CINXE.COM

Document Zbl 1506.62262 - zbMATH Open

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1506.62262 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">&times;</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools &amp; Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1506.62262?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open &mdash; the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span>&nbsp;<a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1506.62262" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span>&nbsp;</span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>&nbsp;&nbsp;</span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>&nbsp;&nbsp;</span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>&nbsp;&nbsp;</span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>&nbsp;&nbsp;</span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>&nbsp;&nbsp;</span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>&nbsp;&nbsp;</span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>&nbsp;&nbsp;</span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>&nbsp;&nbsp;</span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>&nbsp;&nbsp;</span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>&nbsp;&nbsp;</span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>&nbsp;&nbsp;</span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>&nbsp;&nbsp;</span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>&nbsp;&nbsp;</span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>&nbsp;&nbsp;</span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>&nbsp;&nbsp;</span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>&nbsp;&nbsp;</span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>&nbsp;&nbsp;</span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a&nbsp;<span class="item">&</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical and (default)</a></li> <li><a href="#"><span class="token">a&nbsp;<span class="item">|</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab&nbsp;&nbsp;&nbsp;</span><span>&nbsp;</span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>&nbsp;&nbsp;</span><span>&nbsp;</span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab&nbsp;c<span class="item">"</span></span><span>&nbsp;</span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab&nbsp;c<span class="item">)</span></span><span>&nbsp;</span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki &amp; ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &amp;</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b &amp; au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&amp;</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/bu.fan" title="Author Profile">Bu, Fan</a>; <a href="/authors/aiello.allison-e" title="Author Profile">Aiello, Allison E.</a>; <a href="/authors/xu.jason" title="Author Profile">Xu, Jason</a>; <a href="/authors/volfovsky.alexander" title="Author Profile">Volfovsky, Alexander</a></div> <h2 class="title"> <strong>Likelihood-based inference for partially observed epidemics on dynamic networks.</strong> <i>(English)</i> <a class="label nowrap" href="/1506.62262">Zbl 1506.62262</a> </h2> <div class="source"> <a href="/serials/474" title="Journal Profile">J. Am. Stat. Assoc.