Delis Rivera pulls up to her first stop of the day, a three-decker in Dorchester, walks up to the front door, and knocks. No answer.

As a community outreach worker for Tufts Health Plan, Rivera tries to connect with some of the insurer’s hardest-to-reach members — people with chronic health problems who are eligible for a generous insurance package but aren’t taking advantage of it. She can spend weeks knocking on doors without finding anyone.

On this morning, she gets lucky. At the back door, 62-year-old retired housekeeper Reyna Brea answers. Rivera introduces herself in Spanish and, as beans cook on the kitchen stove, they tick through Brea’s health concerns — arthritis, diabetes, depression — and what services might help.

Rivera is among a new cadre of health care workers enlisted in an ambitious Massachusetts experiment to improve care and reduce costs for some of the state’s sickest and poorest residents.

The first-in-the-nation program, called One Care, is for people on both Medicare and Medicaid, those living with disabilities, and with little or no income. Their care is managed by one of two nonprofit health insurers, Tufts and Commonwealth Care Alliance.

By better managing care — helping patients get to regular medical appointments so they don’t end up in hospital emergency rooms, for example — insurers and the state are trying to achieve one of medicine’s greatest goals: better care at lower costs.

The state pays insurers a set budget to manage care for their One Care members, people who are among the roughly 5 percent of Americans accounting for nearly half of all health spending. The program covers far more services than the typical insurance plan, from expanded dental and mental health benefits to hot meals, laundry services, and bus passes to help patients get to appointments. The thinking is that spending on such benefits now will help prevent more expensive medical problems later.

But the effort, launched in 2013, has proven to be far more complicated and costly than expected. And it has revealed a hard truth: to manage care for these people, first you have to find them.

To that end, Tufts and Commonwealth Care Alliance have hired dozens of outreach workers whose job is to find and connect with members.

Rivera, who joined Tufts two years ago, spends many work days at a desk, making calls and sending letters. But calls and letters often go unanswered. So twice a week, Rivera hits the road in her Nissan Maxima, carrying a list of addresses, and looks for members. When no one answers, she leaves a letter at the door.

Rivera’s visit to Reyna Brea’s home in Dorchester showed what happens when outreach efforts work. Brea opened up about her health problems, and she scheduled an appointment with a case manager to discuss them in more detail.

But Rivera would have no more success that day. She pulled up to another triple-decker in the Fields Corner neighborhood, a home she has visited many times before, always looking for the same man — for two years. Again, he did not answer.

“We have about 30 percent [of members] who we simply can’t find,’’ said Dr. Christopher “Kit’’ Gorton, president of public plans at Watertown-based Tufts Health Plan. “We’re unable to reach them in any way.’’

The people Tufts and Commonwealth Care Alliance are trying to reach have incredibly complex lives. About two-thirds of them have a mental health diagnosis. Many have chronic diseases, or a history of substance abuse, or both. Many are homeless, or bouncing from one temporary home to another.

Lester Belden, a 47-year-old program member in Worcester, has struggled with alcohol abuse and bipolar disorder for years. Initially, he didn’t want strangers coming to his home to manage his health problems. “I had trust issues,’’ he said.

But Belden eventually warmed to the idea, and through Tufts Health Plan he gained an advocate who is by his side twice a week, reminding him to take his pills, driving him to the local food pantry, making sure he isn’t locked in a room with a bottle of vodka.

Belden said the constant attention has helped. After years frequenting the emergency room every two or three months because of a flare-up of his mental illness or after drinking too much, Belden said he has not been to a hospital for six months.

Like him, 82 percent of the people enrolled in One Care say they are satisfied with the program, state officials say, and there is some evidence the program is cutting medical spending for people who frequent hospitals. For members at Commonwealth Care Alliance, inpatient hospital stays fell 7.5 percent in 2014, while emergency department visits fell 6.4 percent. For the health plan’s 10 most expensive members, costs dropped 38 percent that year.

“The success of our approach is really predicated on our ability to engage with people,’’ said Dr. Toyin Ajayi, chief medical officer at Boston-based Commonwealth Care.

But finding and holding on to program members is a significant hurdle. The state assigns people on its Medicaid program, known as MassHealth, to One Care, but many opt out because they don’t understand what it is, or don’t want to change doctors, or are simply not interested. Fewer than 13,000 of the more than 100,000 people eligible for One Care are enrolled in the program.

The intense outreach that the program requires is expensive, and it contributed to financial losses of $54 million in the program’s first 18 months, from October 2013 to March 2015. The numbers were so bad that Fallon Health of Worcester dropped out of the program, disrupting care for more than 5,000 people.

The state and federal governments later raised reimbursement rates, and the program’s finances have stabilized, according to state and health plan officials.

State officials say that despite One Care’s struggles, they are committed not only to extending the program for at least another two years, but to trying other models like it.

If Massachusetts finds the right formula for managing care for people with the most complex medical needs, it could serve as an example to the country.

“We’re trying to build this for the nation,’’ said Christopher D. Palmieri, chief executive of Commonwealth Care Alliance. “The whole world is moving toward paying for value and trying to achieve the highest quality within the most reasonable costs. This model is designed to do that.’’

Priyanka Dayal McCluskey can be reached at priyanka.mccluskey@globe.com. Follow her on Twitter @priyanka_dayal.