  Reproduction numbers are widely used for the estimation and prediction of
epidemic spreading processes over networks. However, reproduction numbers do
not enable estimation and prediction in individual communities within networks,
and they can be difficult to compute due to the aggregation of infection data
that is required to do so. Therefore, in this work we propose a novel concept
of distributed reproduction numbers to capture the spreading behaviors of each
entity in the network, and we show how to compute them using certain parameters
in networked SIS and SIR epidemic models. We use distributed reproduction
numbers to derive new conditions under which an outbreak can occur. These
conditions are then used to derive new conditions for the existence,
uniqueness, and stability of equilibrium states. Finally, in simulation we use
synthetic infection data to illustrate how distributed reproduction numbers
provide more fine-grained analyses of networked spreading processes than
ordinary reproduction numbers.
