This site asserts that syphilis, polio, hepatitis, and encephalitis come to general world from the Americas but I am unable to prove that statement could someone confirm or deny it?
Hepatitis and encephalitis describe an inflammation of a specific organ (respectively the liver and the brain), not a specific virus or infectious agent, and can have multiple causes. Like many other infectious agents, the hepatitis viruses have been identified and described rather recently and early accounts are sometimes vague on the symptoms, thus making it difficult to identify a specific disease. I don't know if there are clear accounts of hepatitis infection in historical sources.
Polio is definitely considered to have been present in the old world in pre-Columbian times, possibly on account of its rather specific symptoms. It somewhat paradoxically became a bigger problem much later, with the progress of hygiene, which might account for the confusion.
Syphilis is the most interesting case. It has long been thought that it originated in the Americas and spread to Europe in the sixteenth Century (there is no evidence of large scale epidemics before that time) but new evidence called this into question. Potential confusion with leprosy, nonvenereal endemic syphilis (bejel) and yaws further complicate the interpretation of historical sources. There is some archeological evidence of a disease similar to syphilis in the old world, all the way back to Antiquity (specifically: bone damage in infants born to an infected mother) but such cases are in any case much rarer in Europe than in the Americas and whether this is actually the same disease is still debated. (I found a useful summary of the debate with various reputable sources on Wikipedia but unfortunately only in the French-language version of the website).
Syphilis has been in the America's for awhile, so far no evidence has been found of in pre-colombian Europe. Excerpt:
"Using these criteria, they examined 687 skeletons from archaeological sites in the United States and Ecuador ranging in age from 400 to 6,000 years. Populations to the south (New Mexico, Florida, and Ecuador) proved to have syphilis, while those to the north (Ohio, Illinois, and Virginia) had yaws. By contrast, examination of 1,000 Old World skeletons dated to before contact with the New World revealed no cases of syphilis."
Polio has been around for awhile, see wikipedia.
In terms of age,Hepatitis B is really old,as is the A variety, found in antiquity, Hep C seems to be a relatively recent epidemic, spread by the increased use of blood sharing medical instruments, and substance abuse.
Encephalitis is a catchall term for a variety of ailments found in Europe, Asia, and America.
I imagine that, like many animal species, these viruses have new and old world versions, brought to the Americas with migration of humans, and other primates, mammals,etc. Much in the same way that many North American species originated from Asia, such as Black Bears and Asian Black bears share genetic heritage.
Other notable examples: European Lyme vs. American Lyme (both spirochetes), Wisent Vs. American Bison Vs. Asian Water Buffalo.
Note that this is the same realm of suspect archeological territory, as the findings of Tobacco, Cocaine, and Cannabis in Ancient Egypt, or finding really old Chinese Ships sunken in the Sacramento Delta. Basically, the world has been interconnected longer than most American's and Europeans would care to admit.