Bagless Hepa Vacuum
Let's investigate, the word "HEPA" means "high efficiency particulate air". If a filter is to be rates as "HEPA" it will need to retain all particles as small as .3 microns in size as well as anything larger. Not only that but it will also need to operate at an efficiency level of 99.97%.
Being that the more common particles in the air are approximately 2.4 microns in size, and being that the typical Non-HEPA vacuum cleaner picks up particles from about 30 to 50 microns in size, the smaller particles will simply be blown back into the air.
And so, a bagless HEPA vacuum cleaner will do its part of helping those suffering from allergy and asthma pain by drawing towards its filter those allergens and dust which have settled on floors, carpets and furniture. And remember this is a help as those "settled" allergen and dust particles will eventually get stirred up back into the air where it can be inhaled humans.