### Abstract

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re-stocking from outside sources. Rigorous analysis of the model shows that the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case.

Original language | English (US) |
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Pages (from-to) | 1873-1887 |

Number of pages | 15 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 34 |

Issue number | 15 |

DOIs | |

State | Published - Oct 1 2011 |

Externally published | Yes |

### Keywords

- bovine tuberculosis
- equilibria
- reproduction number
- stability

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

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## Cite this

*Mathematical Methods in the Applied Sciences*,

*34*(15), 1873-1887. https://doi.org/10.1002/mma.1486