Near-critical SIR epidemic on a random graph with given degrees.

Area of Science:

• Epidemiology
• Mathematical Biology
• Network Science

Background:

• Disease emergence and elimination are critical public health concerns.
• Mathematical models analyze epidemic dynamics, particularly near critical thresholds (basic reproductive ratio = 1).

Purpose of the Study:

• To investigate near-critical epidemic behavior in susceptible-infective-recovered models on random graphs.
• To determine the probability and size of large epidemics just above the epidemic threshold.

Main Methods:

• Analysis of susceptible-infective-recovered epidemic models on random (multi)graphs with specified degree sequences.
• Focus on the regime where the basic reproductive ratio approaches 1 slowly as population size increases.
• Application of regularity conditions on degree sequences and bounded third moment assumptions.

Main Results:

• Derived the probability of a large epidemic occurring in near-critical conditions.
• Quantified the expected size of large epidemics under these conditions.
• Established results for sparse binomial random graphs and improved bounds for giant component size in random graphs.

Conclusions:

• Provides a framework for understanding epidemic spread near critical thresholds.
• Offers precise predictions for epidemic probability and size in various random graph settings.
• Confirms a conjecture regarding the size of the giant component in random graphs.