  We investigate the running vacuum model (RVM) in the framework of scalar
field theory.This dynamical vacuum model provides an elegant global explanation
of the cosmic history, namely the universe starts from a non-singular initial
de Sitter vacuum stage, it passes smoothly from an early inflationary era to a
radiation epoch ("graceful exit") and finally it enters the dark matter and
dark energy (DE) dominated epochs, where it can explain the large entropy
problem and predicts a mild dynamical evolution of the DE. Within this
phenomenologically appealing context, we formulate an effective {\it classical}
scalar field description of the RVM through a field $\phi$, called the {\it
vacuumon}, which turns out to be very helpful for an understanding and
practical implementation of the physical mechanisms of the running vacuum
during both the early universe and the late time cosmic acceleration. In the
early universe the potential for the vacuumon may be mapped to a potential that
behaves similarly to that of the scalaron field of Starobinsky-type inflation
at the {\it classical} level, whilst in the late universe it provides an
effective scalar field description of DE. The two representations, however, are
not physically equivalent since the mechanisms of inflation are entirely
different. Moreover, unlike the scalaron, vacuumon is treated as a classical
background field, and not a fully fledged quantum field, hence cosmological
perturbations will be different between the two pictures of inflation.
