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The basic reproduction number in time-heterogeneous environments.

Hisashi Inaba1

  • 1Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan. inaba@ms.u-tokyo.ac.jp.

Journal of Mathematical Biology
|May 16, 2019
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Summary
This summary is machine-generated.

This study introduces a new definition for the basic reproduction number in structured populations within time-heterogeneous environments. The research demonstrates its role as a critical threshold for population persistence and extinction, impacting epidemic modeling.

Keywords:
Basic reproduction numberCone spectral radiusGeneration evolution operatorOrbital spectral radius

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Area of Science:

  • Mathematical Biology
  • Population Dynamics
  • Epidemiology

Background:

  • The basic reproduction number is crucial for understanding population dynamics and disease spread.
  • Existing definitions may not adequately capture complexities in time-heterogeneous environments.

Purpose of the Study:

  • To propose and validate a new, biologically natural definition of the basic reproduction number for structured populations in time-heterogeneous environments.
  • To establish this number as a threshold for population extinction and persistence.
  • To analyze its implications for periodic systems and epidemic models.

Main Methods:

  • Utilizing the generation evolution operator and the mathematical definition of cone spectral radius.
  • Analyzing linear and nonlinear population dynamics.
  • Investigating age-structured SIS epidemic models with time-periodic parameters.

Main Results:

  • The proposed basic reproduction number is mathematically equivalent to the spectral radius of the generation evolution operator.
  • This number acts as a threshold for population extinction in both linear and nonlinear time-heterogeneous systems.
  • A supercritical condition for the basic reproduction number predicts the existence of positive periodic solutions in periodic systems.

Conclusions:

  • The new definition of the basic reproduction number is robust and applicable to complex population dynamics.
  • It provides a unified framework for assessing population persistence and extinction in variable environments.
  • The study advances understanding of epidemic dynamics in structured populations with time-varying parameters.