### Abstract

One of the central problems in botanical epidemiology is whether disease spreads within crops in a regular pattern or follows a random process. In this study, we consider a row of n plants in which m are infected. We then develop a rigorous mathematical approach to investigate the total number of ways to obtain k isolated individuals among m infected plants. We give a recurrence relation in three parameters that describes the problem, then we find a closed-form solution, and give two different approaches to tackle the proof. Finally, we find interesting formulae for the expectation and variance of the random variable that represents the number of infected and isolated plants.

Original language | English |
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Pages (from-to) | 981-993 |

Number of pages | 13 |

Journal | Journal of Difference Equations and Applications |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2013 |

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### Keywords

- binomial coefficients
- hypergeometric function
- recurrence relation
- spread of disease

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Analysis

### Cite this

*Journal of Difference Equations and Applications*,

*19*(6), 981-993. https://doi.org/10.1080/10236198.2012.704915