Learn English – When is it appropriate to use “late” when referring to someone who has passed

Although the phrase can mean that, and often does, it's also sometimes applied in a more broad context. To be "swept off your feet" is to be surprised, enthralled, exhilarated. Critics can be swept off their feet by an epic film; operagoers can be swept off their feet by a beautiful aria, etc.

As for how sweeping became associated with love, that's referring to the aspect of sweeping that means a smooth movement, not the act of using a broom. Ballroom dancers can sweep across the dance floor, a powdery snow can sweep across the barren fields. It's that smooth, fluid motion – and the idea of your emotions being carried in that fashion – that brought about the idiom. A strong ocean or river current can literally sweep you off your feet, and young lovers can do the same thing to each other, figuratively and emotionally.

I'm not sure of the actual entymology of the phrase, but 'to run' means in this case 'to continue to have', so while the fever is being maintained it's 'running'.