</a> <a href="/?q=in%3A493989" title="Articles in this Issue">117, No. 537, 510-526 (2022)</a>. </div> <div class="abstract">Summary: We propose a generative model and an inference scheme for epidemic processes on dynamic, adaptive contact networks. Network evolution is formulated as a link-Markovian process, which is then coupled to an individual-level stochastic susceptible-infectious-recovered model, to describe the interplay between the dynamics of the disease spread and the contact network underlying the epidemic. A Markov chain Monte Carlo framework is developed for likelihood-based inference from partial epidemic observations, with a novel data augmentation algorithm specifically designed to deal with missing individual recovery times under the dynamic network setting. Through a series of simulation experiments, we demonstrate the validity and flexibility of the model as well as the efficacy and efficiency of the data augmentation inference scheme. The model is also applied to a recent real-world dataset on influenza-like-illness transmission with high-resolution social contact tracking records.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7661398">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62F15" title="MSC2020">62F15</a> </td> <td class="space"> Bayesian inference </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62D10" title="MSC2020">62D10</a> </td> <td class="space"> Missing data </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62-08" title="MSC2020">62-08</a> </td> <td class="space"> Computational methods for problems pertaining to statistics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62P10" title="MSC2020">62P10</a> </td> <td class="space"> Applications of statistics to biology and medical sciences; meta analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D30" title="MSC2020">92D30</a> </td> <td class="space"> Epidemiology </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABayesian+data+augmentation">Bayesian data augmentation</a>; <a href="/?q=ut%3Aconditional+simulation">conditional simulation</a>; <a href="/?q=ut%3Acontact+networks">contact networks</a>; <a href="/?q=ut%3Acontinuous-time+Markov+chains">continuous-time Markov chains</a>; <a href="/?q=ut%3Amobile+healthcare">mobile healthcare</a>; <a href="/?q=ut%3Astochastic+susceptible-infectious-removed+model">stochastic susceptible-infectious-removed model</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1506.62262" data-ciurl="/ci/07661398" data-biburl="/bibtex/07661398.bib" data-amsurl="/amsrefs/07661398.bib" data-xmlurl="/xml/07661398.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07661398.pdf" title="Zbl 1506.62262 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1080/01621459.2020.1790376" aria-label="DOI for “Likelihood-based inference for partially observed epidemics on dynamic networks”" title="10.1080/01621459.2020.1790376">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1910.04221"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Aiello, A. E.; Simanek, A. M.; Eisenberg, M. C.; Walsh, A. R.; Davis, B.; Volz, E.; Cheng, C.; Rainey, J. J.; Uzicanin, A.; Gao, H.; Osgood, N., &ldquo;Design and Methods of a Social Network Isolation Study for Reducing Respiratory Infection Transmission: The eX-FLU Cluster Randomized Trial, Epidemics, 15, 38-55 (2016)&nbsp;&middot; <a href="https://doi.org/10.1016/j.epidem.2016.01.001" class="nowrap">doi:10.1016/j.epidem.2016.01.001</a></td> </tr><tr> <td>[2]</td> <td class="space">Anderson, R. M.; May, R. M., Infectious Diseases of Humans: Dynamics and Control (1992), Oxford: Oxford University Press, Oxford</td> </tr><tr> <td>[3]</td> <td class="space">Andrieu, C.; Doucet, A.; Holenstein, R., &ldquo;Particle Markov Chain Monte Carlo Methods, Journal of the Royal Statistical Society, Series B, 72, 269-342 (2010) &middot; <a href="/1411.65020" class="nowrap">Zbl 1411.65020</a>&nbsp;&middot; <a href="https://doi.org/10.1111/j.1467-9868.2009.00736.x" class="nowrap">doi:10.1111/j.1467-9868.2009.00736.x</a></td> </tr><tr> <td>[4]</td> <td