### Abstract

This paper proposes two modified susceptible-infected-recovered-susceptible models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, it is proved that if the basic reproduction number (Formula presented.) of the model is less than one, then the disease-free equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the condition that the threshold value (Formula presented.) is less than one. Otherwise, if (Formula presented.) is more than one, the endemic equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the assumption that the total population (Formula presented.) will tend to a specific plane. In the study of the heterogeneous network model, this paper discusses the existences of the disease-free equilibrium and endemic equilibrium of the model. It is proved that if the threshold value (Formula presented.) is less than one, then the disease-free equilibrium is globally asymptotically stable. Otherwise, if (Formula presented.) is more than one, the system is permanent. A series of numerical experiments are given to illustrate the theoretical results. We also numerically predict the effect of vaccination ratio on the size of HBV-infected mainland Chinese population.

Original language | English (US) |
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Pages (from-to) | 2308-2330 |

Number of pages | 23 |

Journal | Applicable Analysis |

Volume | 94 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2 2015 |

### Keywords

- global stability
- heterogeneous network
- homogenous network
- local stability
- SIRS model

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Applicable Analysis*,

*94*(11), 2308-2330. https://doi.org/10.1080/00036811.2014.983487