Antiviral Cellular Therapy for Enhancing T-cell Reconstitution Before or After Hematopoietic Stem Cell Transplantation

Inclusion Criteria Patients who have received any type of allogeneic transplant or who have a primary immunodeficiency disorder will be eligible if they have CMV, adenovirus, and/or EBV infection/disease with failure of treatment after 7 days of standard therapy OR if unable to tolerate standard therapy. * Patients must meet one of the following criteria: * Recipient of prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cell or single or double cord blood within the previous 18 months, OR * Have a diagnosed primary immunodeficiency disorder (as defined by clinical and laboratory evaluations) and not undergone HSCT. * Treatment of the following persistent or relapsed infections despite standard therapy: * CMV: Treatment of persistent or relapsed CMV disease or infection after standard therapy. For CMV infection, standard therapy is defined as antiviral therapy with ganciclovir, foscarnet or cidofovir for at least 14 days. * Adenovirus: Treatment of persistent or relapsed adenovirus infection or disease despite standard therapy. Standard therapy is defined as antiviral therapy with cidofovir or brincidofovir. * EBV: Treatment of persistent or relapsed EBV infection despite standard therapy. For EBV infection, standard therapy is defined as rituximab given at 375 mg/m2 in patients for 1-4 doses with a CD20+ tumor. Additional Inclusion Criteria: * Patients with simultaneous infections with CMV, EBV and/or Adenovirus infections are eligible if one or more infection(s) is persistent or relapsed despite standard therapy as defined above. Patients with multiple infections with one or more reactivation and one or more controlled infection are eligible to enroll. * Clinical status at enrollment that allows tapering of steroids to equal or less than 0.5 mg/kg/day prednisone (or equivalent) prior to infusion of the VST doses. * Negative pregnancy test in female patients if applicable (childbearing potential who have received a reduced intensity conditioning regimen). * Written informed consent and/or signed assent line from patient, parent or guardian. Exclusion Criteria * Patients receiving ATG, Campath, Basiliximab or other immunosuppressive monoclonal antibodies targeting T-cells within 28 days of screening for enrollment. * Patients who have received donor lymphocyte infusion (DLI) or other experimental cellular therapies within 28 days. * Current therapy with ruxolitinib or other JAK inhibitors within the previous 3 days. * Patients with other uncontrolled infections, defined as bacterial or fungal infections with clinical signs of worsening despite standard therapy. For bacterial infections, patients must be receiving definitive therapy and have no signs of progressing infection for 72 hours prior to enrollment. For fungal infections, patients must be receiving definitive systemic anti-fungal therapy and have no signs of progressing infection for 1 week prior to enrollment. * Progressing infection is defined as hemodynamic instability, worsening physical signs, or radiographic findings attributable to infection. Persisting fever without other signs or symptoms will not be interpreted as progressing infection. * Patients with active and uncontrolled relapse of malignancy (if applicable).