Use the verb pursue when you're chasing after someone — the way that paparazzi pursue a celebrity.
Pursue originally meant "to follow with hostile intent," from a Latin root word, prosequi, "follow, accompany, follow after, or follow up." Today, when you pursue someone, it's not necessarily with bad intentions. A police officer, for example, might pursue a criminal, and your dog might pursue your cat around the house just for fun. If you pursue a career in journalism, you are working to achieve it. Likewise, you pursue a strategy or a hobby if you do it consistently.
Primary Meanings of pursue
follow in or as if in pursuit
carry out or participate in an activity; be involved in
v follow in or as if in pursuit
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go through (an area) in search of prey
chase, chase after, dog, give chase, go after, tag, tail, track, trail
go after with the intent to catch
follow stealthily or recur constantly and spontaneously to
chase an animal up a tree
search the trail of (game)
hound, hunt, trace
pursue or chase relentlessly
pursue until captured
v carry out or participate in an activity; be involved in
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engage in or perform
engage at close quarters
engage in political activities
work toward the passage of some legislation by exchanging political favors such as trading votes