My third differential would be eczema skin which presents as dry,
itchy, red patches on the skin. It is a form of chronic dermatitis (
rash).Allergic reaction is one of the important causes of eczema. Allergic reaction can occur to strong soaps, Solvents, chemicals, certain foods, food additives, plants, metals, cosmetics, even urine and faeces of some animals (dust mites).
I hope it helps. Take care and regards.