Zombie propagation
Sat 21 March 2020
Zombie favorite food warning (Photo credit: wikipedia)
I recently read a paper [1] trying to model a disease propagation. I wanted to play with this model.
The model
The model is know as "SIR" as it divide the population into 3 groups:
- S: suceptible to become a zombie
- I: infected, and thus zombified
- R: recovered, and (hopefully) immune
It then model the move of individuals between groups. The principle of dividing a population into such group and modeling the transitions of individual is known as a compartmental model and many results have already been published about such models. I won't insist on the R0 (basic reproduction number).
For fun, and because it is already done by scientific researchers, I will add other groups:
- Q: quarantined
- D: dead
- AC: asymptomatic carrier (exposed, can infect other but not zombified)
I'll use the follwing transition between groups:
Zombie transition diagram
Note that:
- this is only a toy model
- transitions rates are proportional to population of some groups (more zombies and more suceptibles ⇒ more mettings between both groups, zombies can be put in quarantine faster if there is more non-zombie able to do it)
- ζ = (1)/(λ) in some research paper, λ being a recovery time.
- only zombie can die, e.i. other death causes are not modeled
- once recovered, there is a slight chance to not be immune (model either mutation in the desease or people needing a booster shot)
This model is thus entirely described by the following equations:
(∂S)/(∂t) = − νS(t)I(t) + αR(t) − β(I(t) + AC(t)) × S(t)
(∂I)/(∂t) = β(I(t) + AC(t)) × S(t) − ζI(t) − γ(S(t) + AC(t) + R(t)) × I(t) − μI(t)
(∂R)/(∂t) = ζ(I(t) + Q(t)) + τAC(t) − αR(t)
(∂Q)/(∂t) = γ(S(t) + AC(t) + R(t)) × I(t) − (μ + ζ)Q(t)
(∂D)/(∂t) = μ(I(t) + Q(t))
(∂AC)/(∂t) = νS(t)I(t) − τ(AC(t)
The game
I use python to code this model. I use scipy.integrate.odeint. I simulated some scenarii [2]
This model clearly needs improvements (quarantine is useless in this model). In this model the asymptomous carrier, even if they are few, can have a visible influence on the outcome.
We can now collect possible outcomes:
| [1] | C Witkowski and B Blais (2013), "Bayesian analysis of epidemics-zombies, influenza, and other diseases" |
| [2] | a jupyter notebook containing my code is available here (also in pdf) |
