def __init__(self, a, b, K):
from operator import mul
self._a = a
self._b = b
self._K = K
theta = 1. / b
self.gamma_dist = gamma_dist(a, 0, theta)
def produce_gomes_exfig1(coeffs: list, add_hist=False, n_bins=3, x_values=50, plot_upper_limit=3.0):
"""
Produce figure equivalent to Extended Data Fig 1 of Aguas et al pre-print as a check
"""
# Prelims
x_values = np.linspace(0.1, plot_upper_limit, x_values) # Can't go to zero under some coeffs
gamma_plot = plt.figure()
axis = gamma_plot.add_subplot(111)
# For each requested coefficient of variation
for coeff in coeffs:
gamma_distri = gamma_dist(coeff ** -2.0, scale=coeff ** 2.0)
# Line graph of the raw function
y_values = [gamma_distri.pdf(i) for i in x_values]
axis.plot(x_values, y_values, color="k")
# Numeric integration over parts of the function domain
if add_hist:
lower_terminals, _, _, normalised_heights, bin_width = get_gamma_data(
plot_upper_limit, n_bins, coeff
)
axis.bar(lower_terminals, normalised_heights, width=bin_width, align="edge")
# Return the figure
return gamma_plot
def get_gamma_data(tail_start_point, n_bins, coeff):
"""
Compute a discretised version of a gamma distribution with mean 1 and a given coefficient of variation (sd/mean).
The discretisation involves numerical solving to ensure the coefficient of variation is preserved. However, the mean
of the discrete ditribution may be different to 1.
:param tail_start_point: the last lower terminal
:param n_bins: number of bins (including tail)
:param coeff: requested coefficient of variation
:return:
"""
# First address a few singular scenarios
if n_bins == 1 and coeff > 0.0:
raise ValueError("Cannot compute a positive coefficient of variation using a single bin")
elif n_bins > 1 and coeff == 0.0:
raise ValueError("Cannot compute a null coefficient of variation using multiple bins")
elif n_bins == 1:
return [0.0], [float("inf")], 1.0, 1.0, float("inf")
n_finite_bins = n_bins - 1
# Prelims
lower_terminal, lower_terminals, upper_terminals, heights = 0.0, [], [], []
bin_width = tail_start_point / n_finite_bins
for i_bin in range(n_finite_bins):
# Record the upper and lower terminals
lower_terminals.append(lower_terminal)
upper_terminals.append(lower_terminal + bin_width)
gamma_distri = gamma_dist(coeff ** -2.0, scale=coeff ** 2.0)
heights.append(
gamma_distri.cdf(upper_terminals[-1]) - gamma_distri.cdf(lower_terminals[-1])
)
# Move to the next value
lower_terminal = upper_terminals[-1]
# the last height is the remaining area under the curve (tail)
heights.append(1.0 - sum(heights))
# Find mid-points as the representative values
mid_points = [(lower + upper) / 2.0 for lower, upper in zip(lower_terminals, upper_terminals)]
# add the last representative point such that modelled_CV = input_CV
last_point, last_height = find_last_representative_point(mid_points, heights, coeff)
mid_points.append(last_point)
heights[-1] = last_height
# rescale heights
heights = [h / sum(heights) for h in heights]
lower_terminals.append(upper_terminals[-1])
upper_terminals.append(float("inf"))
# Return everything just in case
return lower_terminals, upper_terminals, mid_points, heights, bin_width
# Model Parameters to sample from Gamma Distributions
gamma_inv, gamma_inv_shape = 7, 0.1 # From report (mean, shape)
sigma_inv, sigma_inv_shape = 5.1, 0.1 # From report (mean, shape)
r0, r0_shape = 2.28, 0.1 # From report (mean, shape)
# For gamma pdf plots
x = np.linspace(1E-6, 10, 1000)
