Infant eczema 90% of the time is due to a food allergy or food sensitivity in my clinical experience.
If the eczema started at 2 months of age, and the baby is breastfed, then the food sensitivity is coming from what the mother is consuming in her diet.
If the baby is bottle fed, then the skin lesions are likely a result of either cow's milk or soy (depending on the formula).
Other aggravating factors could be a Candidiasis (especially if the mom had antibiotics during pregnancy).
Other factors could be low essential fatty acids, or a deficiency in any number of vitamins or minerals.
Getting the baby tested through electrodermal screening for food sensitivites, I have found to be extremely helpful.
Common foods that the breastfeeding mom may need to eliminate are eggs, refined sugar and dairy. At times, I have also seen wheat, gluten, soy and citrus that needed to be eliminated in order for the infant's skin to clear up.
For bottle fed babies, that's a bit trickier. Baby's that don't do well on cow's milk formula, may do okay on hydrolyzed cow's formula, but not always. If your baby does not tolerate any kind of cow's dairy or soy formula, then you need to see a naturopathic doctor (or someone who is knowledgable) to devise an individualized formula for your baby from rice protein.
When your child starts to eat a wide variety of food, then the formula is no longer necessary.
When introducing food, wait to introduce cow's dairy products AFTER 2 years of age. Same with eggs. Introduce wheat after 1 year. (Provided that the child is not sensitive or allergic to these foods).
Internally, I have found L-glutamine, children's enzymes, lactobifidus and Flax-oil to be beneficial. (That is, adding it to their formula).
Topically, there are also creams that can be used, but their effects are very temporary, like cortisone cream. However, unlike cortisone, they don't thin the skin. Creams like Calendula or Neem Tree oil can be beneficial.
The majority of the time, eliminating the offending foods will help clear up the child's eczema extremely well.