### Abstract

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ord inary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extinct or grows. Another important threshold parameter is the reproduction number R, which is the spectral radius of a related positive matrix. As is well known, the spectral bound and R - 1 have the same sign provided that the matrices have a particular form. The relation between spectral bound and reproduction number extends to models with infinite-dimensional state space and then holds between the spectral bound of a resolvent-positive closed linear operator and the spectral radius of a positive bounded linear operator. We also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete-time models. We illustrate the general theory by applying it to an epidemic model with distributed susceptibility, population models with age structure, and, using evolution semigroups, to time-heterogeneous population models.

Original language | English (US) |
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Pages (from-to) | 188-211 |

Number of pages | 24 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2009 |

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### Keywords

- Age-structure
- Evolution semigroups
- Evolutionary systems
- Exponential growth bound
- Integrated semigroups
- Laplace transform
- M matrices
- Next generation operator
- Operator semigroups
- Quasi-positive matrices
- Resolvent-positive operators
- Spectral radius
- Stability
- Time heterogeneity and periodicity

### ASJC Scopus subject areas

- Applied Mathematics