Some memes behave in a similar way. They are sometimes called "retromemes". The idea of a retromeme is an old concept in memetics.
Image macros will be used to illustrate the effect here. Some image macro text behaves in a similar way to a retrovirus.
For example, here's "OH NOES!":
...and here's "O RLY?":
Does a "retromeme" necessarily need to be deleterious to its host? I would say "no". It is true that the deleterious connotations of virus are there, but there are some beneficial viruses. The concept doesn't just apply to deleterious symbionts, but - as often heppens - epidemiology has the best terminology - so it makes reasonable sense to use it.
What is the difference between retromemes and memetic hitchhiking? Certainly retromemes illustrate memetic hitchhiking. A retromeme is very similar to the payload in memetic hitchhiking. However, there do seem to be a few differences. Sometimes the payload and the delivery mechanism used in memetic hitchhiking have a strong association with each other. Since the payload is typically a deleterious meme, it sometimes remains attached to the delivery mechanism - and is not found elsewhere. With a retromeme, it seems reasonable to expect that the payload will have attached itself to multiple different hosts - otherwise it is not behaving in a virus-like manner - and spreading horizontally between hosts.
Of course 'retromeme' is also sometimes used to refer to certain kinds of "steampunk" memes - but this post is not about that usage.
The examples above are of words, but there are other kinds of retromemes. For example, “Miley Cyrus twerks” has become a visual retromeme: