## Abstract

In this paper, we study the dynamics of an SIR epidemic model with a logistic process and a distributed time delay. We first show that the attractivity of the disease-free equilibrium is completely determined by a threshold R _{0}. If R _{0}≤1, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if R _{0}>1, then the disease-free equilibrium is unstable, and meanwhile there exists uniquely an endemic equilibrium. We then prove that for any time delay h>0, the delayed SIR epidemic model is permanent if and only if there exists an endemic equilibrium. In other words, R _{0}>1 is a necessary and sufficient condition for the permanence of the epidemic model. Numerical examples are given to illustrate the theoretical results. We also make a distinction between the dynamics of the distributed time delay system and the discrete time delay system.

Original language | English |
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Pages (from-to) | 3696-3707 |

Number of pages | 12 |

Journal | Communications in Nonlinear Science and Numerical Simulation |

Volume | 17 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2012 |

## Keywords

- Asymptotic stability
- Permanence
- SIR epidemic model
- Time delay