HOUSTON — Two weeks ago, Houston Methodist Hospital opened a special unit to treat critically ill coronavirus patients. The city had reported only a handful of confirmed cases at that point, but the hospital’s 24-bed coronavirus ICU filled up in only about a week, far faster than doctors anticipated.

Alexandra Carnahan, 26, one of the nurses assigned to the unit, was surprised by the number of patients who were in their 30s or 40s, with no prior health problems. Now they were intubated and in critical condition, suffering from COVID-19, the respiratory disease caused by the coronavirus.

These patients can’t have visitors, and most are heavily sedated. So Carnahan and other nurses hold their hands while checking vitals, “to maintain that human connection,” she said. The nurses remind them where they are and try to explain why their families can’t be there, though it’s difficult to know whether the patients can hear them.

A few days ago, one of them, a man in his 50s, unexpectedly squeezed Carnahan’s hand while she and another nurse bathed him, she said. He apparently could hear her voice, and seemed to be reacting. Carnahan told him everything was going to be OK. Afterward, she pulled off her protective mask and face shield, stepped into another room, and cried.

The man was around her parents’ age.

“Seeing that is frightening,” Carnahan said, “knowing that it could easily be us or our loved ones in the hospital bed.”

So far, Houston, the nation’s fourth largest city, has not recorded a swell of confirmed coronavirus cases similar to those straining health care systems in New York and Detroit, giving some residents hope that stay-at-home orders issued by local officials came early enough to stop the virus from spreading too widely here. But interviews with Houston ICU doctors and nurses — as well as the daily rise in the number of critically ill patients now showing up in their hospitals — paint a more sobering picture of what’s happening in southeast Texas.

While limited testing has prevented officials here and across the state from getting a complete accounting of how many people have been infected with the coronavirus, hospitals in greater Houston have seen a steep increase — 40 percent over four days — in the number of patients believed to be suffering from the virus, according to a daily census of hospital admissions collected by the SouthEast Texas Regional Advisory Council, which helps coordinate emergency response efforts.

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The 25-county region surrounding Houston had reported fewer than 950 confirmed coronavirus cases among its 9.3 million residents as of Monday. But on that same day, there were 996 people hospitalized in the region with confirmed or suspected cases of COVID-19, according to the advisory council count. That figure is not cumulative and does not include patients with COVID-19 who were hospitalized and released before Monday.

Dr. Peter Hotez, an infectious disease expert and dean of the National School of Tropical Medicine at the Baylor College of Medicine in Houston, said the disconnect between the number of positive test results and the number of people hospitalized for COVID-19 suggests that a significant majority of coronavirus cases are going undetected here.

“We know that about 1 in 5 or 1 in 10 COVID-19 patients needs to be hospitalized, so what you’re seeing in hospital admissions would suggest that the actual number of cases could be 5 to 10 times higher in Texas than what’s been reported to this point,” he said.