About the Vampire Apocalypse Calculator
This calculator models a vampire outbreak with a discrete exponential growth formula — the same mathematical structure behind compound interest and population doubling problems, adapted to a feed-and-convert mechanic. Set a starting human population, a starting vampire population, how often each vampire feeds, and what fraction of victims turn into vampires, and it estimates how long humans would last.
The formula
Each feeding cycle (one interval), every living vampire feeds on exactly one human. That human turns into a new vampire with probability p (the turn rate) or is simply removed from the human population otherwise. This gives two recurrences:
- Vampire population: Vn+1 = Vn × (1 + p) — a compounding growth factor of (1+p) every cycle, so Vn = V0 × (1+p)n.
- Human population: Hn+1 = Hn − Vn — each vampire removes one human from the pool every cycle.
Solving for the cycle count n where cumulative humans consumed equals the starting human population gives a closed form: n = ln(1 + H₀·p / V₀) / ln(1 + p). Multiplying n by the feeding interval converts cycles into a time span. At the end, the total vampire population equals V₀ + p·H₀, since every consumed human either turned (probability p) or was permanently removed (probability 1−p).
Special case: 100% turn rate
When every victim turns (p = 1), the vampire population exactly doubles each feeding cycle — the fastest growth this model allows, and the classic version of the "vampire math problem" used in physics and math education to illustrate how fast exponential growth can be. A single vampire feeding once a month with a 100% turn rate against a population in the hundreds of millions would, under this model, exhaust that population in only a few years.
Special case: 0% turn rate
When no victims turn (p = 0), the vampire population never grows — it stays at V₀ forever — and humans are consumed at a constant rate of V₀ per cycle. Extinction time becomes simple division: H₀ ÷ V₀ cycles.
Assumptions and limits
The model assumes every vampire feeds exactly once per interval without fail, the turn rate is constant and applies uniformly to every feeding, and there is no hiding, hunting back, resource limits, or population growth on the human side. Real outbreaks (fictional or epidemiological) would be far messier — this is a clean illustration of exponential dynamics, not a demographic forecast.