graphic></disp-formula></p><p>The rapid decline of the daily new cases <italic>R</italic>(<italic>t</italic>) and the infective rate in Θ(<italic>t</italic>) after 30 April 2003 (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>a) were predominantly driven by the spatial proximity <italic>T</italic>(<italic>t</italic>) (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>b and c); and intermittently accompanied by the interaction between (infected) doctors <italic>D</italic> and the urban population density <italic>P</italic> (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>e and f). This trend was interrupted (see the diamond line in Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>a) by a significant transmission between rural and urban areas on 15–18 May 2003 (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>h).</p></sec><sec id="s3b3"><title>Driving forces of spatial clustering</title><p>Inference about the driving forces of spatial clustering was based on the rule
<disp-formula id="FDN023M4"><label>4</label><graphic xlink:href="fdn02309.jpg" position="float"></graphic></disp-formula></p><p>In other words, the larger spatial clusters <italic>C</italic>(<italic>t</italic>) (associated with the <italic>a</italic><sub>4</sub> component in Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>) had a consistent trend and were controlled by spatial proximity <italic>T</italic>(<italic>t</italic>) (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>b and c), whereas the smaller clusters <italic>C</italic>(<italic>t</italic>) (<italic>d</italic><sub>4</sub> component in Fig. <xref ref-type="fig" rid="FDN023F2">2</xref>) were still building up, driven by significant <italic>P</italic> (population density) and <italic>D</italic> (number of doctors) during the specified time period (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>e and f).</p></sec><sec id="s3b4"><title>Spatial clustering and intervention</title><p>Inference concerning spatial clustering and intervention was based on the rule
<disp-formula id="FDN023M5"><label>5</label><graphic xlink:href="fdn02310.jpg" position="float"></graphic></disp-formula></p><p>That is, the larger spatial clusters represented by the <italic>a</italic><sub>4</sub> component of <italic>C</italic>(<italic>t</italic>) in Fig. <xref ref-type="fig" rid="FDN023F2">2</xref> dominated the trend throughout the entire epidemic period, whereas the smaller clusters reflected in the <italic>d</italic><sub>4</sub> component of <italic>C</italic>(<italic>t</italic>) became more active during the decline period. The spatial and temporal patterns of transmission were controlled dynamically by intervention measures, as reflected in the epidemic parameters Θ(<italic>t</italic>) (Fig. <xref ref-type="fig" rid="FDN023F3">3</xref>a).<sup><xref ref-type="bibr" rid="FDN023C32">32</xref></sup> E.g. the ‘(<italic>C</italic>, <italic>R</italic>, Θ)(<italic>t</italic>)’ denotes that the triple within the bracket change with time.</p><p>In addition to the above direct pair-wise relationship between the variables of the SARS epidemic system, some composite inferences can be also drawn. A few examples follow. The
<disp-formula id="FDN023M6"><label>6</label><graphic xlink:href="fdn02311.jpg" position="float"></graphic></disp-formula>demonstrates the spatiotemporal (<italic>s</italic>, <italic>t</italic>) dynamics of the observed spatial clusters (<italic>C</italic>) and the corresponding determinants <italic>F</italic>(<italic>t</italic>) on a daily basis. The
<disp-formula id="FDN023M7"><label>7</label><graphic xlink:href="fdn02312.jpg" position="float"></graphic></disp-formula>represents the fact that the observed time series of the infected cases <italic>R</italic>(<italic>t</italic>) was driven by the spatial clustering variable <italic>C</italic>(<italic>t</italic>) and the epidemic determinants <italic>F</italic>(<italic>t</italic>). The role of the traffic <italic>T</italic> in <italic>F</italic> was to control the larger <italic>C</italic><sub>0</sub>(<italic>s</italic>) [see Eq. (<xref ref-type="disp-formula" rid="FDN023M2">2</xref>)] and <italic>C</italic>(<italic>t</italic>) [Eq. (<xref ref-type="disp-formula" rid="FDN023M4">4</xref>)]. The <italic>P</italic> and <italic>D</italic> in <italic>F</italic> control the smaller <italic>C</italic>(<italic>s</italic>) [Eq. (<xref ref-type="disp-formula" rid="FDN023M2">2</xref>)] and <italic>C</italic>(<italic>t</italic>) [Eq. (<xref ref-type="disp-formula" rid="FDN023M4">4</xref>)]. The role of <italic>C</italic>(<italic>s</italic>, <italic>t</italic>) is to control the <italic>R</italic>(<italic>t</italic>) [Eq. (<xref ref-type="disp-formula" rid="FDN023M5">5</xref>)]. The larger <italic>C</italic>(<italic>s</italic>) occurred during the peak <italic>R</italic>(<italic>t</italic>)-values and the smaller <italic>C</italic>(<italic>s</italic>) occurred during the decline phase of the epidemic, see Eqs. (<xref ref-type="disp-formula" rid="FDN023M2">2</xref>) and (<xref ref-type="disp-formula" rid="FDN023M5">5</xref>) or Eqs. (<xref ref-type="disp-formula" rid="FDN023M2">2</xref>) and (<xref ref-type="disp-formula" rid="FDN023M3">3</xref>). Accordingly, the peak and decline times of the epidemic spread were the two best times to efficiently conduct travel-related control at the ring roads <italic>T</italic> and the doctor (<italic>D</italic>)- and population (<italic>P</italic>)- related measures in the community, given a limited budget for epidemic control.