DALLAS >> Airlines are piling up billions of dollars in additional losses as the pandemic chokes off air travel, but a recent uptick in passengers, however modest, has provided some hope.

American Airlines on Thursday reported a loss of $2.4 billion and Southwest Airlines lost $1.16 billion in the third quarter, typically a very strong period of air travel that includes most of the summer vacation season. Revenue plunged at both airlines.

Combined with earlier losses reported by Delta and United, the four largest U.S. airlines have lost at least $10 billion in each of the last two quarters. It’s an unprecedented nosedive that has caused the once highly profitable airlines to forage for billions of dollars in government aid and private borrowing to hang on until more travelers return.

The airlines, however, are offering upbeat forecasts about Thanksgiving and Christmas travel, even as many states report an increase in confirmed virus cases. Southwest feels confident enough that it will stop blocking middle seats on its planes Dec. 1.

Southwest and Delta are among a handful of airlines still limiting capacity to reassure passengers who are nervous about packed planes during a pandemic. Southwest said it will drop that policy because of “science-based findings from trusted medical and aviation organizations” about how COVID-19 is spread. Airline groups and aircraft manufacturers, relying in part on research by the military, say that strong cabin air flow and high-efficiency filters make planes safer than other indoor settings.

There is a financial angle too, as flights become more full. Southwest said it turned away $20 million in ticket sales in September and estimated it could lose $40 million in November. Delta plans to stop blocking seats in the first half of next year. American and United have long tried to fill every seat.

Southwest President Tom Nealon said that a recent increase in confirmed COVID-19 cases doesn’t seem to be hurting holiday bookings. That’s a sharp contrast to July, when a spike in the virus across the Sun Belt choked off a rise in travel.

“Perhaps because you’re not seeing as many hospitalizations and deaths, although both are rising right now, I get that,” Nealon said. “But perhaps people haven’t been as concerned about it as they were in the past.”

Officials at many airlines believe travel won’t fully return until the pandemic is under control and a vaccine is widely available. That could mean many more months of depressed revenue for the airlines.

Southwest CEO Gary Kelly on Thursday urged Washington to approve more pandemic relief, including a six-month extension of $25 billion in aid to airlines. Without it, he said, his airline can’t afford to maintain jobs and full pay.

Southwest plans to cut pay for nonunion workers by 10% in January and has demanded unions accept lower pay or risk furloughs.

Air travel in the U.S. has recovered slowly in recent months, topping 1 million daily passengers on Sunday for the first time since March. However, air travel in October is still down 65% from a year ago. Business travelers, who fly more often and pay higher fares, are still mostly absent.

“Really the difference is going to be when business travelers start hitting the airways again, something that is modestly starting up but is nothing close to what we need,” American CEO Doug Parker told CNBC.

Delta CEO Ed Bastian said recently that 90% of his airline’s corporate customers are letting some employees travel, but only a small number, and business travel is down 85% at his airline. Southwest believes business travel could still be just 50% to 60% of normal by the end of next year, and up to 20% might be lost forever, Nealon said.

With business travelers sidelined, airlines are more dependent on price-conscious leisure flyers. The average Southwest customer paid 20% less than a year ago.

The airlines are slashing costs to survive on a fraction of their usual revenue. They have cut workforces by convincing thousands of employees to leave — and in the case of American and United, furloughing 32,000 workers between them this month. The airlines are still hoping for another $25 billion from Congress and the White House that American said would allow it to recall furloughed employees.

American, based in Fort Worth, Texas, said its loss after one-time gains and losses was $5.54 per share. That was better than Wall Street feared, as analysts surveyed by Zacks Investment Research forecast a loss of $5.62 per share. Revenue tumbled 73% to $3.17 billion, above the analysts’ prediction of $2.8 billion.

American is burning through cash — $44 million a day in the third quarter — at a much faster rate than Delta, United or Southwest. American signaled Thursday that it will raise another $1 billion by issuing new shares and will defer delivery of 18 Boeing jets to conserve cash.

Southwest’s adjusted loss came to $1.99 per share, also better than expected. Analysts had forecast a loss of $2.44 per share for the Dallas carrier. Revenue dropped 68% to $1.79 billion; analysts predicted $1.68 billion.

Alaska Air Group Inc. reported a loss of $431 million. The Seattle company said that removing one-time gains and costs, the loss came to $3.23 per share, which was wider than the $2.86 per share loss predicted by analysts. Revenue dropped to $701 million, slightly better than Wall Street expected.

Earlier this month, Delta Air Lines posted a $5.4 billion loss and United Airlines lost $1.8 billion for the third quarter.

Airline shares moved higher Thursday. Southwest ended the day up 5.3%, Delta rose 7.1%, United 4.8%, and American 3.2%.