  Mathematical modeling of disease spreading plays a crucial role in
understanding, controlling and preventing epidemic outbreaks. In a microscopic
description of the propagation of a disease over the complex network of human
contacts, the probability that an individual is in a given state (susceptible,
infectious, recovered etc) depends on the state of its neighbors in the
network. Thus it depends on the state of pairs of nodes, which in turns depends
on triples, in a hierarchy of dynamical dependencies. In order to produce
models that are at the same time reliable and manageable, one has to understand
how to truncate such a hierarchy, and how the chosen order of approximation
affects the ability of the model to forecast the real temporal evolution of an
epidemics. In this paper we provide a systematic analysis of the reliability
(under different hypotheses on the quantity and quality of available data) of
the predictions obtained by truncating the hierarchy either at the level of
individuals or at the level of pairs. We find that pair-based models are
reliable tools both for estimating the epidemiological parameters and for
forecasting the temporal evolution of the epidemics, under all conditions taken
into account in our work. However, a pair-based approach provides a much better
prediction of an individual-based one, only if better data, namely information
on the state of node pairs, are available. Overall, our results suggest that
more refined mathematical models need to be informed by improved contact
tracing techniques to better support decision on policies and containment
measures to adopt.
