Smithfield said the complaint was without merit. “The health and safety of our employees is our top priority at all times,” said Keira Lombardo, the company’s executive vice president for corporate affairs and compliance. She cited a policy of not commenting on pending litigation, but she said the accusations “include claims previously made against the company that have been investigated and determined to be unfounded.”

Smithfield has shuttered a plant in Wisconsin and a plant in Martin City, Mo., in addition to a South Dakota slaughterhouse that employs hundreds of workers who have been infected by the virus. Tyson Foods has closed plants in Indiana, Washington and Iowa, one of which has reopened, and plants owned by other companies in Minnesota and Illinois have experienced outbreaks.

The Centers for Disease Control and Prevention, which toured the South Dakota facility last week, recommended Thursday that Smithfield establish more social distancing barriers and possibly slow down the production line there to create more space between workers.

Beyond seeking to make workers safer, the complaint about the plant in Milan, Mo., is testing whether public nuisance law dating back hundreds of years can be used to protect workers on the job. The plaintiffs argue that Smithfield, by failing to take adequate safety measures, risks a coronavirus outbreak that could quickly spread to the entire community.

“It exists in any state — the idea of bringing the public nuisance,” said Karla Gilbride, a lawyer with Public Justice, a legal advocacy group that has worked with the Smithfield workers in Milan for several years and is helping to bring the complaint.

“If, whether it’s a private company or a private citizen, they’re operating something on their property and whatever they’re doing is unsafe and poses a danger to the entire community,” Ms. Gilbride said, “then the public has a right to safety and health.”