Hello,
It can be due to an allergic reaction to the toothpaste or irritated hair follicles present around the mouth. Avoid any kind of cosmetics. Topical antibiotics such as mupirocin or neomycin containing ointment are also useful. In case the symptoms persist then oral antibiotics and topical steroids may be needed. Please consult a dermatologist in that case.
I hope it helps. Take care and regards.
I know people who react to toothpaste. You might want to try more natural types, like Tom's. Or even try using plain baking soda instead of toothpaste and see if that gives relief before buying a new product.