Disease Outbreak Response Decision-making Under Uncertainty: A retrospective analysis of measles in Sao Paulo¶
%matplotlib inline
import pandas as pd
import numpy as np
import numpy.ma as ma
from datetime import datetime
import matplotlib.pyplot as plt
import pdb
from IPython.core.display import HTML
def css_styling():
styles = open("styles/custom.css", "r").read()
return HTML(styles)
css_styling()
data_dir = "data/"
Import outbreak data
measles_data = pd.read_csv(data_dir+"measles.csv", index_col=0)
measles_data.NOTIFICATION = pd.to_datetime(measles_data.NOTIFICATION)
measles_data.BIRTH = pd.to_datetime(measles_data.BIRTH)
measles_data.ONSET = pd.to_datetime(measles_data.ONSET)
measles_data = measles_data.replace({'DISTRICT': {'BRASILANDIA':'BRAZILANDIA'}})
Sao Paulo population by district
sp_pop = pd.read_csv(data_dir+'sp_pop.csv', index_col=0)
_names = sp_pop.index.values
_names[_names=='BRASILANDIA'] = 'BRAZILANDIA'
sp_pop.set_index(_names, inplace = True)
sp_pop.head()
Plot of cumulative cases by district
measles_onset_dist = measles_data.groupby(['DISTRICT','ONSET']).size().unstack(level=0).fillna(0)
measles_onset_dist.cumsum().plot(legend=False, grid=False)
total_district_cases = measles_onset_dist.sum()
Top 5 districts by number of cases
totals = measles_onset_dist.sum()
totals.sort(ascending=False)
totals[:5]
Age distribution of cases, by confirmation status
by_conclusion = measles_data.groupby(["YEAR_AGE", "CONCLUSION"])
counts_by_cause = by_conclusion.size().unstack().fillna(0)
ax = counts_by_cause.plot(kind='bar', stacked=True, xlim=(0,50), figsize=(15,5))
Vaccination data¶
vaccination_data = pd.read_csv('data/BrazilVaxRecords.csv', index_col=0)
vaccination_data.head()
vaccination_data.VAX[:18]
vax_97 = np.r_[[0]*(1979-1921+1), vaccination_data.VAX[:17]]
n = len(vax_97)
FOI_mat = np.resize((1 - vax_97*0.9), (n,n)).T
# Mean age of infection for those born prior to vaccination coverage, assuming R0=16
A = 4.37
(1 - vax_97*0.9)[:-1]
np.tril(FOI_mat).sum(0)
natural_susc = np.exp((-1/A) * np.tril(FOI_mat).sum(0))[::-1]
vacc_susc = (1 - vax_97*0.9)[::-1]
vacc_susc[0] = 0.5
vacc_susc
sia_susc = np.ones(len(vax_97))
birth_year = np.arange(1922, 1998)[::-1]
by_mask = (birth_year > 1983) & (birth_year < 1992)
sia_susc[by_mask] *= 0.2
downsample = lambda x: np.array([x[s].mean() for s in age_slices])
Stochastic Disease Transmission Model¶
As a baseline for comparison, we can fit a model to all the clinically-confirmed cases, regardless of lab confirmation status. For this, we will use a simple SIR disease model, which will be fit using MCMC.
This model fits the series of 2-week infection totals in each district $i$ as a set of Poisson models:
$$Pr(I(t)_{i} | \lambda(t)_i) = \text{Poisson}(\lambda(t)_i) $$
Where the outbreak intensity is modeled as:
$$\lambda(t)_i = \beta [I^{(w)}(t-1)_i]^{\alpha} S(t-1)_i$$
$$\alpha \sim \text{Exp}(1)$$
We will assume here that the transmission rate is constant over time (and across districts):
$$\beta \sim \text{Gamma}(1, 0.1)$$
To account for the influence of infected individuals from neighboring districts on new infections, the outbreak intensity was modeled using a spatial-weighted average of infecteds across districts, where populations were weighted as an exponential function of the distance between district centroids:
$$w_{d} = \text{exp}(-\theta d)$$
$$\theta \sim \text{Exp}(1)$$
Confirmation Sub-model¶
Rather than assume all clinical cases are true cases, we can adjust the model to account for lab confirmation probability. This is done by including a sub-model that estimates age group-specific probabilities of confirmation, and using these probabilities to estimate the number of lab-confirmed cases. These estimates are then plugged into the model in place of the clinically-confirmed cases.
We specified a structured confirmation model to retrospectively determine the age group-specific probabilities of lab confirmation for measles, conditional on clinical diagnosis. Individual lab confirmation events $c_i$ were modeled as Bernoulli random variables, with the probability of confirmation being allowed to vary by age group:
$$c_i \sim \text{Bernoulli}(p_{a(i)})$$
where $a(i)$ denotes the appropriate age group for the individual indexed by i. There were 16 age groups, the first 15 of which were 5-year age intervals $[0,5), [5, 10), \ldots , [70, 75)$, with the 16th interval including all individuals 75 years and older.
