  The majority of research on epidemics relies on models which are formulated
in continuous-time. However, real-world epidemic data is gathered and processed
in a digital manner, which is more accurately described by discrete-time
epidemic models. We analyse the discrete-time NIMFA epidemic model on directed
networks with heterogeneous spreading parameters. In particular, we show that
the viral state is increasing and does not overshoot the steady-state, the
steady-state is globally exponentially stable, and we provide linear systems
that bound the viral state evolution. Thus, the discrete-time NIMFA model
succeeds to capture the qualitative behaviour of a viral spread and provides a
powerful means to study real-world epidemics.
