Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks

A growing number of studies are investigating the effect of contact structure on the dynamics of epidemics in large-scale complex networks. Whether findings thus obtained apply also to networks of small size, and thus to many real-world biological applications, is still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of network structure (local, small-world, random and scale-free), there is a linear threshold determined by the probability of infection transmission between connected nodes and the probability of infection persistence in an infected node. The threshold is significantly lower for scale-free networks compared to local, random and small-world ones only if super-connected nodes have a higher number of links both to and from other nodes. The starting point, the node at which the epidemic starts, does not affect the threshold conditions, but has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation.