# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate. (English) Zbl 1171.34033
Authors’ abstract: Recently, {\it S. Ruan} and {\it W. Wang} [J. Differ. Equations 188, No. 1, 135--163 (2003; Zbl 1028.34046)] 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-Takens 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-Takens bifurcations of limit cycles and homoclinic loops of order 2.

##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 92D30 Epidemiology 34C23 Bifurcation (ODE) 34C20 Transformation and reduction of ODE and systems, normal forms 34C37 Homoclinic and heteroclinic solutions of ODE 34C05 Location of integral curves, singular points, limit cycles (ODE)
Full Text: