Flea eggs can lay dormant on the ground for years, only hatching when the ground vibrates. Other than a quake, the subtle vibration means an animal walking by. Food!
Flea eggs are transparent. Folks can't see them and pick them off. What most folks think are eggs is actually flea feces - which is entirely made of blood. If you wet the little black pellets, they return to their red liquid state.
Chemicals are not necessary. Diatomaceous earth is all natural, earth friendly and will end a flea infestation both inside and outside. A solution of Neem oil (cousin to the Tee Tree) works better than any chemical repellant there is. It's also an excellent coat conditioner, as well as the best treatment/preventative for any time of fungus such as ringworm.
Fleas can be a real nuisance, but you must use some kind of chemical-based treatment. To get the problem under control, make an appointment with a local groomer and ask them to give your cat a flea dip. You can give the cat a flea dip yourself if you prefer. Talk to your vet about a safe dip to use - many of the over-the-counter dips and shampoos are too strong for kittens, or even cats in general.
Be sure that all of your pets are treated simultaneously. Do not forget that fleas live in the carpet and ground/grass 80% of the time, so the environment must be treated at the same time as the kitty. Either buy a spray or bug bomber for the house. An alternative is to call a pest control company and let them professionally treat your home.
You may want to talk to your vet about anti-flea products, such as Program, Advantage, Frontline, or Revolution. Typically, these products only have to be administered to your pet once a month. Such medications prevent fleas from laying eggs, so they cannot reproduce and spread. Be sure to keep a flea comb handy. It's a good idea to run the comb through your cat's hair frequently to inspect for flea dirt. (Some people keep their Persians shaved so they can look for flea eggs easily.)