The most common and most likely answer to this crossword clue is the 7 letter word ANTHRAX
I've seen this crossword clue in Independent
A
N
T
H
R
A
X
More clues leading to the result ANTHRAX
- Disease in granny's heart and chest, lacking oxygen
- Is it stranger than artist having unknown disease?
- A new location for 16 doesn’t have old disease
- Possible problem with lower tax — an upset around here, oddly
- American chopper with pesticide finally missing area of plant disease
- Case for admission, chest not over deadly disease
- Sickness worker Henry mentioned pains
Buddy explains!

The straight part
We know that "Sickness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case sickness = anthrax.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ANTHRAX mean?
ANTHRAX noun- An acute infectious disease of herbivores, especially sheep and cattle, caused by Bacillus anthracis.
- The human disease that can occur in humans through contact with infected herbivores, tissue from infected animals, or high concentrations of anthrax spores, but is not usually spread between humans, with symptoms including lesions on the skin or in the lungs, often fatal.