Monotone dependence of the spectral bound on the transition rates in linear compartment models.
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.
Hadeler
K P
KP
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA. hadeler@uni-tuebingen.de
Thieme
H R
HR
D000465:Algorithms; D000818:Animals; D050936:Basic Reproduction Number; D003141:Communicable Diseases / Q000635:transmission; D006801:Humans; D016014:Linear Models; D008954:Models, Biological; D010935:Plant Diseases / Q000382:microbiology / Q000706:statistics & numerical data; D011157:Population Dynamics; D011158:Population Growth