Monotone dependence of the spectral bound on the transition rates in linear compartment models |
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Authors: | K P Hadeler H R Thieme |
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Institution: | Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA, hadeler@uni-tuebingen.de. |
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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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Keywords: | Quasipositive matrix Irreducible matrix Perron-Frobenius Basic reproduction number Bifurcation of stationary points Invasion thresholds Adaptive dynamics Epidemics |
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