On the Equivalence of Instantaneous and Mechanistic Reproduction Numbers

2026-06-13

AI Summary

This study clarifies the equivalence between the instantaneous reproduction number used in epidemiological surveillance and the mechanism-based reproduction number derived from compartmental models. Under the assumption of homogeneous mixing, the authors rigorously establish this equivalence for the first time by integrating renewal equation theory with SEIR model analysis, and further derive the corresponding generation interval distribution inherent to the SEIR framework. The work demonstrates that the generation interval is not a model-free construct but implicitly encodes specific compartmental structures, thereby elucidating the underlying modeling assumptions required for accurate real-time estimation of the reproduction number. These findings provide a solid theoretical foundation for epidemic dynamic modeling and enhance the interpretability of reproduction number estimates in public health practice.

Abstract

The effective reproduction number ($R_t$) is widely used to track epidemic dynamics in real time. The standard estimation framework uses "instantaneous $R_t$," defined via the renewal equation, which relates new infections to past infections through a generation interval distribution. Compartmental models like SEIR yield a seemingly distinct quantity, "mechanistic $R_t$," based on the effective contact rate and duration of infectiousness. We prove these two definitions are equivalent under homogeneous mixing, the standard assumption in compartmental modeling. We also derive the generation interval distribution implied by SEIR dynamics. A practical consequence is that generation intervals, often treated as assumption-light inputs to renewal equation estimators, in fact encode specific compartmental structure.

Problem

• effective reproduction number
• instantaneous reproduction number
• mechanistic reproduction number
• generation interval
• compartmental models

Innovation

• effective reproduction number
• generation interval
• renewal equation
• compartmental models
• SEIR