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According to a paper by Costas J. Efthimiou and Sohang Gandhi vampires can't possibly exist: they've done the maths (PDF file) and "if vampires truly feed with even a tiny fraction of the frequency that they are depicted to in the movies and folklore, then the human race would have been wiped out quite quickly after the first vampire appeared."

Basically, Efthimiou and Gandhi assume that if vampires feed once a month and if they turn each victim into a vampire the simple geometric progression will ensure that the growth of the vampire population will outstrip the human birthrate very quickly, with catastrophic consequences. Their conclusion:

Apparently, whomever devised the vampire legend had failed his college algebra and philosophy courses.

So it's bad luck Buffy. Nice try, Professor Van Helsing. Beware the awesome power of the geometric progression.

Or possibly not. Clive Thompson reckons that he's spotted a way to resolve the situation. He's calculated that there's a sweet spot at which it's just about possible for a determined slayer who never takes a holiday to keep the vampire population at a manageable level.

I can't help but think that the real reason we're not knee deep in vampires is much simpler: we're obviously living in a universe where turning into a vampire requires that the victim not only be bitten, but that they in turn drink the blood of a vampire. (See, for example, Ann Rice's Lestat novels. But do yourself a favour and stop after The Vampire Lestat or The Queen of the Damned: do not, on any account, read The Tale of the Body Thief or any of the later novels in the series. Trust me on this.)

Or, alternatively, the vampires – conscious of the need to avoid depleting the stock of cattle too far – can subsist on animal blood for long spells. Such creatures would 'turn' a victim only very occasionally, perhaps when they met a particularly delightful potential companion.