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A note on persistence about structured population models.

Kazuki Kawachi1

  • 1Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan. kkawachi@ms.u-tokyo.ac.jp

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|August 11, 2012
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This study shows that structured epidemic models are persistent when the basic reproduction ratio exceeds one. This means the infected population proportion stays above a certain level, regardless of the starting conditions.

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Area of Science:

  • Mathematical Epidemiology
  • Structured Population Dynamics

Background:

  • Persistence is a crucial concept in epidemic modeling, indicating sustained disease presence.
  • Structured population models, considering factors like age or duration, offer more realistic disease dynamics.
  • Previous studies often faced challenges in demonstrating persistence in complex, infinite-dimensional models.

Purpose of the Study:

  • To investigate and demonstrate uniform strong persistence in chronic age-structured and age-duration-structured epidemic models.
  • To establish a method for proving persistence in complex structured population dynamics.

Main Methods:

  • Application of Thieme's technique for proving uniform strong persistence.
  • Utilizing Fréchet-Kolmogorov L(1)-compactness criteria to overcome challenges in infinite-dimensional spaces.
  • Analysis of two specific structured epidemic models: chronic age-structured and age-duration-structured.

Main Results:

  • Both epidemic models exhibit uniform strong persistence when the basic reproduction ratio (R0) is greater than one.
  • The proportion of the infected subpopulation is bounded away from zero, independent of initial conditions over time.
  • Demonstrated the successful application of compactness criteria in infinite-dimensional settings for persistence proofs.

Conclusions:

  • Uniform strong persistence is a key characteristic of these structured epidemic models under specific conditions (R0 > 1).
  • The study provides a practical and effective methodology for proving persistence in a broader range of structured population models.
  • Highlights the importance of mathematical tools like Thieme's technique and compactness criteria in epidemiological research.