The document discusses data assimilation methods in parameter estimation for a tuberculosis transmission model in India, highlighting the importance of accurate model parameter estimation to understand TB dynamics. It compares various estimation techniques, such as the Ensemble Kalman Filter and Bayesian Melding, and presents results that capture seasonality in TB data. The findings indicate that the model estimates suggest TB is endemic in India, with varying infection rates across states.

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