### 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) |
---|---|

Pages (from-to) | 188-211 |

Number of pages | 24 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### 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

### Cite this

**Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity.** / Thieme, Horst.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity

AU - Thieme, Horst

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Age-structure

KW - Evolution semigroups

KW - Evolutionary systems

KW - Exponential growth bound

KW - Integrated semigroups

KW - Laplace transform

KW - M matrices

KW - Next generation operator

KW - Operator semigroups

KW - Quasi-positive matrices

KW - Resolvent-positive operators

KW - Spectral radius

KW - Stability

KW - Time heterogeneity and periodicity

UR - http://www.scopus.com/inward/record.url?scp=67649378819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649378819&partnerID=8YFLogxK

U2 - 10.1137/080732870

DO - 10.1137/080732870

M3 - Article

VL - 70

SP - 188

EP - 211

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 1

ER -