Brooks County attorney David Garcia said those people will be arrested and prosecuted.
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Garcia made the announcement this week after community members notified his office of seeing people who tested positive at grocery stores and businesses.
"If you're going to go out and endanger other people, and we find out about it, we will prosecute you," Garcia told KIII. "People have not really embraced the dangers of COVID-19. It's dangerous. It's killing people, and it's making people very sick. So either do it because you're concerned about others or do it because you're going to be punished if you don't."
Garcia also said if people continue to ignore safety precautions, they could face legal action, noting there are a number of statutes that can be used once someone tests positive.
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While many residents appeared to support the idea, others were concerned it won't be effective.
"I don't see how something like that can be enforced," said resident Roxie Fuller. "I don't know that they can."
"We need to be responsible for our actions, and if we're not going to be responsible for our actions because we're trying to protect other people, well there's a punishment for it, too," Garcia said, adding that at the end of the day, it's up to everyone to follow the safety guidelines for the good of their neighbors.
Brooks County is around 80 miles north of the Mexico border and has a population of approximately 7,200. Falfurrias, the county seat, is 275 miles southwest of Houston.