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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA

    K. P. Hadeler & H. R. Thieme

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  1. K. P. Hadeler
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  2. H. R. Thieme
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Correspondence to H. R. Thieme.

Additional information

K. P. Hadeler was supported by NSF Grant DMS-0502349.

H. R. Thieme was supported by NSF Grants DMS-0314520 and DMS-0715451.

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Hadeler, K.P., Thieme, H.R. Monotone dependence of the spectral bound on the transition rates in linear compartment models. J. Math. Biol. 57, 697–712 (2008). https://doi.org/10.1007/s00285-008-0185-z

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  • DOI: https://doi.org/10.1007/s00285-008-0185-z

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