Networks, epidemics and vaccination through contact tracing.

We consider a (social) network whose structure can be represented by a simple random graph having a pre-specified degree distribution. A Markovian susceptible-infectious-removed (SIR) epidemic model is defined on such a social graph. We then consider two real-time vaccination models for contact tracing during the early stages of an epidemic outbreak. The first model considers vaccination of each friend of an infectious individual (once identified) independently with probability p. The second model is related to the first model but also sets a bound on the maximum number an infectious individual can infect before being identified. Expressions are derived for the influence on the reproduction number of these vaccination models. We give some numerical examples and simulation results based on the Poisson and heavy-tail degree distributions where it is shown that the second vaccination model has a bigger advantage compared to the first model for the heavy-tail degree distribution.