William Olson, the chief of operations for eight Oregon hospitals, grew worried when he was shown a heat map of coronavirus cases and flulike symptoms among patients across seven Western states.

The maps captured trends for patients of Providence, which owns 51 hospitals and shared the results early Monday with its hospital executives. Seattle’s outbreak, depicted in blue dots, was already ballooning. And now Portland had its own blue specks. If the pattern held, his hospitals were about to be severely tested. “That was the...