</p></sec></sec></sec><sec id="s4"><title>Discussion</title><sec id="s4a"><title>Main finding of this study</title><p>The present work proposed a methodological framework to determine the main features of SARS transmission during the Beijing epidemic of 2003 using various datasets and space-time statistics techniques. Epidemic determinants and the relevant statistics are pair-wise linked in a space-time context, which means that the study of determinant associations can considerably enhance the mainstream epidemic analysis and improve the understanding of previously undetected relationships within the epidemic system.</p><p>Several interesting findings were obtained. Spatial neighbourhood is a major component of epidemic spread modelling, whereas coupling spatial transmission with population density and healthcare workers was sporadically significant. Changes in the spatial spread of SARS over time indicated which intervention measures are likely to be most effective at different phases of the outbreak. When the epidemic is growing via diffused infection, efficient intervention measures include isolating cases and reducing inter-regional movement. When cases of the disease appear in clusters, resources should be directed towards curbing transmission within localities that have a high incidence of infection. Our results suggested that improving control measures at the local level would have been the most efficient approach, before the end of April, 2003. Local intervention strategies include quarantine of family members. Once these measures take effect, transmission is reduced substantially and becomes dominated by more remote contacts. The evidence that diffusion of infections drove the declining stage of the outbreak may indicate that the measures were successfully controlling local transmission.</p><p>Temporal variables (e.g. number of infectives, season and weather), spatial variables (e.g. risk exposure, surveillance network, travel warnings and isolation) and other factors (e.g. immunity, population, doctors, hospitals and transportation) are mapped on the SARS epidemic system. This allows transferring operations from a less detectable or operable domain to a domain with features that improve determinant identification, generate accurate predictions and sound inferences, and contribute to strategies aiming at controlling disease spread. E.g. the free travel or isolation over regions (space) lead to an increase or decrease of the total number of infectives (time); and the seasonal temperature change (time) alters the spatial pattern of risk exposure (space).</p><p>More than one statistics have to be calculated in epidemic data analysis, so more resources are needed to improve the accuracy of such calculations. Fortunately, there is a plethora of computer software and open source codes that are easily accessible on the Internet and greatly facilitate data analysis computations. In this study, the datasets are managed by the GIS software ARCGIS (<uri xlink:type="simple" xlink:href="http://www.esri.com/software/arcgis/">http://www.esri.com/software/arcgis/</uri>). The hierarchical clustering is implemented by the Crimestat software (<uri xlink:type="simple" xlink:href="http://www.icpsr.umich.edu/CRIMESTAT/">http://www.icpsr.umich.edu/CRIMESTAT/</uri>). Moran's coefficient <italic>I</italic><sub><italic>M</italic></sub> is calculated using the GeoDA software (<uri xlink:type="simple" xlink:href="http://www.geoda.uiuc.edu/">http://www.geoda.uiuc.edu/</uri>). Wavelet decomposition and SSEIR are computed using Matlab (<uri xlink:type="simple" xlink:href="http://www.icpsr.umich.edu/CRIMESTAT/">http://www.icpsr.umich.edu/CRIMESTAT/</uri>), and the BW test is performed using software compiled by the authors. Primary training on GIS and spatial statistics are needed in order to operate the above software.</p></sec><sec id="s4b"><title>What is already known on this topic</title><p>Several epidemiologic assessments of the SARS-CoV have concluded that this coronavirus is sufficiently transmissible to cause a very large epidemic, but not so contagious as to be uncontrollable with good, basic public health measures.