In early March, hand sanitizer started flying off the shelves. Then it became nearly impossible to find toilet paper in many stores.

Now, as several of the nation's largest meat processing plants have been forced to temporarily close due to the coronavirus pandemic, meat producers are warning of potential shortages if the crisis continues to disrupt operations for an extended period of time.

On Sunday, Smithfield Farms, one of the nation's largest pork producers, announced it was closing one of its meat processing plants in South Dakota. But it wasn't just any plant. This particular facility in Sioux Falls is responsible for 4-5% of all U.S. pork production (according to figures provided by the company). It will be closed for at least two weeks.

The closure comes after at least 300 of the facility's 3,700 employees tested positive for the novel coronavirus. The Virginia-based company is owned by China's WH Group, which is the world's largest pork producer, supplying meat for brands like Smithfield and Nathan's Famous.

“The closure of this facility, combined with a growing list of other protein plants that have shuttered across our industry, is pushing our country perilously close to the edge in terms of our meat supply. It is impossible to keep our grocery stores stocked if our plants are not running," Kenneth M. Sullivan, Smithfield's president and chief executive officer, said in a statement released Sunday. "These facility closures will also have severe, perhaps disastrous, repercussions for many in the supply chain, first and foremost our nation’s livestock farmers."

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Now, a primary concern for Sullivan and other meat processors is that many American pig farmers have nowhere to send their animals for processing and may face circumstances similar to that of produce farmers who, after seeing demand from shuttered restaurants plummet, are struggling to get their surplus to new markets.

So should consumers be worried about an impending shortage of pork and other meats?

An employee handles sides of pork on a conveyor belt at a Smithfield Foods Inc. pork processing facility. Smithfield Foods is one of the nation's largest suppliers of pork products, including chops and bacon. Bloomberg via Getty Images

"Prior to March, a large percentage of pork products were produced and sold to restaurants," Julie Niederhoff, associate professor of Supply Chain Management at Syracuse University, told TODAY. "This left a fairly stocked pork supply chain where the the temporary closure of one plant isn't likely to impact consumers nearly as much as it impacts farmers."

Niederhoff is currently predicting a minor short term impact on the price and availability of pork. But that's only if the plant is closed for a few weeks. A longer closure could be detrimental if there's a domino effect throughout the company, she said.

"Nearly 60% of pork is processed in 15 plants all in close geographic proximity to this Smithfield plant," Niederhoff said. If COVID-19 forces more plants in the area to close, "consumers would definitely feel it."

Many farmers and industry workers are already feeling it. According to the U.S. Department of Agriculture, meat manufacturing beats out dairy, grains, beverages and produce to account for the largest sector of food and beverage manufacturing in the U.S. The industry is comprised of nearly 500,000 workers.

On Monday JBS USA announced it was closing its Greeley, Colorado, beef plant indefinitely and sending 6,000 employees home to self-quarantine. At least two people who worked at the plant have died from COVID-19 since March and dozens more have tested positive. According to Kerns and Associates, the plant processes about 5% of the total daily beef slaughter in the U.S.

Tyson Foods Inc. and Cargill Inc. have also recently closed plants after workers tested positive for coronavirus. Meanwhile, 50 workers at a Perdue plant walked out on the job after claiming they were exposed to the virus. These events aren't limited to one geographical area, either. Meat processing plant closures are happening in Georgia, Pennsylvania and Iowa, among other states.

According to a report published in March by the Food and Agriculture Organization of the United Nations, the U.S. isn't the only country potentially facing different types of food shortages. Many countries are dealing with decreasing labor forces and logistics problems in their supply chains. For example in Argentina, road blocks are not only preventing the spread of the virus, they're preventing trucks carrying crops from making deliveries. Other countries like Vietnam and Russia are hoarding crops that would otherwise be exported.

"Countries must find best ways to strike a balance between the need to keep production going and the necessity of protecting the workers," read the report.

So far, the U.S. government's response has not been too alarmist. On Friday, President Donald Trump said in a press briefing he's expecting Secretary of Agriculture Sonny Perdue to "use all of the funds and authorities at his disposal to make sure that our food supply is stable and safe."

Jackie Filson, a spokesperson for Food & Water Watch (a nonprofit that campaigns for corporate transparency), told TODAY that any stimulus money should go directly to farmers and growers, not large companies like Smithfield Foods.

"Companies wouldn't be having to contemplate full-scale plant closures if they had prepared adequately for the possibility of a pandemic and responded appropriately by providing employees with personal protective equipment right away," said Filson. She believes the USDA has been turning a "blind eye" to how food production companies operate today, choosing profits over worker safety.

But Keira Lombardo, executive vice president of corporate affairs and compliance at Smithfield Foods, insisted that the company has been following CDC guidelines, including offering 14-day, COVID-19-related leave with pay. She added that the company has also taken additional steps.

"These (measures) include adding extra hand sanitizing stations, boosting personal protective equipment, continuing to stress the importance of personal hygiene, enhancing cleaning and disinfection, expanding employee health benefits, implementing thermal scanning, increasing social distancing, installing plexiglass and other physical barriers and restricting all nonessential visitors," Lombardo said.

Despite Smithfield's latest efforts, on Monday — the same day the company announced it was donating $30 million worth of meat to food banks — six employees at a plant in Pennsylvania reportedly tested positive for coronavirus, Pittsburgh's KDKA reported.

Some experts say it's likely that more employees who process all types of food, not just meat products, will test positive in the coming weeks.

"This crisis emphasizes the need to modernize our entire agriculture and food system with state-of-the art technologies that decrease reliance on a precarious labor force," said Patrick J. Stover, vice-chancellor and dean of agriculture and life sciences at Texas A&M AgriLife.

Scenes of emptying shelves may worry consumers, but Stover told TODAY there is not a shortage of food itself. Empty shelves seen currently are the result of panic buying and he predicts that will taper off as the crisis abates on some parts of the country.

There is, however, the major problem of how to process and deliver food to consumers when so much of the production system has become centralized. With increasingly larger plants housing a vast number of workers who often do their jobs in close quarters, the likelihood of those workers falling ill within the same timeframe is much greater than if those workers were operating in smaller facilities in more isolated parts of the country.

Still, for now, Stover doesn't think U.S. consumers should be worried.

"The strength of the U.S. food system is its diversity, which supports consumer choice and individual health needs," said Stover. "In countries where the African Swine Fever decimated pork production, poultry and other meat sources filled the gap. Consumers enjoy variety in the food they eat, so longterm changes in meat preferences are unlikely."