SEAFORD, Del. —  One June evening, Tywanda Sullivan’s 3-year-old son gasped for air. His tiny chest heaved. Her adrenaline spiking, Sullivan dialed the hospital: My boy, she said — his inhaler isn’t working.

“Can you put him on?” a nurse asked.

Sullivan held the phone to his lips. Rae’Shawn coughed and coughed.

“Hang up,” the nurse said. “Call 911.”

An ambulance, Sullivan feared, would take an hour to reach him in this rural Delaware town.  So, she buckled Rae’Shawn into his car-seat. His eyes drooped in the rearview mirror. “Stay awake, Rae’Shawn,” she said. “Stay awake.”

An Affordable Care Act grant has led to a dramatic drop in asthma-related incidents like Rae’Shawn’s in Delaware — and significant Medicaid savings — but that funding may disappear in a matter just of months. Doctors knew the problem: too many kids were landing in the E.R., fighting to breathe. Most visits were entirely preventable. Culprits ranged from insidious airway-irritants — mom’s perfume, an old mattress — to communication breakdowns between physician and parent.

Three years ago, Nemours Children’s Health Systems — a national nonprofit hospital system with 16 clinics in Delaware — clamored for one of Obamacare’s first  innovation grants. Nemours pledged to use the money for a holistic asthma program with a two-pronged mission: reduce hospital visits among children — and, by extension, cut state Medicaid spending.The government obliged with $3.7 million.

By medical measurements, Nemours’ asthma program, which relies on counseling and home visits, has worked. Child emergency room visits dropped 40-percent in the first two years, and asthma-triggered hospitalizations dipped 30-percent. The Medicaid savings target is $4.7 million.

But economically, the asthma program is doomed to fail.

That’s because the hospitals lose a dollar of revenue for every dollar Medicaid saves, Nemours chief executive David Bailey said. Thanks to an unfortunate quirk of our health-care system, hospitals often aren’t reimbursed for money-saving  programs under Medicaid, including those that help low-income children. This is in stark contrast, health-care experts say, to the various preventative care incentives aimed at the elderly under Medicare, which give hospitals a cut of what they save the government.

When the grant money dries up in July, the asthma program will likely disband. And Seaford kids, Bailey said, will lose medical support they need.

The end of Nemours’ asthma program means Sullivan, 32, will lose a resource she calls life-saving. She’s the mother of two boys with severe asthma. Nine-year-old Ronshay speaks up when his chest feels tight. Rae’Shawn quietly struggles until someone notices.

She can’t remember how many times they’ve rushed to the emergency room. Eight? More?

Trips became rare after Sullivan met a Nemours community health worker who functions as both therapist and nurse. Months ago, Monique Cornish swept the family’s bungalow for asthma triggers: a potent Yankee candle, bleach disinfectant, Sullivan’s favorite Victoria’s Secret body spray.

Cornish and Sullivan text weekly: Can my boys play outside? Will the pollen send them over?

There was much left to learn, to fear. Just when Sullivan allowed herself to relax, this flare-up shattered her peace.

Now, she heard Rae’Shawn gurgle in the backseat.

An estimated 6.8 million American kids have asthma. Delaware spends between $25 million and $30 million treating patients each year with the chronic lung disease. Child asthma inflates public health-care spending in Seaford, population roughly 7,200, where the median annual household income is just below $33,000. Most kids, including Sullivan’s sons, are covered by Medicaid plans.

Nemours’s modest Delaware initiative highlights a persistent flaw in our national health-care system. Under the widely used fee-for-service payment model, hospitals have little financial incentive to reduce patient visits. Sick kids, including those suffering from controllable diseases like asthma, are sources of income.

“For years, physicians and hospitals have complained they don’t get to spend time with patients because fee-for-service promotes volume over value,”  said Katherine Hayes, director of health policy at the Bipartisan Policy Center. “It’s a matter of getting as many patients in and out as you can.”

Many hospitals are trying to switch to value-based payment systems, which reward care quality, Hayes said. Some are striking better deals with insurance providers. Some call for the Obama administration to push harder for hospital payment reform.

“Here’s one way to look at it,” Hayes said. “Would a patient rather operate under a system based on how well they treat you, the quality of care they provide and how quickly you get better?

Last year, Nemours served 300,000 patients nationwide. It’s compensated for each treatment delivered. If a population’s health improves, the bottom line suffers.

“We now know that we can truly improve the care of children,” said Bailey, a pediatric gastroenterologist by training. “As a doctor, I couldn’t be happier. As a business person, how do I do it? How do I pay for that?”

