32.1: Epidemic Dynamics - Discrete Case

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%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()

The dynamics of infection and the spread of an epidemic can be modeled as a linear dynamical system.

We count the fraction of the population in the following four groups:

Susceptible: the individuals can be infected next day
Infected: the infected individuals
Recovered (and immune): recovered individuals from the disease and will not be infected again
Deceased: the individuals died from the disease

We denote the fractions of these four groups in \(x(t)\). For example \(x(t)=(0.8,0.1,0.05,0.05)\) means that at day \(t\), 80% of the population are susceptible, 10% are infected, 5% are recovered and immuned, and 5% died.

We choose a simple model here. After each day,

5% of the susceptible individuals will get infected
3% of infected inviduals will die
10% of infected inviduals will recover and immuned to the disease
4% of infected inviduals will recover but not immuned to the disease
83% of the infected inviduals will remain

A = np.matrix([[0.95, 0.04, 0, 0],[0.05, 0.83, 0, 0],[0, 0.1, 1, 0],[0,0.03,0,1]])
sym.Matrix(A)

Do This

If we start with \(x(0)=(1,0,0,0)\) for day 0. Use the for loop to find the distribution of the four groups after 50 days.

x0 = np.matrix([[1],[0],[0],[0]])
x  = x0
for i in range(50):
    x = A*x
print(x)

[[0.15041595]
 [0.05576501]
 [0.61063003]
 [0.18318901]]

Do This

Write a program to apply the above transformation matrix for 200 iterations and plot the results.

#Put your answer to the above question here