Universal Statistics of Incubation Periods and Other Detection Times via Diffusion Models

Area of Science:

• * Mathematical modeling and statistical analysis of diffusion processes.
• * Stochastic processes and their applications in physical and biological sciences.

Background:

• * Incubation time distributions often exhibit right skewness, a phenomenon observed across various scientific disciplines.
• * Existing models for diffusion processes provide a framework for understanding time-dependent phenomena.

Purpose of the Study:

• * To provide a rigorous mathematical explanation for the statistical features of incubation times.
• * To demonstrate the universality of these features in one-dimensional diffusion models.
• * To analyze the behavior of exit time distributions under vanishing noise conditions.

Main Methods:

• * Development of a general one-dimensional diffusion model.
• * Mathematical derivation of the incubation time distribution's right skewness.
• * Analysis of limiting exit time distributions in the small noise regime.

Main Results:

• * A rigorous proof of the characteristic right skewness for general one-dimensional diffusion models.
• * Demonstration that vanishing noise leads to limiting exit time distributions that are either Gaussian or Gumbel.
• * Validation of these findings against existing empirical data and theoretical models.

Conclusions:

• * The observed statistical features of incubation times are universal properties of diffusion models.
• * The findings have broad applicability to various detection or halting time scenarios, including psychological response times.
• * The study provides a robust theoretical foundation for understanding time distributions in stochastic systems.