• Dose Limiting toxicity: (DLT) rate by CTCAE v5 [ Time Frame: 6 weeks ]
  Safety of HLA-matched VSTs: Safety including acute GVHD (Graft-versus-host disease) grades III-IV of the last dose of VSTs, secondary graft failure, or grades 3-5 non-hematological adverse events.
• Overall Response Rate: (The viral load of the virus (or viruses) by viral PRC (replication-competent retrovirus) within 42 days) [ Time Frame: 6 weeks ]
  Antiviral Responses: Antiviral responses within 42 days after the first dose of VSTs.

• Dose Limiting toxicity: (DLT) rate by CTCAE v5 [ Time Frame: 6 weeks ]
  Safety of HLA-matched VSTs: Safety including acute GVHD grades III-IV of the last dose of VSTs, secondary graft failure, or grades 3-5 non-hematological adverse events.
• Overall Response Rate: (The viral load of the virus (or viruses) by viral PRC within 42 days) [ Time Frame: 6 weeks ]
  Antiviral Responses: Antiviral responses within 42 days after the first dose of VSTs.

The purpose of this study is to use VSTs (virus-specific T cells) from a donor that is a partial HLA (human leukocyte antigen) match with the patient to treat viral infections after an allogeneic hematopoietic stem cell transplant (HSCT).

The patient must have had a myeloablative or non-myeloablative allogeneic HSCT using either bone marrow, single/double umbilical cord blood, or peripheral blood stem cells (PBSC). After a transplant, while the immune system grows back, the patient is at risk for infection. Some viruses can stay in the body for life and are normally controlled by a healthy immune system, but if the immune system is weakened, like after a transplant, they can cause life threatening infections. He/she must have had an infection with one or more of the following viruses -Epstein Barr virus (EBV), cytomegalovirus (CMV), adenovirus (AdV), Human polyomavirus type I (BKV), and human polyomavirus type II (JCV)- that has persisted or recurred despite standard therapy.

In this study, the investigators want to use white blood cells that have been trained to treat viral infections. In an earlier study the investigators showed that treatment with such specially trained T cells has been successful when the cells are made from the transplant donor. However as it takes 1-2 months to make the cells, that approach is not practical for patients who already have an infection. In a subsequent study, the investigators were able to create multivirus-specific T cells (VSTs) from the blood of healthy donors and created a bank of these cells. The investigators then successfully used these banked cells to treat virus infections after a stem cell transplant. In this study the investigators have further modified their production method to decrease the potential side effects and the investigators want to find out if they can use these banked VSTs to fight infections caused by the viruses mentioned above.

The virus-specific T cells (VSTs) given to the patient will be thawed and injected into their intravenous line. To prevent an allergic reaction if the patient had a prior reaction to blood products like blood transfusions or platelets, prior to receiving the VSTs he/she may be given diphenhydramine (Benadryl) and acetaminophen (Tylenol). The patient will remain in the clinic for at least one hour after the infusion.

All participants on this study will be infused with the same number (dose) of cells. If the patient has persistent infection after the first dose, the investigators would discuss this with the patient and allow them to receive up to four more treatments if there were no complications with prior infusions. These additional treatments might be with cells from the same donor, or if the investigators feel that there is another donor whose cells might be better for the patient, the investigators would use cells from a different donor. This second product will be administered at the same dose level 14 days after the patient's initial infusion, and any additional infusions should be at least 14 days apart. After each VST infusion, the patient will be monitored as described above.

After the patient receives the cells the patient's transplant doctor will monitor the levels of the virus the subject is infected with in their blood.

The patient will continue to be followed by their doctor(s) after the injection. They will be seen in the clinic by research staff for follow up every week for 6 weeks. To learn more about the way the VSTs are working in the patient's body, up to an extra 30-40 ml (6-8 teaspoons) of blood may be taken before the infusion and then at 2, 4, and 6 weeks. Blood should come from the central intravenous line, and should not require extra needle sticks.

Any leftover samples of blood may be used to help future research. The specimens may be kept for a long time. These specimens and information about the patient's circumstances may be shared with other cancer researchers. Although there will be a record identifying under what circumstances these specimens were obtained, under all circumstances the patient's identity will be kept confidential.

Study Duration : The patient will be on the study for approximately 42 days after the patient's infusion. If the patient receives additional doses of the T cells as described above, the patient will be followed for 42 days after their last dose of T-cells.

Inclusion Criteria:

1. Patients has to have received a prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow, single/double cord blood or PBSC

2. Cells administered as:

   1. Treatment of relapsed or persistent reactivation or infection for EBV, CMV, adenovirus, and/or BK virus despite standard therapy. Multiple infections are eligible to enroll.
   2. Early treatment for single or multiple infections with EBV, CMV, adenovirus, and/or BK virus following treatment failure or in patients who are unable to tolerate standard therapy
   3. Treatment of JC virus infection
3. Steroids less than or equal to 1.0 mg/kg/day methylprednisolone (or equivalent)
4. Hgb ≥ 7.0 gm/dl
5. Available VSTs must be partially HLA matched and verified
6. Negative pregnancy test (if female of childbearing potential after reduced intensity conditioning)
7. Patient or parent/guardian capable of providing informed consent.

Exclusion Criteria:

1. Received ATG (anti-thymocyte globulin), Campath or other T cell immunosuppressive monoclonal antibodies in the last 28 days.
2. Patients with other uncontrolled infections
3. Patients who are less than 28 days removed from their allogeneic hematopoietic stem cell transplant or who have received donor lymphocyte infusions (DLI) within 28 days.
4. Evidence of GVHD >= grade 2
5. Active and uncontrolled relapse of malignancy
6. Requirement for FiO2 (fraction of inspired oxygen) > 50% oxygen to maintain arterial oxygen saturation > 90%.Note: patients requiring oxygen at FiO2<=50% to maintain arterial oxygen saturation >90% are eligible to receive VSTs if the reason for this oxygen requirement is believed attributable to the virus being treated