How did early childhood baptisms started in Catholicism? Jesus got baptized at an age old enough to decide for himself.

Today, I always see infants getting baptized. According to Catholicism, when, where and how did early childhood (infant) baptisms got started?

Was there a mandate from the Vatican that imposes infants need to be baptized as soon as possible?
The earliest recorded instance of child baptism is in the Acts of the Apostles when Lydia, a seller of purple of the city of Thyatira was baptized along with her household (Acts 16:14).  Other instances of child baptism recorded in the New Testament are in Acts 16:33, Acts 18:8, and 1 Corinthians 1:16.  

This is all in accordance with the commandment given by Peter that all, including children, be baptized in the name of Jesus Christ for the remission of sins (Acts 2:38-39).