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

K. P. Hadeler, Horst Thieme

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)697-712
Number of pages16
JournalJournal of Mathematical Biology
Volume57
Issue number5
DOIs
StatePublished - Nov 2008

Fingerprint

Spectral Bound
Compartment Model
Infectious Disease Transmission
Population Dynamics
Linear Models
Linear Model
Monotone
Strictly
Population
Positive Matrices
Monotone Function
Infectious Diseases
Stationary States
Population Model
Population dynamics
infectious diseases
population dynamics
Nonlinear Systems
Linear Systems
Eigenvalue

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.

In: Journal of Mathematical Biology, Vol. 57, No. 5, 11.2008, p. 697-712.

Research output: Contribution to journalArticle

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