Anupam Kumar, a critical-care doctor who specializes in treating damaged lungs, sees about a dozen extremely sick patients with Covid-19 on a typical shift.

Like most doctors, he is anxious he could become infected with the new coronavirus that causes Covid-19 and carry it home to his family. But unlike most other U.S. health-care providers, he has another worry: Infection could also lead to the deportation of him and his family.

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