Since the age interval choices were arbitrary, and the confirmation probabilities of adjacent groups likely correlated, we modeled the correlation structure directly, using a multivariate logit-normal model. Specifically, we allowed first-order autocorrelation among the age groups, whereby the variance-covariance matrix retained a tridiagonal structure.
$$\begin{aligned} \Sigma = \left[{ \begin{array}{c} {\sigma^2} & {\sigma^2 \rho} & 0& \ldots & {0} & {0} \\ {\sigma^2 \rho} & {\sigma^2} & \sigma^2 \rho & \ldots & {0} & {0} \\ {0} & \sigma^2 \rho & {\sigma^2} & \ldots & {0} & {0} \\ \vdots & \vdots & \vdots & & \vdots & \vdots\\ {0} & {0} & 0 & \ldots & {\sigma^2} & \sigma^2 \rho \\ {0} & {0} & 0 & \ldots & \sigma^2 \rho & {\sigma^2} \end{array} }\right] \end{aligned}$$
From this, the confirmation probabilities were specified as multivariate normal on the inverse-logit scale.
$$ \text{logit}(p_a) = \{a\} \sim N(\mu, \Sigma)$$
Priors for the confirmation sub-model were specified by:
$$\begin{aligned} \mu_i &\sim N(0, 100) \\ \sigma &\sim \text{HalfCauchy}(25) \\ \rho &\sim U(-1, 1) \end{aligned}$$
Age classes are defined in 5-year intervals.
age_classes = [0,5,10,15,20,25,30,35,40,100]
measles_data.dropna(subset=['YEAR_AGE'], inplace=True)
measles_data['YEAR_AGE'] = measles_data.YEAR_AGE.astype(int)
measles_data['AGE_GROUP'] = pd.cut(measles_data.AGE, age_classes, right=False)
Lab-checked observations are extracted for use in estimating lab confirmation probability.
CONFIRMED = measles_data.CONCLUSION == 'CONFIRMED'
CLINICAL = measles_data.CONCLUSION == 'CLINICAL'
DISCARDED = measles_data.CONCLUSION == 'DISCARDED'
Extract confirmed and clinical subset, with no missing county information.
lab_subset = measles_data[(CONFIRMED | CLINICAL) & measles_data.COUNTY.notnull()].copy()
age = lab_subset.YEAR_AGE.values
ages = lab_subset.YEAR_AGE.unique()
counties = lab_subset.COUNTY.unique()
y = (lab_subset.CONCLUSION=='CONFIRMED').values
_lab_subset = lab_subset.replace({"CONCLUSION": {"CLINICAL": "UNCONFIRMED"}})
by_conclusion = _lab_subset.groupby(["YEAR_AGE", "CONCLUSION"])
counts_by_cause = by_conclusion.size().unstack().fillna(0)
ax = counts_by_cause.plot(kind='bar', stacked=True, xlim=(0,50), figsize=(15,5), grid=False)
lab_subset.shape
y.sum()
Proportion of lab-confirmed cases older than 20 years
(measles_data[CONFIRMED].YEAR_AGE>20).mean()
age_classes
#Extract cases by age and time.
age_group = pd.cut(age, age_classes, right=False)
age_index = np.array([age_group.categories.tolist().index(i) for i in age_group])
age_groups = age_group.categories
age_groups
age_slice_endpoints = [g[1:-1].split(',') for g in age_groups]
age_slices = [slice(int(i[0]), int(i[1])) for i in age_slice_endpoints]
# Get index from full crosstabulation to use as index for each district
dates_index = measles_data.groupby(
['ONSET', 'AGE_GROUP']).size().unstack().index
unique_districts = measles_data.DISTRICT.dropna().unique()
excludes = ['BOM RETIRO']
N = sp_pop.drop(excludes).ix[unique_districts].sum().drop('Total')
N
N_age = N.iloc[:8]
N_age.index = age_groups[:-1]
N_age[age_groups[-1]] = N.iloc[8:].sum()
N_age
Compile bi-weekly confirmed and unconfirmed data by Sao Paulo district
sp_counts_2w = lab_subset.groupby(
['ONSET', 'AGE_GROUP']).size().unstack().reindex(dates_index).fillna(0).resample('2W', how='sum')
# All confirmed cases, by district
confirmed_data = lab_subset[lab_subset.CONCLUSION=='CONFIRMED']
confirmed_counts = confirmed_data.groupby(
['ONSET', 'AGE_GROUP']).size().unstack().reindex(dates_index).fillna(0).sum()
all_confirmed_cases = confirmed_counts.reindex_axis(measles_data['AGE_GROUP'].unique()).fillna(0)
# Ensure the age groups are ordered
I_obs = sp_counts_2w.reindex_axis(measles_data['AGE_GROUP'].unique(),
axis=1).fillna(0).values.astype(int)
Check shape of data frame
- 28 bi-monthly intervals, 9 age groups
