Flea bites are not contagious and will not spread, causing more flea bites.
The only thing that will spread more flea bites is the presence of more adult fleas who are looking for a blood meal.
As long as the person or animal who has received flea bites has been removed from the environment that is infested with fleas, he should not sustain any more bites.
If you or your pets are continually receiving more flea bites, it is necessary to treat for flea infestation using flea medication, flea traps, etc. in order to kill the adult fleas were biting, interrupt the lifecycle of the flea that they cannot reproduce.