Document Zbl 1550.92045

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1550.92045 - 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;">×</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 & 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="/?q=rf%3A7472025+ai%3Ahan.zhixia"> <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 — 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> <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="rf:7472025 ai:han.zhixia" 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> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </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 <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </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 & 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 &</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 & 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>&</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/zhao.jiandong" title="Author Profile">Zhao, Jiandong</a>; <a href="/authors/zhang.tonghua" title="Author Profile">Zhang, Tonghua</a>; <a href="/authors/han.zhixia" title="Author Profile">Han, Zhixia</a></div> <h2 class="title"> <strong>Dynamic behavior of a stochastic SIRS model with two viruses.</strong> <i>(English)</i> <a class="label nowrap" href="/1550.92045">Zbl 1550.92045</a> </h2> <div class="source"> <a href="/serials/2718" title="Journal Profile">Int. J. Nonlinear Sci. Numer. Simul.</a> <a href="/?q=in%3A478114" title="Articles in this Issue">22, No. 7-8, 809-825 (2021)</a>. </div> <div class="abstract">Summary: To study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D30" title="MSC2020">92D30</a> </td> <td class="space"> Epidemiology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34F05" title="MSC2020">34F05</a> </td> <td class="space"> Ordinary differential equations and systems with randomness </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H10" title="MSC2020">60H10</a> </td> <td class="space"> Stochastic ordinary differential equations (aspects of stochastic analysis) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adisease-free+equilibrium">disease-free equilibrium</a>; <a href="/?q=ut%3Amoment+average+in+time">moment average in time</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Astochastic+SIRS+model">stochastic SIRS model</a></div>  <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">×</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;">  </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">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1550.92045" data-ciurl="/ci/07486825" data-biburl="/bibtex/07486825.bib" data-amsurl="/amsrefs/07486825.bib" data-xmlurl="/xml/07486825.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07486825.pdf" title="Zbl 1550.92045 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/10.1515/ijnsns-2019-0208" aria-label="DOI for “Dynamic behavior of a stochastic SIRS model with two viruses”" title="10.1515/ijnsns-2019-0208">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">W. O. Kermazk and A. G. McKendrick, “A contribution to the mathematical theory of epidemic,” Proc. R. Soc. Lond. A, vol. 115, pp. 700-721, 1927. · <a href="/53.0517.01" class="nowrap">JFM 