One of my cats, who's since passed away, was asthmatic. Air purifiers throughout the house and a motion activated box fan plus furnace filter near the litterboxes resulted in an immediate and noticeable improvement in his quality of life. I've noticed a similar improvement in my own health.
I've since ended up with a variety of air cleaners:
* IKEA FÖRNUFTIG[0] is a small and relatively quiet unit. It can be wall-mounted, so it can take up virtually no space. The unit is reasonably priced. Filters are cheap.
* IKEA STARKVIND[1] is a much larger unit (also available in end table form[2] to save space), but also relatively quiet on the lower speeds. It's an interesting unit - integrates into Home Assistant (the unit speaks Zigbee), and has a PM2.5 air quality sensor. This unit is a lot more expensive than the FÖRNUFTIG, but the filters are reasonably priced.
* The box fan plus single furnace filter is incredibly noisy, but really good at dealing with cat litter dust. There is a huge range of price/quality when it comes to filters[, I just use the cheaper ones since I'm focusing on large dust particles.
* I have a couple of units that use Bionaire aer1 filters[3]. The units I have are quiet and reasonably sized, though they get louder as the filter fills up. The filters are expensive, and one of the units takes two of them which doesn't help matters. There is a variety of filters available.
There's a huge spectrum of tradeoffs between noise, size of the unit, filtration effectiveness, replacement filter cost, and extra features. I'm not convinced I've found the sweet spot yet.
[0] https://www.ikea.com/ca/en/p/foernuftig-air-purifier-white-5...
[1] https://www.ikea.com/ca/en/p/starkvind-air-purifier-black-40...
[2] https://www.ikea.com/ca/en/p/starkvind-table-with-air-purifi...
[3] https://www.bionairecanada.com/en_CA/service-and-support/aer...