Ruxolitinib With and Without CTLA-4 Ig Abatacept for the Prophylaxis of Graft-Versus-Host Disease and Cytokine Release Syndrome After T-cell Replete Haploidentical Peripheral Blood Hematopoietic Cell Transplantation

Inclusion Criteria: Patients must meet the following criteria within 30 days prior to Day -3 unless otherwise noted. * Diagnosis of one of the hematological malignancies listed below: * Acute myelogenous leukemia (AML) in complete morphological remission, complete remission with incomplete hematologic recovery, and complete remission with partial hematologic recovery (based on ELN Criteria47). * Acute lymphocytic leukemia (ALL) in complete morphological remission (MRD negative by flow cytometry with sensitivity to ≤ 10-4). * Myelodysplastic syndrome with ≤ 10% blasts in bone marrow. * Non-Hodgkin lymphoma (NHL) or Hodgkin lymphoma (HD) in second or greater complete or partial remission. * Myelofibrosis with ≤ 10% blasts in bone marrow. Up to five patients with myelofibrosis will be permitted in Regimen 1 and up to five in Regimen 2. * AML in partial response. One patient will be enrolled in Regimen 1 given the prospect of potential benefit. * Planned treatment is T cell-replete peripheral blood haploidentical donor transplantation. * Available HLA-haploidentical donor who meets the following criteria: * Blood-related family member, including (but not limited to) sibling, offspring, cousin, nephew, or parent. Younger donors should be prioritized. * At least 18 years of age. * HLA-haploidentical donor/recipient match by at least low-resolution typing per institutional standards. * In the investigator's opinion, is in general good health and medically able to tolerate leukapheresis required for harvesting hematopoietic stem cells. * No active hepatitis. * Negative for HTLV and HIV. * Not pregnant. * Donor selection will be in compliance with FDA guidelines as provided in 21 CFR 1271 for donor eligibility https://www.fda.gov/downloads/BiologicsBloodVaccines/GuidanceComplianceRegulatoryInformation/Guidances/Tissue/UCM091345.pdf * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2. * Adequate organ function as defined below: * Total bilirubin ≤ 1.5 x IULN. * AST (SGOT) and ALT (SGPT) ≤ 3.0 x IULN. * Creatinine ≤ 1.5 x IULN OR creatinine clearance ≥ 45 mL/min/1.73 m2 by Cockcroft-Gault Formula. * Oxygen saturation ≥ 90% on room air. * LVEF ≥ 40%. * FEV1 and FVC ≥ 40% predicted, DLCOc ≥ 40% predicted. If DLCO is \< 40%, patients will still be considered eligible if deemed safe after a pulmonary evaluation. * Able to receive GVHD prophylaxis with tacrolimus, mycophenolate mofetil (if applicable), and cyclophosphamide. * At least 18 years of age at the time of study consent * The effects of ruxolitinib and abatacept on the developing human fetus are unknown. Additionally, tacrolimus may increase risk of hypertension, preeclampsia, preterm birth, and low birth weight; and mycophenolate mofetil is considered to be teratogenic. For this reason, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study and for the duration of the study. * Able to understand and willing to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). Exclusion Criteria: * Prior allogeneic transplant (regardless of whether donor was related, unrelated, or cord). Prior autologous transplant is not exclusionary. * Presence of donor specific antibodies (DSA) with Mean Fluorescence Intensity (MFI) of ≥ 4000 as assessed by the single antigen bead assay. * Known HIV or active hepatitis B or C infection. Known current history of active tuberculosis. * Known hypersensitivity to one or more of the study agents. * Planning to receive antithymocyte globulin as part of the pre-transplant conditioning regimen. * Currently receiving or has received any investigational drugs within the 14 days prior to the first dose of study drug (Day -3). * Pregnant and/or breastfeeding. Women of childbearing potential must have a negative pregnancy test within 14 days of Day -3. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, autoimmune disease, symptomatic congestive heart failure, unstable angina pectoris, or unstable cardiac arrhythmias. * Immunosuppressive doses of steroids. Subjects with steroids for adrenal insufficiency will not be excluded.