Is desire really only a function of the mind? Isn't it often physical... maybe even primarily physical? It ultimately has to be as everything is physical, even the mind.
One thing that's important, per my understanding of Epicurus, is to not conflate desires with pleasures. I interpret this, rightly or wrongly, as desires are not feelings and therefore not pains. Pleasure/pain is one biological function, desire is another.
I also think that it would be instructive to review modern biochemistry in this regard; it may or may not clarify Epicurus, but we have more information than he did and so it's worth reviewing the ideas from a contemporary standpoint. It's of note that our contemporary standpoint often validates Epicurus' thinking to a large degree, which is one reason why I for one find EP so appealing.