assert I_obs.shape == (28, len(age_groups))
Prior distribution on susceptible proportion:
$$p_s \sim \text{Beta}(2, 100)$$
from pymc import rbeta
plt.hist(rbeta(2, 100, 10000))
I_obs
obs_date = '1997-12-01' #'1997-06-15'
obs_index = sp_counts_2w.index <= obs_date
I_obs_t = I_obs[obs_index]
np.sum(I_obs_t, (0)) / float(I_obs_t.sum())
from pymc import rgamma
plt.hist(rgamma(16,1,size=10000))
75./age.mean()
from pymc import MCMC, Matplot, AdaptiveMetropolis, Slicer, MAP
from pymc import (Uniform, DiscreteUniform, Beta, Binomial, Normal, CompletedDirichlet,
Poisson, NegativeBinomial, negative_binomial_like, poisson_like,
Lognormal, Exponential, binomial_like,
TruncatedNormal, Binomial, Gamma, HalfCauchy, normal_like,
MvNormalCov, Bernoulli, Uninformative,
Multinomial, rmultinomial, rbinomial,
Dirichlet, multinomial_like)
from pymc import (Lambda, observed, invlogit, deterministic, potential, stochastic,)
def measles_model(obs_date, confirmation=True, spatial_weighting=False, all_traces=True):
n_periods, n_age_groups = I_obs.shape
### Confirmation sub-model
if confirmation:
# Specify priors on age-specific means
age_classes = np.unique(age_index)
mu = Normal("mu", mu=0, tau=0.0001, value=[0]*len(age_classes))
sig = HalfCauchy('sig', 0, 25, value=1)
var = sig**2
cor = Uniform('cor', -1, 1, value=0)
# Build variance-covariance matrix with first-order correlation
# among age classes
@deterministic
def Sigma(var=var, cor=cor):
I = np.eye(len(age_classes))*var
E = np.diag(np.ones(len(age_classes)-1), k=-1)*var*cor
return I + E + E.T
# Age-specific probabilities of confirmation as multivariate normal
# random variables
beta_age = MvNormalCov("beta_age", mu=mu, C=Sigma,
value=[1]*len(age_classes))
p_age = Lambda('p_age', lambda t=beta_age: invlogit(t))
@deterministic(trace=False)
def p_confirm(beta=beta_age):
return invlogit(beta[age_index])
# Confirmation likelihood
lab_confirmed = Bernoulli('lab_confirmed', p=p_confirm, value=y,
observed=True)
'''
Truncate data at observation period
'''
obs_index = sp_counts_2w.index <= obs_date
I_obs_t = I_obs[obs_index]
# Index for observation date, used to index out values of interest
# from the model.
t_obs = obs_index.sum() - 1
if confirmation:
@stochastic(trace=all_traces, dtype=int)
def I(value=(I_obs_t).astype(int), n=I_obs_t, p=p_age):
# Binomial confirmation process
return np.sum([binomial_like(x, x.sum(), p) for x in value])
else:
I = I_obs_t
assert I.shape == (t_obs +1, n_age_groups)
# Transmission parameter
beta = HalfCauchy('beta', 0, 25, value=8)
# # Downsample annual series to observed age groups
# downsample = lambda x: np.array([x[s].mean() for s in age_slices])
# A = Lambda('A', lambda beta=beta: 75./(beta - 1))
# lt_sum = downsample(np.tril(FOI_mat).sum(0)[::-1])
# natural_susc = Lambda('natural_susc', lambda A=A: np.exp((-1/A) * lt_sum))
natural_susc = Beta('natural_susc', 1, 1, value=[0.02]*n_age_groups)
@deterministic
def p_susceptible(natural_susc=natural_susc):
return downsample(sia_susc) * downsample(vacc_susc) * natural_susc
# Estimated total initial susceptibles
S_0 = Binomial('S_0', n=N_age.astype(int), p=p_susceptible)
S = Lambda('S', lambda I=I, S_0=S_0: S_0 - I.cumsum(0))
# Check shape
assert S.value.shape == (t_obs+1., n_age_groups)
S_t = Lambda('S_t', lambda S=S: S[-1])
# Force of infection
@deterministic
def lam(beta=beta, I=I, S=S):
return (beta * I.sum(1) * (S / N_age.values).T).T
# Check shape
assert lam.value.shape == (t_obs+1, n_age_groups)
# Poisson likelihood for observed cases
@potential
def new_cases(I=I, lam=lam):
return negative_binomial_like(I[1:], lam[:-1], I[:-1]+1)
return locals()
iterations = 500000
burn = 400000
M = MCMC(measles_model('1997-06-15', confirmation=True))
M.use_step_method(AdaptiveMetropolis, M.natural_susc)
M.sample(iterations, burn)
M.sample(iterations, burn)
Matplot.summary_plot(M.p_susceptible)
Matplot.plot(M.beta)
Matplot.summary_plot(M.p_age)
