Journal
SIAM JOURNAL ON APPLIED MATHEMATICS
Volume 70, Issue 1, Pages 188-211Publisher
SIAM PUBLICATIONS
DOI: 10.1137/080732870
Keywords
quasi-positive matrices; M matrices; resolvent-positive operators; operator semigroups; evolutionary systems; evolution semigroups; integrated semigroups; spectral radius; exponential growth bound; stability; Laplace transform; age-structure; time heterogeneity and periodicity; next generation operator
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Funding
- NSF [DMS-0715451]
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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 R, which is the spectral radius of a related positive matrix. As is well known, the spectral bound and R - 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.
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