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"is overrun" is correct and can be used in written English.
It is commonly used to describe a situation where something or somewhere is overwhelmed or taken over by a large number of people or things. Example: "The city was overrun with tourists during the summer festival." In this example, "is overrun" is used to convey that the city was filled with an excessive number of tourists. Another example: "The bookstore is always overrun with customers on Saturdays." Here, "is overrun" is used to describe the bookstore being filled with many customers, likely making it difficult to navigate or find a quiet space.
Dictionary
Exact(58)
Misrata's hospital is overrun with casualties.
Britain is overrun by Cleggmania.
Schiano's roster is overrun with freshmen and sophomores.
"The American political system is overrun by money.
Nineteen Eighty-four is overrun with "obvious didacticism".
The Internet is overrun with footage from the United States military in Iraq and Afghanistan.
And Seattle Grace is overrun with bikers involved in a horrific accident.
If your mine in Congo is overrun by rebels, you have lost a mine.
Related: GPs are exhausted, A&E is overrun and hospitals are broke.
Kemal's obsession with the past overwhelms him, and his own identity is overrun by his compulsion.
Hollywood, of course, is overrun with the type — how does Bening tell the difference?
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com