53.0517.01</a></td> </tr><tr> <td>[2]</td> <td class="space">H. W. Hethcote, “The mathematics of infectious disease,” SIAM Rev., vol. 42, no. 4, pp. 599-653, 2000,doi:10.1137/s0036144500371907. · <a href="/0993.92033" class="nowrap">Zbl 0993.92033</a> · <a href="https://doi.org/10.1137/s0036144500371907" class="nowrap">doi:10.1137/s0036144500371907</a></td> </tr><tr> <td>[3]</td> <td class="space">T. Zhang and Z. Teng, “Global behavior and permanence of SIRS epidemic model with time delay,” Nonlinear Anal. Real World Appl., vol. 9, pp. 1409-1424, 2008, doi:10.1016/j.nonrwa.2007.03.010. · <a href="/1154.34390" class="nowrap">Zbl 1154.34390</a> · <a href="https://doi.org/10.1016/j.nonrwa.2007.03.010" class="nowrap">doi:10.1016/j.nonrwa.2007.03.010</a></td> </tr><tr> <td>[4]</td> <td class="space">Z. Zhang, Y. Suo, J. Peng and W. Lin, “Singular perturbation approach to stability of a SIRS epidemic system,” Nonlinear Anal. Real World Appl., vol. 10, pp. 2688-2699, 2009, doi:10.1016/j.nonrwa.2008.07.009. · <a href="/1162.92039" class="nowrap">Zbl 1162.92039</a> · <a href="https://doi.org/10.1016/j.nonrwa.2008.07.009" class="nowrap">doi:10.1016/j.nonrwa.2008.07.009</a></td> </tr><tr> <td>[5]</td> <td class="space">R. Xu, Z. Ma and Z. Wang, “Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity,” Comput. Math. Appl., vol. 59, pp. 3211-3221, 2010, doi:10.1016/j.camwa.2010.03.009. · <a href="/1193.34115" class="nowrap">Zbl 1193.34115</a> · <a href="https://doi.org/10.1016/j.camwa.2010.03.009" class="nowrap">doi:10.1016/j.camwa.2010.03.009</a></td> </tr><tr> <td>[6]</td> <td class="space">Y. Muroya, Y. Enatsu and Y. Nakata, “Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays,” Nonlinear Anal. Real World Appl., vol. 12, pp. 1897-1910, 2011, doi:10.1016/j.nonrwa.2010.12.002. · <a href="/1225.93091" class="nowrap">Zbl 1225.93091</a> · <a href="https://doi.org/10.1016/j.nonrwa.2010.12.002" class="nowrap">doi:10.1016/j.nonrwa.2010.12.002</a></td> </tr><tr> <td>[7]</td> <td class="space">Y. Zhang and X.-Q. Zhao, “A reaction-diffusion Lyme disease model with seasonality,” SIAM J. Appl. Math., vol. 73, no. 6, pp. 2077-2099, 2013, doi:10.1137/120875454. · <a href="/1288.35299" class="nowrap">Zbl 1288.35299</a> · <a href="https://doi.org/10.1137/120875454" class="nowrap">doi:10.1137/120875454</a></td> </tr><tr> <td>[8]</td> <td class="space">X. Zhang and Y. Zhang, “Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity,” Discrete Continuous Dyn. Syst. - Ser. B (DCDS-B), vol. 23, no. 6, pp. 2625-2640, 2018, doi:10.3934/dcdsb.2018124. · <a href="/1404.35058" class="nowrap">Zbl 1404.35058</a> · <a href="https://doi.org/10.3934/dcdsb.2018124" class="nowrap">doi:10.3934/dcdsb.2018124</a></td> </tr><tr> <td>[9]</td> <td class="space">S. Bowong, A. Temgoua, Y. Malong and J. Mbang, “Mathematical study of a class of epidemiological models with multiple infectious stages,” Int. J. Nonlinear Sci. Numer. Stimul., 2019, doi:10.1515/ijnsns-2017-0244. · <a href="/1550.34119" class="nowrap">Zbl 1550.34119</a> · <a href="https://doi.org/10.1515/ijnsns-2017-0244" class="nowrap">doi:10.1515/ijnsns-2017-0244</a></td> </tr><tr> <td>[10]</td> <td class="space">H. Song and Y. Zhang, “Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways,” Electron. J. Qual. Theor. Differ. Equ., vol. 