The dose of doxycycline usually prescribed is actually not high enough to even have any antibiotic activity. Since the antibiotics are prescribed at such low doses there are minimal long-term side effects. When using low doses of antibiotics at anti-inflammatory doses, studies have shown long-term use to be safe.
https://www.healthgrades.com/right-care/rosacea/is-long-term-antibiotic-use-for-rosacea-safe
Thanks.