Effect of Vaccination Strategies on the Herd Immunity of Growing 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, and the strategy used in selecting individuals for vaccination. Social networks however undergo constant growth, and it may be argued that network growth may change the level of herd immunity present in social networks. In this paper, we analyse the effect of growth and immunization strategies on herd immunity of social networks. Considering three classical topologies - Random, scale-free and small-world, we compare the influence of immunization strategies on each of them and then discuss how network growth can nullify or amplify these differences. We show that betweenness based vaccination is best strategy of immunization, regardless of topology, in static networks, but its prominence over other strategies diminishes in dynamically growing topologies. 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. We compare the relative influence of growth on each class of networks vaccinated under different strategies.