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

A fractional approach to study the pure-temporal Epidemic Type Aftershock Sequence (ETAS) process for earthquakes modeling