### 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 *R*_{0} (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*) = − *ν**S*(*t*)*I*(*t*) + *α**R*(*t*) − *β*(*I*(*t*) + *AC*(*t*)) × *S*(*t*)

(∂*I*)/(∂*t*) = *β*(*I*(*t*) + *AC*(*t*)) × *S*(*t*) − *ζ**I*(*t*) − *γ*(*S*(*t*) + *AC*(*t*) + *R*(*t*)) × *I*(*t*) − *μ**I*(*t*)

(∂*R*)/(∂*t*) = *ζ*(*I*(*t*) + *Q*(*t*)) + *τ**AC*(*t*) − *α**R*(*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*) = *ν**S*(*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) |