Atopic Dermatitis (Eczema): This is a common condition in infants, characterized by itchy, inflamed skin, and often associated with allergies, including allergic rhinitis which could explain the runny nose. The fact that the rash has worsened over time is consistent with eczema, especially if the infant has been exposed to irritants or allergens.

Viral Exanthem: Many viral infections can cause rashes in infants, and some, like roseola, can be accompanied by upper respiratory symptoms like a runny nose. The rash and runny nose could be part of a viral syndrome.
Allergic Contact Dermatitis: If the infant has been exposed to new products or substances that come into contact with the skin, an allergic reaction could occur, causing a rash. This could be localized to the face if the allergen is in a product used on the face.
Irritant Contact Dermatitis: Similar to allergic contact dermatitis but caused by an irritant rather than an allergen. This could be due to excessive use of certain products or exposure to substances that irritate the skin.

Congenital Syphilis: Although less common, congenital syphilis can cause a rash and other systemic symptoms, including those affecting the nose. It's crucial to consider this in the differential diagnosis due to its severe consequences if left untreated.
Meningitis: While less directly related to the symptoms described, in very young infants, the presentation of meningitis can be non-specific, including rash and irritability. It's a life-threatening condition that requires immediate diagnosis and treatment.
Kawasaki Disease: This condition can cause a rash, among other symptoms like fever and conjunctivitis. It's a significant diagnosis to consider because of its potential to cause cardiac complications if not treated promptly.

Staphylococcal Scalded Skin Syndrome (SSSS): A rare but serious condition caused by staphylococcal toxins that can cause widespread skin lesions and can start with localized skin infection. It's more common in infants and young children.
Zika Virus: In areas where Zika virus is prevalent, it could be considered in the differential diagnosis, especially if there's a history of travel to endemic areas. Zika can cause a rash and other systemic symptoms in infants.
Leiner's Disease (Severe Combined Immunodeficiency with Eczema): A rare condition characterized by severe eczema, recurrent infections, and failure to thrive, due to an underlying immunodeficiency.