num_samples = 1000
#### Gamma distributed samples for gamma_inv ####
#### --> This might be wrong?!?!?
k = 1
loc = gamma_inv
theta = gamma_inv_shape
gamma_inv_dist = gamma_dist(k, loc, theta)
gamma_inv_samples = gamma_inv_dist.rvs(num_samples)
# Plot gamma samples and pdf
count, bins, ignored = plt.hist(gamma_inv_samples, 50, density=True)
plt.plot(x,
gamma_inv_dist.pdf(x),
'r',
label=r'$k=%.1f,\ \theta=%.1f$' % (k, theta))
#### Gamma distributed samples for sigma_inv ####
k = 1
loc = sigma_inv
theta = sigma_inv_shape
sigma_inv_dist = gamma_dist(k, loc, theta)
sigma_inv_samples = sigma_inv_dist.rvs(num_samples)
def plotSIR_sampledParams(beta_samples, gamma_inv_samples, filename, *prob_params):
fig, (ax1,ax2) = plt.subplots(1,2, constrained_layout=True)
###########################################################
################## Plot for Beta Samples ##################
###########################################################
count, bins, ignored = ax1.hist(beta_samples, 30, density=True)
if prob_params[0] == 'uniform':
ax1.set_xlabel(r"$\beta \sim \mathcal{N}$", fontsize=15)
if prob_params[0] == 'gaussian':
mu = prob_params[1]
sigma = prob_params[2] + 0.00001
ax1.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
np.exp( - (bins - mu)**2 / (2 * sigma**2) ), linewidth=2, color='r')
ax1.set_xlabel(r"$\beta \sim \mathcal{N}$", fontsize=15)
if prob_params[0] == 'gamma':
g_dist = gamma_dist(prob_params[2], prob_params[1], prob_params[3])
# Plot gamma samples and pdf
x = np.arange(0,1,0.001)
ax1.plot(x, g_dist.pdf(x), 'r',label=r'$k = 1, \mu=%.1f,\ \theta=%.1f$' % (prob_params[1], prob_params[2]))
for tick in ax1.xaxis.get_major_ticks():
tick.label.set_fontsize(15)
for tick in ax1.yaxis.get_major_ticks():
tick.label.set_fontsize(15)
plt.xlim(0, 1.0)
ax1.grid(True, alpha=0.3)
ax1.set_title(r"Histogram of $\beta$ samples", fontsize=20)
###############################################################
################## Plot for Gamma^-1 Samples ##################
###############################################################
count, bins, ignored = ax2.hist(gamma_inv_samples, 30, density=True)
if prob_params[0] == 'gaussian':
mu = prob_params[3]
sigma = prob_params[4] + 0.00001
ax2.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
np.exp( - (bins - mu)**2 / (2 * sigma**2) ), linewidth=2, color='r')
ax2.set_xlabel(r"$\gamma^{-1} \sim \mathcal{N}$", fontsize=15)
if prob_params[0] == 'uniform':
ax2.set_xlabel(r"$\gamma^{-1} \sim \mathcal{U}$", fontsize=15)
if prob_params[0] == 'gamma':
g_dist = gamma_dist(prob_params[5], prob_params[4], prob_params[6])
# Plot gamma samples and pdf
x = np.arange(1,15,0.1)
ax2.plot(x, g_dist.pdf(x), 'r',label=r'$k = 1, \mu=%.1f,\ \theta=%.1f$' % (prob_params[3], prob_params[4]))
for tick in ax2.xaxis.get_major_ticks():
tick.label.set_fontsize(15)
for tick in ax2.yaxis.get_major_ticks():
tick.label.set_fontsize(15)
plt.xlim(1, 17)
ax2.grid(True, alpha=0.3)
plt.title(r"Histogram of $\gamma^{-1}$ samples", fontsize=20)
fig.subplots_adjust(left=.12, bottom=.14, right=.93, top=0.93)
fig.set_size_inches(20/2, 8/2, forward=True)
# Store plot
plt.savefig(filename + ".png", bbox_inches='tight')
def rollout_SIR_sim_stoch(*prob_params, **kwargs):
'''
Run a single simulation of stochastic SIR dynamics
'''
verbose = kwargs['verbose']
N = kwargs['N']
days = kwargs['days']
# Sample from Uniform Distributions
if prob_params[0] == 'uniform':
if prob_params[1] == prob_params[2]:
beta = prob_params[1]
else:
beta = np.random.uniform(prob_params[1],prob_params[2])
if prob_params[3] == prob_params[4]:
gamma_inv = prob_params[3]
else:
gamma_inv = np.random.uniform(prob_params[3],prob_params[4])
# Sample from Gaussian Distributions
if prob_params[0] == 'gaussian':
beta_mean = prob_params[1]
beta_std = prob_params[2]
gamma_inv_mean = prob_params[3]
gamma_inv_std = prob_params[4]
# Sample from Gaussian Distributions
beta = 0
while beta < 0.02:
beta = np.random.normal(beta_mean, beta_std)
gamma_inv = 0
while gamma_inv < 0.1:
gamma_inv = np.random.normal(gamma_inv_mean, gamma_inv_std)
# Sample from Gaussian Distributions
if prob_params[0] == 'gamma':
beta_loc = prob_params[1]
beta_scale = prob_params[2]
beta_shape = prob_params[3]
gamma_inv_loc = prob_params[4]
gamma_inv_scale = prob_params[5]
gamma_inv_shape = prob_params[6]
# Sample from Gamma Distributions
if beta_scale == 0:
beta = beta_loc
else:
beta_dist = gamma_dist(beta_scale, beta_loc, beta_shape)
beta = beta_dist.rvs(1)[0]
if gamma_inv_scale == 0:
gamma_inv = gamma_inv_loc
else:
gamma_inv_dist = gamma_dist(gamma_inv_scale, gamma_inv_loc, gamma_inv_shape)
gamma_inv = gamma_inv_dist.rvs(1)[0]
# Derived values
gamma = 1.0 / gamma_inv
r0 = beta * gamma_inv
if verbose:
print('***** SIMULATING SIR MODEL DYNAMICS *****')
print('***** Hyper-parameters *****')
print('N=',N,'days=', days, 'r0=',r0, 'gamma_inv (days) = ',gamma_inv)
print('***** Model-parameters *****')
print('beta=',beta, 'gamma=',gamma)
# Create Model
model_kwargs = {}
model_kwargs['r0'] = r0
model_kwargs['inf_period'] = gamma_inv
model_kwargs['I0'] = kwargs['I0']
model_kwargs['R0'] = kwargs['R0']
model = SIR(N,**model_kwargs)
S,I,R,t = model.project(days,'ode_int')
return S, I, R, t, beta, gamma_inv