Induction Chemotherapy With Afatinib, Ribavirin, and Weekly Carboplatin/Paclitaxel for Stage IVA/IVB HPV Associated Oropharynx Squamous Cell Cancer (OPSCC)

Inclusion Criteria: * Department of Pathology at MSKCC confirmation of diagnosis of oropharynx squamous cell cancer, stage IVA/IVB, that is HPV associated. Evidence of HPV can be p16 immunohistochemistry and/or HPV in situ hybridization positive test result on tumor tissue, either at MSKCC or other CLIA-approved lab. * Age ≥ 18 years of age * Karnofsky Performance Status ≥ 80 * Adequate organ function, as follows: Adequate bone marrow reserve: absolute neutrophil count (ANC) ≥ 1.5 X 109/L, platelets ≥ 160 X 109/L, hemoglobin ≥ 12 g/dL Hepatic: total bilirubin within normal limits (≤ 1.0 mg/dL); alkaline phosphatase (AP), aspartate transaminase (AST) and alanine transaminase (ALT) ≤ 1.5 X ULN (upper limit of normal) Renal: Serum creatinine ≤ 1.3 mg/dL. Patients with serum creatinine \> 1.3 mg/dL may be eligible if creatinine clearance (CrCl) ≥ 55 mL/min based on the standard Cockroft and Gault formula. * Patients of childbearing potential must have a negative serum pregnancy test within 14 days of treatment. Patients must agree to use a reliable method of birth control during and for 6 months following the last dose of study drug. * Ability to swallow oral medication. * Expansion Cohort only: At least one unstained slide from pre-treatment diagnostic biopsy or fine needle aspirate must be available for correlative immunohistochemistry study. Exclusion Criteria: * Prior chemotherapy or radiation for tonsillar or base of tongue squamous cell cancer * History of hemolytic anemia or thalassemia * Active infection or serious underlying medical condition that would impair the patient's ability to receive protocol treatment. * Current therapeutic anticoagulation with Coumadin (warfarin) * Current or prior treatment with ribavirin * Known active Hepatitis B or C * Any prior documented history of transient ischemic attack (TIA) or cerebrovascular accident (CVA) * New York Heart Association (NYHA) Grade II or greater congestive heart failure * Clinically significant peripheral vascular disease * Inability to discontinue any of the following potent P-gp inhibitors (cyclosporine, erythromycin, ketoconazole, itraconazole, quinidine, phenobarbital salt with quinidine, ritonavir, valspodar, verapamil) or inducers (St John's wort, rifampicin). * Known pre-existing interstitial lung disease. * Presence of poorly controlled gastrointestinal disorders that could affect the absorption of the trial drug (e.g. Crohn's disease, ulcerative colitis, malabsorption, or CTC grade ≥2 diarrhea of any etiology) based on treating physician assessment.