Universal statistics of incubation periods and other detection times via diffusion models

TL;DR

This paper explains the universal statistical features of incubation and detection times using diffusion models, providing rigorous proofs and showing their relevance across various fields like epidemiology and psychology.

Contribution

It offers a rigorous mathematical framework for understanding the universal behavior of exit times in diffusion models, including their skewness and limiting distributions.

Findings

Incubation times exhibit characteristic right skewness.

Limiting distributions are Gaussian and Gumbel under small noise.

Features are universal across different detection time scenarios.

Abstract

We suggest an explanation of typical incubation times statistical features based on the universal behavior of exit times for diffusion models. We give a mathematically rigorous proof of the characteristic right skewness of the incubation time distribution for very general one-dimensional diffusion models. Imposing natural simple conditions on the drift coefficient, we also study these diffusion models under the assumption of noise smallness and show that the limiting exit time distributions in the limit of vanishing noise are Gaussian and Gumbel. Thus they match the existing data as well as the other existing models do. The character of our models, however, allows us to argue that the features of the exit time distributions that we describe are universal and manifest themselves in various other situations where the times involved can be described as detection or halting times, for example, response times studied in psychology.