A simple mathematical description of the spread of a disease in a population is the so-called SIR model, which divides the (fixed) population of N individuals into three "compartments" which may vary as a function of time, t:
The SIR model describes the change in the population of each of these compartments in terms of two parameters, β and γ. β describes the effective contact rate of the disease: an infected individual comes into contact with β·N other individuals per unit time (of which the fraction that are susceptible to contracting the disease is S/N). γ is the mean recovery rate: that is, 1/γ is the mean period of time during which an infected individual can pass it on.
The differential equations describing this model were first derived by Kermack and McKendrick [Proc. R. Soc. A, 115, 772 (1927)]:
The following Python code integrates these equations for a disease characterised by parameters β = 0.2, 1/γ = 10days in a population of N=1000 (perhaps flu in a school). The model is started with a single infected individual on day 0: I(0)=1.
In [11]:
%matplotlib inline
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
# Total population, N.
N = 1000
# Initial number of infected and recovered individuals, I0 and R0.
I0, R0 = 1, 0
# Everyone else, S0, is susceptible to infection initially.
S0 = N - I0 - R0
# Contact rate, beta, and mean recovery rate, gamma, (in 1/days).
beta, gamma = 0.2, 1./10
# A grid of time points (in days)
t = np.linspace(0, 200, 200)
# The SIR model differential equations.
def deriv(y, t, N, beta, gamma):
S, I, R = y
dSdt = -beta * S * I / N
dIdt = beta * S * I / N - gamma * I
dRdt = gamma * I
return dSdt, dIdt, dRdt
# Initial conditions vector
y0 = S0, I0, R0
# Integrate the SIR equations over the time grid, t.
ret = odeint(deriv, y0, t, args=(N, beta, gamma))
S, I, R = ret.T
# Plot the data on three separate curves for S(t), I(t) and R(t)
plt.figure(figsize=(16, 4), dpi=72)
plt.plot(t, S/1000, 'b', label='Susceptible')
plt.plot(t, I/1000, 'r', label='Infected')
plt.plot(t, R/1000, 'g', label='Recovered with immunity')
plt.xlabel('Time (days)')
plt.ylabel('Percentage of individuals')
plt.xlim(0,200)
plt.ylim(0,1)
plt.legend(loc='center right', shadow=True)
plt.show()