Step 1: Identify the classic distribution. In infants, scabies characteristically involves the palms, soles, axillae, umbilicus, genitalia and scalp/face - sites that are typically spared in older children and adults.
Step 2: The morphology fits - intense pruritus (worse at night) with erythematous papules, vesicles, pustules and sometimes burrows or nodules. Vesiculation on palms and soles is especially typical of infantile scabies.
Step 3: The history of similar lesions in family members is a strong clue, since scabies (Sarcoptes scabiei) is highly contagious and spreads among close contacts.
Step 4: Why the others are wrong - Atopic dermatitis in infants favours the cheeks and extensor surfaces and spares the umbilicus; it is not classically vesicular on palms/soles nor transmitted to contacts. Psoriasis shows well-demarcated scaly plaques, is uncommon and non-pruritic-vesicular in infants. Seborrhoeic dermatitis gives greasy yellow scales on scalp/face/folds and is non-pruritic.
Key fact: Itchy palmoplantar and umbilical vesicles in an infant with affected contacts = scabies.