  Using the simple ETAS branching model of seismicity, which assumes that each
earthquake can trigger other earthquakes, we quantify the role played by the
cascade of triggered seismicity in controlling the rate of aftershock decay as
well as the overall level of seismicity in the presence of a constant external
seismicity source. We show that, in this model, the fraction of earthquakes in
the population that are aftershocks is equal to the fraction of aftershocks
that are indirectly triggered and is given by the average number of triggered
events per earthquake. Previous observations that a significant fraction of
earthquakes are triggered earthquakes therefore imply that most aftershocks are
indirectly triggered by the mainshock.
