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This study introduces a quantitative framework to compare spread models, defining spread rates and using transition constants to measure relative changes. The framework reveals a dense spectrum of rate relationships and applies to population dynamics.

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Area of Science:

  • Mathematical modeling
  • Complex systems science
  • Theoretical ecology

Background:

  • Quantitative comparisons of spread phenomena (epidemics, populations, spatial patterns) are crucial.
  • Existing models often lack a unified framework for direct rate-based comparison.
  • Understanding intergenerational transformations in propagation is key.

Purpose of the Study:

  • To develop a quantitative framework for comparing spread models.
  • To define and extend spread rates for systematic analysis.
  • To introduce the transition constant for measuring relative rate changes.

Main Methods:

  • Development of a quantitative framework for 1-spread models on d-ary trees.
  • Definition of asymptotic spread rate and generalized spread rates using input-output vectors.
  • Utilizing ξ-matrices and shift equivalence to derive relations between spread rates.
  • Introduction of the transition constant for comparing ξ-matrix-equivalent models.

Main Results:

  • Explicit relations between spread rates of different systems are derived.
  • The transition constant quantifies relative dilation/attenuation of asymptotic spread rates.
  • The set of attainable transition constants is dense in [1,∞), indicating diverse quantitative rate relationships.
  • Application to population systems yields distribution rates capturing growth and migration.

Conclusions:

  • The developed framework provides a robust method for quantitative comparison of spread models.
  • The transition constant offers a novel metric for understanding relative spread dynamics.
  • The framework's application to population systems demonstrates its versatility in analyzing complex dynamics.