It could be rosacea or even a fungal infection which would leave a dry, scaly skin on healing. On more symptoms like finding out if there is any itching, oozing, bleeding after scratching etc. and appearance of pus in the eruption we can come to whether it is Acne rosacea.
Homeopathic Medicine has very good remedies to see that the rosacea subsides and also can prevent recurrence by looking at the causative factors. Even if it is a fungal infection it can be cured with medicines like Caladium, Sarsaparilla, Sulph, Graphites, Merc sol. Etc.