2009/01/01 by Horst R. Thieme · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Applied mathematics #Bounded function #Differential equation #Eigenvalues and eigenvectors #Evolution and Genetic Dynamics #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Matrix (chemical analysis) #Operator (biology) #Ordinary differential equation #Physics #Population #Population model #Quantum mechanics #Resolvent #Spectral radius #Upper and lower bounds
paper · doi:10.1137/080732870
crossref issued 2009/01/01 · crossref published 2009/01/01 · crossref published-print 2009/01/01 · openalex publication_date 2009/01/01 · crossref created 2009/05/13 · crossref deposited 2023/05/25 · openalex created_date 2025/10/10 · crossref indexed 2026/07/30 · openalex updated_date 2026/08/04
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.