Document Zbl 1225.93091

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We obtain sufficient conditions which ensure the global asymptotic stability of the endemic equilibrium of the system. The obtained results improve those in <span class="zbmathjax-textit">R. Xu</span> and <span class="zbmathjax-textit">Z. Ma</span> [Chaos Solitons Fractals 41, No. 5, 2319–2325 (2009; <a href="/1198.34098">Zbl 1198.34098</a>)], and are very useful for a large class of SIRS models.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5912735">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%3A93C10" title="MSC2020">93C10</a> </td> <td class="space"> Nonlinear systems in control theory </td> </tr><tr> <td class="mono"> 93B40 </td> <td class="space"> Computational methods in systems theory (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D20" title="MSC2020">92D20</a> </td> <td class="space"> Protein sequences, DNA sequences </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34H05" title="MSC2020">34H05</a> </td> <td class="space"> Control problems involving ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ASIRS+epidemic+model">SIRS epidemic model</a>; <a href="/?q=ut%3Anonlinear+incidence">nonlinear incidence</a>; <a href="/?q=ut%3Aglobal+asymptotic+stability">global asymptotic stability</a>; <a href="/?q=ut%3Adistributed+delays">distributed delays</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1198.34098">Zbl 1198.34098</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 1225.93091" data-ciurl="/ci/05912735" data-biburl="/bibtex/05912735.bib" data-amsurl="/amsrefs/05912735.bib" data-xmlurl="/xml/05912735.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05912735.pdf" title="Zbl 1225.93091 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.nonrwa.2010.12.002" aria-label="DOI for “Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays”" title="10.1016/j.nonrwa.2010.12.002">DOI</a> <a class="btn btn-default btn-xs" type="button" href="http://hdl.handle.net/20.500.11824/439" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anderson, R. M.; May, R. M., Population biology of infectious diseases: part I, Nature, 280, 361-367 (1979)</td> </tr><tr> <td>[2]</td> <td class="space">Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47, 4107-4115 (2001) · <a href="/1042.34585" class="nowrap">Zbl 1042.34585</a></td> </tr><tr> <td>[3]</td> <td class="space">Capasso, V.; Serio, G., A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42, 41-61 (1978) · <a href="/0398.92026" class="nowrap">Zbl 0398.92026</a></td> </tr><tr> <td>[4]</td> <td class="space">Cooke, K. L., Stability analysis for a vector disease model, Rocky Mt. J. Math., 9, 31-42 (1979) · <a href="/0423.92029" class="nowrap">Zbl 0423.92029</a></td> </tr><tr> <td>[5]</td> <td class="space">Huo, Hai-Feng; Ma, Zhan-Ping, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul., 15, 459-468 (2010) · <a href="/1221.34197" class="nowrap">Zbl 1221.34197</a></td> </tr><tr> <td>[6]</td> <td class="space">Jin, Y.; Wang, W.; Xiao, S., An SIRS model with a nonlinear incidence rate, Chaos Solitons Fractals, 34, 1482-1497 (2007) · <a href="/1152.34339" class="nowrap">Zbl 1152.34339</a></td> </tr><tr> <td>[7]</td> <td class="space">Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69, 1871-1886 (2007) · <a href="/1298.92101" class="nowrap">Zbl 1298.92101</a></td> </tr><tr> <td>[8]</td> <td class="space">Liu, W. M.; Levin, S. A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23, 187-204 (1986) · <a href="/0582.92023" class="nowrap">Zbl 0582.92023</a></td> </tr><tr> <td>[9]</td> <td class="space">Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17, 1141-1145 (2004) · <a href="/1071.34082" class="nowrap">Zbl 1071.34082</a></td> </tr><tr> <td>[10]</td> <td class="space">McCluskey, C. C., Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11, 55-59 (2010) · <a href="/1185.37209" class="nowrap">Zbl 1185.37209</a></td> </tr><tr> <td>[11]</td> <td class="space">Mena-Lorca, J.; Hethcote, H. W., Dynamic models of infectious diseases as regulators of population size, J. Math. Biol., 30, 693-716 (1992) · <a href="/0748.92012" class="nowrap">Zbl 0748.92012</a></td> </tr><tr> <td>[12]</td> <td class="space">Muroya, Y.; Enatsu, Y.; Nakata, Y., Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate, J. Math. Anal. Appl. (2010)</td> </tr><tr> <td>[13]</td> <td class="space">Pang, G.; Chen, L., A delayed SIRS epidemic model with pulse vaccination, Chaos Solitons Fractals, 34, 1629-1635 (2007) · <a href="/1152.34379" class="nowrap">Zbl 1152.34379</a></td> </tr><tr> <td>[14]</td> <td class="space">Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42, 931-947 (2000) · <a href="/0967.34070" class="nowrap">Zbl 0967.34070</a></td> </tr><tr> <td>[15]</td> <td class="space">Wang, W., Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15, 423-428 (2002) · <a href="/1015.92033" class="nowrap">Zbl 1015.92033</a></td> </tr><tr> <td>[16]</td> <td class="space">Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208, 419-429 (2007) · <a href="/1119.92042" class="nowrap">Zbl 1119.92042</a></td> </tr><tr> <td>[17]</td> <td class="space">Xu, R.; Ma, Z., Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals, 41, 2319-2325 (2009) · <a href="/1198.34098" class="nowrap">Zbl 1198.34098</a></td> </tr><tr> <td>[18]</td> <td class="space">Yang, Y.; Xiao, D., Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B, 13, 195-211 (2010) · <a href="/1187.34121" class="nowrap">Zbl 1187.34121</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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