<sup><xref ref-type="bibr" rid="FDN023C3">3</xref>,<xref ref-type="bibr" rid="FDN023C5">5</xref></sup> It is a common modelling practice to use a single or several separate data statistics to investigate individual epidemic properties and determinants,<sup><xref ref-type="bibr" rid="FDN023C6">6</xref></sup> and to quantify policies addressing problems caused by the disease.<sup><xref ref-type="bibr" rid="FDN023C34">34</xref></sup> Among other things, the spatial pattern, time process and driving forces of an epidemic can be explored by hierarchical clustering, SSEIR, and BW, respectively. Also, epidemiologic curves have been generated that display the day-to-day rate of SARS growth<sup><xref ref-type="bibr" rid="FDN023C19">19</xref></sup> but without accounting for the ‘spatial pattern-time process-driving forces’ associations of the epidemic. Only a limited number of studies have attempted to partially represent and evaluate determinant associations and links between separate statistics under conditions of uncertainty.</p></sec><sec id="s4c"><title>What this study adds</title><p>The present work is the first systematic effort to represent the aforementioned associations and links in a rigorous manner and to assess their considerable role in understanding the various aspects of epidemic distribution across space-time. Advanced mathematical modelling of an epidemic could potentially unveil the relationships between various determinants, but this modelling is often impossible due to the unavailability of the necessary scientific knowledge, including a lack of understanding of the fundamental disease mechanisms.<sup><xref ref-type="bibr" rid="FDN023C35">35</xref>,<xref ref-type="bibr" rid="FDN023C36">36</xref></sup> Nevertheless, the present study shows that using the available datasets an informative modelling approach can be developed that is based on the pair-wise linking of epidemic determinants and the formulation of a network that systematically connects the relevant statistics. In addition, unlike earlier SARS modelling studies,<sup><xref ref-type="bibr" rid="FDN023C5">5</xref></sup> the present approach accounts in an efficient manner for the spatial variation of the data available. This approach is particularly useful when the disease mechanisms are not clear and the corresponding mathematical model is difficult to develop for reasons mentioned above. Moreover, the approach can be successfully implemented to simulate the response of SARS transmission to various epidemic control factors, identify target areas and determine the critical time and relevant factors. Given the response by individuals, institutions and governments to a new and dangerous disease, the findings of the present study can be useful in the implementation of efficient epidemic intervention strategies and effective population protection policies.</p></sec><sec id="s4d"><title>Limitations of this study</title><p>The present modelling approach is concerned mainly with statistical associations rather than causal laws. The latter are available primarily in cases in which the fundamental disease mechanisms (molecular biology, etc.) are well understood. In this sense, SARS modelling could benefit from future developments in basic scientific research in disease etiology and mathematical theories of epidemic propagation, as well as from sophisticated surveillance capabilities and meta-analysis. Furthermore, the present SARS modelling could be improved by considering epidemic variation and dependence in a composite space-time continuum under conditions of multi-sourced uncertainty. This may involve a combination of random field theories with temporal GIS techniques.</p></sec><sec id="s4e"><title>Future applications</title><p>By means of the novel methodological framework considered in the present work, future epidemic studies may not only benefit from ‘meta data analysis’ but also from ‘meta model analysis’. A wide variety of inter-disciplinary datasets that are relevant to the epidemic spread can be integrated by means of this framework. The statistics functions and spatial analysis techniques described above can be readily applied to the study of epidemics other than SARS. Furthermore, the methodological framework is not limited to the statistical tools and datasets considered in this paper, but is more general and flexible, in the sense that it may include additional information sources, statistics functions and models; and it can allow the performance of other types of logical inferences, thus generating new findings and conclusions concerning the epidemic under investigation. By generalizing the current approach within the context of random field theories with temporal GIS techniques, one may consider additional issues related to the next generation of (stochastic) SARS models allowing for space-time variations, multi-sourced uncertainty, seasonality and varying transmission modes, the case of SARS virus long-time persistence, and examining the case of global eradication rather than local control in terms of early epidemic detection measures.