The goal of this clinical study is to learn if Pegasys (pegylated interferon) or Roferon (interferon) can make the Trivalent Inactivated Influenza vaccine (TIV) more effective in increasing the body's immune reaction against the flu virus in patients with Chronic Lymphocytic Leukemia (CLL).

The Study Drugs:

Pegylated interferon and interferon are designed to stop the growth of viruses. Pegylated interferon has a molecule that makes it last longer in the body. Standard interferon does not have this molecule that makes it last longer.

TIV is designed to prevent the flu.

Screening Test:

Before you can start treatment on this study, you will have a "screening test" to help the doctor decide if you are eligible to take part in this study. Blood (about 4 teaspoons) will be drawn for routine tests. If you have had a routine blood test in the last 4 weeks, this blood will not need to be drawn. Women who are able to have children must have a negative blood (about 1 teaspoon) blood test within 2 weeks of study entry.

Study Groups:

If you are found to be eligible to take part in this study, you will be randomly assigned (as in the roll of dice) to 1 of 3 groups.

If you are assigned to Group 1, you will receive TIV alone. If you are assigned to Group 2, you will receive pegylated interferon with TIV. If you are assigned to Group 3, you will receive interferon with TIV.

You will have an equal chance of being assigned to any of the 3 groups. Your doctor, the study doctor, study staff, and you will know which group you are in.

Study Drug Administration:

You will receive TIV through a needle into a muscle in your arm (the arm you do not write with) on Days 1 and 28.

• If you are in Group 1, you will only receive TIV
• If you are in Group 2, you will receive pegylated interferon right before you receive TIV. It will be given through a needle under the skin. It will be given near the TIV injection site.
• If you are in Group 3, you will receive interferon right before you receive TIV. It will be given through a needle under the skin. It will be given near the TIV injection site.

After you are given the injection(s), you will be watched for 15 minutes in the clinic before you are allowed to go home.

You will keep a diary for 1 week (7 days) after the injection(s). In it, you will record any symptoms you may be experiencing, and you will also record your body temperature.

Study Visits:

On Day 1, blood (about 4 teaspoons) will be drawn to check your immune system's response to the flu virus.

On Day 8, you will be asked to return to the clinic and your symptom/temperature diary will be reviewed. If you cannot come to the clinic, a member of the research staff will call you to review this information. This phone call will take 15-30 minutes.

On Day 28, blood (about 4 teaspoons) will be drawn to check your immune system response to the flu virus.

On Day 56, blood (about 4 teaspoons) will be drawn to check you immune system response to the flu virus. If you live out of town and cannot return to M. D. Anderson for this visit, you may have your blood drawn at a local clinic. You will then mail the blood to the study doctor in a pre-paid envelope.

On Month 6, you will be called and asked about any side effects you may have experienced. This phone call will take 15-30 minutes.

Length of Study:

You will be considered off study after the Month 6 phone call. You will be taken off-study early if intolerable side effects or an allergic reaction occurs.

This is an investigational study. TIV, pegylated interferon, and interferon are FDA approved and commercially available. At this time, the use of pegylated interferon and standard interferon with TIV is only being used for research.

Up to 60 patients will take part in this study. All will be enrolled at M. D. Anderson.

Inclusion Criteria:

1. Capable of understanding the protocol requirements and risks and providing written informed consent.
2. Patients with histologically or cytologically confirmed diagnosis of chronic lymphocytic leukemia according to established guidelines.
3. Patients with Rai stages 0 to 4.
4. Age >/= 18 years old.
5. If patients have been treated with antineoplastic therapy, it must have been finished 3 months or longer prior to enrollment.
6. Patients with complete or partial remission and those with stable (CLL) disease will be considered.
7. Patients who have received influenza vaccine in past 4 months will also be considered.
8. Patients willing to receive recombinant cytokine.
9. Patient willing to receive commercially available influenza vaccine that will not provide protection against the following years of influenza strains.
10. Patients must have adequate hepatic function defined as follows: total bilirubin \