bayesian estimation of reproductive number for tuberculosis in india
TRANSCRIPT
Data Assimilation Methods in Parameter Estimation: An Application to Tuberculosis Transmission Model IIT Mandi, Himachal Pradesh Pankaj Narula and Arjun Bhardwaj Supervisors: Dr. Sarita Azad Dr. Ankit Bansal
International Conference on Mathematical Techniques in Engineering
Applications (ICMTEA 2013)
Outline
Epidemiology of Tuberculosis (TB)
Model Formulation and Parameters
Research Interests
Previous Work on TB
Estimation Methods
Results
Epidemiology of Tuberculosis
TB is one of the most widespread infectious diseases, and a leading cause of global mortality.
Particularly, TB in India accounts for 25% of the world’s incident cases.
RNTCP is being implemented by Government of India in the country with DOTS strategy.
The SIR Epidemic Model
S
Susceptible: can catch the disease
I
Infectious: have caught the disease and can spread it to susceptible
R
Recovered: have recovered from the disease and are immune.
dS dt
= – b S I
dI dt
= b S I – γI
dR dt
= γ I
S + I + R = 1
Parameters of the Model
= The infection rate
= The Removal rate
= Fraction of infectious persons.
Basic reproduction number obtained as:
Average secondary number of infections caused by an infective in total susceptible population. An epidemic occur if .
Fraction of population needs to be vaccinated 1 − 1/R0.
0
pR
b
bp
0 1R
Model Formulation
SILS Model of TB
Recovery rate is assumed to be 0.85.
Aim is to estimate β and p.
( )
(1 )
dS ISI L
dt N
dI pISI tL
dt N
dL p ISI tL
dt N
b
b
b
Research Interests
Mathematical models, deterministic or statistical, are important tools to understand TB dynamics and analyse voluminous data collected by various agencies like WHO, RNTCP.
Challenge is to accurately estimate model parameters.
Parameters like infection rate measure the disease burden and evaluate the measures for control.
Previous work On TB
Parameter Estimation of Tuberculosis
Transmission Model using Ensemble
Kalman Filter; Vihari et al. (2013)
Bayesian Melding Estimation of a
Stochastic SEIR Model, Hotta et al. (2010)
Tuberculosis in intra-urban settings: a
Bayesian approach; Souza et al. (2007)
Methods of Parameter Estimation
Least Square
Maximum Likelihood Method
Ensemble Kalman Filter
Bayesian Melding
Maximum Likelihood Method
To estimate a density function
whose parameters are
as an ML estimate of
( )p x
1
( ) ( / )n
i
i
L P x
1 2( , ,....., )t
m
arg max[ ( )]L
Ensemble Kalman Filter (EnKf)
The EnKf is a MC approximation of the
Kalman filter.
It avoids evolving the covariance matrix of
the pdf of the state vector.
The basic idea is to predict the values first
and then to adjust it by actual value.
Ensemble Kalman Filter
Forecast Step
Ensemble Mean
Error Matrices
Analysis Step
1j j jp p
t t tk k 1,2,3,....,j m
1 1
1
1j
mpp
t t
j
k km
1
1
1 1 1 1
1 1 1 1
[ ....... ]
[ ....... ]
q
t
q
t
ppp p p
k t t t t
ppp p p
y t t t t
E k k k k
E y y y y
( [ ] )k p j fj j jt t tt t tk k K y v y
Bayesian Melding Method
Bayesian melding which observes the
existence of two priors, explicit and
implicit, on every input and output.
The technique works good with
stochastic and deterministic models with
in high dimensional parameter estimation.
Bayesian Melding Method
These priors are coupled via logarithmic
pooling.
It calibrates the knowledge and
uncertainty of inputs and outputs of the
model.
The technique ignores the Borel paradox.
Results
We have used BIP. Bayes.Melding package
to estimate trend of various parameters.
We have used Fitmodel for Monte Carlo
simulations, 2000 samples were discarded.
Prior distributions for parameters are
taken to be normal.
Results
The parameter estimation framework
presented here captures seasonality well
in the data which could not be expected
from standard-likelihood methods.
The estimates presented here are verified
from three different approaches.
Results
Comparison of parameters values
A- our results (EnKf)(2011) B- Our results (Bayesian Melding) C-Christopher Dye(2012) * 8 secondary infections per year.
Parameters A(2011) B C
β 1.72 1.84* 3.5
p 0.6 0.30 0.45
R0 1.29 0.69 0.78
Results Comparison of estimated values of β for
highest infected state Manipur from three
different approaches Bayesian Melding
Mean value = 1.90
EnKf
Mean value = 2.31
Least square
Mean value = 2.32
Results Estimated values of parameters for India
Ro< 1 which shows the disease is endemic in
the country
Seasonal trend
in the plot of
β and Ro
00.35 0.94R
1.3 2.54b
Results
Ranges of R0
TB transmission across various Indian states
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