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Takes place in a future where all sickness and disease has been eliminated leaving people with weak immune systems. The main character is in charge of sending all time hoppers from the past to a moon colony to prevent the spread of disease. Travel backwards in time is presumed impossible when a time hopper who claims to have a bi-directional device is caught. This time hopper looks a lot like the main character but is sent to the moon colony anyway. The main character tinkers with his device, hops backward in time, tries to return to his time and is confronted with a duplicate of himself and gets sent to the moon colony despite declaring he posesses a bi-directional device.

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"Absolutely Inflexible" by Robert Silverberg, first published in Fantastic Universe, July 1956; the reprint in New Worlds Science Fiction #72, June 1958 is available at the Internet Archive. Here is a plot summary from Majipoor.com:

It's a nifty little time-travel paradox story. In the 28th Century, the Earth has been rid of disease, so germ-ridden time travelers from the past are quarantined at a prison on the moon where they can't infect the world's unprotected population.

Takes place in a future where all sickness and disease has been eliminated leaving people with weak immune systems. The main character is in charge of sending all time hoppers from the past to a moon colony to prevent the spread of disease.

"But can't I live on Earth and stay in this space suit?" the time-jumper asked, panicky now that he saw his interview with Mahler was coming to an end. "That way I'd be sealed off from contact at all times."

"Please don't make this any harder for me," Mahler said. "I've explained to you why we must be absolutely inflexible about this. There cannot—must not—be any exceptions. It's two centuries since last there was any occurrence of disease on Earth. In all this time we've lost most of the resistance acquired over the previous countless generations of disease. I'm risking my life coming so close to you, even with the space suit sealing you off."

Travel backwards in time is presumed impossible when a time hopper who claims to have a bi-directional device is caught.

"Took this away from our latest customer," Fournet said. "He told the medic who examined him that it was a two-way rig, and I thought I'd bring it to show you."

Mahler came to full attention quickly. A two-way rig? Unlikely, he thought. But it would mean the end of the dreary jumper prison on the Moon if it were true. Only how could a two-way rig exist?

This time hopper looks a lot like the main character but is sent to the moon colony anyway.

The guards brought the jumper into Mahler's office. He was fairly tall, Mahler saw, and young. It was difficult to see his face clearly through the dim plate of the protective space suit all jumpers were compelled to wear, but Mahler could tell that the young time-jumper's face had much of the lean, hard look of Mahler's own. It seemed that the jumper's eyes widened in surprise as he entered the office, but Mahler was not sure.

The main character tinkers with his device, hops backward in time,

He picked up the time-jumper's rig and examined it. A two-way rig would be the solution, of course. As soon as the jumper arrives, turn him around and send him back. They'd get the idea soon enough. Mahler found himself wishing it were so; he often wondered what the jumpers stranded on the Moon must think of him.

[. . .]

He touched his left hand gingerly to the indicated place. There was a little crackle of electricity. He let go, quickly, and started to replace the time-rig on his desk when the desk abruptly faded out from under him.

tries to return to his time and is confronted with a duplicate of himself and gets sent to the moon colony despite declaring he posesses a bi-directional device.

Suddenly Mahler saw the insane circle complete. He recalled the jumper, the firm, deep-voiced, unafraid time jumper who had arrived claiming to have a two-way rig and who had marched off to the Moon without arguing. Now Mahler knew who that jumper was.

But how did the cycle start? Where did the two-way rig come from in the first place? He had gone to the past to bring it to the present to take it to the past to—

His head swam. There was no way out. He looked at the man behind the desk and began to walk toward him, feeling a wall of circumstance growing around him, while he, in frustration, tried impotently to beat his way out.

It was utterly pointless to argue. Now with Absolutely Inflexible Mahler. It would just be a waste of breath. The wheel had come full circle, and he was as good as on the Moon. He looked at the man behind the desk with a new, strange light in his eyes.

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