Zombie Invasion Calculator

Simulates a zombie outbreak using the SZR (Susceptible-Zombie-Removed) epidemic model — the same compartmental math used for real disease outbreaks — to project how a population splits between humans, zombies, and the removed over time.

Quick Facts

Model
SZR compartmental ODE system
Same Susceptible-Infected-Removed structure used in disease-outbreak epidemiology, integrated here with 4th-order Runge-Kutta.
Known result
Disease-free state is technically unstable
Whenever resurrection rate ζ > 0, containment shown here is short-term — the basic model's zombie-free equilibrium is never permanently stable.

Your Results

Calculated
Humans remaining
-
Susceptible population, day N
Zombies remaining
-
Zombie population, day N
Tipping point
-
First day zombies outnumber humans
Outcome
-
Model verdict

Ready

Set your population and rates, then run the simulation.

Understanding the Zombie Invasion Calculator

This tool simulates a fictional zombie outbreak using the SZR (Susceptible-Zombie-Removed) model — a set of three coupled differential equations built on the same compartmental structure used in real mathematical epidemiology (the SIR model for infectious disease). The zombie version was first formalized by Munz, Hudea, Imad and Smith in their widely cited 2009 paper "When Zombies Attack! Mathematical Modelling of an Outbreak of Zombie Infection." It is a genuine, well-established piece of applied mathematics, borrowed here for an entertaining scenario rather than a real forecast.

The formulas

The population is split into three compartments that always sum to your starting population: S (susceptible humans), Z (zombies), and R (removed — the dead who have not yet reanimated). Written as fractions of the total population, the basic model is:

  • dS/dt = −β·S·Z — humans become zombies through contact, at transmission rate β.
  • dZ/dt = β·S·Z + ζ·R − α·S·Z — zombies gain new members from infected humans and from reanimated removed individuals (rate ζ), and lose members to human resistance (destroy rate α).
  • dR/dt = α·S·Z − ζ·R — destroyed zombies join the removed pool; the removed pool drains back into zombies at rate ζ.

This calculator numerically integrates those equations with fourth-order Runge-Kutta (RK4) over your chosen number of days, assuming births and non-outbreak deaths are negligible during the timeframe (a standard simplification for short, fast-moving outbreak models).

The destroy-rate vs. transmission-rate race

The outcome hinges on the balance between β (how fast zombies convert humans) and α (how fast humans destroy zombies). When α is comfortably larger than β, the zombie population is pushed toward zero within the simulated window. When β is larger than α, zombies grow at an accelerating rate once their numbers are large enough to matter, and eventually overtake the human population — the "tipping point" this calculator reports.

Reading your results

Humans remaining and Zombies remaining are the S and Z compartments at the end of your simulated period. Tipping point is the first simulated day on which zombies outnumber humans — if that never happens within your chosen window, the calculator says so instead of guessing. Outcome summarizes the trajectory in plain language: contained, spreading, dominated, or wiped out.

Frequently Asked Questions

What formula does this zombie calculator use?
It uses the SZR (Susceptible-Zombie-Removed) model, a set of three differential equations from mathematical epidemiology first applied to zombies by Munz, Hudea, Imad and Smith in their 2009 paper "When Zombies Attack!". Humans (S) become zombies (Z) through contact at rate β, zombies are destroyed by humans at rate α, and the removed dead (R) can reanimate back into zombies at rate ζ.
Can humans ever truly win?
In the short term, yes: set the destroy rate (α) higher than the transmission rate (β) and the zombie population is pushed toward zero within the simulated window. Mathematically, though, the zombie-free equilibrium of the basic model is technically unstable whenever the resurrection rate ζ is greater than zero, so a "contained" result should be read as safe for now, not a permanent guarantee.
Why do the population numbers always add up the same?
The model conserves total population: humans (S) plus zombies (Z) plus removed (R) always equals your starting population, because every person who leaves one compartment enters another. Births and natural deaths unrelated to the outbreak are assumed negligible over the simulated timeframe.
What do the rate inputs actually mean?
Each rate is a fraction per day. Transmission rate β is the share of human-zombie encounters that convert a human each day. Destroy rate α is the share of zombies humans eliminate each day. Resurrection rate ζ is the share of the removed population that reanimates each day. All three are illustrative assumptions you control — no real-world estimates exist.

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