From an article on the Issues Etc website. By Dennis Kastens on infant baptism.
Also from this era are headstones for children who received emergency baptism with ages ranging from 11 months to 12 years. Since the patristic sources of the third century, as those earlier, give us to understand that the children of Christian parents were baptized in infancy, we must conclude that these emergency baptisms were administered to children of non-Christians. The inscriptions themselves confirm this conclusion. In the Roman catacomb of Priscilla is reference to a private emergency baptism that was administered to the one-and-three-quarter-year-old Apronianus and enabled him to die as a believer. The inscription reads:
Dedicated to the departed.
Florentius made this inscription
for his worthy son Apronianus who
lived one year and nine months and five days.
As he was truly loved by his grandmother
and she knew that his death was imminent,
she asked the church that he might depart from the world as a believer.