Warning: I have almost no knowledge of biology past the high school level.
Viruses generally have three components: the DNA, the virus protein coat, and an outer membrane "decorated" with these surface marker glycoproteins. I am thinking that a virus would want to infect as many hosts as possible, so that it would reproduce as much as possible, why would a virus just infect one group of organisms.
What evolutionary advantage do viruses have in host specificity?
It is true for any living creature, that it would be great for it if it could thrive in all environments. Any creature would do better if it had a greater ecological niche while remaining as competitive in each of these niches. However, competition, predation and other biotic and abiotic factors lead species to specialize in specific niches. Of course, some species are more generalist and some are more specialist but I won't go into these details.
When it comes to parasites, such as viruses, the story is the same. A host is an environment. Being less specific would be great but the immune system is no easy detail to get around. Viruses are often quite specific to a given species, just because it evolved to be efficient for a given host but tend not to be that efficient in other hosts.
Note that parasites are not only species specific but also often tissue specific and specific to the specifics genetics of the host (e.g. malaria).
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Thank you @DeNovo for helpful comment
I would more specifically address the advantage/benefit, in that even though a virus may be host specific (or not, since it really doesn't matter in terms of gain), even the most virulent of viruses throughout history have not killed off entire species, save the 'virus' of being man (whole other story), yet for some reason some within a species survive! They may have a specific protein, are a carrier - the list is long and not entirely understood. In any case, they often handoff whatever adaptable mutation they have to their offspring, thereby making them immune to such viruses. Survival of the fittest?
Viruses infect a cell via a receptor. Without the specific receptor infection is not possible. A virus will mutate to enter a cell via another receptor (HIV-2). So, I say viruses are receptor specific.