class="space">Auranen, K.; Arjas, E.; Leino, T.; Takala, A. K., &ldquo;Transmission of Pneumococcal Carriage in Families: A Latent Markov Process Model for Binary Longitudinal Data, Journal of the American Statistical Association, 95, 1044-1053 (2000) &middot; <a href="/1008.62106" class="nowrap">Zbl 1008.62106</a>&nbsp;&middot; <a href="https://doi.org/10.1080/01621459.2000.10474301" class="nowrap">doi:10.1080/01621459.2000.10474301</a></td> </tr><tr> <td>[5]</td> <td class="space">Bailey, N. T., The Mathematical Theory of Infectious Diseases and Its Applications (1975), Bucks: Charles Griffin &amp;amp; Company Ltd, Bucks &middot; <a href="/0334.92024" class="nowrap">Zbl 0334.92024</a></td> </tr><tr> <td>[6]</td> <td class="space">Barrat, A.; Cattuto, C.; Tozzi, A. E.; Vanhems, P.; Voirin, N., &ldquo;Measuring Contact Patterns With Wearable Sensors: Methods, Data Characteristics and Applications to Data-Driven Simulations of Infectious Diseases, Clinical Microbiology and Infection, 20, 10-16 (2014)&nbsp;&middot; <a href="https://doi.org/10.1111/1469-0691.12472" class="nowrap">doi:10.1111/1469-0691.12472</a></td> </tr><tr> <td>[7]</td> <td class="space">Becker, N. G.; Britton, T., &ldquo;Statistical Studies of Infectious Disease Incidence, Journal of the Royal Statistical Society, Series B, 61, 287-307 (1999) &middot; <a href="/0913.62102" class="nowrap">Zbl 0913.62102</a>&nbsp;&middot; <a href="https://doi.org/10.1111/1467-9868.00177" class="nowrap">doi:10.1111/1467-9868.00177</a></td> </tr><tr> <td>[8]</td> <td class="space">Bell, D.; Nicoll, A.; Fukuda, K.; Horby, P.; Monto, A., &ldquo;World Health Organization Writing Group. Non-Pharmaceutical Interventions for Pandemic Influenza, National and Community Measures,&rdquo;, Emerging Infectious Diseases, 12, 88-94 (2006)</td> </tr><tr> <td>[9]</td> <td class="space">Britton, T., &ldquo;Stochastic Epidemic Models: A Survey, Mathematical Biosciences, 225, 24-35 (2010) &middot; <a href="/1188.92031" class="nowrap">Zbl 1188.92031</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.mbs.2010.01.006" class="nowrap">doi:10.1016/j.mbs.2010.01.006</a></td> </tr><tr> <td>[10]</td> <td class="space">Britton, T., &ldquo;Epidemic Models on Social Networks—With Inference,&rdquo;, Statistica Neerlandica (2020) &middot; <a href="/1541.92079" class="nowrap">Zbl 1541.92079</a></td> </tr><tr> <td>[11]</td> <td class="space">Britton, T.; O’Neill, P. D., &ldquo;Bayesian Inference for Stochastic Epidemics in Populations With Random Social Structure, Scandinavian Journal of Statistics, 29, 375-390 (2002) &middot; <a href="/1036.62017" class="nowrap">Zbl 1036.62017</a>&nbsp;&middot; <a href="https://doi.org/10.1111/1467-9469.00296" class="nowrap">doi:10.1111/1467-9469.00296</a></td> </tr><tr> <td>[12]</td> <td class="space">Cauchemez, S.; Ferguson, N. M., &ldquo;Likelihood-Based Estimation of Continuous-Time Epidemic Models From Time-Series Data: Application to Measles Transmission in London, Journal of the Royal Society Interface, 5, 885-897 (2008)&nbsp;&middot; <a href="https://doi.org/10.1098/rsif.2007.1292" class="nowrap">doi:10.1098/rsif.2007.1292</a></td> </tr><tr> <td>[13]</td> <td class="space">Cauchemez, S.; Temime, L.; Valleron, A.-J.; Varon, E.; Thomas, G.; Guillemot, D.; Boëlle, P.-Y., &ldquo;S. pneumonia Transmission According to Inclusion in Conjugate Vaccines: Bayesian Analysis of a Longitudinal Follow-Up in schools, BMC Infectious Diseases, 6, 14 (2006)&nbsp;&middot; <a href="https://doi.org/10.1186/1471-2334-6-14" class="nowrap">doi:10.1186/1471-2334-6-14</a></td> </tr><tr> <td>[14]</td> <td class="space">Clementi, A. E.; Macci, C.; Monti, A.; Pasquale, F.; Silvestri, R., &ldquo;Flooding Time of Edge-Markovian Evolving Graphs, SIAM Journal on Discrete Mathematics, 24, 1694-1712 (2010) &middot; <a href="/1221.68170" class="nowrap">Zbl 1221.68170</a>&nbsp;&middot; <a href="https://doi.org/10.1137/090756053" class="nowrap">doi:10.1137/090756053</a></td> </tr><tr> <td>[15]</td> <td class="space">Cui, J.; Zhang, Y.; Feng, Z., &ldquo;Influence of Non-Homogeneous Mixing on Final Epidemic Size in a Meta-Population Model, Journal of Biological Dynamics, 13, 31-46 (2019) &middot; <a href="/1447.92410" class="nowrap">Zbl 1447.92410</a>&nbsp;&middot; <a href="https://doi.org/10.1080/17513758.2018.1484186" class="nowrap">doi:10.1080/17513758.2018.1484186</a></td> </tr><tr> <td>[16]</td> <td class="space">Dong, W., Pentland, A., and Heller, K. A. (2012), &ldquo;Graph-Coupled HMMs for Modeling the Spread of Infection,&rdquo; arXiv no. 1210.4864.</td> </tr><tr> <td>[17]</td> <td class="space">Eames, K.; Tilston, N.; White, P.; Adams, E.; Edmunds, W., &ldquo;The Impact of Illness and the Impact of School Closure on Social Contact Patterns, Health Technology Assessment, 14, 267-312 (2010)&nbsp;&middot; <a href="https://doi.org/10.3310/hta14340-04" class="nowrap">doi:10.3310/hta14340-04</a></td> </tr><tr> <td>[18]</td> <td class="space">Edmunds, W. J.; Kafatos, G.; Wallinga, J.; Mossong, J., &ldquo;Mixing Patterns and the Spread of Close-Contact Infectious Diseases, Emerging Themes in Epidemiology, 3, 10 (2006)&nbsp;&middot; <a href="https://doi.org/10.1186/1742-7622-3-10" class="nowrap">doi:10.1186/1742-7622-3-10</a></td> </tr><tr> <td>[19]</td> <td class="space">Edmunds, W. J., O’Callaghan, C., and Nokes, D. (1997), &ldquo;Who Mixes With Whom? A Method to Determine the Contact Patterns of Adults That May Lead to the Spread of Airborne Infections,&rdquo; Proceedings of the Royal Society B: Biological Sciences, 264, 949-957. DOI: .</td> </tr><tr> <td>[20]</td> <td class="space">Fan, K.; Eisenberg, M.; Walsh, A.; Aiello, A.; Heller, K., Hierarchical Graph-Coupled HMMs for Heterogeneous Personalized Health Data, the 21th ACM SIGKDD International Conference (2015)&nbsp;&middot; <a href="https://doi.org/10.1145/2783258.2783326" class="nowrap">doi:10.1145/2783258.2783326</a></td> </tr><tr> <td>[21]</td> <td class="space">Fan, K., Li, C., and Heller, K. (2016), &ldquo;A Unifying Variational Inference Framework for Hierarchical Graph-Coupled HMM With an Application to Influenza Infection,&rdquo; in Thirtieth AAAI Conference on Artificial Intelligence.</td> </tr><tr> <td>[22]</td> <td class="space">Ferguson, N. M.; Laydon, D.; Nedjati-Gilani, G.; Imai, N.; Ainslie, K.; Baguelin, M.; Bhatia, S.; Boonyasiri, A.; Cucunubá, Z.; Cuomo-Dannenburg, G.; Dighe, A., &ldquo;Impact of Non-Pharmaceutical Interventions (NPIs) to Reduce COVID19 Mortality and Healthcare Demand,&rdquo;, Imperial College COVID-19 Response Team, 10, 77482 (2020)</td> </tr><tr> <td>[23]</td> <td class="space">Finkenstädt, B. F.; Grenfell, B. T., &ldquo;Time Series Modelling of Childhood Diseases: A Dynamical Systems Approach, Journal of the Royal Statistical Society, Series C, 49, 187-205 (2000) &middot; <a href="/0944.62100" class="nowrap">Zbl 0944.62100</a>&nbsp;&middot; <a href="https://doi.org/10.1111/1467-9876.00187" class="nowrap">doi:10.1111/1467-9876.00187</a></td> </tr><tr> <td>[24]</td> <td class="space">Fintzi, J.; Cui, X.; Wakefield, J.; Minin, V. N., &ldquo;Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data, Journal of Computational and Graphical Statistics, 26, 918-929 (2017)&nbsp;&middot; <a