By Agence France-Presse
The seeming virus "flare-up" killed one of the boys, a 15-year-old, according to research findings published in The Lancet Infectious Diseases medical journal.
The boy tested positive for Ebola in Monrovia, Liberia, on November 19, 2015, and died four days later, it said.
His father and eight-year-old brother also tested positive for Ebola RNA -- molecules that convey DNA instructions -- and experienced symptoms such as tiredness, headache and fever. Both recovered.
The mother was also tested but had no traces of Ebola RNA, said the research team.
However, she carried Ebola antibodies, indicating she had previously been infected and that her body had launched an immune response to fight off the virus.
Further tests "suggested that she survived an acute Ebola virus infection in July 2014," said a press statement.
The virus appears to have "re-emerged" after the woman gave birth to a fourth son, "making the infection flare up in October 2015, then transmitting the virus to her family," it added.
Liberia was declared free of human-to-human Ebola virus transmission in May 2015.
The virus carried by the father and two sons appears to have been a "continuation" of the 2014 outbreak, when the mother was infected while taking care of her brother who died of Ebola.
The couple's baby boy also had Ebola antibodies, which he likely got through his mother's milk, the researchers said.
This was the first known case of transmission from a "persistently infected" Ebola survivor, the team reported.
It was already known that the virus, which caused a major outbreak in Sierra Leone, Liberia and Guinea in 2013-15 in which more than 11,300 people died, can persist for many months in semen, in the eye, and in breast milk of women infected while pregnant.
The World Health Organization has said that transmission by people who beat infection is "rare".
"Exactly how the virus was passed from family member to family member remains unclear, but the authors suggest it could be close physical contact or contact with body fluids," said the journal statement.
Ebola was first identified in what was then Zaire -- now the Democratic Republic of Congo -- in 1976.
Since then, there have been several outbreaks, of which the West African outbreak was the deadliest by far.
Often contracted from eating infected bushmeat, the virus can be transmitted from person to person via the blood, certain body fluids, or organs of an infected or recently deceased person.
Transfer can happen by touching a sick or dead person, and likely also sexual intercourse, research has shown.
On average, the virus kills about half of the people it infects.