Heterogeneous network epidemics: real-time growth, variance and extinction of infection

TL;DR

This paper provides analytical formulas for the early dynamics of epidemic spread on networks, highlighting how degree distribution moments influence infection prevalence and extinction probability, applicable even with few initial cases.

Contribution

It introduces closed-form expressions for mean and variance of infections over time on configuration model networks, valid for small initial outbreaks, enhancing understanding of early epidemic behavior.

Findings

Mean infection depends on first two moments of degree distribution.

Variance in infection depends on first three moments.

Analytic extinction probabilities are computationally efficient.

Abstract

Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multi-type branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distribution - in particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual.