32.2: Epidemic Dynamics - Continuous Model

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

Instead of using the discrete markov model, we can also use a continuous model with ordinary differential equations.

For example, we have that

\[\dot{x}_1 = {dx_1(t)\over dt} = -0.05x_1(t)+ 0.04 x_2(t) \nonumber \]

It means that the changes in the susceptible group depends on susceptible and infected individuals. It increase because of the recovered people from infected ones and it decreases because of the infection.

Similarly, we have the equations for all three groups.

\[\dot{x}_2 = {dx_2(t)\over dt} = 0.05x_1(t)-0.17 x_2(t) \\
\dot{x}_3 = {dx_3(t)\over dt}= 0.1 x_2(t) \\
\dot{x}_4 = {dx_4(t)\over dt} = 0.03 x_2(t) \nonumber \]

Do This

We can write it as system of ODEs as \(\dot{x}(t) = Bx(t)\). Write down the matrix \(B\) in numpy.matrix

# Put your answer to the above question here.

Do This

Plot all the distribution for 200 days. Then compare it with the discrete version.

x0 = np.matrix([[1],[0],[0],[0]])
n = 200
x_all = np.matrix(np.zeros((4,n)))

### Your code starts here ###

### Your code ends here ###
for i in range(4):
    plt.plot(x_all[i].T)

D2, C2 = np.linalg.eig(B)
np.allclose(C2*np.diag(D2)*C2**(-1), B)
C = C2