Influence of vaccination strategies and topology on the herd immunity of complex networks

It is well known that non-vaccinated individuals may be protected from contacting a disease by vaccinated individuals in a social network through community protection (herd immunity). Such protection greatly depends on the underlying topology of the social network, the strategy used in selecting individuals for vaccination, and the interplay between these. In this paper, we analyse how the interplay between topology and immunization strategies influences the herd immunity of social networks. First, we introduce an area under curve measure which can quantify the levels of herd immunity in a social network. Then, using this measure, we analyse the above mentioned interplay in three ways: (1) by comparing vaccination strategies across topologies, (2) by analysing the influence of selected topological metrics, and (3) by considering the influence of network growth on herd immunity. For qualitative comparison, we consider three classical topologies (scale-free, random, and small-world) and three vaccination strategies (natural, random, and betweenness-based immunization). We show that betweenness-based vaccination is the best strategy of immunization in static networks, regardless of topology, but its prominence over other strategies diminishes in dynamically growing topologies. We find that the network features that lead to 'small-worldness' in networks (low diameter and high clustering) discourage herd immunity, regardless of the vaccination strategy, while preferential mixing (high assortativity) encourages it. In terms of growth, we demonstrate that herd immunity of random networks actually increases with growth, if the proportion of survivors to a secondary infection is considered, while the community protection in scale-free and small-world networks decreases with growth. Our work highlights the complex balance between social network structure and vaccination strategies in influencing community protection, and contributes a numerical measure to quantify this.