A lot of people believe that, as a pathogen adapts to a new host, it will reduce its virulence. Usually people phrase this something like, “The virus doesn’t want to kill off all its hosts.” More sophisticated people might say something similar in less teleogenic terms, like John Timmer: “Typically, viruses that rapidly kill their host have a very short history, as they rapidly run out of places to reproduce“. 
Sometimes this does happen, as with spumaviruses.  Sometimes it sort of but not really happens, as with myxomaviruses.  (Myxoma viruses are a poster child for the reduced-virulence theory, because early after their introduction into Australian rabbits, myxomavirus virulence did rapidly drop. But the virulence dropped from over 99% to “just” 70% mortality — higher than Ebola or smallpox — and stayed at that level for decades.) And sometimes, viruses evolve to increased virulence.
In the example of evolution to increasing viral virulence that I’ve usually used, the details are a little unusual and may involve human intervention via a chicken vaccine.  But here’s a new example, in a more natural system: Rabbit Haemorrhagic Disease Virus in Australian rabbits. 
By studying the change in virulence of recent field isolates at a single field site we show, for the first time, that RHDV virulence has increased through time, likely because of selection to overcome developing genetic resistance in Australian wild rabbits. High virulence also appears to be favoured as rabbit carcasses, rather than diseased animals, are the likely source of mechanical insect transmission.
It’s important to understand the mechanisms of pathogenesis and, especially, transmission, because that’s what drives pathogen evolution.