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

Research output: Contribution to journalArticle

164 Citations (Scopus)

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 language English (US) 188-211 24 SIAM Journal on Applied Mathematics 70 1 https://doi.org/10.1137/080732870 Published - 2009

Fingerprint

Spectral Bound
Reproduction number
Population Structure
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
• Stability
• Time heterogeneity and periodicity

ASJC Scopus subject areas

• Applied Mathematics

Cite this

In: SIAM Journal on Applied Mathematics, Vol. 70, No. 1, 2009, p. 188-211.

Research output: Contribution to journalArticle

@article{8ce6dc0fda0248eeafa5f4a8305d22de,
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.",
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",
author = "Horst Thieme",
year = "2009",
doi = "10.1137/080732870",
language = "English (US)",
volume = "70",
pages = "188--211",
journal = "SIAM Journal on Applied Mathematics",
issn = "0036-1399",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "1",

}

TY - JOUR

T1 - Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity

AU - Thieme, Horst

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Age-structure

KW - Evolution semigroups

KW - Evolutionary systems

KW - Exponential growth bound

KW - Integrated semigroups

KW - Laplace transform

KW - M matrices

KW - Next generation operator

KW - Operator semigroups

KW - Quasi-positive matrices

KW - Resolvent-positive operators

KW - Spectral radius

KW - Stability

KW - Time heterogeneity and periodicity

UR - http://www.scopus.com/inward/record.url?scp=67649378819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649378819&partnerID=8YFLogxK

U2 - 10.1137/080732870

DO - 10.1137/080732870

M3 - Article

VL - 70

SP - 188

EP - 211

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 1

ER -