BERGAMO, Italy—Ambulances here have stopped using sirens. The frequent blaring only adds to local fears. Besides, there are few other vehicles on the road in Italy’s national lockdown.

Most are headed to the Papa Giovanni XXIII Hospital, a large, modern medical facility in a prosperous Italian city that has been overwhelmed by the coronavirus disease. There aren’t enough ventilators to intubate all patients with Covid-19 who have severe breathing trouble. The intensive-care unit is taking almost no patients older than 70,...