NEW DELHI—For many countries, battling the coronavirus has been a sprint to bring infections under control, then a struggle to keep it that way. In India, it has turned out to be a marathon.

No one knows that better than Dr. Santhosh Kumar, a 48-year-old infectious disease specialist who treated some of the country’s first infections back in February. He is still fighting the virus, these days building a 10,000-strong corps of health-care workers and volunteers in southern India to tackle yet another wave of infections.

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