SAN BERNARDINO, California (CNN) -- Three-year-old Briant Rodriguez remained in a hospital bed fatigued and malnourished Monday, 15 days after being kidnapped at gunpoint and then dumped on the streets of Mexico.

Liberato Vega, 30, left, and Israel Moreno, 28, are suspects in the kidnapping of 3-year-old Briant Rodriguez.

Police don't know much about what happened to the boy, who was taken after gunmen tied his family and ransacked his California home on May 3.

They don't know why the gunmen broke into the home, why they kidnapped the bubbly 3-year-old or how he ended up wandering the streets of Mexicali with a shaved head, rather than the long hair he had sported before the kidnapping.

But police believe they do know who is responsible.

In a news conference on Monday, the San Bernardino Sheriff's Department and FBI identified the two suspect gunmen as Liberato Vega, 30, and Israel Moreno, 28.

The two men, who authorities said were illegal immigrants with criminal records, are believed to have burst into the San Bernardino home, ransacked the house, tied up 3-year-old Briant Rodriguez's family and snatched the child two weeks ago.

The boy was returned to his family Saturday in the border town of Calexico.

"It was a very emotional and unforgettable experience," said San Bernardino County Sheriff's Sgt. Doug Hubbard, who was with the boy's mother when he was returned. "Enough said there -- before I get emotional."

Officials said the boy was still in the hospital Monday and was being treated for malnourishment and fatigue.

"We're very happy that he's alive," San Bernardino County Sheriff Rod Hoops said this weekend. "A 3-year-old goes missing in this country for two weeks -- sometimes it has an unhappy ending.

"This one did not."

Now, police are trying to piece together exactly what may have happened during the 13 days the boy was missing.

Detectives from the San Bernardino Sheriff's Department and Federal Bureau of Investigations agents were able to identify the alleged kidnappers based on unspecified physical evidence at the San Bernardino home and interviews with the boy's family and neighbors.

Hubbard said the family has adamantly denied knowing who may be responsible for the kidnapping or having any association with those responsible.

FBI Supervisory Special Agent Don Roberts said it was, however, "likely that this family was the target" of the kidnappers.

"But why, we don't know," Roberts said, adding that the kidnappers had clearly planned the crime.

Officials said they believe Vega and Moreno are the two men they captured on video at a home-improvement store near the Rodriguez home before the crime buying tape like the kind that was used to bind the family.

Hubbard said there are no-bail warrants issued for both men for home invasion robbery and kidnapping and Roberts said the FBI also obtained federal warrants for unlawful flight to avoid prosecution for both men.

"We will leave no stone unturned until these two are returned to San Bernardino to face justice," Roberts said. "We will not take a break until that day happens."

Authorities are hunting for both men, who have criminal records, in Mexico where they were both born. Roberts said both men have previously been deported from the United States, where they were living illegally. Their previous residences are empty, police said.

Vega, who police said they considered to be the primary suspect, has had seven encounters with authorities in San Bernardino, according to court records, including four convictions in 2005 for a variety of charges, including driving under the influence. He also has a dismissed burglary case and two open cases against him for driving under the influence, providing false information to police and driving without a license.

Police described Vega as a Hispanic male, 5 feet 9 inches weighing 130 pounds.

Moreno was convicted in 2008 for driving under the influence and has another similar case open against him. Police described Moreno as a Hispanic male, 5 feet 6 inches weighing 150 pounds.

Police have said they are also looking for 21-year-old Claudia Acosta, Vega's girlfriend, who may be traveling with him. They aren't sure how she's related to the kidnapping.

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