/
SIR.py
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SIR.py
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# ---
# jupyter:
# jupytext:
# formats: ipynb,py:light
# text_representation:
# extension: .py
# format_name: light
# format_version: '1.5'
# jupytext_version: 1.11.2
# kernelspec:
# display_name: Python 3
# language: python
# name: python3
# ---
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import root
# + code_folding=[2, 8]
## these functions translate the continuous-time rates into discrete time at weekly frequency
def b2beta(g,
time_unit_in_years):
beta = 1-np.exp(-g/time_unit_in_years)
return beta
def g2gamma(b,
time_unit_in_years):
gamma = 1-np.exp(-b/time_unit_in_years)
return gamma
# +
## estimates from Shiller and Pound (1989)
types = ['INSRAND','INSRPI','INDRAND','INDRPI']
gs_est = [1.84,1.39,3.72,1.73]
## all median
bs_est = [20.66,7.35,12.74,3.71]
## all median
# + code_folding=[0]
def SIR(beta,
gamma,
T,
x0):
s0,i0,r0 = x0
s_path = [s0]
i_path = [i0]
r_path = [r0]
for t in range(T-1):
s = s_path[t]-beta*s_path[t]*i_path[t]
s_path.append(s)
i = i_path[t]+beta*s_path[t]*i_path[t]-gamma*i_path[t]
i_path.append(i)
r = r_path[t]+gamma*i_path[t]
r_path.append(r)
return s_path,i_path, r_path
# -
## some parameters to testing
beta = 0.1
gamma = 0.03
i0 = 0.01
s0 = 1-i0
r0 = 0.0
x0 = (s0,i0,r0)
T = 204
times = range(T)
# +
### Solve final values/ Steady State fraction
def final_R_eq(r_fraction,
beta,
gamma):
return 1-r_fraction- np.exp(-beta/gamma*r_fraction)
def solve_r_ss(beta,
gamma):
eq = lambda R_SS: final_R_eq(R_SS,
beta,
gamma)
r_ss = root(eq,
x0 =0.6).x
return r_ss
# + code_folding=[]
## plot different simulated paths
if __name__ == "__main__":
## configurations of the plot
lw = 4
lbsize = 15
## plot
fig, axs = plt.subplots(2,2,
figsize=(15, 12),
facecolor='w',
edgecolor='k')
fig.subplots_adjust(hspace = 0.3, wspace=.1)
axs = axs.ravel()
nb = len(types)
nbw1y = 52
for x in range(nb):
g = gs_est[x]
gamma = g2gamma(g,nbw1y)
b = bs_est[x]
beta = b2beta(b,nbw1y)
r_ss = solve_r_ss(beta,gamma)
s,i,r = SIR(beta, # infection rate
gamma, # recovery rate
T,
x0)
title = types[x]+':\n'+r'$\beta={}$'.format(round(beta,2))+'\n'+r'$\gamma ={}$'.format(round(gamma,2))
axs[x].set_title(title,fontsize=lbsize)
axs[x].hlines(r_ss,
0.0,
T,
color='r',
linestyle ='dashed',
lw=lw,
label=r'$R_{+\infty}$')
axs[x].plot(times,s,'-',lw=lw,label='S')
axs[x].plot(times,i,'--',lw=lw,label='I')
axs[x].plot(times,r,'-.',lw=lw,label='R')
axs[x].set_xlim(0.0,T)
axs[0].legend(loc=0,prop={'size': 15})
axs[x].tick_params(axis='x', labelsize=lbsize)
axs[x].tick_params(axis='y', labelsize=lbsize)
plt.savefig("./draft/chapter/figures/sir_simulate.png")
# -
solve_r_ss(0.03,0.08)