It is quite probable that Pierius was not fleeing from Alexandria because of the persecutions but because of his possible troubled relations with Theonas' successor and the new patriarch, Peter. He was appointed bishop of Alexandria in either 300 or 303, and is not mentioned by Jerome as Pierius' ecclesiastical superior. The persecutions of Christians in Egypt were not that severe, and Peter is known to be in conflict with some of his clergy and other Egyptian bishops (Wipszycka 2015: 75-95).
Pierius is also mentioned by Eusebius of Caesarea in the Ecclesiastical History (7., but there is no mention there of Pierius' move to Rome.