I thought I thoroughly understood these particles until I tried writing a phrase “sick with the flu” earlier today. I typed インフルで病む and went about my business scribbling the rest of my practice journal routines. Then later it occurred to me that maybe it should have been インフルと病む。
I now suspect と is the correct particle and that で would be used if I were to say HOW one is sick with the flu (say, だれかの嚔でやむ – from someone’s sneeze… though I have a sneaking suspicion I’d have to shoehorn in a になる somewhere, but that’s beside the point of this post)
Belthazar beat me to it… but yea 病む is transitive when talking about a specific disease. When you can’t tell what particle goes with a verb you can try to check dictionaries for example sentences.
It’s true that 病む takes a direct object, so things like [disease]を病む are seen. So that’s regarding the grammar of the question.
But as for the vocab and usages, I can’t find any instances of インフルエンザを病む (or インフルを anything) in the BCCWJ though. And only a few on Google generally.
I think with the flu, the most common verb to use is かかる, and with かかる being intransitive, you would say インフルエンザにかかる.
EDIT: And a weird aside, 胸を病む seems to be an idiomatic expression meaning “to have consumption (pulmonary tuberculosis).” I kept seeing that in example sentences for the “suffer from a disease” meaning, and I was like “chest is not a disease,” but it’s an expression it seems.