Actually the disease vectors mostly traveled the other way, from China to Europe. The most famous example would be the Black Death, which traveled from Asia to Europe along the Silk Road, thanks in large part to it being carried by the Oriental Rat Flea ( Xenopsylla cheopis ). Smallpox is another example, thought to have originally evolved in India around 3000 years ago. By 165 AD it had (likely) traveled to Rome and (likely) caused the Antonine Plague, which killed up to 1/3 of the population.
Biologically speaking, the longer a population has existed in a certain area, and the denser the population, the more diseases evolve along side of them. For example Africa is the cradle of humanity. Humans have existed there for a very long time. Africa has many diseases that are endemic, such as Ebola, Legionaries Disease, Malaria, etc. These diseases originally evolved in Africa and in some cases (such as Ebola) still exist in large part only there. In other cases (such as Malaria) they have traveled. In Victorian times Europeans who traveled to Africa would almost invariable catch some sort of tropical disease, their life expectancy in Africa wasn't much better than the life expectancy of a native North American after European contact.
Asia was populated before Europe, and for a large part of it's history has had a fairly dense population. They had a longer time for diseases to evolve along side of them. You can still see this today when you look at modern disease outbreaks. SARS and H5N1 (the bird flu) both originated in China and spread outwards from there.
Malaria and sickle-cell anemia are particularly interesting. Malaria evolved in Africa and has existed there since pre-historic times. Each year it causes hundreds of thousands or even millions of deaths (in 2015 there were around 429,000 deaths from malaria). Sickle-cell anemia is a disease that also evolved in Africa. It's a genetic disease that causes your red blood cells to become deformed. The median life expectancy of an individual with sickle-cell disease in 45 years old, and that's with modern medicine and blood transfusions. Before modern medicine most died in childhood, you'd be lucky to live to see your 20th birthday.
Genetically you have to have 2 copies of the gene for sickle cell anemia to express, one from your mother and one from your father. If you have only one copy of the gene then you're a carrier, but the disease doesn't affect you. If you're a carrier then genetically speaking there's a roughly 25% chance that your children will express the full sickle-cell disease, and a 50% chance that your children will be carriers (this is a simplification).
But, here's the kicker, if you're a carrier then you have a much,much lower chance of contracting malaria. Having a single bad gene causes your red blood cells to deform slightly, not enough to adversely effect your health, but enough to make contracting Malaria, a blood-born disease, difficult.
So sickle-cell anemia is a disease that evolved in response to another disease. There's a 25% chance that it will kill your children, but a 50% chance that they will thrive and not catch malaria. There's also a 25% chance that your children won't carry the sickle-cell gene at all, but these children are then highly susceptible to malaria. Since most deaths from malaria occur in children (72% of malarial deaths are children younger than 5) if you don't have a copy of the gene then you are more likely to die before you're old enough to have children. If you have 2 copies then you also die before having children. But if you have a single copy you live for much longer and have a high chance of having children and passing the gene on to them.
Sickle-cell anemia is a disease that almost exclusively affects only Africans. Europeans don't carry the gene, nor do Asians. This tells us that Malaria and Africa have existed together for a very long time.
Other factors that affect disease transmission along the silk road: Travel along the silk road would take a long time. Generally speaking it would take around 2 months to go from one end to the other (of course this varied greatly. Marco Polo took 24 years to complete a round trip if his story is to be believed). Most diseases don't have an incubation period that long. For example, if you catch the flu you don't still have it 2 months later. After a week or two you're cured, or dead. There are very few diseases that can survive for several months inside a human and still be contagious months later. In effect the long travel time acted as a sort of quarantine period, allowing the disease to run its course before you arrive at your destination. The reason that the Black Death traveled is that it had a reservoir population. The rats and the fleas could carry the disease and spread it back and forth between rats, people, and fleas which greatly aided it's transmission. Malaria also has a reservoir in the mosquito, which is why it has traveled so far from Africa.