{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:33:59Z","timestamp":1787340839585,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135\u2013163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov\u2013Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov\u2013Takens bifurcations of limit cycles and homoclinic loops of order 2.<\/jats:p>","DOI":"10.1137\/070700966","type":"journal-article","created":{"date-parts":[[2008,12,3]],"date-time":"2008-12-03T18:08:33Z","timestamp":1228327713000},"page":"621-639","source":"Crossref","is-referenced-by-count":80,"title":["Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate"],"prefix":"10.1137","volume":"69","author":[{"given":"Yilei","family":"Tang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Deqing","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shigui","family":"Ruan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weinian","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,12,3]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2004.01.003"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/040604947"},{"key":"R3","unstructured":"R. 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