Do you mean transmissibility? I get what you mean, but I've never heard this word used this way. (Virulence is, more or less, the non-fatal version of mortality -- how much damage the disease does -- so not that.)
Be that as it may, once the disease is established in a new species it tends to get less harmful because of exactly what you're talking about -- but plenty of diseases through history have been in the short run both fast-spreading and deadly, especially right after they jump into a new population. Which is exactly what H5N1 is doing right now (on all three counts).