86, pp. 1-19, 2019, doi:10.14232/ejqtde.2019.1.86. · <a href="/1449.35080" class="nowrap">Zbl 1449.35080</a> · <a href="https://doi.org/10.14232/ejqtde.2019.1.86" class="nowrap">doi:10.14232/ejqtde.2019.1.86</a></td> </tr><tr> <td>[11]</td> <td class="space">Y. Xing, L. Zhang and X. Wang, “Modelling and stability of epidemic model with free-living pathogens growing in the environment,” J. Appl. Anal. Comput., vol. 10, no. 1, pp. 55-70, 2020, doi:10.11948/20180269. · <a href="/1460.92225" class="nowrap">Zbl 1460.92225</a> · <a href="https://doi.org/10.11948/20180269" class="nowrap">doi:10.11948/20180269</a></td> </tr><tr> <td>[12]</td> <td class="space">E. Beretta and V. Capassoin. “Global stability results for a multigroup SIR epidemic model,” in Mathematical Ecology, T. G. Hallam, L. J. Gross and S. A. Levin, Eds., New Jersey, World Scientific, 1988, pp. 317-340. · <a href="/0684.92015" class="nowrap">Zbl 0684.92015</a></td> </tr><tr> <td>[13]</td> <td class="space">W. Huang, K. L. Cooke and C. Castillo-Chavez, “Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission,” SIAM J. Appl. Math., vol. 52, pp. 835-854, 1992, doi:10.1137/0152047. · <a href="/0769.92023" class="nowrap">Zbl 0769.92023</a> · <a href="https://doi.org/10.1137/0152047" class="nowrap">doi:10.1137/0152047</a></td> </tr><tr> <td>[14]</td> <td class="space">R. Sun, “Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence,” Comput. Math. Appl., vol. 60, pp. 2286-2291, 2010, doi:10.1016/j.camwa.2010.08.020. · <a href="/1205.34066" class="nowrap">Zbl 1205.34066</a> · <a href="https://doi.org/10.1016/j.camwa.2010.08.020" class="nowrap">doi:10.1016/j.camwa.2010.08.020</a></td> </tr><tr> <td>[15]</td> <td class="space">R. Sun and J. Shi, “Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates,” Appl. Math. Comput., vol. 218, pp. 280-286, 2011, doi:10.1016/j.amc.2011.05.056. · <a href="/1223.92057" class="nowrap">Zbl 1223.92057</a> · <a href="https://doi.org/10.1016/j.amc.2011.05.056" class="nowrap">doi:10.1016/j.amc.2011.05.056</a></td> </tr><tr> <td>[16]</td> <td class="space">J. Zhao, L. Wang and Z. Han, “Stability analysis of two new SIRS models with two viruses,” Int. J. Comput. Math., vol. 95, no. 10, pp. 2026-2035, 2018, doi:10.1080/00207160.2017.1364369. · <a href="/1499.34291" class="nowrap">Zbl 1499.34291</a> · <a href="https://doi.org/10.1080/00207160.2017.1364369" class="nowrap">doi:10.1080/00207160.2017.1364369</a></td> </tr><tr> <td>[17]</td> <td class="space">X. Yu, S. Yuan and T. Zhang, “Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling,” Nonlinear Analysis: Hybrid Systems, vol. 34, pp. 209-225, 2019, doi:10.1016/j.nahs.2019.06.005. · <a href="/1435.34056" class="nowrap">Zbl 1435.34056</a> · <a href="https://doi.org/10.1016/j.nahs.2019.06.005" class="nowrap">doi:10.1016/j.nahs.2019.06.005</a></td> </tr><tr> <td>[18]</td> <td class="space">X. Yu, S. Yuan and T. Zhang, “Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment,” Appl. Math. Comput., vol. 347, pp. 249-264, 2019, doi:10.1016/j.amc.2018.11.005. · <a href="/1428.92097" class="nowrap">Zbl 1428.92097</a> · <a href="https://doi.org/10.1016/j.amc.2018.11.005" class="nowrap">doi:10.1016/j.amc.2018.11.005</a></td> </tr><tr> <td>[19]</td> <td class="space">W. Zhang, S. Zhao, X. Meng and