Mental health pushes the physical health. And I believe research will find some magical, mystical substance in dog saliva that we benefit from when they lick our face. Remember that some cats can be very affectionate. Also, puppies may be too much work for someone with advanced disease. Plus, with older animals, you can see whether their size and temperament agree with you.
Most birds pose a minimal risk for transmitting disease.
PAWS recommends that people at risk do not keep or handle reptiles. In fact, HIV positive people contract infections more often from contaminated food, water, soil, or even other people than from pets. Precautions apply for children as well as adults. However, children may want to snuggle more with their pets. Some pets, like cats, may bite or scratch to get away from children.