### Abstract

Based on a Ross–Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

Original language | English (US) |
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Pages (from-to) | 2255-2280 |

Number of pages | 26 |

Journal | Journal of Mathematical Biology |

Volume | 79 |

Issue number | 6-7 |

DOIs | |

State | Published - Dec 1 2019 |

### Keywords

- Basic reproduction number
- Disease persistence
- Habitat fragmentation
- Human movement
- Line-sum symmetric matrix
- Vector-borne disease

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

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## Cite this

*Journal of Mathematical Biology*,

*79*(6-7), 2255-2280. https://doi.org/10.1007/s00285-019-01428-2