  The concurrency of edges, quantified by the number of edges that share a
common node at a given time point, may be an important determinant of epidemic
processes in temporal networks. We propose theoretically tractable Markovian
temporal network models in which each edge flips between the active and
inactive states in continuous time. The different models have different amounts
of concurrency while we can tune the models to share the same statistics of
edge activation and deactivation (and hence the fraction of time for which each
edge is active) and the structure of the aggregate (i.e., static) network. We
analytically calculate the amount of concurrency of edges sharing a node for
each model. We then numerically study effects of concurrency on epidemic
spreading in the stochastic susceptible-infectious-susceptible and
susceptible-infectious-recovered dynamics on the proposed temporal network
models. We find that the concurrency enhances epidemic spreading near the
epidemic threshold while this effect is small in many cases. Furthermore, when
the infection rate is substantially larger than the epidemic threshold, the
concurrency suppresses epidemic spreading in a majority of cases. In sum, our
numerical simulations suggest that the impact of concurrency on enhancing
epidemic spreading within our model is consistently present near the epidemic
threshold but modest. The proposed temporal network models are expected to be
useful for investigating effects of concurrency on various collective dynamics
on networks including both infectious and other dynamics.
