Solution of an infection model near threshold

Area of Science:

• Epidemiology
• Statistical Mechanics
• Mathematical Modeling

Background:

• The susceptible-infected-recovered (SIR) model is a fundamental framework for understanding epidemic dynamics.
• Analyzing outbreak size distribution near the threshold infectivity is crucial for predicting epidemic behavior and impact.
• Previous work by Ben-Naim and Krapivsky established key scaling laws for outbreak sizes.

Purpose of the Study:

• To derive the distribution of total outbreak size in the susceptible-infected-recovered model for large populations.
• To investigate epidemic dynamics in the critical region near threshold infectivity.
• To recover and confirm existing scaling results for outbreak sizes.

Main Methods:

• The study employs a mapping of the epidemic model to the first passage time problem of a random walker.
• The random walker is subjected to a time-dependent drift, specifically one that increases linearly with time.
• Asymptotic analysis is used to derive the distribution of total outbreak size in the limit of large population size (N).

Main Results:

• The distribution of total outbreak size is derived for large populations (N).
• The effective maximal size of an outbreak scales as N^(2/3).
• The average outbreak size scales as N^(1/3), consistent with prior research, with an explicit scaling function provided.

Conclusions:

• The study confirms and provides a detailed derivation for the scaling laws of epidemic outbreak sizes near threshold infectivity.
• The findings offer a deeper understanding of epidemic behavior in large populations, particularly concerning outbreak magnitude.
• The connection to random walker problems provides a powerful analytical tool for epidemic modeling.