### Abstract

For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.

Original language | English (US) |
---|---|

Pages (from-to) | 697-712 |

Number of pages | 16 |

Journal | Journal of Mathematical Biology |

Volume | 57 |

Issue number | 5 |

DOIs | |

State | Published - Nov 2008 |

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### Keywords

- Adaptive dynamics
- Basic reproduction number
- Bifurcation of stationary points
- Epidemics
- Invasion thresholds
- Irreducible matrix
- Perron-Frobenius
- Quasipositive matrix

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

**Monotone dependence of the spectral bound on the transition rates in linear compartment models.** / Hadeler, K. P.; Thieme, Horst.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 57, no. 5, pp. 697-712. https://doi.org/10.1007/s00285-008-0185-z

}

TY - JOUR

T1 - Monotone dependence of the spectral bound on the transition rates in linear compartment models

AU - Hadeler, K. P.

AU - Thieme, Horst

PY - 2008/11

Y1 - 2008/11

N2 - For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.

AB - For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.

KW - Adaptive dynamics

KW - Basic reproduction number

KW - Bifurcation of stationary points

KW - Epidemics

KW - Invasion thresholds

KW - Irreducible matrix

KW - Perron-Frobenius

KW - Quasipositive matrix

UR - http://www.scopus.com/inward/record.url?scp=49549088357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49549088357&partnerID=8YFLogxK

U2 - 10.1007/s00285-008-0185-z

DO - 10.1007/s00285-008-0185-z

M3 - Article

C2 - 18488225

AN - SCOPUS:49549088357

VL - 57

SP - 697

EP - 712

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 5

ER -