Document Zbl 1348.93226

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As in the majority of the work on infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the disease-free equilibrium is the unique equilibrium over the network. Otherwise, an endemic equilibrium state emerges, where some infection remains within the network. Using notions from positive systems theory, we provide novel proofs for the global asymptotic stability of the equilibrium points in both cases over strongly connected networks based on the value of the basic reproduction number, a fundamental quantity in the study of epidemics. When the network topology is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of an endemic state, and study the stability of the disease-free equilibrium. Finally, we demonstrate that the $n$-intertwined Markov model can be viewed as a best-response dynamical system of a concave game among the nodes. This characterization allows us to cast new infection spread dynamics; additionally, we provide a sufficient condition for global convergence to the disease-free equilibrium, which can be checked in a distributed fashion.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6652798">Cited in <strong>23</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D20" title="MSC2020">93D20</a> </td> <td class="space"> Asymptotic stability in control theory </td> </tr><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%3A60J10" title="MSC2020">60J10</a> </td> <td class="space"> Markov chains (discrete-time Markov processes on discrete state spaces) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A91A80" title="MSC2020">91A80</a> </td> <td class="space"> Applications of game theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Astability+analysis">stability analysis</a>; <a href="/?q=ut%3Anetworks">networks</a>; <a href="/?q=ut%3Adirected+graphs">directed graphs</a>; <a href="/?q=ut%3Anonlinear+control+systems">nonlinear control systems</a>; <a href="/?q=ut%3Ainterconnected+systems">interconnected systems</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 1348.93226" data-ciurl="/ci/06652798" data-biburl="/bibtex/06652798.bib" data-amsurl="/amsrefs/06652798.bib" data-xmlurl="/xml/06652798.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06652798.pdf" title="Zbl 1348.93226 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/10.1016/j.automatica.2016.07.037" aria-label="DOI for “Stability of epidemic models over directed graphs: A positive systems approach”" title="10.1016/j.automatica.2016.07.037">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1407.6076"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>[2]</td> <td class="space">Başar, T.; Olsder, G. 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B., Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 520-534 (1965) · <a href="/0142.17603" class="nowrap">Zbl 0142.17603</a></td> </tr><tr> <td>[21]</td> <td class="space">Shuai, Z.; van den Driessche, P., Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73, 4, 1513-1532 (2013) · <a href="/1308.34072" class="nowrap">Zbl 1308.34072</a></td> </tr><tr> <td>[22]</td> <td class="space">Sontag, E. D.; Wang, Y., On characterizations of the input-to-state stability property, Systems & Control Letters, 24, 5, 351-359 (1995) · <a href="/0877.93121" class="nowrap">Zbl 0877.93121</a></td> </tr><tr> <td>[24]</td> <td class="space">Van Mieghem, P.; Omic, J.; Kooij, R., Virus spread in networks, IEEE/ACM Transactions on Networking, 17, 1, 1-14 (2009)</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>

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