Many workers in the logistics and delivery industry get at most a handful of paid sick days a year. That can push them to work through coughs and fevers. Now, with so many people and businesses relying on a functioning delivery system, the pressure to go in has only intensified, workers said.

A package sorter in a UPS facility near San Francisco said she had been going into work every day, even though her “throat feels like broken glass.” The worker, who requested anonymity, said she was petrified that she would lose her job if she called in sick. “I can’t afford to be homeless,” she said.

Another UPS package handler, Jeff Donigian, who works in Manchester, N.H., said he had been showing up to work feeling ill and coughing. He told a supervisor; he was told to go in anyway.

A UPS driver with a compromised immune system said she had been hauling packages around Northern California despite feeling sick for a week. With an avalanche of orders, her supervisor told her that UPS could not afford to lose her, even for a couple of days.

“While it is possible to find an exception, our work force has been provided information and supplies to manage health risks,” said Steve Gaut, a UPS spokesman.

He said that the company had “substantially increased cleaning and disinfecting surfaces throughout our facilities,” and that the vehicles and equipment were disinfected every day. Masks are being made available to drivers who made deliveries to health care and assisted-living facilities, and the company is “distributing an additional 250,000 bottles of hand sanitizer.”

Mr. Gaut declined to comment on the experiences of the drivers or workers handling packages, but emphasized that UPS did not want sick employees to go into work, adding that the company provides paid time off for those who are infected.