Hello, my name is***** and I have over 20 years of experience as a veterinarian. I understand that you are concerned about his lethargy and the amount of sneezing that your young fellow has been doing the past couple of days. I understand that his housemate is also sneezing and that both cats snuck out for a short period of time a couple weeks ago. A sneeze simply means that something is irritating his nose. It can be dust, pollens, an irritating, strong smell, increased tearing draining into his nose, a viral, bacterial or fungal infection or a foreign body, a tooth root infection or nasal mass irritating the nasal mucosa. If this were a one time thing then dust, pollen, a strong smell, or increased tearing would be most likely. Since this is lingering and a housemate is also affected I would be more inclined to think of an infection (fungal, bacterial or viral).If he has no, or a little clear, nasal discharge, is eating and drinking normally, and his nose conformation is unchanged (no swelling or bumps) then it may be a viral infection, in which case this should pass on its own. You could try giving both kitties an amino acid supplement called L-lysine at a dose of 500mg orally twice daily. If this infection is due to the Herpes virus (one of the more common causes of respiratory infections
in cats) this amino acid interferes with virus replication and will shorten the infection's duration and severity. Good supplements to try are made by the Viralys brand which comes in a powder to add to the food
or a tasty gel. In the meantime using a humidifier in the room they sleep in most of the time, and/or taking them into the bathroom with you when you run a hot bath or shower to breathe in steam/humid air can help. If his housemate improves but he continues to sneeze beyond the usual 2-3 weeks that these infections run their course the next step would be sedation, dental radiographs to look for bad tooth roots, a nasal scope and skull radiographs to look for a foreign body or mass. It may be helpful to collect samples for bacterial and fungal cultures as well.But I suspect that when they snuck out they were stressed (weakening their immune systems) and may have been exposed to virus particles left behind in the environment. If that's the case they should improve with time and supportive care, just like we do with a cold.Please let me know if you have any further questions.