What is your brother's age. Is he 15 or are you 15? If your brother is under 2 years of age, his options for medication will be limited.
Also, can you please provide me the name of the antibiotics prescribed?
Antibiotics are used for bacterial infections. They do not help viral infections - other than to prevent secondary bacterial infections from developing.
It is possible he may have has a virus. If this is the case, antibiotics would not be expected to have any effect.
If it is bacterial, perhaps, the antibiotics prescribed were not appropriate for the bacterial infection. Some physicians will actually take a nasal swab for culture to identify the actual organisms so as to prescribe the most appropriate antibiotic. Until the results of the culture come back, they prescribe a prophylactic antibiotic to cover typical organisms. This is really a better approach then just prescribing antibiotics and hoping it works and if not then prescribing another.
I would recommend your brother be seen by an otolaryngologist or ENT. These types of physicians specialize in ear, nose, and throat infections. If your brother has not gotten appreciably better - it may be time to seek such a consultation.
I will be happy to provide you a link he you find a physician in your area:
The first is from the American Medical Association. The link is: https://extapps.ama-assn.org/doctorfinder/recaptcha.jsp
The second, which I prefer, requires registration. It is from the American Board of Medical Specialties (ABMS). The link is: https://www.certificationmatters.org/is-your-doctor-board-certified/search-now.aspx. For this one, you do not need to know the name of a particular physician - just enter the city and state and specialty (i.e. otolaryngology, ENT, etc.) into the field then register.
My sense is your brother may be having allergies and could potentially benefit from taking an antihistamine like levocetirizine (Xyzal) and a nasal corticosteroid like fluticasone (Veramyst) providing your brother's physician deems them appropriate. And the cough is most likely due to nasal drainage.
It is also possible the cough is being triggered by environmental stimuli (e.g. exposure to smoking, new pet, etc.). Do any of these apply?
I hope that I have answered your question to your satisfaction; if I have, I would very much appreciate your "accepting" the response provided; bonuses are always appreciated. If you have anything else you would like to ask prior to accepting, I would be more than happy to answer.