Viral Exanthem: Given the patient's presentation with fever, headache, and subsequent development of a rash, a viral exanthem is a plausible diagnosis. The rash's characteristics (not itchy, not painful) and its timing (following fever) support this possibility. Viral infections can cause meningioencephalitis and are often accompanied by skin manifestations.

Bacterial Meningitis with Rash: Although the patient received ceftriaxone, which is effective against many bacteria causing meningitis, some bacteria like Neisseria meningitidis can cause a rash (e.g., petechial or purpuric rash). The absence of pain or itch does not rule out bacterial causes.
Drug Rash (due to ceftriaxone): It's possible that the skin lesions are a reaction to the antibiotic treatment the patient received. Drug rashes can occur at any time during or even after the completion of antibiotic therapy.
Vasculitis: Conditions like leukocytoclastic vasculitis can present with skin lesions following a febrile illness. The lesions are often not itchy or painful but can be associated with systemic symptoms.

Meningococcemia: This is a life-threatening condition caused by Neisseria meningitidis. It can present with fever, headache, and a rash (often petechial or purpuric). Although the patient has been treated with ceftriaxone, which is appropriate for meningococcemia, the diagnosis must be considered due to its high mortality rate if not promptly treated.
Rocky Mountain Spotted Fever (RMSF): This tick-borne illness can cause fever, headache, and a characteristic rash. It's critical to consider RMSF due to its severity and the need for specific treatment (doxycycline).

Secondary Syphilis: This condition can cause a wide range of skin manifestations, including rashes that are not typically itchy or painful. Given the patient's presentation, although less likely, syphilis should be considered, especially if other common causes are ruled out.
Erythema Multiforme: A skin condition characterized by target lesions, often following infections or drug reactions. It's less likely given the description of the rash but should be considered if the rash's appearance is atypical or if there are mucosal involvement signs.