What a great question! In short, the answer is 399 B.C.
A suitable place to begin is in the winter of 399. After the Roman victory over the Volscian town of Anxur, Livy, the great historian of Rome, states that the winter was remarkably cold and snowy to the point where roads were blocked and the Tiber became unnavigable.
The summer did not provide much relief, as the summer was also so extreme in heat that it was deadly for all living things. Due to this extreme swing in temperature, the senate voted to consult the Sibylline Books, something that the Romans periodically did.
The outcome was the first celebration of a lectisternium to seek peace with the gods. For eight days the Romans made sacrifices to and spread out couches for Apollo, Latona, Diana, Hercules, Mercury, and Neptune. Livy emphasizes the expansive celebration of the lectisternium by noting that all throughout the city doors stood wide open to all; even prisoners were let free, just in case the gods had befriended them.