Zombie Apocalypse Calculator (SZR outbreak model of Munz et al., 2009)

Understanding the zombie apocalypse calculator — concept and principle

This calculator runs the classic SZR outbreak model from a tongue-in-cheek epidemiology paper by Munz, Hudea, Imad and Smith (2009). You set the starting humans and zombies and the rates of biting, zombie defeat, resurrection and natural death, and it shows who is left after a number of days. It is a fun way to see how real infectious-disease models work.

Each zombie infects βS people a day but is destroyed at rate αS; here infection wins.

How it is calculated

S′ = −βSZ − δS

Z′ = βSZ + ζR − αSZ

R′ = δS + αSZ − ζR

S = susceptible humans, Z = zombies, R = removed (dead).
β: infection when a zombie meets a human; α: zombies destroyed in such meetings; ζ: removed rising as zombies; δ: natural deaths. No births are included during the short outbreak.
The equations are solved with the 4th-order Runge–Kutta method in steps of 0.01 day.

The principle

This is the same idea as the SIR models used for real epidemics. New infections depend on how often zombies and humans meet, which is proportional to S × Z. If β is larger than α, each meeting is more likely to create a zombie than to destroy one, so zombies multiply while humans last. Because the dead can rise again (ζ), the removed group keeps feeding the zombies. The paper's conclusion is that in the basic model, unless zombies are destroyed quickly and aggressively, the outbreak ends with everyone a zombie or dead.

Worked example

Default inputs (the paper's example): 500 humans, 1 zombie, β = 0.0095, α = 0.005, ζ = 0.0001, δ = 0.0001, 10 days.

At the start βS = 0.0095 × 500 = 4.75 new zombies per zombie per day, against αS = 2.5 destroyed, so zombies grow fast.
By day 2 about 362 humans remain; by day 4 about 14.
Humans fall below 1 % of the start (5 people) on day 4.5.
After 10 days: humans 0 (0 %), zombies 238, removed 263.
Zombies are still rising slowly through resurrection, so the peak is 238 on day 10. Outcome: Zombies take over.

Each point is the calculator re-run with only “Infection rate β (per zombie-human meeting)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Infection rate β (per zombie-human meeting)	Humans left at the end
0.00237	499
0.00475	486
0.00713	2
0.0095	0
0.0119	0
0.0143	0
0.019	0

How to read the result

Try raising α above β: humans then win most encounters and the outbreak can die out. Lowering β, like reducing contact in a real epidemic, slows the spread and pushes the peak later. The model is deterministic and assumes everyone mixes evenly, so it shows trends, not exact headcounts.

Common mistakes

Treating the result as a forecast; it is a teaching model.
Forgetting that rates are per day and per meeting.
Running too few days to see the end state.
Reading fractions of a zombie literally; the model treats people as a continuous quantity.

Keep in mind: This is a playful model. Real epidemics depend on many more factors, so do not use it to judge real disease risk.

Frequently asked questions

Who wins in the zombie model?

With the paper's default values, the zombies, within a few days.

How can humans survive in the model?

By destroying zombies faster than they infect people, which means α larger than β.

Is this a real research paper?

Yes, it was published in 2009 as a light-hearted introduction to epidemic modelling.

Sources: Munz P., Hudea I., Imad J., Smith R. J. (2009), "When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection", in Infectious Disease Modelling Research Progress, Nova Science, pp. 133–150

Checked 9 Nov 2026. Munz P., Hudea I., Imad J., Smith R. J. (2009), “When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection”, in Infectious Disease Modelling Research Progress (Nova Science), pp. 133–150 — basic SZR model and the example parameters α = 0.005, β = 0.0095, ζ = 0.0001, δ = 0.0001, N = 500.

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