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In this file photo, travelers walk through John F. Kennedy Airport on February 28, 2013 in New York City.

Uruguay's banking crisis had already claimed Rosana Araujo's savings when she and her family, tourist visas in hand, boarded a Miami-bound plane in Montevideo. She stayed in Miami after her visa expired, supporting herself on random jobs, cleaning homes or babysitting.

"I feel lucky because I got here in a plane," she said. "I feel lucky because I didn't have to cross a desert."

While the national discussion on immigration focuses on securing the U.S.-Mexico border with a multibillion dollar wall, travelers such as Araujo who overstay their visits have become the main source of illegal immigration in the United States.

The Homeland Security Department said 527,127 people more than the population of Atlanta who entered the U.S. by plane or ship, not land, and were supposed to leave the country in the 2015 fiscal year overstayed. Demographers estimate that two-thirds of foreigners who arrived in 2013 and are now in the U.S. illegally were admitted with valid travel documents, the New York-based Center for Migration Studies Found.

The administration's directives expanding immigration enforcement affect all immigrants in the country illegally, but they focus on those who cross on land, which is the smaller share of newly arrived immigrants in the country illegally. Nonetheless, the U.S. is preparing to build a wall along the 2,000-mile southern border estimated to cost somewhere between $8 billion and $20 billion. An additional 5,000 Border Patrol agents are to be added at additional cost.

The report estimated that the number of people crossing the border illegally fell from 400,000 in 2000 to 140,000 in 2013. Customs and Border Protection said 12,193 people were caught trying to enter the United States illegally from Mexico in March for the second straight monthly decline in arrests at the border. The agency hadn't reported fewer arrests in a month in at least 17 years.

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Robert Warren, author of the report, says President Donald Trump's immigration orders and plans on the border don't reflect current migration patterns.

"Overstays have been steady for the past 10 or 12 years, but the illegal entries into this country are at a level we haven't seen in 20 or 30 years," the demographer said. "Building a wall across the entire southern border is a statement of policy failure."

The new executive orders say little about visa overstays, in part because the U.S. government can't confirm the total number of travelers who remain in the country after their visas expire. Airlines and vessels report departures to the Department of Homeland Security. But foreigners who leave in vehicles through the ports of entry along the borders are not accounted for because of "major physical infrastructure, logistical and operational hurdles" a January 2016 Department of Homeland Security report said.

Recent high-profile cases of immigrants who are detained or go public with their stories show a different, more common side of illegal immigration.

In July 2015, Zully Palacios flew from her native Peru into Houston and made her way to Vermont to work at a lodge. She faces deportation after immigration officers detained her. About 100 people gathered in Burlington to protest her arrest last month, saying she was unfairly targeted for advocating for dairy-farm workers.

"We are all defending one another. We all should have the right to live peacefully without hurting anybody. We are coming to this country to contribute," said Palacios, who was freed on bond.

Statistics aside, overstays aren't easily tracked. Immigration officials don't monitor travelers to detect when they overstay their visits, and even when return tickets are required for entry, "the planes don't present a 'no-show' list," Orlando immigration lawyer Carlos Colombo said.

"A lot of people fall into limbo, and fall out of status because they thought they were going to be able to do one thing and they find out they actually can't," Colombo said.

People who overstay their visits worry about penalties they might face if they leave and try to return. If a foreigner overstayed his visa for more than one year, he could be banned from returning for 10 years. Immigration officials at Miami and other airports have sent back travelers who had previously overstayed their visits.

But the determined find weaknesses in the immigration system. Some have gone back home, applied for and received new visas from the State Department and successfully returned to the U.S., something lawyers say is caused by a lack of information exchange among government agencies.

Hyun Kim arrived in the U.S. from South Korea 18 years ago with his parents, who simply decided to stay. He's now 20, and more concerned about being allowed to work than being caught by officials. Until after Trump's victory, he didn't know to apply for the Deferred Action for Childhood Arrivals program, which allows some immigrants brought into the country as children to be shielded from deportation and allowed to work.

"My biggest fear is not deportation. It's how I am supposed to make a living here," said Kim, who hopes to go to college for computer science or engineering but works as a waiter in Virginia. "I never imagine living in Korea. But I can't attend university because I can't afford it."

The obstacles those who overstay face depend on the state where they live. Some can't get a driver's license or in-state tuition rates. Many, such as Araujo of Uruguay, work random jobs cleaning homes or babysitting and avoid airports and law enforcement.

In a recent trip to New York, she traveled by car with friends from Miami instead of flying. She wanted to ride the Staten Island Ferry to cruise by the Statue of Liberty but held off when she saw Homeland Security vehicles parked near the terminal and heard officials were screening passengers.

She hopes she'll have another opportunity someday to lay her eyes on the iconic symbol of freedom.

Associated Press writer Elliot Spagat in San Diego contributed to this report.

Published at 9:10 AM PDT on Apr 13, 2017 | Updated at 9:18 AM PDT on Apr 13, 2017

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Border Plan Overlooks Driver of Illegal Immigration: Visas - NBC 7 San Diego