SPECTRAL BOUND AND REPRODUCTION NUMBER FOR INFINITE-DIMENSIONAL POPULATION STRUCTURE AND TIME HETEROGENEITY

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extin...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 2009-01, Vol.70 (1), p.188-211
Main Author: THIEME, HORST R.
Format: Article
Language:English
Subjects:
Banach space
Differential equations
Disease models
Linear transformations
Mathematical models
Matrices
Matrix
Ordinary differential equations
Population
Population growth
Population structure
Reproduction
Semigroups
Studies
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Summary:Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extinct or grows. Another important threshold parameter is the reproduction number which is the spectral radius of a related positive matrix. As is well known, the spectral bound and – 1 have the same sign provided that the matrices have a particular form. The relation between spectral bound and reproduction number extends to models with infinite-dimensional state space and then holds between the spectral bound of a resolvent-positive closed linear operator and the spectral radius of a positive bounded linear operator. We also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete-time models. We illustrate the general theory by applying it to an epidemic model with distributed susceptibility, population models with age structure, and, using evolution semigroups, to time-heterogeneous population models.
ISSN:0036-1399
1095-712X
DOI:10.1137/080732870