Modeling a Zombie Apocalypse

This example demonstrates how to solve a system of first order ODEs using SciPy. Note that a Nth order equation can also be solved using SciPy by transforming it into a system of first order equations. In a this lighthearted example, a system of ODEs can be used to model a “zombie invasion”, using the equations specified in Munz et al. 2009.

The system is given as:

dS/dt = P - B_S_Z - d_S dZ/dt = B_S_Z + G_R - A_S_Z dR/dt = d_S + A_S_Z - G_R

with the following notations:

• S: the number of susceptible victims
• Z: the number of zombies
• R: the number of people “killed”
• P: the population birth rate
• d: the chance of a natural death
• B: the chance the “zombie disease” is transmitted (an alive person becomes a zombie)
• G: the chance a dead person is resurrected into a zombie
• A: the chance a zombie is totally destroyed

This involves solving a system of first order ODEs given by: dy/dt = f(y, t)

Where y = [S, Z, R].

The code used to solve this system is below:

# zombie apocalypse modeling
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
plt.ion()
plt.rcParams['figure.figsize'] = 10, 8

P = 0           # birth rate
d = 0.0001  # natural death percent (per day)
B = 0.0095  # transmission percent  (per day)
G = 0.0001  # resurect percent (per day)
A = 0.0001  # destroy percent  (per day)

# solve the system dy/dt = f(y, t)
def f(y, t):
    Si = y[0]
    Zi = y[1]
    Ri = y[2]
    # the model equations (see Munz et al. 2009)
    f0 = P - B*Si*Zi - d*Si
    f1 = B*Si*Zi + G*Ri - A*Si*Zi
    f2 = d*Si + A*Si*Zi - G*Ri
    return [f0, f1, f2]

# initial conditions
S0 = 500.                   # initial population
Z0 = 0                      # initial zombie population
R0 = 0                      # initial death population
y0 = [S0, Z0, R0]   # initial condition vector
t  = np.linspace(0, 5., 1000)       # time grid

# solve the DEs
soln = odeint(f, y0, t)
S = soln[:, 0]
Z = soln[:, 1]
R = soln[:, 2]

# plot results
plt.figure()
plt.plot(t, S, label='Living')
plt.plot(t, Z, label='Zombies')
plt.xlabel('Days from outbreak')
plt.ylabel('Population')
plt.title('Zombie Apocalypse - No Init. Dead Pop.; No New Births.')
plt.legend(loc=0)

# change the initial conditions
R0 = 0.01*S0   # 1% of initial pop is dead
y0 = [S0, Z0, R0]

# solve the DEs
soln = odeint(f, y0, t)
S = soln[:, 0]
Z = soln[:, 1]
R = soln[:, 2]

plt.figure()
plt.plot(t, S, label='Living')
plt.plot(t, Z, label='Zombies')
plt.xlabel('Days from outbreak')
plt.ylabel('Population')
plt.title('Zombie Apocalypse - 1% Init. Pop. is Dead; No New Births.')
plt.legend(loc=0)

# change the initial conditions
R0 = 0.01*S0   # 1% of initial pop is dead
P  = 10        # 10 new births daily
y0 = [S0, Z0, R0]

# solve the DEs
soln = odeint(f, y0, t)
S = soln[:, 0]
Z = soln[:, 1]
R = soln[:, 2]

plt.figure()
plt.plot(t, S, label='Living')
plt.plot(t, Z, label='Zombies')
plt.xlabel('Days from outbreak')
plt.ylabel('Population')
plt.title('Zombie Apocalypse - 1% Init. Pop. is Dead; 10 Daily Births')
plt.legend(loc=0)

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