A Study of Mogamulizumab to Prevent Adult T-cell Leukemia/Lymphoma in People With HTLV-1

Inclusion Criteria: Screening Cohort (US patients only): * Age ≥18 years when informed consent is obtained * Has freely given written informed consent to participate in the study Treatment Cohorts (Cohorts 1 and 2): * Positive for anti-HTLV-1 antibody in the serum using an FDA approved assay for US patients (Avioq HTLV-I/II Microelisa System). UK patients should use UK Accreditation Service (UKAS) accredited tests, Abbot Architect ELISA Serology Screening assay and confirmatory serology Western Blot (performed at Public Health England, Virus Ref Dept, Colindale). * High-risk phenotype (PVL≥8% of PBMC) * Age ≥18 years when informed consent is obtained * Primary organ functions are stable * Neutrophil count: ≥ 1000/mm3, unless patient has diagnosis of ethnic neutropenia * Platelets: ≥100,000/mm3 * Hemoglobin: ≥9.0 g/dL * Serum aspartate aminotransferase (AST): ≤1.5x upper limit of normal (ULN) * Alanine aminotransferase (ALT): ≤1.5x ULN * Total bilirubin: ≤1.5x ULN * Serum creatinine (Cr): ≤1.5x ULN * Blood oxygen saturation (SpO2): ≥90% * Electrocardiogram (ECG): No abnormal findings requiring treatment are observed * Has freely given written informed consent to participate in the study * For females of reproductive potential: use of effective contraception during treatment and for at least 3 months after completion of mogamulizumab therapy. For males who have sexual intercourse with females of reproductive potential: use of effective contraception during treatment and for at least 3 months after completion of mogamulizumab therapy. Exclusion Criteria: In order to protect subjects and avoid any problems in evaluating the study drug, patients who meet any of the following criteria should be excluded from enrollment in the study, in either screening or treatment cohorts: * Patients with a history of any of the following: * Neutrophil count: ≤1000/mm3, unless patient has diagnosis of ethnic neutropenia * Acute or chronic hepatitis or hepatic cirrhosis, other than patients with positive antibodies and negative PCR as noted in criteria #11 and #12 below. * Tuberculosis or with active tuberculosis * Myocardial infarction within 12 months prior to the date of enrollment * Allergic reaction to administration of antibody drug products * Other cancers. Patients with a history of a localized solid tumor who received definitive, curative treatment and who have been without evidence of disease for 5 years prior to enrollment will be able to enroll in the study. Patients with radically resected basal cell carcinoma of the skin, squamous cell carcinoma (except malignant melanoma), noninvasive cervix carcinoma, carcinoma in situ in the gastrointestinal tract or corpus of the uterus, localized thyroid cancer, and localized renal cell carcinoma will be able to enroll in the study if they are determined to be completely cured, even if within 5 years of enrollment. * Prior treatment with immunosuppressants or interferon alpha products within 6 months prior to the date of enrollment * Serious complications (heart failure, lung disease, renal failure, hepatic failure, uncontrolled diabetes mellitus, etc.) * History of an active autoimmune disease that has required systemic treatment in past 2 years (ie, with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment and is allowed. * Any ailment that could be exacerbated by the administration of KW-0761, in the judgment of the Principal Investigator or co-Investigator * Diagnosis of ATL * Women who are pregnant, breastfeeding, who may be pregnant, or wish to bear children while receiving treatment or within 3 months of last dose of mogamulizumab * Patients who have taken multivitamins (Alinamin, vitamin C, etc.) or supplements such as fucoidan, catechin, and pentosan polysulfate within 2 weeks prior to the date of enrollment * Prior treatment with other study drugs within 4 months prior to giving informed consent * Complications of spinal cord compressive lesions such as cervical spine disease, disc herniation, and ossification of the yellow ligament * Uncontrolled psychiatric disorder, epilepsy, or dementia * Positive test for Hepatitis B surface antigen or HBV-DNA (using real-time PCR). Positive Hepatitis B core antibody is permitted if HBV-DNA PCR is negative and the patient remains on prophylaxis during study. * Positive test for Hepatitis C virus antibody, unless Hepatitis C PCR is negative. * Positive test for HIV antibody, unless undetectable HIV RNA \> 6 months and CD4 within normal limits per institutional standard. * Patients considered unqualified to participate in the study by the Principal Investigator or co-Investigator