When a person is addicted to something, lets say a drug, and they are deprived of it for too long, they start getting sick, have anxiety attacks and crave for it extremely and uncontrollably.
Is there a word in English for that exact condition?
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When someone stops taking a drug they've developed a dependence on, they're said to exhibit (drug) withdrawal symptoms, collectively called withdrawal syndrome when they occur together (e.g., alcohol withdrawal syndrome).
Or shortened, as user Michael Harvey says in a comment, the addict is said to suffer from withdrawal.