Pyrethrin/Pyrethrum is hardly new, it was a "natural" plant based insecticide, and is actually one of the oldest insecticides out there. But drizzle is right, it's significantly safer than many other insecticides, although in large enough quantities it's still toxic to humans and other animals.
The insecticide properties of Pyrethrum have been known since the 1800's, and the ground up flowers that produce it were sold to those back then who could afford it for insect control. Wholesale production of synthetic Pyrethrin started when the mad scramble to replace DDT with safer, but sometimes less effective, alternatives. (A huge screw-up BTW, pulling DDT from the third world did more damage by killing more people via insect disease, than DDT ever did ecological damage. Education to proper limited use would have sufficed. Rachel Carson and her book, "Silent Spring" could be considered as one of the largest indirect mass-murderers in history...)
Another good "natural" insecticide that is very safe for people and pets is D-Limonene which is a major component in citrus oil. It's actually used as a food scent/flavoring additive, and is also used as a scenting agent in air fresheners, cleaning products, and plastic fruit etc. So people actually eat the stuff! To make you sick, you have to drink enough of it straight to give yourself gastrointestinal distress.
It works OK as an area insecticide or a residual barrier insecticide, but it is absolute death on bugs when used as a wash or a wet spray. It essentially is a very potent degreaser, and it "degreases" insects to death. Most all bugs have natural waxes and oils that help hold their insides in, and their shells together, and the D-Limonene just strips them right off, and they die almost instantly.
We found out D-Limonene when we adopted our dog as a puppy from a wretched flophouse of a home, and he was covered in fleas. He was too young and small to use traditional flea treatments. The local pet supply mega-store recommended a bath with D-Limonene as it was the only safe treatment for animals that small or young. I've helped friends flea bathe their dogs with traditional insecticide soaps, and the fleas jumped, swam, and hid in the dog's armpits and ears for several minutes, and we never seemed to get them all. We dipped our puppy in a bath of warm water and D-Limonene, and all the fleas just floated up to the top, stone dead, almost instantly. And we had a lemon-fresh puppy to boot.
Also, as ChocolateLab33 said, humans are a secondary food source for fleas, they usually only bite out of desperation, they're too evolved as an animal parasite to make it on a (relatively) fur-less human that bathes daily. If you clean aggressively, and remove the vacuum bag etc. They'll probably die out on their own. The only caveat is that they might stick around if there's sufficient mouse or rodent population in the building you were unaware of.