The sample sentence given needs to be rephrased to be able to express the intended meaning because with the way it is worded, we cannot tell which patient had which illness.
If what we want to say is: "each patient only had one illness...," we may phrase the sentence the following way:
One patient had a cold, another one had a flu, the next one had a stomachache, and the last one had a headache.
If what we want to say is each patient had all illnesses listed, we may phrase the sentence the following way:
Each patient had a cold, flu, stomachache, and headache.
Nowhere in the above sentences is the word "respective" or "respectively" found as these words are intended for the following usage:
RESPECTIVELY is an adverb which means "in the order given."
Example sentence: I gave the bag and book to Trish and Sam, respectively.
(i.e. I gave the bag to Trish and I gave the book to Sam.)
RESPECTIVE is an adjective which means "separate."
Example sentence: The animal trainer motioned the lion and tiger to go into their respective cages.
(i.e. The lion went into his own cage and the tiger went into his own cage.)
I hope the above helps. :)