href="https://doi.org/10.1080/10618600.2017.1328365" class="nowrap">doi:10.1080/10618600.2017.1328365</a></td> </tr><tr> <td>[25]</td> <td class="space">Funk, S.; Salathé, M.; Jansen, V. A., &ldquo;Modelling the Influence of Human Behaviour on the Spread of Infectious Diseases: A Review, Journal of the Royal Society Interface, 7, 1247-1256 (2010)&nbsp;&middot; <a href="https://doi.org/10.1098/rsif.2010.0142" class="nowrap">doi:10.1098/rsif.2010.0142</a></td> </tr><tr> <td>[26]</td> <td class="space">Geweke, J., Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments, 196 (1991), Minneapolis, MN: Federal Reserve Bank of Minneapolis, Research Department, Minneapolis, MN</td> </tr><tr> <td>[27]</td> <td class="space">Gibson, G. J.; Renshaw, E., &ldquo;Estimating Parameters in Stochastic Compartmental Models Using Markov Chain Methods, Mathematical Medicine and Biology, 15, 19-40 (1998) &middot; <a href="/0916.92024" class="nowrap">Zbl 0916.92024</a>&nbsp;&middot; <a href="https://doi.org/10.1093/imammb/15.1.19" class="nowrap">doi:10.1093/imammb/15.1.19</a></td> </tr><tr> <td>[28]</td> <td class="space">Gillespie, D. T., &ldquo;A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, Journal of Computational Physics, 22, 403-434 (1976)&nbsp;&middot; <a href="https://doi.org/10.1016/0021-9991(76)90041-3" class="nowrap">doi:10.1016/0021-9991(76)90041-3</a></td> </tr><tr> <td>[29]</td> <td class="space">Guttorp, P.; Minin, V. N., Stochastic Modeling of Scientific Data (2018), Boca Raton, FL: Chapman and Hall/CRC, Boca Raton, FL</td> </tr><tr> <td>[30]</td> <td class="space">He, D.; Ionides, E. L.; King, A. A., &ldquo;Plug-and-Play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study, Journal of the Royal Society Interface, 7, 271-283 (2010)&nbsp;&middot; <a href="https://doi.org/10.1098/rsif.2009.0151" class="nowrap">doi:10.1098/rsif.2009.0151</a></td> </tr><tr> <td>[31]</td> <td class="space">Hethcote, H. W., &ldquo;The Mathematics of Infectious Diseases, SIAM Review, 42, 599-653 (2000) &middot; <a href="/0993.92033" class="nowrap">Zbl 0993.92033</a>&nbsp;&middot; <a href="https://doi.org/10.1137/S0036144500371907" class="nowrap">doi:10.1137/S0036144500371907</a></td> </tr><tr> <td>[32]</td> <td class="space">Ho, L. S. T.; Crawford, F. W.; Suchard, M. A., &ldquo;Direct Likelihood-Based Inference for Discretely Observed Stochastic Compartmental Models of Infectious Disease, The Annals of Applied Statistics, 12, 1993-2021 (2018) &middot; <a href="/1405.62177" class="nowrap">Zbl 1405.62177</a>&nbsp;&middot; <a href="https://doi.org/10.1214/18-AOAS1141" class="nowrap">doi:10.1214/18-AOAS1141</a></td> </tr><tr> <td>[33]</td> <td class="space">Ho, L. S. T.; Xu, J.; Crawford, F. W.; Minin, V. N.; Suchard, M. A., &ldquo;Birth/Birth-Death Processes and Their Computable Transition Probabilities With Biological Applications, Journal of Mathematical Biology, 76, 911-944 (2018) &middot; <a href="/1384.60094" class="nowrap">Zbl 1384.60094</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s00285-017-1160-3" class="nowrap">doi:10.1007/s00285-017-1160-3</a></td> </tr><tr> <td>[34]</td> <td class="space">Hobolth, A.; Stone, E. A., &ldquo;Simulation From Endpoint-Conditioned, Continuous-Time Markov Chains on a Finite State Space, With Applications to Molecular Evolution, The Annals of Applied Statistics, 3, 1204 (2009) &middot; <a href="/1196.62139" class="nowrap">Zbl 1196.62139</a>&nbsp;&middot; <a href="https://doi.org/10.1214/09-AOAS247" class="nowrap">doi:10.1214/09-AOAS247</a></td> </tr><tr> <td>[35]</td> <td class="space">Höhle, M.; Jørgensen, E., Estimating Parameters for Stochastic Epidemics (2002), [The Royal Veterinary and Agricultural University]: [The Royal Veterinary and Agricultural University], Dina</td> </tr><tr> <td>[36]</td> <td class="space">Hoti, F.; Erästö, P.; Leino, T.; Auranen, K., &ldquo;Outbreaks of Streptococcus pneumonia Carriage in Day Care Cohorts in Finland—Implications for Elimination of Transmission, BMC Infectious Diseases, 9, 102 (2009)&nbsp;&middot; <a href="https://doi.org/10.1186/1471-2334-9-102" class="nowrap">doi:10.1186/1471-2334-9-102</a></td> </tr><tr> <td>[37]</td> <td class="space">Ionides, E. L.; Nguyen, D.; Atchadé, Y.; Stoev, S.; King, A. A., &ldquo;Inference for Dynamic and Latent Variable Models via Iterated, Perturbed Bayes Maps, Proceedings of the National Academy of Sciences of the United States of America, 112, 719-724 (2015) &middot; <a href="/1359.62345" class="nowrap">Zbl 1359.62345</a>&nbsp;&middot; <a href="https://doi.org/10.1073/pnas.1410597112" class="nowrap">doi:10.1073/pnas.1410597112</a></td> </tr><tr> <td>[38]</td> <td class="space">Kermack, W. O.; McKendrick, A. G., &ldquo;A Contribution to the Mathematical Theory of Epidemics,&rdquo; (1927) &middot; <a href="/53.0517.01" class="nowrap">JFM 53.0517.01</a></td> </tr><tr> <td>[39]</td> <td class="space">Kiss, I. Z.; Berthouze, L.; Taylor, T. J.; Simon, P. L., &ldquo;Modelling Approaches for Simple Dynamic Networks and Applications to Disease Transmission Models, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 1332-1355 (2012) &middot; <a href="/1365.92124" class="nowrap">Zbl 1365.92124</a>&nbsp;&middot; <a href="https://doi.org/10.1098/rspa.2011.0349" class="nowrap">doi:10.1098/rspa.2011.0349</a></td> </tr><tr> <td>[40]</td> <td class="space">Kiti, M. C.; Tizzoni, M.; Kinyanjui, T. M.; Koech, D. C.; Munywoki, P. K.; Meriac, M.; Cappa, L.; Panisson, A.; Barrat, A.; Cattuto, C.; Nokes, D. J., &ldquo;Quantifying Social Contacts in a Household Setting of Rural Kenya Using Wearable Proximity Sensors, EPJ Data Science, 5, 21 (2016)&nbsp;&middot; <a href="https://doi.org/10.1140/epjds/s13688-016-0084-2" class="nowrap">doi:10.1140/epjds/s13688-016-0084-2</a></td> </tr><tr> <td>[42]</td> <td class="space">Masuda, N.; Holme, P., Temporal Network Epidemiology (2017), Singapore: Springer, Singapore &middot; <a href="/1383.92006" class="nowrap">Zbl 1383.92006</a></td> </tr><tr> <td>[43]</td> <td class="space">Melegaro, A.; Jit, M.; Gay, N.; Zagheni, E.; Edmunds, W. J., &ldquo;What Types of Contacts Are Important for the Spread of Infections? Using Contact Survey Data to Explore European Mixing Patterns, Epidemics, 3, 143-151 (2011)&nbsp;&middot; <a href="https://doi.org/10.1016/j.epidem.2011.04.001" class="nowrap">doi:10.1016/j.epidem.2011.04.001</a></td> </tr><tr> <td>[45]</td> <td class="space">Mossong, J.; Hens, N.; Jit, M.; Beutels, P.; Auranen, K.; Mikolajczyk, R.; Massari, M.; Salmaso, S.; Tomba, G. S.; Wallinga, J.; Heijne, J., &ldquo;Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases, PLoS Medicine, 5, e74 (2008)&nbsp;&middot; <a href="https://doi.org/10.1371/journal.pmed.0050074" class="nowrap">doi:10.1371/journal.pmed.0050074</a></td> </tr><tr> <td>[46]</td> <td class="space">Neal, P. J.; Roberts, G. O., &ldquo;Statistical Inference and Model Selection for the 1861 Hagelloch Measles Epidemic, Biostatistics, 5, 249-261 (2004) &middot; <a href="/1096.62123" class="nowrap">Zbl 1096.62123</a>&nbsp;&middot; <a href="https://doi.org/10.1093/biostatistics/5.2.249" class="nowrap">doi:10.1093/biostatistics/5.2.249</a></td> </tr><tr> <td>[47]</td> <td class="space">Ogura, M.; Preciado, V. M., &ldquo;Stability of Spreading Processes Over Time-Varying Large-Scale Networks, IEEE Transactions on Network Science and Engineering, 3, 44-57 (2016)&nbsp;&middot; <a href="https://doi.org/10.1109/TNSE.2016.2516346" class="nowrap">doi:10.1109/TNSE.2016.2516346</a></td> </tr><tr> <td>[48]</td> <td class="space">Ogura, M.; Preciado, V. M.; Masuda, N.; Holme, P., Temporal Network Epidemiology, Optimal Containment of Epidemics in Temporal and Adaptive Networks, 241-266 (2017), Singapore: Springer, Singapore &middot; <a href="/1401.92198" class="nowrap">Zbl 1401.92198</a></td> </tr><tr> <td>[49]</td> <td class="space">O’Neill, P. D., &ldquo;Bayesian Inference for Stochastic Multitype Epidemics in Structured Populations Using Sample Data, Biostatistics, 10, 779-791 (2009) &middot; <a href="/1437.62566" class="nowrap">Zbl 1437.62566</a></td> </tr><tr> <td>[50]</td> <td class="space">Ozella, L.; Gesualdo, F.; Tizzoni, M.; Rizzo, C.; Pandolfi, E.; Campagna, I.; Tozzi, A. E.; Cattuto, C., &ldquo;Close Encounters Between Infants and Household Members Measured Through Wearable Proximity Sensors, PLoS One, 13, e0198733 (2018)&nbsp;&middot; <a href="https://doi.org/10.1371/journal.pone.0198733" class="nowrap">doi:10.1371/journal.pone.0198733</a></td> </tr><tr> <td>[51]</td> <td class="space">Pooley, C.; Bishop, S.; Marion, G., &ldquo;Using Model-Based Proposals for Fast Parameter Inference on Discrete State Space, Continuous-Time Markov Processes, Journal of the Royal Society Interface, 12, 20150225 (2015)&nbsp;&middot; <a href="https://doi.org/10.1098/rsif.2015.0225" class="nowrap">doi:10.1098/rsif.2015.0225</a></td> </tr><tr> <td>[52]</td> <td class="space">Shaw, L. B.; Schwartz, I. B., &ldquo;Fluctuating Epidemics on Adaptive Networks, Physical Review E, 77, 066101 (2008)&nbsp;&middot; <a href="https://doi.org/10.1103/PhysRevE.77.066101" class="nowrap">doi:10.1103/PhysRevE.77.066101</a></td> </tr><tr> <td>[53]</td> <td class="space">Tavaré, S., &ldquo;The Linear Birth-Death Process: An Inferential Retrospective, Advances in Applied Probability, 50, 253-269 (2018) &middot; <a href="/1436.62100" class="nowrap">Zbl 1436.62100</a>&nbsp;&middot; <a href="https://doi.org/10.1017/apr.2018.84" class="nowrap">doi:10.1017/apr.2018.84</a></td> </tr><tr> <td>[54]</td> <td class="space">Tsang, T. K.; Fang, V. J.; Ip, D. K. M.; Perera, R. A. P. M.; So, H. C.; Leung, G. M.; Peiris, J. S. M.; Cowling, B. J.; Cauchemez, S., &ldquo;Indirect Protection From Vaccinating Children Against Influenza in Households, Nature Communications, 10, 106 (2019)&nbsp;&middot; <a href="https://doi.org/10.1038/s41467-018-08036-6" class="nowrap">doi:10.1038/s41467-018-08036-6</a></td> </tr><tr> <td>[55]</td> <td class="space">Tunc, I.; Shkarayev, M. S.; Shaw, L. B., &ldquo;Epidemics in Adaptive Social Networks With Temporary Link Deactivation, Journal of Statistical Physics, 151, 355-366 (2013) &middot; <a href="/1329.92141" class="nowrap">Zbl 1329.92141</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s10955-012-0667-7" class="nowrap">doi:10.1007/s10955-012-0667-7</a></td> </tr><tr> <td>[56]</td> <td class="space">US Centers for Disease Control and Prevention (CDC) (2018)</td> </tr><tr> <td>[57]</td> <td class="space">Van Kerckhove, K.; Hens, N.; Edmunds, W. J.; Eames, K. T., &ldquo;The Impact of Illness on Social Networks: Implications for Transmission and Control of Influenza, American Journal of Epidemiology, 178, 1655-1662 (2013)&nbsp;&middot; <a href="https://doi.org/10.1093/aje/kwt196" class="nowrap">doi:10.1093/aje/kwt196</a></td> </tr><tr> <td>[58]</td> <td class="space">Van Segbroeck, S.; Santos, F. C.; Pacheco, J. M., &ldquo;Adaptive Contact Networks Change Effective Disease Infectiousness and Dynamics, PLoS Computational Biology, 6, e1000895 (2010)&nbsp;&middot; <a href="https://doi.org/10.1371/journal.pcbi.1000895" class="nowrap">doi:10.1371/journal.pcbi.1000895</a></td> </tr><tr> <td>[59]</td> <td class="space">Vanhems, P.; Barrat, A.; Cattuto, C.; Pinton, J.