Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models

Horst R. Thieme

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

The analysis of the discrete-time dynamics of structured iteroparous populations involves a basic yearly turnover operator B= A+ H with a structural transition operator A and a mating and fertility operator H. A and H map a normal complete cone X+ of an ordered normed vector space X into itself and are (positively) homogenous and continuous on X+, A is additive and H is order-preserving. Assume that r(A) < 1 for the spectral radius of A. Let HR1 with R1=∑j=0∞Aj be the next generation operator and T= r(B), the spectral radius of B, be the (basic) turnover number and R= r(HR1) be the (basic) reproduction number. We explore conditions for a turnover/reproduction trichotomy, namely one (and only one) of the following three possibilities to hold: (i) 1 < T≤ R, (ii) 1 = T= R, (iii) 1 > T≥ R. In some cases, one may also like to consider the lower reproduction number R= lim λ1+r(HRλ), Rλ=∑j=0∞λ-(n+1)An. R is also useful to study the case r(A) = 1 to explore conditions for the dichotomy 1 = T≥ R or 1 < T≤ R≤ ∞.

Original languageEnglish (US)
Title of host publicationAdvances in Discrete Dynamical Systems, Difference Equations and Applications - 26th ICDEA, 2021
EditorsSaber Elaydi, Mustafa R.S. Kulenović, Senada Kalabušić
PublisherSpringer
Pages495-539
Number of pages45
ISBN (Print)9783031252242
DOIs
StatePublished - 2023
Event26th International Conference on Difference Equations and Applications, ICDEA 2021 - Sarajevo, Bosnia and Herzegovina
Duration: Jul 26 2021Jul 30 2021

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume416
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference26th International Conference on Difference Equations and Applications, ICDEA 2021
Country/TerritoryBosnia and Herzegovina
CitySarajevo
Period7/26/217/30/21

Keywords

  • Cones
  • Continuity of the spectral radius
  • Eigenvector
  • Extinction
  • Feller kernel
  • Generation growth factor
  • Homogeneous operators
  • Integral projection models
  • Integro-difference equations
  • Mating function
  • Net reproductive value
  • Ordered vector spaces
  • Pair-formation function
  • Population growth factor
  • Rank structure
  • Resolvent
  • Stability

ASJC Scopus subject areas

  • General Mathematics

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