Document Zbl 1266.92057

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Appl. Math.</a> <a href="/?q=in%3A318830" title="Articles in this Issue">2013, Article ID 314958, 9 p. (2013)</a>. </div> <div class="abstract">Summary: A SIR epidemic model is proposed to understand the impact of limited medical resources on the infectious disease transmission. The basic reproduction number is identified. Existence and stability of equilibria are obtained under different conditions. Bifurcations, including backward and Hopf bifurcations, are analyzed. Our results suggest that the model considering the impact of limited medical resource may exhibit vital dynamics, such as bistability and periodicity when the basic reproduction number $\mathbb R_0$ is less than unity, which implies that the basic reproductive number itself is not enough to describe whether the disease will prevail or not and a subthreshold number is needed. It is also shown that a sufficient number of sickbeds and other medical resources are very important for disease control and eradication. Considering the costs, we provide a method to estimate a suitable treatment capacity for a disease in a region.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6175803">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92C60" title="MSC2020">92C60</a> </td> <td class="space"> Medical epidemiology </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="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 1266.92057" data-ciurl="/ci/06175803" data-biburl="/bibtex/06175803.bib" data-amsurl="/amsrefs/06175803.bib" data-xmlurl="/xml/06175803.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06175803.pdf" title="Zbl 1266.92057 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/10.1155/2013/314958" aria-label="DOI for “Rich dynamics of an epidemic model with saturation recovery”" title="10.1155/2013/314958">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">M. E. Alexander and S. M. Moghadas, “Periodicity in an epidemic model with a generalized non-linear incidence,” Mathematical Biosciences, vol. 189, no. 1, pp. 75-96, 2004. · <a href="/1073.92040" class="nowrap">Zbl 1073.92040</a> · <a href="https://doi.org/10.1016/j.mbs.2004.01.003" class="nowrap">doi:10.1016/j.mbs.2004.01.003</a></td> </tr><tr> <td>[2]</td> <td class="space">X. Shi, J. Cui, and X. Zhou, “Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1088-1106, 2011. · <a href="/1215.34102" class="nowrap">Zbl 1215.34102</a> · <a href="https://doi.org/10.1016/j.na.2010.09.038" class="nowrap">doi:10.1016/j.na.2010.09.038</a></td> </tr><tr> <td>[3]</td> <td class="space">H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599-653, 2000. · <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>[4]</td> <td class="space">Z. Hu, Z. Teng, and H. Jiang, “Stability analysis in a class of discrete SIRS epidemic models,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2017-2033, 2012. · <a href="/1254.92082" class="nowrap">Zbl 1254.92082</a> · <a href="https://doi.org/10.1016/j.nonrwa.2011.12.024" class="nowrap">doi:10.1016/j.nonrwa.2011.12.024</a></td> </tr><tr> <td>[5]</td> <td class="space">J. Pang, J. Cui, and J. Hui, “Rich dynamics of epidemic model with sub-optimal immunity and nonlinear recovery rate,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 440-448, 2011. · <a href="/1225.37108" class="nowrap">Zbl 1225.37108</a> · <a href="https://doi.org/10.1016/j.mcm.2011.02.033" class="nowrap">doi:10.1016/j.mcm.2011.02.033</a></td> </tr><tr> <td>[6]</td> <td class="space">H. Wan and H. Zhu, “The backward bifurcation in compartmental models for West Nile virus,” Mathematical Biosciences, vol. 227, no. 1, pp. 20-28, 2010. · <a href="/1194.92067" class="nowrap">Zbl 1194.92067</a> · <a href="https://doi.org/10.1016/j.mbs.2010.05.006" class="nowrap">doi:10.1016/j.mbs.2010.05.006</a></td> </tr><tr> <td>[7]</td> <td class="space">W. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58-71, 2006. · <a href="/1093.92054" class="nowrap">Zbl 1093.92054</a> · <a href="https://doi.org/10.1016/j.mbs.2005.12.022" class="nowrap">doi:10.1016/j.mbs.2005.12.022</a></td> </tr><tr> <td>[8]</td> <td class="space">X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433-443, 2008. · <a href="/1144.92038" class="nowrap">Zbl 1144.92038</a> · <a href="https://doi.org/10.1016/j.jmaa.2008.07.042" class="nowrap">doi:10.1016/j.jmaa.2008.07.042</a></td> </tr><tr> <td>[9]</td> <td class="space">http://www.stats.gov.cn/tjsj/qtsj/.</td> </tr><tr> <td>[10]</td> <td class="space">W. Wang and S. Ruan, “Bifurcation in an epidemic model with constant removal rate of the infectives,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 775-793, 2004. · <a href="/1054.34071" class="nowrap">Zbl 1054.34071</a> · <a href="https://doi.org/10.1016/j.jmaa.2003.11.043" class="nowrap">doi:10.1016/j.jmaa.2003.11.043</a></td> </tr><tr> <td>[11]</td> <td class="space">J. D. Murray, Mathematical Biology, Springer, New York, NY, USA, 1998. · <a href="/0704.92001" class="nowrap">Zbl 0704.92001</a> · <a href="https://doi.org/10.1007/b98869" class="nowrap">doi:10.1007/b98869</a></td> </tr><tr> <td>[12]</td> <td class="space">P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29-48, 2002. · <a href="/1015.92036" class="nowrap">Zbl 1015.92036</a> · <a href="https://doi.org/10.1016/S0025-5564(02)00108-6" class="nowrap">doi:10.1016/S0025-5564(02)00108-6</a></td> </tr><tr> <td>[13]</td> <td class="space">L. Perko, Differential Equations and Dynamical Systems, Springer, New York, NY, USA, 1996. · <a href="/0854.34001" class="nowrap">Zbl 0854.34001</a> · <a href="https://doi.org/10.1007/978-1-4684-0249-0" class="nowrap">doi:10.1007/978-1-4684-0249-0</a></td> </tr><tr> <td>[14]</td> <td class="space">W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359-380, 1987. · <a href="/0621.92014" class="nowrap">Zbl 0621.92014</a> · <a href="https://doi.org/10.1007/BF00277162" class="nowrap">doi:10.1007/BF00277162</a></td> </tr><tr> <td>[15]</td> <td class="space">J. Lin, V. Andreasen, and S. A. Levin, “Dynamics of influenza A drift: the linear three-strain model,” Mathematical Biosciences, vol. 162, no. 1-2, pp. 33-51, 1999. · <a href="/0947.92017" class="nowrap">Zbl 0947.92017</a> · <a href="https://doi.org/10.1016/S0025-5564(99)00042-5" class="nowrap">doi:10.1016/S0025-5564(99)00042-5</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>

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