</p></sec></sec><sec id="s5"><title>Funding</title><p>The work was funded by the NSFC (40471111, 70571076), the MOST (2006AA12Z215, 2007AA12Z241) and CAS (KZCX2-YW-308).</p></sec></body><back><ack><title>Acknowledgements</title><p>We would like to thank A. J. McMichael, N. Becker and K. Glass for their help in this research, and the Beijing Center for Disease Control and E. Zhong and D. Zhuang for preparing the SARS information and relevant data.</p></ack><app-group><app><title>Appendix 1: notation</title><sec id="s6"><title></title><def-list><def-item><term><italic>s</italic></term><def><p>spatial location.</p></def></def-item><def-item><term><italic>t</italic></term><def><p>time period.</p></def></def-item><def-item><term><italic>R</italic>(<italic>s</italic>, <italic>t</italic>)</term><def><p>daily (<italic>t</italic>) reported new cases of SARS in each of the 18 districts (<italic>s</italic>) of the city of Beijing (China).</p></def></def-item><def-item><term>Θ(<italic>t</italic>)</term><def><p>epidemic parameters.</p></def></def-item><def-item><term><italic>H</italic>(<italic>s</italic>)</term><def><p>residential locations of all 11 108 people who had been in close contact with infected persons.</p></def></def-item><def-item><term><italic>F</italic>(<italic>s</italic>)</term><def><p>determinants of SARS.</p></def></def-item><def-item><term><italic>P</italic></term><def><p>population counts in 246 census units.</p></def></def-item><def-item><term>Hosp</term><def><p>location of hospitals.</p></def></def-item><def-item><term><italic>D</italic></term><def><p>number of doctors in a hospital.</p></def></def-item><def-item><term><italic>C</italic>(<italic>s</italic>), <italic>C</italic>(<italic>t</italic>) and <italic>C</italic></term><def><p>spatial pattern of risk exposure, temporal change in spatial clustering of cases, and spatial pattern in general, respectively.</p></def></def-item><def-item><term><italic>T</italic>(<italic>s</italic>), <italic>T</italic>(<italic>t</italic>) and <italic>T</italic></term><def><p>main traffic lines with spatial location, traffic factor of SARS transmission in time, and traffic in general, respectively.</p></def></def-item><def-item><term>(<italic>C</italic>, <italic>R</italic>, Θ)(<italic>t</italic>)</term><def><p>triplet of entities within the bracket change with time.</p></def></def-item><def-item><term><italic>A</italic>&B</term><def><p>entities <italic>A</italic> and <italic>B</italic>.</p></def></def-item><def-item><term><italic>A</italic> ∼ <italic>B</italic></term><def><p>entities <italic>A</italic> and <italic>B</italic> are connected.</p></def></def-item><def-item><term><italic>A</italic> ∼ <italic>m</italic> ∼ <italic>B</italic></term><def><p><italic>A</italic> is linked to <italic>B</italic> via <italic>m.</italic></p></def></def-item><def-item><term><italic>A</italic>→<italic>B</italic></term><def><p><italic>B</italic> is inferred from <italic>A</italic>.</p></def></def-item></def-list></sec><sec id="s7"><title>Appendix 2: the SEIR model</title><p>On the basis of the discussion in Kermack and McKendrick<sup><xref ref-type="bibr" rid="FDN023C7">7</xref></sup> and Anderson,<sup><xref ref-type="bibr" rid="FDN023C25">25</xref></sup> the following SEIR model has been proposed.
<disp-formula id="FDN023M8"><label>8</label><graphic xlink:href="fdn02313.jpg" position="float"></graphic></disp-formula>where <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) are the number of exposed, infectious and removed individuals at time <italic>t</italic>, ℓ(<italic>t</italic>) average number of contacts per infectious person (depends on time <italic>t</italic> because the control effort changes with <italic>t</italic>), <italic>G</italic> the rate at which exposed (latent) individuals become infectious. <italic>A</italic> the rate at which infectious individuals are removed (recover or are isolated), <italic>R</italic><sub>0</sub> =ℓ(0)/<italic>a</italic> ≈ (<italic>b</italic> + <italic>c</italic>)/<italic>a</italic> the basic reproduction number for this model, <italic>R</italic><sub>0</sub> ≈ <italic>b</italic>/<italic>a</italic> approximated eventual reproduction number, 1/<italic>g</italic> average incubation period, 1/<italic>g</italic> average infection period and <italic>a</italic> = 0.252, <italic>b</italic> = 0.008, <italic>c</italic> = 0.588, <italic>d</italic> = 0.368, <italic>e</italic> = 54 and <italic>g</italic> = 0.200.</p></sec><sec id="s8"><title>Appendix 3: nearest-neighbour hierarchical clustering</title><p>Our datasets included the geographical locations of the residences of the 11 108 identified contacts. In the absence of clustering, the mean distance of an identified contact from its nearest neighbour and the standard error of this mean are given by, respectively,
