  Continuous-time Markov process models of contagions are widely studied, not
least because of their utility in predicting the evolution of real-world
contagions and in formulating control measures. It is often the case, however,
that discrete-time approaches are employed to analyze such models or to
simulate them numerically. In such cases, time is discretized into uniform
steps and transition rates between states are replaced by transition
probabilities. In this paper, we illustrate potential limitations to this
approach. We show how discretizing time leads to a restriction on the values of
the model parameters that can accurately be studied. We examine numerical
simulation schemes employed in the literature, showing how synchronous-type
updating schemes can bias discrete-time formalisms when compared against
continuous-time formalisms. Event-based simulations, such as the Gillespie
algorithm, are proposed as optimal simulation schemes both in terms of
replicating the continuous-time process and computational speed. Finally, we
show how discretizing time can affect the value of the epidemic threshold for
large values of the infection rate and the recovery rate, even if the ratio
between the former and the latter is small.
