The hair is falling out because you need nutrients. Infections create major stress in the body which wastes nutrients. Hair falling out is a protein issue which is of course dependent on proteins but also on co factors such as the b vitamin family. B12 is of particular note but all the b vits are critical. Antioxidants, minerals, whole foods...
There is some evidence that some nutrients may help the pathogens but you are clearly deficient. With nutrients you will strengthen the body which will help fight the infection, detoxify and stabilize the endocrine system etc.