<disp-formula id="FDN023M9"><label>9</label><graphic xlink:href="fdn02314.jpg" position="float"></graphic></disp-formula>where <italic>A</italic> is the area of the region and <italic>N</italic> is the number of identified contacts in that region. We used the CrimeStat software package,<sup><xref ref-type="bibr" rid="FDN023C37">37</xref></sup> which defines a threshold distance
<disp-formula id="FDN023M10"><label>10</label><graphic xlink:href="fdn02315.jpg" position="float"></graphic></disp-formula>and identifies all points having neighbours within this threshold. The software selects first-order clusters sequentially, beginning with the point having the largest number of neighbours. The second-order clusters were obtained by calculating the centre of each first-order cluster, and then repeating the above procedure on these centre points. The first-order clusters included at least eight points and the second-order clusters included at least four points.</p></sec><sec id="s9"><title>Appendix 4: Moran's <italic>I</italic> coefficient</title><p>To avoid confusion with Panel 1, we denoted Moran's <italic>I</italic> coefficient<sup><xref ref-type="bibr" rid="FDN023C35">35</xref></sup> as
<disp-formula id="FDN023M11"><label>11</label><graphic xlink:href="fdn02316.jpg" position="float"></graphic></disp-formula>where</p><def-list><def-item><term><italic>x</italic><sub><italic>i</italic></sub>(t)</term><def><p>= number of new SARS cases observed in district <italic>i</italic> on day <italic>t</italic>,</p></def></def-item><def-item><term><italic>x̄</italic>(<italic>t</italic>)</term><def><p>= mean number of new SARS cases per district on day <italic>t</italic>,</p></def></def-item></def-list><disp-formula><graphic xlink:href="fdn02317.jpg" position="float"></graphic></disp-formula><p><italic>N</italic> = number of districts.</p></sec><sec id="s10"><title>Appendix 5: BW join-count statistic</title><p>We started with the proposed disease dispersal network, which consisted of connections (or ‘joins’) between districts. Each day a district was coded black (B) if it reported a SARS case on that day; otherwise it was coded white (W). Every joint in the network linked two B districts (BB), two W districts (WW) or a B and a W district (BW). The observed number of BW joins was compared with the expected number, and a standard normal deviation (z-scores) was used to test the significance.<sup><xref ref-type="bibr" rid="FDN023C39">39</xref>,<xref ref-type="bibr" rid="FDN023C40">40</xref></sup> High negative values of the statistics indicated evidence of clustering of cases on the network and high positive values indicated evidence of spacing. The statistics are as follows:
<disp-formula id="FDN023M12"><label>12</label><graphic xlink:href="fdn02318.jpg" position="float"></graphic></disp-formula><disp-formula id="FDN023M13"><label>13</label><graphic xlink:href="fdn02319.jpg" position="float"></graphic></disp-formula><disp-formula id="FDN023M14"><label>14</label><graphic xlink:href="fdn02320.jpg" position="float"></graphic></disp-formula><disp-formula id="FDN023M15"><label>15</label><graphic xlink:href="fdn02321.jpg" position="float"></graphic></disp-formula><disp-formula id="FDN023M16"><label>16</label><graphic xlink:href="fdn02322.jpg" position="float"></graphic></disp-formula></p><def-list><def-item><term><italic>N</italic><sub>BW</sub></term><def><p>number of BW joins,</p></def></def-item><def-item><term><italic>x</italic><sub><italic>i</italic></sub></term><def><p>=
<disp-formula><graphic xlink:href="fdn02323.jpg" position="float"></graphic></disp-formula></p></def></def-item><def-item><term><italic>w</italic><sub><italic>ij</italic></sub></term><def><p>=
<disp-formula><graphic xlink:href="fdn02324.jpg" position="float"></graphic></disp-formula></p></def></def-item><def-item><term><italic>w</italic><sub><italic>i.</italic></sub></term><def><p>average of <italic>w</italic><sub><italic>ij</italic></sub> over the second subscript,</p></def></def-item><def-item><term><italic>N</italic></term><def><p>total number of districts,</p></def></def-item><def-item><term><italic>N</italic><sub><italic>B</italic></sub>, <italic>N</italic><sub><italic>W</italic></sub></term><def><p>numbers of BB and WW joins, respectively.</p></def></def-item></def-list></sec></app></app-group><ref-list><title>References</title><ref id="FDN023C1"><label>1</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Satija</surname><given-names>N</given-names></name><name><surname>Lal</surname><given-names>SK</given-names></name></person-group><article-title>The molecular biology of SARS coronavirus</article-title><source>Ann NY Acad 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