Broadway, a hallmark New York industry that drew 14.8 million patrons and grossed $1.8 billion last season, is vulnerable to economic damage from the outbreak for multiple reasons: Its audience skews older, and older people seem especially vulnerable to this virus; its audience is heavily made up of tourists, and travel is drying up; and its events involve large numbers of people packed into tight spaces — a situation risky enough that it is being banned in some countries.

On Wednesday, the owners of two theaters said that a part-time usher who worked for them had tested positive for the virus. No other worker had fallen ill, but the owners asked audience members and employees who were present at the same performances to monitor their health.

Broadway’s leaders say they are determined to keep their theaters open if at all possible, and anticipate that they would collectively close only if ordered to do so by a government agency. That is not unthinkable: Some performance venues have been closed in Austria, Germany and Italy, among other places.

On Thursday, Mayor Bill de Blasio said he was trying to avoid closing theaters, but also said they might need to cut down on audience size if they wanted to stay open.