-F.; Khanafer, N.; Régis, C.; Kim, B.-A.; Comte, B.; Voirin, N., &ldquo;Estimating Potential Infection Transmission Routes in Hospital Wards Using Wearable Proximity Sensors, PLoS One, 8, e73970 (2013)&nbsp;&middot; <a href="https://doi.org/10.1371/journal.pone.0073970" class="nowrap">doi:10.1371/journal.pone.0073970</a></td> </tr><tr> <td>[60]</td> <td class="space">Voirin, N.; Payet, C.; Barrat, A.; Cattuto, C.; Khanafer, N.; Régis, C.; Kim, B.-A.; Comte, B.; Casalegno, J.-S.; Lina, B.; Vanhems, P., &ldquo;Combining High-Resolution Contact Data With Virological Data to Investigate Influenza Transmission in a Tertiary Care Hospital, Infection Control &amp; Hospital Epidemiology, 36, 254 (2015)&nbsp;&middot; <a href="https://doi.org/10.1017/ice.2014.53" class="nowrap">doi:10.1017/ice.2014.53</a></td> </tr><tr> <td>[61]</td> <td class="space">Volz, E., &ldquo;SIR Dynamics in Random Networks With Heterogeneous Connectivity, Journal of Mathematical Biology, 56, 293-310 (2008) &middot; <a href="/1143.92036" class="nowrap">Zbl 1143.92036</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s00285-007-0116-4" class="nowrap">doi:10.1007/s00285-007-0116-4</a></td> </tr><tr> <td>[62]</td> <td class="space">Volz, E., and Meyers, L. A. (2007), &ldquo;Susceptible-Infected-Recovered Epidemics in Dynamic Contact Networks,&rdquo; Proceedings of the Royal Society B: Biological Sciences, 274, 2925-2934. DOI: .</td> </tr><tr> <td>[63]</td> <td class="space">Volz, E.; Meyers, L. A., &ldquo;Epidemic Thresholds in Dynamic Contact Networks, Journal of the Royal Society Interface, 6, 233-241 (2008)</td> </tr><tr> <td>[64]</td> <td class="space">Wallinga, J.; Edmunds, W. J.; Kretzschmar, M., &ldquo;Perspective: Human Contact Patterns and the Spread of Airborne Infectious Diseases, TRENDS in Microbiology, 7, 372-377 (1999)&nbsp;&middot; <a href="https://doi.org/10.1016/S0966-842X(99)01546-2" class="nowrap">doi:10.1016/S0966-842X(99)01546-2</a></td> </tr><tr> <td>[65]</td> <td class="space">Who (2004)</td> </tr><tr> <td>[66]</td> <td class="space">Who (2020)</td> </tr><tr> <td>[67]</td> <td class="space">WHOW Group, &ldquo;Nonpharmaceutical Interventions for Pandemic Influenza, National and Community Measures, Emerging Infectious Diseases, 12, 88 (2006)</td> </tr><tr> <td>[68]</td> <td class="space">Xu, J.; Guttorp, P.; Kato-Maeda, M.; Minin, V. N., &ldquo;Likelihood-Based Inference for Discretely Observed Birth-Death-Shift Processes, With Applications to Evolution of Mobile Genetic Elements, Biometrics, 71, 1009-1021 (2015) &middot; <a href="/1419.62481" class="nowrap">Zbl 1419.62481</a>&nbsp;&middot; <a href="https://doi.org/10.1111/biom.12352" class="nowrap">doi:10.1111/biom.12352</a></td> </tr><tr> <td>[69]</td> <td class="space">Xu, J.; Minin, V. N., &ldquo;Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing,&rdquo;, 2015, 952 (2015)</td> </tr><tr> <td>[70]</td> <td class="space">Yang, C. H.; Jung, H., &ldquo;Topological Dynamics of the 2015 South Korea MERS-CoV Spread-on-Contact Networks, Scientific Reports, 10, 4327 (2020)&nbsp;&middot; <a href="https://doi.org/10.1038/s41598-020-61133-9" class="nowrap">doi:10.1038/s41598-020-61133-9</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH&nbsp;Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> &copy; 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms &amp; Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none">&nbsp;<span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10