Data Assimilation Methods in Parameter Estimation: An Application to Tuberculosis Transmission Model. Conference presentation at ICMTEA-2013.

1. 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 MathematicalTechniques in Engineering
Applications (ICMTEA 2013)

2. Outline
 Epidemiology of Tuberculosis (TB)
 Model Formulation and Parameters
 Research Interests
 PreviousWork onTB
 Estimation Methods
 Results

3. 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.

4. 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

5. 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
p
R
b


b

p
0 1R 

6. Model Formulation
 SILS Model ofTB
Recovery rate is assumed to be 0.85.
Aim is to estimate β and p.
( )
(1 )
dS IS
I L
dt N
dI pIS
I tL
dt N
dL p IS
I tL
dt N
b

b

b


  

  
 
  

7. Research Interests
 Mathematical models, deterministic or
statistical, are important tools to
understand TB dynamics and analyse
voluminous data collected by various
agencies likeWHO, RNTCP.
 Challenge is to accurately estimate
model parameters.
 Parameters like infection rate measure
the disease burden and evaluate the
measures for control.

8. 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)

9. Methods of Parameter Estimation
 Least Square
 Maximum Likelihood Method
 Ensemble Kalman Filter
 Bayesian Melding

10. 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   
argmax[ ( )]L 



11. 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.

12. Ensemble Kalman Filter
 Forecast Step
 Ensemble Mean
 Error Matrices
 Analysis Step
1
j j jp p
t t tk k    1,2,3,....,j m
1 1
1
1 j
m
pp
t t
j
k k
m
 

 
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 t
k k K y v y   

13. 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.

14. 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.

15. 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.

16. 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.

17. 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

18. 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

19. 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 

20. Results
Ranges of R0
TB transmission across various Indian states

21. THANKYOU