The accounts in Josephus indicate that the temple (and Jerusalem) lay desolate FIFTY years (Apion 1:21), which fifty years were a part of the SEVENTY years of SERVITUDE (Antiquities 11:1:1). Apion 1:19 does NOT contradict this; it confirms that the fifty year's period of desolation occurred "during the interval of seventy years". The prophecies in Jeremiah do not indicate seventy years of "desolation", nor of "deportation" or 'captivity' (whether of a specific group, or viewed in the aggregate), but of servitude, i.e. "vassalage". Note that the prophecies as recorded in Jeremiah do not even require deportation and exile at all: pls cf. Jer. 27:7,8,11-13,17; 38:17,18.
The SEVENTY years of SERVITUDE (Antiquities 11:1:1) run from 609 B.C.E. (following the Assyrian Empire's crushing defeat at Haran and the subsequent vassalage of Judea to Babylon) until 539 B.C.E. (when Babylon itself was defeated). The FIFTY years of DESOLATION for the Temple (Apion 1:21) run from 587 B.C.E. (when Jerusalem and its Temple were destroyed) until 537 B.C.E. (when the foundations of the Temple were laid [second year of Cyrus]). The several deportations occurred at various times during the seventy years, but the SERVITUDE of Judea as a nation need not begin with any of the deportations as recorded in the Bible.