### 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) |
---|---|

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 2011 |

Externally published | Yes |

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### Keywords

- bovine tuberculosis
- equilibria
- reproduction number
- stability

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Mathematical Methods in the Applied Sciences*,

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

**Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis.** / Agusto, Folashade B.; Lenhart, Suzanne; Gumel, Abba; Odoi, Agricola.

Research output: Contribution to journal › Article

*Mathematical Methods in the Applied Sciences*, vol. 34, no. 15, pp. 1873-1887. https://doi.org/10.1002/mma.1486

}

TY - JOUR

T1 - Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

AU - Agusto, Folashade B.

AU - Lenhart, Suzanne

AU - Gumel, Abba

AU - Odoi, Agricola

PY - 2011/10

Y1 - 2011/10

N2 - 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.

AB - 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.

KW - bovine tuberculosis

KW - equilibria

KW - reproduction number

KW - stability

UR - http://www.scopus.com/inward/record.url?scp=80053560629&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053560629&partnerID=8YFLogxK

U2 - 10.1002/mma.1486

DO - 10.1002/mma.1486

M3 - Article

AN - SCOPUS:80053560629

VL - 34

SP - 1873

EP - 1887

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 15

ER -