Bayesian Estimation of Reproductive Number for Tuberculosis in India

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

THANKYOU