Trial of Scheduled Versus Treatment Administration of Donor-Derived Viral Specific T-cells for Viral Infections After Stem Cell Transplant

SCHEDULED ARM: Inclusion Criteria: * Recipient must be at least 21 days after stem cell infusion * Clinical status must allow tapering of any steroids to \< 0.5mg/kg prednisone or other steroid equivalent * No critical illness making VST infusion hazardous Exclusion Criteria: * Active acute GVHD grades II-IV. * Uncontrolled relapse of malignancy. * Infusion of ATG or alemtuzumab within 2 weeks prior to VST infusion. Alemtuzumab levels will be collected in the second week following stem cell infusion in patients who received alemtuzumab as part of their conditioning regimen. The level must be less than or equal to 0.15 prior to infusion of VSTs. In patients with level greater than 0.15, alemtuzumab levels can be checked serially until a level ≤ 0.15 is obtained. They would become eligible for scheduled VST infusion at that point. TREATMENT ARM Inclusion Criteria: * Blood adenovirus PCR ≥1,000 * Blood CMV PCR ≥ 500 * Blood EBV PCR ≥ 9,000 * Plasma BKV PCR \>1,000 * Evidence of invasive adenovirus infection. Adenovirus infection will be defined as the presence of adenoviral positivity as detected by PCR or culture from one site such as stool or blood or urine or nasopharynx. Adenovirus disease will be defined as the presence of adenoviral positivity as detected by culture or PCR from more than 2 sites such as stool or blood or urine or nasopharynx. * Evidence of invasive CMV infection, defined as pneumonitis, retinitis, colitis, hepatitis * Evidence of EBV-associated lymphoproliferation (EBV-LPD) defined as proven EBV-LPD by biopsy or probable EBV-LPD defined as an elevated EBV DNA level in the blood associated with clinical symptoms (adenopathy or fever or masses on imaging) but without biopsy confirmation. * Evidence of symptomatic BK virus infection, defined as hemorrhagic cystitis or BK nephropathy. * No active acute GVHD grades II-IV * No uncontrolled relapse of malignancy * No infusion of ATG or alemtuzumab within 2 weeks of VST infusion. * Clinical status must allow tapering of any steroids to \< 0.5mg/kg prednisone or other steroid equivalent