Driving one evening in Wuhan, China, last month, Zhang Xiaochun pulled her car to the side of the road. She was on the verge of a breakdown.

She’d been working nonstop for days at the center of China’s coronavirus outbreak, where she is a doctor. Both of her parents had Covid-19, the disease caused by the virus, as did many of her colleagues. The number of sick and dying was climbing. And on this day, Dr. Zhang had forgotten about her 9-year-old daughter, who was home alone and scared—and who, at that moment, wasn’t picking...