L-lyseine is used to slow down the replication of viruses. Antibiotics work on bacteria. If your cat has a viral infection, antibiotics won't do anything to help.
Our cat is also on L-lyseine and will be for life. The most common causes of respiratory infections in cats are herpes and chlamydia. The virus lies dormant in their sinuses and when it flares up, they have runny eyes and snotty noses. The virus will never completely go away, which is why you keep them on the supplement all the time. It has certainly helped our cat. It's quite inexpensive and we get about a month's supply at a time. We don't have to see the vet, I just call for a refill.