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**Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure.**
*(English)*
Zbl 1195.34130

This paper deals with a SIRS epidemic model with stage structure, formulated as a nonlinear differential system including a discrete time delay. The main goal of this work is to study the effects of the delay on the dynamic behaviour of the system. Taking the delay as a parameter, it is shown that when the delay passes through a critical value, the positive equilibrium of the model loses its stability and a Hopf bifurcation occurs. Previously, existence of such a positive equilibrium is established and its stability is discussed by means of the analysis of the roots of a characteristic equation associated to a linearized equation around the equilibrium.

Formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained, by using the normal form theory. The paper ends with some numerical simulations to illustrate the above theoretical results.

Formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained, by using the normal form theory. The paper ends with some numerical simulations to illustrate the above theoretical results.

Reviewer: Eva Sanchez (Madrid)

### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

92D30 | Epidemiology |

34K18 | Bifurcation theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34K19 | Invariant manifolds of functional-differential equations |

34K17 | Transformation and reduction of functional-differential equations and systems, normal forms |

### Keywords:

Hopf bifurcation; discrete time delay; stability switch; epidemiology; SIRS model with stage structure
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\textit{T. Zhang} et al., Nonlinear Anal., Real World Appl. 11, No. 1, 293--306 (2010; Zbl 1195.34130)

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