The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.

Barbour, A D (1994). *Threshold phenomena in epidemic theory.* In: Kelly, F P. Probability, statistics and optimisation. Chichester: Wiley, 101-116.

## Abstract

The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.

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## Additional indexing

Item Type: | Book Section, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 1994 |

Deposited On: | 09 Apr 2010 09:25 |

Last Modified: | 05 Apr 2016 13:27 |

Publisher: | Wiley |

Series Name: | Wiley Series in Probability and Mathematical Statistics |

ISBN: | 0-471-94829-2 |

Additional Information: | This is a preprint of an article published in [Barbour, A. D. Threshold phenomena in epidemic theory. Probability, statistics and optimisation, 101--116], Wiley Series in Probability and Mathematical Statistics Copyright © 1994 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1320745 |

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