T. Zhang, “Evolutionary analysis of adaptive dynamics model under variation of noise environment,” Appl. Math. Model., vol. 84, pp. 222-239, 2020, doi:10.1016/j.apm.2020.03.045. · <a href="/1481.92124" class="nowrap">Zbl 1481.92124</a> · <a href="https://doi.org/10.1016/j.apm.2020.03.045" class="nowrap">doi:10.1016/j.apm.2020.03.045</a></td> </tr><tr> <td>[20]</td> <td class="space">J. Xu, T. Zhang and K. Song, “A stochastic model of bacterial infection associated with neutrophils,” Appl. Math. Comput., vol. 373, p. 125025, 2020, doi:10.1016/j.amc.2019.125025. · <a href="/1433.92017" class="nowrap">Zbl 1433.92017</a> · <a href="https://doi.org/10.1016/j.amc.2019.125025" class="nowrap">doi:10.1016/j.amc.2019.125025</a></td> </tr><tr> <td>[21]</td> <td class="space">X. Mao, G. Marion and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stoch. Process. Their Appl., vol. 97, pp. 95-110, 2002, doi:10.1016/s0304-4149(01)00126-0. · <a href="/1058.60046" class="nowrap">Zbl 1058.60046</a> · <a href="https://doi.org/10.1016/s0304-4149(01)00126-0" class="nowrap">doi:10.1016/s0304-4149(01)00126-0</a></td> </tr><tr> <td>[22]</td> <td class="space">C. Zhu and G. Yin, “On hybrid competitive Lotka-Volterra ecosystems,” Nonlinear Anal, vol. 71, pp. e1370-e1379, 2009, 10.1016/j.na.2009.01.166. · <a href="/1238.34059" class="nowrap">Zbl 1238.34059</a></td> </tr><tr> <td>[23]</td> <td class="space">J. Yu, D. Jiang and N. Shi, “Global stability of two-group SIR model with random perturbation,” J. Math. Anal. Appl., vol. 360, pp. 235-244, 2009, doi:10.1016/j.jmaa.2009.06.050. · <a href="/1184.34064" class="nowrap">Zbl 1184.34064</a> · <a href="https://doi.org/10.1016/j.jmaa.2009.06.050" class="nowrap">doi:10.1016/j.jmaa.2009.06.050</a></td> </tr><tr> <td>[24]</td> <td class="space">C. Ji, D. Jiang and N. Shi, “Multigroup SIR epidemic model with stochastic perturbation,” Physica A, vol. 390, pp. 1747-1762, 2011, doi:10.1016/j.physa.2010.12.042. · <a href="https://doi.org/10.1016/j.physa.2010.12.042" class="nowrap">doi:10.1016/j.physa.2010.12.042</a></td> </tr><tr> <td>[25]</td> <td class="space">Z. Han and J. Zhao, “Stochastic SIRS model under regime switching,” Nonlinear Anal. Real World Appl., vol. 14, pp. 352-364, 2013, doi:10.1016/j.nonrwa.2012.06.008. · <a href="/1267.34079" class="nowrap">Zbl 1267.34079</a> · <a href="https://doi.org/10.1016/j.nonrwa.2012.06.008" class="nowrap">doi:10.1016/j.nonrwa.2012.06.008</a></td> </tr><tr> <td>[26]</td> <td class="space">X. Fan and Z. Wang, “Stability analysis of an SEIR epidemic model with stochastic perturbation and numerical simulation,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 14, no. 2, pp. 113-121, 2013, doi:10.1515/ijnsns-2012-0054. · <a href="/1401.92181" class="nowrap">Zbl 1401.92181</a> · <a href="https://doi.org/10.1515/ijnsns-2012-0054" class="nowrap">doi:10.1515/ijnsns-2012-0054</a></td> </tr><tr> <td>[27]</td> <td class="space">T. Tang, Z. Teng and Z. Li, “Threshold behavior in a class of stochastic SIRS epidemic models with nonlinear incidence,” Stoch. Anal. Appl., vol. 33, no. 6, pp. 994-1019, 2015, doi:10.1080/07362994.2015.1065750. · <a href="/1343.92521" class="nowrap">Zbl 1343.92521</a> · <a href="https://doi.org/10.1080/07362994.2015.1065750" class="nowrap">doi:10.1080/07362994.2015.1065750</a></td> </tr><tr> <td>[28]</td> <td