### Abstract

Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, ℛ_{0}, exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep, ℛ̃_{0}. If ℛ_{0} > 1 and ℛ̃_{0} > 1, bluetongue persists in cattle and midges even though it may eradicate the sheep, relying on cattle as a reservoir. If ℛ0 > 1 > ℛ̃_{0}, bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.

Original language | English (US) |
---|---|

Pages (from-to) | 1160-1184 |

Number of pages | 25 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

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### Keywords

- Bluetongue
- Delay
- Disease reservoir
- Persistence

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*46*(2), 1160-1184. https://doi.org/10.1137/120878197

**Uniform persistence in a model for bluetongue dynamics.** / Gourley, Stephen A.; Röst, Gergely; Thieme, Horst.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 46, no. 2, pp. 1160-1184. https://doi.org/10.1137/120878197

}

TY - JOUR

T1 - Uniform persistence in a model for bluetongue dynamics

AU - Gourley, Stephen A.

AU - Röst, Gergely

AU - Thieme, Horst

PY - 2014

Y1 - 2014

N2 - Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, ℛ0, exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep, ℛ̃0. If ℛ0 > 1 and ℛ̃0 > 1, bluetongue persists in cattle and midges even though it may eradicate the sheep, relying on cattle as a reservoir. If ℛ0 > 1 > ℛ̃0, bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.

AB - Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, ℛ0, exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep, ℛ̃0. If ℛ0 > 1 and ℛ̃0 > 1, bluetongue persists in cattle and midges even though it may eradicate the sheep, relying on cattle as a reservoir. If ℛ0 > 1 > ℛ̃0, bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.

KW - Bluetongue

KW - Delay

KW - Disease reservoir

KW - Persistence

UR - http://www.scopus.com/inward/record.url?scp=84902674115&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902674115&partnerID=8YFLogxK

U2 - 10.1137/120878197

DO - 10.1137/120878197

M3 - Article

AN - SCOPUS:84902674115

VL - 46

SP - 1160

EP - 1184

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -