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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling
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Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations.

Dimitri Breda1, Toshikazu Kuniya2, Jordi Ripoll3

  • 1CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle scienze 206, 33100 Udine, Italy.

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Summary

This study analyzes a collocation method for calculating the basic reproduction number in structured populations. The method is proven to be accurate and numerically stable for these complex population dynamics.

Keywords:
Basic reproduction numberNext-generation operatorPseudospectral collocationSpectral approximationSpectral radiusStability analysis of equilibriaStructured population dynamics

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Area of Science:

  • Mathematical Biology
  • Numerical Analysis
  • Operator Theory

Background:

  • The basic reproduction number is crucial for understanding disease spread in structured populations.
  • Computing this number often involves complex, infinite-dimensional operators.
  • A novel collocation approach has been proposed for this computation.

Purpose of the Study:

  • To provide a comprehensive theoretical and numerical analysis of the collocation approach for computing the basic reproduction number.
  • To establish the mathematical foundations for discretizing the problem.
  • To rigorously assess the accuracy and convergence properties of the method.

Main Methods:

  • Analysis of theoretical properties of infinite-dimensional operators associated with population dynamics models.
  • Proof of operator compactness under mild regularity conditions.
  • Detailed error and convergence analyses of the collocation method.
  • Numerical validation through diverse test cases.

Main Results:

  • The relevant operators are proven to be compact, enabling reformulation as an eigenvalue problem.
  • The collocation method achieves the expected spectral accuracy.
  • Numerical tests confirm the analytical findings and reveal method-specific characteristics.

Conclusions:

  • The collocation approach is theoretically sound and numerically accurate for computing the basic reproduction number in structured populations.
  • The method's spectral accuracy is rigorously demonstrated.
  • The findings provide a validated computational tool for epidemiological and ecological modeling.