## 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.

### 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
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### References:

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