Equivalence formulas for generalized spread rates

This paper develops a quantitative framework for comparing spread models motivated by rate-based comparisons commonly used in epidemic, population, and spatial propagation studies. We consider 1-spread models on d-ary trees and define the spread rate as the asymptotic proportion of type occurrences in infinite spread patterns. This notion is further extended to generalized spread rates through input-output type vectors, allowing intergenerational transformations to be systematically described. Using ξ-matrices together with elementary and finite shift equivalence, we derive explicit relations between spread rates of different systems. Central to this comparison is the transition constant, a relational quantity that measures the relative dilation or attenuation of asymptotic spread rates between two spread models whose ξ-matrices are finite shift equivalent. We show that the set of attainable transition constants is dense in [1,∞), revealing a wide spectrum of quantitative rate relationships beyond qualitative equivalence. Finally, the framework is applied to population systems by incorporating a movement function, leading to the notion of distribution rates that capture both population growth and spatial migration.