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"What's happened in this pandemic is we still have that bounty of livestock," Sullivan said. "The linchpin in the middle is the harvest facilities, and there's a relatively small number ... of plants that actually harvest all those animals and turn it into food."
"We're having COVID cases amongst our employees, and when you get COVID cases in a plant, we're having absenteeism and we're having difficulty operating the plant," he continued.
The National Pork Producers Council wants the Department of Agriculture to make $1 billion in pork purchases, but Sullivan said there would be no meat for the USDA to buy if processing plants close.
"We have to be able to process those pigs in the plants," he said. "When the plants don't operate, we don't even have the meat to sell."
The closure of a Smithfield Foods plant in South Dakota, one of the largest pork processing plants in the U.S., after hundreds of employees tested positive for coronavirus caused Sullivan to warn of a possible meat shortage.
Smithfield closed the plant after pressure from South Dakota Gov. Kristi Noem and Sioux Falls Mayor Paul TenHaken.
The plant, which employs about 3,700 people in the state's largest city, has become a hot spot for infections.
Smithfield announced a three-day closure last week so it could sanitize the plant and install physical barriers to enhance social distancing. But on Sunday, it announced the plant's indefinite closure. It will resume operations in Sioux Falls after receiving further directions from local, state and federal officials.
The American Association of Meat Processors expressed concern over the Sioux Falls plant's closure.
"Normally the closure of a plant would not have a huge impact on supply, but the demand on meat and poultry during this pandemic has been vast," AAMP Executive Director Chris Young told FOX Business. "There is a concern that more companies could end up in the same situation as Smithfield. ... Many companies are screening employees and others who enter their plants on a daily basis, as well as trying to follow CDC guidelines for social distancing when they can."
The Associated Press contributed to this report.