Hello from JustAnswer.
Could you attach a picture of the red patches?
Thank you for the picture.
I agree with your doctor that this looks like rosacea. If the oral doxycycline is not helping, then the medicine that I would usually consider next would be topical metronidazole, a different antibiotic. However, this medicine is a prescription medicine in most countries. The best that can be obtained over the counter would be a retinoid, such as tretinoin or retinol. The tretinoin is stronger if you can get it, but is only over the counter in certain countries. The retinol is more widely available.
I would also note that if you are out in the sub much during your holiday, using a strong sunscreen would be important because sun exposure can make it much worse.
If I can provide any additional information, please let me know.
In many people with rosacea, the long term course can be progressive, so that the symptoms will get gradually worse over time. Even in people with intermittent symptoms, the flare-ups can be progressive, so that the exacerbations can get worse over time.