MALMO, Sweden — On paper, Jeta and her sons, Ardian and Dardan, don’t exist. Sweden’s Migration Agency denied them asylum status two years ago, and the Albanian natives have been hiding here ever since. “I’m scared all the time,” says Dardan, 16. “I don’t dare to talk to the other kids in my class, because I don’t know if they’ll report me to the police.”

The three, who asked that their real names be withheld, are among at least 10,000 rejected asylum seekers living without papers in Sweden. Their numbers are expected to grow. The Migration Agency, which initially struggled last year to cope with the volume of asylum applications, is now working its way through that paperwork. Of the record 163,000 asylum seekers who arrived in Sweden in 2015, up to 80,000 will be required to leave, Sweden’s interior minister, Anders Ygeman, said last week.

While Syrians receive automatic asylum, most migrants from countries deemed safe — such as the Balkans nations, Algeria and Morocco — are turned down. Jeta, who applied for asylum in 2013, said she told immigration officials she feared for her safety because her husband was part of a criminal gang trafficking in women and drugs. But, she says, the Migration Agency official she spoke with told her, “That’s something your country has to sort out, not us.”

Like all unsuccessful asylum applicants, Jeta and her sons received plane tickets back to their home country. But she was determined to stay. An Albanian woman living legally in Sweden took them in, then drove them about two hours from the small Swedish city where they’d been living to Malmö, a larger city, where they were more likely to go undetected. The friend dropped them at a church she’d heard was assisting undocumented immigrants.

According to the Migration Agency’s 2015 figures, of 22,494 unsuccessful asylum applicants, 9,718 left voluntarily while 3,336 refused and were referred to the police. Some of those individuals are from countries reluctant to take them back, such as Afghanistan and Morocco. As a result, the Migration Agency continues to house them until the Swedish government can negotiate a solution. At least 1,606 began asylum processes in other European nations. Another 7,590 simply vanished.

“We have no idea where they are,” says Sverker Spaak, who oversees the Migration Agency’s refugee returns program. “Some may leave the country and apply for asylum in another country under a different name. And some go into hiding.”

The police lack the resources to scour the country for refugees. Still, many of those who stay live in fear of arrest. “I know I can trust the teachers,” says Dardan. “But I don’t dare be with the other students, so I’m usually just by myself in a corner.” Like his mother and brother, he uses a fake name outside their home. It’s a complicated existence, says Jeta, “but it’s better for the boys to have two identities and live than having one identity and be dead.”

The Malmö church helped Jeta enroll her sons in school and obtain health coverage. An Albanian woman living legally in Sweden found them an apartment. (Swedish law guarantees underage undocumented immigrants the same right to education and health care as other children, while adults have a right to emergency health care, as well as abortions and treatment of contagious diseases. A few cities, such as Malmö, even provide rent subsidies.) Jeta and the boys are among 18 undocumented migrants assisted by the church in recent years, says Lena, a church social worker, who asked that her real name be withheld.

Government officials consider such assistance subversive, but not illegal, unless it’s carried out for financial gain, says Michael Williams, vice chairman of FARR, the Swedish Network of Refugee Support Groups. Immigration officials rarely conduct raids on religious institutions. That said, most churches and charities that assist refugees do so quietly, although the convent Alsike, outside Stockholm, has become known for its nuns’ vocal pro-immigrant activism.

Some individuals also take in rejected asylum seekers. Christina and her husband, who asked that their last names be withheld, have housed two refugees in the last few years. One stayed for a year-and-a-half. “The way I see it, we don’t hide them,” she says. “They hide in our house.”

Few undocumented immigrants, however, completely curtail their movements. Jeta takes Ardian to school each weekday and collects him each afternoon. Once a week, she visits the church for Swedish classes and social activities for immigrants.

The Swedish government says it’s confident that the 60,000 to 80,000 asylum seekers who face expulsion will cooperate. It even plans to charter flights for them. But based on the Migration Agency’s own estimates, up to 40,000 will instead choose a life in legal limbo, and vanish. For those refugees, there are few economic prospects. Jeta can’t work legally, and when her sons reach 18, they will have no right to schooling or routine health care.

“Day and night, every second, I think about the future,” says Dardan. “But I can’t figure out what it’s supposed to be like.”