Keeping the asthma program afloat would require slashing budgets in other areas, Bailey said. “Our payment reform isn’t as fast as our innovation reform,” he said. “So many hospitals, especially non-profits, are in this same boat. My fear is that, across the country, we’ll see all kinds of great ideas slip backwards.”

Last month, Bailey visited the Center for Medicaid and Medicare Innovation’s office in Baltimore. He asked for a grant extension. The asthma program is clearly effective, he told them: Seaford kids are much healthier. E.R. visits are plummeting. But without the ACA grant money, the nonprofit group can’t financially sustain those gains.

The officials seemed sympathetic, he said, but denied his request. That’s largely because Medicaid is state-run. Shared savings programs under Obamacare split Medicare dollars, federal money. The innovation grants, according to CMS, are one-time monetary boosts. The hospitals are thereafter responsible for maintaining programs, a representative said.

How Medicaid reimburses Nemours is dictated by Delaware. And Delaware’s Medicaid plans are managed by private insurance companies.“There are no bad actors here,” Bailey said. “It’s a problem when you have systems of care and systems of payment in flux and they’re not in sync.”

Nemours’s asthma program employs doctors, nurses, psychologists and community health-care workers who visit homes, inspect them for airway irritants and teach parents how to prevent asthma attacks.

The team asks local school bus drivers to cut engines as soon as they park, reducing exhaust in the air. It advises parents, teachers and janitors to clean with vinegar — not bleach. It sends pest exterminators to public housing. It replaces old mattresses, which double in weight after eight years from dust build-up, Bailey said. It teaches children how, when and why to take medication.

And now, Bailey said, the team members are looking for other jobs: “If I were in their shoes, I’d be looking forward as well.”

Sullivan wishes the support didn’t have an expiration date. At first, she didn’t know what to think of Cornish, the community health worker Nemours assigned to her family.

A doctor put them in touch when the asthma program launched. Now, Cornish visits every six weeks. The women sit on Sullivan’s brown suede couch, knees touching, like old friends catching up.

Conrish juggles 52 families. “But she’s always there when you need her,” Sullivan said. “She calls doctors to get answers, fast. She calls teachers and babysitters — tells them about the boys’ asthma, what to watch for … Monique is our voice. Without her, our voice wouldn’t be as loud.”

Rae’Shawn calls her ‘Nique. She teaches him how to express pain. When his asthma flares, the boy now tells his mother, “I’m having belly breathes.”

Cornish instructs Sullivan to text her with questions, any time. Call the hospital the instant she’s scared. Try to stay calm, think clearly. Asthma management is constant, deliberate — a skill to practice and master.

That June evening, as Rae’Shawn coughed, Sullivan sped to the nearest hospital.

She carried the boy into the E.R., where a nurse placed an oxygen mask over his mouth.

His breathing steadied. His chest settled. Your boy, the nurses said, will be fine. No one knew what caused the asthma attack. It started when Sullivan’s mother was babysitting. Confused, she’d called Sullivan: “I think you need to get here. Now.”

Rae’Shawn returned to his normal medicine routine: An inhaler puff in the morning, a puff before playtime.

Six months have passed, severe flare-up free. Sullivan always hears coughing, real or imagined. She works full-time, 30 miles away, phone volume cranked up. The “Oompa Loompa” ringtone makes her heart pound. Is it the daycare calling? The school?

She tells Cornish her worries on a brisk November afternoon. “We’ve been doing everything right. We watch the weather channel before we step outside, like, ‘Is the air safe today?’”

This meeting could be one of their last. Months ago, Cornish broke the news to her gently: When the grant money’s gone, there’s no guarantee my job will still exist.

“Monique — you’re a call away. A text away,” Sullivan says. “When you’re around, It’s like — okay. It’s okay. It’s going to be okay.”

The community health worker takes out her notebook. She tracks the boy’s progress, his gradual improvement.

Rae’Shawn’s Aerochamber mask is getting too small, Sullivan tells her.

“It used to cover his face.”

Cornish smiles. She can replace it.

“He’s growing up!”

Sullivan watched her oldest son flourish under Cornish’s guidance. Ronshay now asks teachers for an inhaler break before gym. He tells other moms to clean with vinegar — not bleach. And next week, he’ll finally get the birthday party of his dreams: An afternoon at Stratosphere, the local trampoline gym.

Hours of bouncing no longer scare Sullivan. She trusts Ronshay to self-monitor his breathing.

She wants the same thing for Rae’Shawn.