class="space">L. Wang, Z. Teng, T. Tang and Z. Li, “Threshold dynamics in stochastic SIRS epidemic models with nonlinear incidence and vaccination,” Comput. Math. Methods Med., vol. 2017, p. 7294761, 2017, doi:10.1155/2017/7294761. · <a href="/1377.92102" class="nowrap">Zbl 1377.92102</a> · <a href="https://doi.org/10.1155/2017/7294761" class="nowrap">doi:10.1155/2017/7294761</a></td> </tr><tr> <td>[29]</td> <td class="space">S. Liu, L. Zhang and Y. Xing, “Dynamics of a stochastic heroin epidemic model,” J. Comput. Appl. Math., vol. 351, pp. 260-269, 2019, doi:10.1016/j.cam.2018.11.005. · <a href="/1405.37090" class="nowrap">Zbl 1405.37090</a> · <a href="https://doi.org/10.1016/j.cam.2018.11.005" class="nowrap">doi:10.1016/j.cam.2018.11.005</a></td> </tr><tr> <td>[30]</td> <td class="space">L. Zhang, S. Liu and X. Zhang, “Asymptotic behavior of a stochastic virus dynamics model with intracellular delay and humoral immunity,” J. Appl. Anal. Comput., vol. 9, no. 4, pp. 1425-1442, 2019, doi:10.11948/2156-907x.20180270. · <a href="/1465.92134" class="nowrap">Zbl 1465.92134</a> · <a href="https://doi.org/10.11948/2156-907x.20180270" class="nowrap">doi:10.11948/2156-907x.20180270</a></td> </tr><tr> <td>[31]</td> <td class="space">X. Yu, S. Yuan and T. Zhang, “Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment,” Appl. Math. Comput., vol. 347, pp. 249-264, 2019, doi:10.1016/j.amc.2018.11.005. · <a href="/1428.92097" class="nowrap">Zbl 1428.92097</a> · <a href="https://doi.org/10.1016/j.amc.2018.11.005" class="nowrap">doi:10.1016/j.amc.2018.11.005</a></td> </tr><tr> <td>[32]</td> <td class="space">S. Liu, L. Zhang, X.-B. Zhang and A. Li, “Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size,” Int. J. Biomath., vol. 12, no. 1, p. 1950005, 2019, doi:10.1142/s1793524519500050. · <a href="/1406.92595" class="nowrap">Zbl 1406.92595</a> · <a href="https://doi.org/10.1142/s1793524519500050" class="nowrap">doi:10.1142/s1793524519500050</a></td> </tr><tr> <td>[33]</td> <td class="space">R. Rifhat, A. Muhammadhaji and Z. Teng, “Asymptotic properties of a stochastic SIRS epidemic model with nonlinear incidence and varying population sizes,” Dyn. Syst. Int. J., vol. 35, no. 1, pp. 56-80, 2020, doi:10.1080/14689367.2019.1620689. · <a href="/1455.92139" class="nowrap">Zbl 1455.92139</a> · <a href="https://doi.org/10.1080/14689367.2019.1620689" class="nowrap">doi:10.1080/14689367.2019.1620689</a></td> </tr><tr> <td>[34]</td> <td class="space">J. Xu, T. Zhang and K. Song, “A stochastic model of bacterial infection associated with neutrophils,” Appl. Math. Comput., vol. 373, p. 125025, 2020, doi:10.1016/j.amc.2019.125025. · <a href="/1433.92017" class="nowrap">Zbl 1433.92017</a> · <a href="https://doi.org/10.1016/j.amc.2019.125025" class="nowrap">doi:10.1016/j.amc.2019.125025</a></td> </tr><tr> <td>[35]</td> <td class="space">N. Tuerxun, B. Wen and Z. Teng, “The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion,” Math. Comput. Simulat., 2020, doi:10.1016/j.matcom.2020.03.008. · <a href="/1524.92115" class="nowrap">Zbl 1524.92115</a> · <a href="https://doi.org/10.1016/j.matcom.2020.03.008" class="nowrap">doi:10.1016/j.matcom.2020.03.008</a></td> </tr><tr> <td>[36]</td> <td class="space">T. Zhang, X. Liu, X. Meng