Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity

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Abstract

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ord inary 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.

Original languageEnglish (US)
Pages (from-to)188-211
Number of pages24
JournalSIAM Journal on Applied Mathematics
Volume70
Issue number1
DOIs
StatePublished - 2009

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Spectral Bound
Reproduction number
Population Structure
Spectral Radius
Population Model
Positive Linear Operators
Threshold Parameter
Positive Matrices
Evolution Semigroup
Age Structure
Closed Operator
Discrete-time Model
Infinite-dimensional Spaces
Epidemic Model
Bounded Linear Operator
Resolvent
Susceptibility
Linear Operator
State Space
Differential equation

Keywords

  • Age-structure
  • Evolution semigroups
  • Evolutionary systems
  • Exponential growth bound
  • Integrated semigroups
  • Laplace transform
  • M matrices
  • Next generation operator
  • Operator semigroups
  • Quasi-positive matrices
  • Resolvent-positive operators
  • Spectral radius
  • Stability
  • Time heterogeneity and periodicity

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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title = "Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity",
abstract = "Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ord inary 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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