### Abstract

Language | English |
---|---|

Pages | 1-36 |

Number of pages | 36 |

Journal | Journal of Mathematical Biology |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2009 |

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### Keywords

- backward bifurcation
- equilibrium and stability analysis
- basic reproduction ratio
- simulation
- three stage model
- bovine respiratory syncytial virus

### Cite this

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*Journal of Mathematical Biology*, vol. 59, no. 1, pp. 1-36. https://doi.org/10.1007/s00285-008-0206-y

**Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model.** / Greenhalgh, D.; Griffiths, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

AU - Greenhalgh, D.

AU - Griffiths, M.

PY - 2009/7

Y1 - 2009/7

N2 - In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R 0 < 1, where R 0 is the basic reproductive number, and a unique supercritical endemic equilibrium for R 0 > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.

AB - In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R 0 < 1, where R 0 is the basic reproductive number, and a unique supercritical endemic equilibrium for R 0 > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.

KW - backward bifurcation

KW - equilibrium and stability analysis

KW - basic reproduction ratio

KW - simulation

KW - three stage model

KW - bovine respiratory syncytial virus

U2 - 10.1007/s00285-008-0206-y

DO - 10.1007/s00285-008-0206-y

M3 - Article

VL - 59

SP - 1

EP - 36

JO - Journal of Mathematical Biology

T2 - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 1

ER -