SIR Model Explanation

Mathematical Model and Python Code

The SIR model is a simple mathematical model to describe the spread of a disease in a population. It consists of three compartments:

• S: Susceptible - individuals who can contract the disease.
• I: Infected - individuals who have contracted the disease and can spread it.
• R: Recovered - individuals who have recovered from the disease and are immune.

Mathematical Model

The SIR model can be represented by the following system of differential equations:

\[ \frac{dS}{dt} = -\beta \frac{S I}{N} \]

\[ \frac{dI}{dt} = \beta \frac{S I}{N} - \gamma I \]

\[ \frac{dR}{dt} = \gamma I \]

Where:

• \(S(t)\): Number of susceptible individuals at time \(t\)
• \(I(t)\): Number of infected individuals at time \(t\)
• \(R(t)\): Number of recovered individuals at time \(t\)
• \(\beta\): Contact rate
• \(\gamma\): Recovery rate
• \(N = S(t) + I(t) + R(t)\): Total population, assumed to be constant

Python Code for SIR Model

# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# 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 = 0.3, 1./10 
# A grid of time points (in days)
t = np.linspace(0, 160, 160)

# 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
plt.figure(figsize=(10, 6))
plt.plot(t, S, 'b', label='Susceptible')
plt.plot(t, I, 'r', label='Infected')
plt.plot(t, R, 'g', label='Recovered')
plt.xlabel('Time /days')
plt.ylabel('Number')
plt.title('SIR Model')
plt.legend()
plt.show()