^^ Aren't you meant to train less frequently so that this isn't an issue :dunnodude:
The HIT system runs off the principle that you essentially overdo it in a session so that you don't have to spend as long training and have to have less days in the gym. Great if you don't have a lot of spare time or whatever, but not exactly the best training paradigm. So not actually meant to eat into your recovery ability and health, if it is you are going about it the wrong way.
Basic stuff to strengthen your immune system would be Getting plenty of vitamins/minerals from natural sources and or supplements.
^^ I've read Mentzer's take, Jones' take, Darden's take and Yates' take. I've also liked the Hardgainer approach which is a version of HIT and DC which is similar to Hardgainer with some stupid crap thrown in.
The big overriding philosophy is to not overtrain and tax your recovery ability. Your immune function is part of this. I know that the next day you are usually ground into the ground, but this shouldn't have stretched you so far as to break you.