WUHAN, China—Coughing badly, Zhu Chunxia sat on a sidewalk in the rain on Monday, awaiting transport to a facility where her apartment complex’s residential committee said she could be treated for the new coronavirus sweeping through this central Chinese city.

The ride never came. Though her doctor was almost certain she was infected with the virus, a throat-swab test she had taken came back negative, which meant the facility wouldn’t take her.

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