So, just what viral is your vet saying? Did you ask him about calicivirus??? A swab test or titer can be done to determine it.
What symptoms is Max displaying? What is the vet treating him with?? The vet IS treating him, right?
You should know that calicivirus is highly contagious between cats, and even after treatment if the cat appears recovered, the cat can remain a carrier to other cats. You need to talk to your vet about this, and make darned sure it's been treated effectively and not to expose other cats.
I'd also strongly advise talking to the breeder and informing him/her about your cats, so they can test and monitor their kittens and cats before they start getting sick and end up misdiagnosed, or, before they adopt out kittens or cats who may be carriers and would get sick after adoption (new owners might not know how to identify this).
While calicivirus is generally treated effectively with antibiotics, secondary infections and serious complications like pneumonia can occur, so it's imperative your vet is telling you what you need to know, and identifying this correctly, or differentiating it with something else. Very young, older, or immuno-compromised cats are at higher risk for infections and complications.
On your lyme's issue, is there any chance you brought ticks into the home that may have attached to your cats? Lyme's is not transmitted between humans and animals, it's the tick that transmits the disease, please be sure your home is tick-free, and if you live in a wooded area, check your clothes and shoes etc when entering your home. Lyme's causes very similar symptoms in animals that Oliver experienced, but testing is necessary to confirm.
Did Oliver receive any vaccinations before you aquired him?
How old is Max, and is he up-to-date on his vaccinations?
Are your cats kept strictly indoors only?