Otdfbt

There's little consistency in the articles we use (or don't use) with names of illnesses.

Ex:

I have a cold

I have the flu

I have diarrhea

Strep or strep throat is one of the many illnesses that do not get an article:

I have strep/strep throat

But this is only true when we're using strep as the name of a condition. It's also the name of a bacterium - actually, the name of the bacterium is streptococcus, but it's occasionally shortened to strep. When we're talking about the bacteria, as opposed to the illness caused by the bacteria, we use articles as per normal:

The streptococcus bacteria were hard to find.

A streptococcus bacterium can be oxidase-negative.

If you say "I have a sore throat," the a refers to your throat, so this is grammatically correct.

In various medical texts "sore throat" without a is used when it refers to sore throat as a condition. This is a matter of the usage of medical terms. Most conditions (pneumonia, ovarian cancer, ulcerative colitis, tendonitis) are without a, but there are several exceptions (a common cold, the flu, etc.).

Strep throat is an infection of the throat by the bacterium Streptococcus, so it is much more specific than sore throat.

If you say "I have a strep throat infection," the a refers to an infection, not a throat or sore throat.

But you usually say "I have strep throat," without a, because here you refer to a condition.

Symptoms are more commonly used without a; it's again a matter of usage (cough is more common than a cough).

The difference between "I have a sore throat" and "I have strep throat" lies in the predicate of the sentence.

When the word "Has/Had/Have" is used, we use an article for singular predicates, no article for plural predicates, and also no article for proper nouns or compound nouns (depending on use).

I have a movie. (Singular)

I have movies. (Plural)

I have a Godzilla movie. (Singular with Proper Noun used as an adjective)

I have Godzilla. (Proper Noun predicate, used in a conversation where the topic is known to be about movies)

When speaking about medical terms, this style is also used:

I have a sore foot.

I have Athlete's Foot.

"Strep throat" in particular is not normally capitalized from what I've seen, and so it is not as easily identified as a proper or compound noun.

One easy way to identify adjectives (and adverbs, prepositional phrases, clauses, etc.) is to eliminate it from the sentence and determine if the sentence still makes sense.

I have a sore throat. --> I have a throat.

This sentence can still be understood. "Sore" is an adjective.

I have strep throat. --> I have throat.

This sentence does not make sense.

Therefore, the word "strep" is not being used as an adjective in the sentence. Instead, it is being used as part of the compound noun "strep throat". Within that compound noun, "strep" is in fact used as an adjective of "throat", but the entire compound noun of "strep throat" is the target of "have".

"I have strep" is a sentence where "strep" is used as a commonly-understood abbreviation of "strep throat".

You can have a sore throat because of a number of different causes; a streptococcus ("strep") infection is just one of them. A strep throat is a sore throat but not always the reverse.

I think this has to do with the difference between mass and count nouns, but I am not sure.

So English makes a distinction between things and stuff. Things come in individual chunks. So, for example, you can count them. I can have three drinks, or five friends, or three left shoes. Stuff is some sort of substance or, we might say, it is continuous. It does not come in individual chunks, unless you measure those out. So for instance, I cannot breathe "three airs." I could maybe breathe three liters of air, measuring it out. Sometimes for the really common stuff, it becomes things because it is always measured in just one way. So for example water and coffee and sugar and cream are all stuffs, but in a restaurant, a waiter might say "table three needs two waters and three coffees, two black, one with two sugars and three creams." He is able to play with this aspect of grammar, because in that context he means two glasses of water, three cups of coffee, one of which needs two packets of sugar and three containers of cream.

Similarly if you were to say "She has two salts" it is not grammatical at first, and in my attempts to make it grammatical in my head I start to hope that we are talking about a chemist who has two different kinds of salt, or so, maybe she is mixing sodium chloride and potassium iodide for example.

Now, throats are things. They are things even when they are sore.

Throats can be sore for a lot of reasons. Maybe you were screaming all night last night at a concert and now you have a sore throat. Maybe your throat was recently operated on in a surgery to remove your tonsils and it is still sore.

Strep is a particular bacterium that causes a particular disease, known as strep throat. Diseases like strep throat and cancer are usually thought of as stuff, not as things. So much like the "She has two salts" example above where I am hoping she is a chemist, if you tell me "He has three cancers", that sounds very unfortunate, because I have to imagine that you are talking about different kinds of cancer, "yes, brain cancer, pancreatic cancer, and prostate cancer." That's a very sick man indeed! The words we use to measure these conditions are somewhat different; I have to speak of cases of cancer or so, because it is such an unusual substance.