Epidemics and vaccination on weighted graphs

TL;DR

This paper models epidemics on weighted graphs with inhomogeneous infection probabilities, deriving thresholds and analyzing vaccination strategies that outperform random neighbor vaccination in reducing epidemic spread.

Contribution

It introduces a framework for analyzing Reed-Frost epidemics on weighted graphs with inhomogeneous transmission probabilities, including new vaccination strategies and threshold expressions.

Findings

Derived epidemic thresholds for weighted graphs with i.i.d. and degree-dependent weights.

Proposed a neighbor vaccination strategy that outperforms random neighbor vaccination.

Showed the basic reproduction number is reduced by the new vaccination approach.

Abstract

A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge $(u,v)$ in the graph is equipped with two weights $W_{(u,v)}$ and $W_{(v,u)}$ that represent the (subjective) strength of the connection and determine the probability that $u$ infects $v$ in case $u$ is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d.\ weights and for weights that are functions of the degrees. For i.i.d.\ weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.