*(English)*Zbl 1073.92040

Summary: We develop and analyze a simple susceptible, infected, vaccinated (SIV) epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized nonlinear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the PoincarĂ© index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate.

Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.

##### MSC:

92D30 | Epidemiology |

34C23 | Bifurcation (ODE) |

34D05 | Asymptotic stability of ODE |

34C60 | Qualitative investigation and simulation of models (ODE) |

37N25 | Dynamical systems in biology |

65C20 | Models (numerical methods) |