BOA VISTA, Brazil — As throngs of Venezuelans trek to Brazil in search of food and medicine, hospitals along Brazil's sparsely populated northern border are struggling to pick up the pieces of Venezuela's failing health care system.

A spiraling economic crisis and hyperinflation have emptied Venezuelan hospitals of needles, bandages and medicine. Desperate for care and often undocumented, patients are overwhelming Brazilian emergency rooms as they turn up by the thousands.

“Demand is growing faster than I can manage. Every month the number of patients grows exponentially. How can I plan for that?” said Marcilene Moura, the director of the general hospital in the Brazilian border state of Roraima. “What happens if this continues? I'm going to run out of supplies by the middle of the year.”

The flood of patients is also causing another crisis in Roraima, a poor and remote state that serves as the gateway into Brazil for Venezuelans: Roraima General does not have facilities to care for the bodies of patients who die there. It has resorted to sending corpses to the police station morgue, where they remain for up to three months while authorities try to identify them.

The morgue, intended to house victims of violent crimes, is so overwhelmed that bodies are sometimes stacked two to a drawer, according to an employee who declined to give her name because she was not authorized to speak to the media.

The overcrowded morgue is just one example of how an unprecedented immigration wave from Venezuela is straining Brazil's health care system. Roraima General admitted 100 patients from Venezuela in September. By December, that number had doubled.

To deal with the influx of patients, the hospital has transformed waiting rooms into patient wards, cramming in as many beds as will fit. But demand is so high that patients strapped to IVs lay in beds that spill out into the corridors.

Moura said that she is blowing through the hospital's annual budget in six months. Forty percent of the patients in her intensive-care unit are foreigners, and each costs her up to $1,500 a day.

Brazilian hospitals end up caring not just for the Venezuelan patients but also for their families, who often show up with no place to stay or money for food. Families of other hospital patients sometimes take them in, but more often they sleep in chairs in the hospital hallways and rely on government services for meals and showers.

That's the case for Rogelio Sanchez, a 39-year-old carpenter from Venezuela who had slept at the hospital for three days after his 18-year-old son, Ronny, broke his clavicle in a motorcycle accident. While the accident took place in Venezuela, three hours from Brazil, Ronny's ambulance dropped him off at the border because the local hospitals did not have supplies for surgery or exams. Rogelio had not left his side. “I don't know anyone in Brazil, but I'm scared there won't be medicine for my son in Venezuela,” he said.

In December, Roraima's governor declared a state of emergency and asked for federal funds to help mitigate the crisis. Brazil's Health Ministry has created a response center to feed newly arrived Venezuelans and direct them to medical and social services. But the state of emergency and the federal resources that come with it expire after 180 days, and Roraima's Health Department worries about what will happen if Venezuela slides further into chaos.

“Soon we will be dealing with a refugee camp,” said Ivan Soriano Andrade, Roraima's adjunct health secretary. “We have a ticking time bomb in our hands.”