and T. Zhang, “Spatio-temporal dynamics near the steady state of a planktonic system,” Comput. Math. Appl., vol. 75, no. 12, pp. 4490-4504, 2018, doi:10.1016/j.camwa.2018.03.044. · <a href="/1417.92222" class="nowrap">Zbl 1417.92222</a> · <a href="https://doi.org/10.1016/j.camwa.2018.03.044" class="nowrap">doi:10.1016/j.camwa.2018.03.044</a></td> </tr><tr> <td>[37]</td> <td class="space">L. Wang and J. Zhao, “A predator-prey model with genetic differentiation both in the predator and prey,” Math. Biosci. Eng., vol. 17, no. 3, pp. 2616-2635, 2020, doi:10.3934/mbe.2020143. · <a href="/1472.92190" class="nowrap">Zbl 1472.92190</a> · <a href="https://doi.org/10.3934/mbe.2020143" class="nowrap">doi:10.3934/mbe.2020143</a></td> </tr><tr> <td>[38]</td> <td class="space">S. Yan, D. Jia, T. Zhang and S. Yuan, “Pattern dynamics in a diffusive predator-prey model with hunting cooperations,” Chaos Solitons Fractals, vol. 130, p. 109428, 2020, doi:10.1016/j.chaos.2019.109428. · <a href="/1489.92128" class="nowrap">Zbl 1489.92128</a> · <a href="https://doi.org/10.1016/j.chaos.2019.109428" class="nowrap">doi:10.1016/j.chaos.2019.109428</a></td> </tr><tr> <td>[39]</td> <td class="space">Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” J. Math. Anal. Appl., vol. 334, pp. 69-84, 2007, doi:10.1016/j.jmaa.2006.12.032. · <a href="/1113.92052" class="nowrap">Zbl 1113.92052</a> · <a href="https://doi.org/10.1016/j.jmaa.2006.12.032" class="nowrap">doi:10.1016/j.jmaa.2006.12.032</a></td> </tr><tr> <td>[40]</td> <td class="space">R. Z. Hasminskij, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980. · <a href="/0441.60060" class="nowrap">Zbl 0441.60060</a></td> </tr><tr> <td>[41]</td> <td class="space">E. Tornatore, S. M. Buccellato and P. Vetro, “Stability of a stochastic SIR system,” Physica A, vol. 354, pp. 111-126, 2005, doi:10.1016/j.physa.2005.02.057. · <a href="https://doi.org/10.1016/j.physa.2005.02.057" class="nowrap">doi:10.1016/j.physa.2005.02.057</a></td> </tr><tr> <td>[42]</td> <td class="space">Q. Lu, “Stability of SIRS system with random perturbations,” Physica A, vol. 388, pp. 3677-3686, 2009, doi:10.1016/j.physa.2009.05.036. · <a href="https://doi.org/10.1016/j.physa.2009.05.036" class="nowrap">doi:10.1016/j.physa.2009.05.036</a></td> </tr><tr> <td>[43]</td> <td class="space">S. Hu, Stochastic Differential Equations, Beijing, Science Press, 2010 (in Chinese).</td> </tr><tr> <td>[44]</td> <td class="space">C. Zhu and G. Yin, “On competitive Lotka-Volterra model in random environments,” J. Math. Anal. Appl., vol. 357, pp. 154-170, 2009, doi:10.1016/j.jmaa.2009.03.066. · <a href="/1182.34078" class="nowrap">Zbl 1182.34078</a> · <a href="https://doi.org/10.1016/j.jmaa.2009.03.066" class="nowrap">doi:10.1016/j.jmaa.2009.03.066</a></td> </tr><tr> <td>[45]</td> <td class="space">A. Friedman, Stochastic Differential Equations and Applications, I and II, New York, Academic Press, 1975. · <a href="/0323.60056" class="nowrap">Zbl 0323.60056</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 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"> © 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 & 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"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1550.92045 - zbMATH Open