  Waiting times between two consecutive infection and recovery events in
spreading processes are often assumed to be exponentially distributed, which
results in Markovian (i.e., memoryless) continuous spreading dynamics. However,
this is not taking into account memory (correlation) effects and discrete
interactions that have been identified as relevant in social, transportation,
and disease dynamics. We introduce a framework to model continuous, discrete,
and hybrid forms of (non-)Markovian susceptible-infected-recovered (SIR)
stochastic processes on networks. The hybrid SIR processes that we study in
this paper describe infections as discrete-time Markovian and recovery events
as continuous-time non-Markovian processes, which mimic the distribution of
cell cycles. Our results suggest that the effective-infection-rate description
of epidemic processes fails to uniquely capture the behavior of such hybrid and
also general non-Markovian disease dynamics. Providing a unifying description
of general Markovian and non-Markovian disease outbreaks, we instead show that
the mean transmissibility produces the same phase diagrams independent of the
underlying inter-event-time distributions.
