  The contact process is an emblematic model of a non-equilibrium system,
containing a phase transition between inactive and active dynamical regimes. In
the epidemiological context, the model is known as the
susceptible-infected-susceptible (SIS) model, and widely used to describe
contagious spreading. In this work, we demonstrate how accurate and efficient
representations of the full probability distribution over all configurations of
the contact process on a one-dimensional chain can be obtained by means of
Matrix Product States (MPS). We modify and adapt MPS methods from many-body
quantum systems to study the classical distributions of the driven contact
process at late times. We give accurate and efficient results for the
distribution of large gaps, and illustrate the advantage of our methods over
Monte Carlo simulations. Furthermore, we study the large deviation statistics
of the dynamical activity, defined as the total number of configuration changes
along a trajectory, and investigate quantum-inspired entropic measures, based
on the second R\'enyi entropy.
