Applications of the Variance of Final Outbreak Size for Disease Spreading in Networks

The assumption that all susceptible individuals are equally likely to acquire the disease during an outbreak (by direct contact with an infective individual) can be relaxed by bringing into the disease spread model a contact structure between individuals in the population. The structure is a random network or random graph that describes the kind of contacts that can result in transmission. In this paper we use an approach similar to the approaches of Andersson (Ann Appl Probab 8(4):1331–1349, 1998) and Newman (Phys Rev E 66:16128, 2002) to study not only the expected values of final sizes of small outbreaks, but also their variability. Using these first two moments, a probability interval for the outbreak size is suggested based on Chebyshev’s inequality. We examine its utility in results from simulated small outbreaks evolving in simulated random networks. We also revisit and modify two related results from Newman (Phys Rev E 66:16128, 2002) to take into account the important fact that the infectious period of an infected individual is the same from the perspective of all the individual’s contacts. The theory developed in this area can be extended to describe other “infectious” processes such as the spread of rumors, ideas, information, and habits.