I live in a tropical climate. There is a native cockroach that lives everywhere outside and occasionally will wander into homes and set up shop. It took five years to kill all breeding colonies in the house after I bought it.
Even now, cockroaches wander in once every few weeks so I need to remain vigilant.






I would ask about the source first.
Are your air vents dirty? Outdoor pollution? Shedding pets? Fabrics?
An air purifier may be an expensive mask rather than resolving the issue at the source.