What is a likely diagnosis for a 13-month-old boy presenting with eczematous rashes, pruritus, loose stools, and colic?

In this scenario, the combination of eczematous rashes, pruritus, loose stools, and colic in a 13-month-old boy is indicative of a non-IgE mediated cow milk protein allergy. This type of allergy occurs when the immune system reacts to cow's milk proteins in a delayed manner, typically over hours to days, leading to gastrointestinal symptoms such as loose stools and colic, along with skin manifestations like eczema.

Non-IgE mediated reactions can cause a range of symptoms, including gastrointestinal issues and skin problems, making them a plausible diagnosis in this case. In infants, cow's milk protein allergy is one of the most common causes of such symptoms, resulting from the body’s inability to tolerate cow's milk proteins, even when no immediate hypersensitivity response (like IgE antibodies) takes place.

The other conditions listed, such as IgE mediated cow milk protein allergy, typically present with more acute symptoms, including hives or anaphylaxis. Atopic dermatitis with food intolerances can also lead to eczema and gastrointestinal symptoms, but in this case, the presentation strongly fits the pattern of a non-IgE mediated allergy, which is more likely to cause the combination of loose stools and colic alongside the skin