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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:
34C60Qualitative investigation and simulation of models (ODE)
92D30Epidemiology
34C23Bifurcation (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
34C37Homoclinic and heteroclinic solutions of ODE
34C05Location of integral curves, singular points, limit cycles (ODE)
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