  In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS)
model is a branching process used world-wide to forecast earthquake intensity
rates and reproduce many statistical features observed in seismicity catalogs.
In this paper, we describe a fractional differential equation that governs the
earthquake intensity rate of the pure temporal ETAS model by using the Caputo
fractional derivative and we solve it analytically. We highlight that the tools
and special functions of fractional calculus simplify the classical methods
employed to obtain the intensity rate and let us describe the change of
solution decay for large times. We also apply and discuss the theoretical
results to the Japanese catalog in the period 1965-2003.
