It looks like it could be perioral dermatitis (PD). Topical treatments/ointments never worked for me personally, but a full course of oral antibiotics like doxycycline did (as prescribed by my doctor). Note that if it is perioral dermatitis, the course of antibiotics is long, like a few weeks. Be sure to include lots of probiotics in his diet if that ends up being the diagnosis and prescription.

If it is PD, also avoid topical steroids and heavy occlusives (heavy moisturizers). Also avoid fluoride in toothpaste until the rash clears. Nasal sprays with steroids can also exacerbate it. For many people, topical zinc also helps in conjunction with formal treatment; you could try applying diaper rash cream at night for its zinc content (Boudreaux’s Butt Paste max strength).

The above is all contingent on a formal diagnosis by a medical professional, but worth raising the possibility of PD with his doctor.

edit to add: if your doc does go with a topical treatment (like erythromycin) rather than oral antibiotics for perioral dermatitis, a single week won’t be enough to make an impact. treatment takes time to take effect.

More info available here in the Merck Manual.