Epidemics and vaccination on weighted graphs

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.

► We study epidemics and vaccination on graphs with weighted edges.
► The weight of an edge represent the infection probability through the edge.
► Expressions for the epidemic threshold are derived.
► Vaccination strategies based on the edge weights are shown to outperform other commonly studied strategies.