(CNN) Four times, the parents of a missing doctor with the US Centers for Disease Control and Prevention have been told that a body has been found. Each time, Tia and Terrell Cunningham are sent into heart-wrenching agony, only to learn that it isn't their son.

"It takes you to a place that the light is not shining in," Terrell Cunningham said. "I won't call it a dark place, but they are lows. This is extremely hard."

The parents spoke Monday afternoon as the search for their son Dr. Timothy J. Cunningham, 35, entered a third week after he disappeared without a trace.

A Harvard-educated doctor, Cunningham is a highly regarded epidemiologist at the CDC, having risen through the ranks to become a team leader in the US Public Health Service Commissioned Corps. He even earned a spot last year in the Atlanta Business Chronicle's 40 Under 40 list, a who's-who of the city's residents.

The disappearance has prompted a high-profile police search and a $10,000 reward for any information about the case.

Authorities have said Cunningham disappeared after leaving work early on February 12, complaining of feeling ill. His worried parents drove down from Maryland, arriving on February 14 only to find his phone, keys, wallet, car and dog at his house.

He has not been seen or heard from since.

"Everything about this disappearance is unusual," his father said.

His mother added, "we really need him back to complete our circle."

The Cunninghams also sought to beat back an Internet rumor that their son was a whistle-blower who had warned about the flu shot being responsible for this year's deadly flu season.

"I must address this issue," his father said. "It is a lie. ... It is not factual. Hopefully, he'll come back and be able to address that."

The parents said their sole focus is finding their son, the middle of their three children.

A $10,000 reward has been offered for information on Dr. Timothy J. Cunningham's case.

Fliers have begun circulating across Atlanta, showing Cunningham's magnetic smile and urging anyone with information to call 911. Friends say he is smart and caring, a man with a big grin who liked doling out big hugs to match his smile.

Pat Upshaw-Monteith, president and CEO of Leadership Atlanta, said she had recently met with Cunningham because he is taking on one of its highest-level volunteer positions. "Everything seemed to be going very, very well for him -- and then for him to disappear, it just doesn't add up," she said.

Cunningham worked in epidemiology, trying to understand health differences across demographics. With more than 16 years of experience in public health, he has co-authored 28 publications on topics ranging from sleep deprivation to pulmonary disease, with a special focus on how health issues affect minorities. He worked on public health emergencies including Superstorm Sandy, the Ebola outbreak and the Zika virus.

CDC spokewoman Kathy Harben said Cunningham "is a highly respected member of our CDC family. ... Our thoughts are with his friends and family during this difficult time."

The parents said they knew that something with their son was amiss on the evening of February 11, after they spoke with him by phone and exchanged a series of text messages. "We've shared that with the detectives, and we've kept that as a private matter," his father said.

"As a parent, you have indicators when things are just not right with your child, and that was the case," he said.

His mother said she received a text message at 5:21 a.m. on the day he was last seen. "Are you awake?" her son asked.

Her phone was on silent mode. "I wish I had that opportunity to answer that text," she said.

When they arrived at their son's house after he went missing, the parents said, they knew that something was wrong because he had left his Tibetan spaniel unattended. The dog, officially named Mister Bojangles Cunningham but known as Bo, had twice accompanied Cunningham to Harvard where he went for his master's and doctoral degrees.

He loved the dog so much, his parents said, he'd drive the 130 miles to Tuskegee, Alabama, to have the pooch's teeth cleaned. "I tell you all that to really understand the relationship of Tim and Bo," the father said. "To work as hard as he has worked -- and to just now disappear -- it's such a challenge for us to understand."

Both parents said they've been sustained by the outpouring of support from strangers and friends alike -- and that their faith has helped them get through these difficult two weeks. "I often say, 'Lord, you have put me in this position. What would you have me to learn?' " his father said. "I'm praying for a positive outcome but having difficulty in understanding the lesson."

Terrell Cunningham retired December 31 after years with the Food and Drug Administration. He is also a retired Air Force colonel. "This was supposed to have been one series of memorable events after another," he said. "This is not how we planned retirement."

He praised the police department for working with the family. He understands the reason why the family was notified about the four bodies that were found, but it doesn't make it any easier as a parent to bear such news.

"It is quite agonizing to wait on the news that it's not our son," he said. "We'd just like to send our sympathies and condolences to those families."

While both try to remain positive, the parents have reflected on favorite memories.

For the father, it was in September for his 60th birthday. He had ordered all three of his children -- Anterio, Tiara and Tim -- join him and their mother in Cabo San Lucas. "I told them: I expect you to be there," he said.

His middle son was always so busy working, his father feared he would not make it. "But he quickly said, 'I'm coming.' "

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It was an all-inclusive resort, and the family indulged in food and revelry. "It was so good to see Tim having so much fun, because he is such a hard worker. Very seldom does he have the time to just let go," his father said. "It was awesome."

His mother recalled hiking Stone Mountain with her boy on a Mother's Day. Near the top of the mountain, the granite slope was slick from rain, and Tia slipped and fell, nearly sliding down the entire slope.

Her son rushed to her. "You scared me, Momma," he said. "Now, I'm going to have to hold your hand."

She responded with, "Tim, you don't want to walk around the park holding your mother's hand."

That Mother's Day, her grown son held her hand the entire way down the mountain.

Now, she longs to hold his hand again.