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Abstract: In this paper we discuss how to determine a degenerate equilibrium of planar analytic systems to be focus-center type. A method of generalized normal sectors is used to determine orbits in exceptional directions near high degenerate equilibria. We obtain a sufficient and necessary condition for the existence of orbits going to origin in a generalized normal sector in class III. Thus, together with some criterions of orbits going to origin in a generic quasi-sector, we can characterize whether the degenerate equilibrium is of focus-center type in every case. The effectiveness of our methods is shown in an example which has a high degenerate equilibrium. [Copyright &y& Elsevier]

Abstract: In this paper, we give the global dynamical behaviors of a reduced SIRS epidemic model with a nonlinear incidence rate . We first discuss the qualitative properties of the equilibria in the interior of the first quadrant, and study the bifurcations including saddle–node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Then we consider equilibria at infinity, determining the number of orbits in exceptional directions for the global tendency. In this discussion, the unspecified degree of polynomials and their high degeneracy prevent us from using the methods of blowing-up or normal sectors in some cases. We lastly discuss the existence and uniqueness of limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper we consider an epidemic model, which is a reduced SIRS model with a constant removal rate of the infective individuals, for its qualitative properties which were not revealed in [W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the refectives, J. Math. Anal. Appl. 291 (2004) 775–793]. We first prove the uniqueness of closed orbits if they exist for this epidemic model. Then we discuss the qualitative properties of equilibria at infinity for global tendencies. Therefore, global dynamical behaviors of this system are obtained in the end. [Copyright &y& Elsevier]