BOSTON — Zeituni Onyango came to the United States seeking asylum from her native Kenya but was turned down and ordered to leave the country in 2004.

Four years later, she is still here. And her nephew is about to become president of the United States.

Onyango’s family connection to Barack Obama has thrown a spotlight on a phenomenon many Americans might find startling: An estimated half-million immigrants are living in the United States in defiance of deportation orders.

U.S. Immigration and Customs Enforcement has stepped up efforts to catch fugitive aliens, as they are known, and now has about 100 “fugitive operations teams” around the country. In the past year, the teams have made 34,000 arrests, more than double the number two years ago. But there are still 560,000 such immigrants in the U.S.

Fugitive aliens include people who, like Obama’s aunt, sought asylum in the United States but were rejected and ordered to leave the country. Others were caught entering or living in this country illegally, and failed to show at their deportation hearings.

Often, illegal immigrants who have been issued deportation notices are given a certain amount of time to get out of the country on their own. They are not forcibly put aboard a plane; these deportations essentially operate on the honor system.

Generally, if these immigrants stay out of trouble — if they don’t get pulled over by police or swept up in a workplace raid, for example — they are in little danger of being thrown out of the country.

That galls many immigration reform advocates, who say the practice breeds disrespect for the law and emboldens immigrants to sneak in and stay.

“We are strong believers of enforcement of our immigration laws, and this is a priority area for getting the message across to this country, that if they’ve been convicted of committing crimes or if they have been ordered deported, that they will be apprehended if they try to hide and continue to stay in the country,” said Jack Martin of the Federation for American Immigration Reform.

Government officials say that they do the best they can with the money and manpower available to them, and that they focus on the most serious cases, including those involving illegal immigrants who have committed crimes in this country.

“ICE has taken tremendous steps at closing these cases and apprehending fugitives,” spokesman Richard Rocha said. “However, we prioritize our efforts on egregious violators and criminal aliens.”

Overall, there are an estimated 11 million illegal immigrants living in the U.S. In the last year, the government arrested and deported a record number of illegal immigrants, nearly 350,000, according to ICE.

Critics of the agency complain of the government’s former “catch and release” policy along the U.S.-Mexico border, in which non-Mexicans caught sneaking across were released into this country with a date to appear for an immigration hearing. Officials ended the practice in 2006. Now, these immigrants are held until their hearings.

After paying smugglers $40,000 to get his family to the United States, Juan, his wife and 3-year-old son were caught the moment they crossed into the U.S. from Mexico. A judge ordered the Bolivian family deported, released them on bond and gave them three months to leave.

Nine years later, they are still living in the United States. But they avoid going out in public and refuse to drive for fear of getting pulled over by police.

The wondering is painful, Juan said. But he said that is better than going back to Bolivia, where their financial prospects are bleak.

“For me, the best chance is to provide education to my children, and that’s something that I can do for them here,” said Juan, a 38-year-old construction worker in Maryland who supports two older daughters in Bolivia. He asked that only his first name be used to protect his family.

Advocates say many immigrants defy deportation orders because they have lived in the United States for years, married, had children and put down roots in their communities.

"Is it worth going around with that feeling that you might be discovered? Or packing up your entire family now and settling all your obligations in the United States, buying airplane tickets for your family, moving back to a country where you haven’t lived for many years, where you are worried about you are going to find a job? There’s another whole set of uncertainties,” said Maureen O’Sullivan, a Boston immigration lawyer.

It is not clear when Onyango, the 56-year-old half-sister of Obama’s late Kenyan-born father, first came to the United States. But she moved into a state-subsidized public housing project in Boston in 2003.

After it was reported days before the election that she was in the country illegally, Onyango left Boston and went to Cleveland, where she hired an immigration attorney to fight her deportation order. She is staying with relatives in Cleveland, said her new attorney, Margaret Wong.

The Obama camp has said the candidate did not know about his aunt’s status. “If she is violating laws, those laws have to be obeyed,” Obama said just before the Nov. 4 election.

Advocates say the only way to reduce the number of illegal immigrants is to overhaul the nation’s immigration laws.

“I ask the new president, I implore the new president to provide legislation that would allow us to become legal, to have papers,” Juan said. “I don’t mind paying fees, but I want to come out of the shadows."