Bayesian Emulation and Calibration of a Dynamic Epidemic Model for - PowerPoint PPT Presentation

Bayesian Emulation and Calibration of a Dynamic Epidemic Model for H1N1 Influenza Marian Farah 1 Paul Birrell 1 , Stefano Conti 2 , Daniela De Angelis 1 , 2 1 MRC Biostatistics Unit, Cambridge, UK 2 Health Protection Agency, London, UK ICERM Bayesian Nonparametrics Workshop September 19, 2012

Dynamic Bayesian Motivation modeling for epidemics Marian Farah Biostatistics Cambridge • Tracking and predicting the behavior of an emerging epidemic is essential for a prompt public health response. Outline Introduction • Inferential goals : Methods Results • What is happening? i.e., real-time estimation of the Discussion epidemic parameters . • What is going to happen next? i.e., forecasting the (short-term) evolution of the epidemic. • What happened? i.e., “reconstructing” the epidemic by estimating its parameters and evolution dynamics . • Noisy time-series data coming from different sources .

Dynamic Bayesian modeling for epidemics Marian Farah Biostatistics Cambridge 1 Introduction: Epidemic modeling Outline Introduction Methods 2 Emulation and calibration of epidemic models Results Discussion 3 Preliminary results 4 Discussion

Dynamic Bayesian modeling for epidemics Marian Farah Biostatistics Cambridge Outline Introduction Introduction Methods Results Discussion

Dynamic Bayesian Epidemic modeling modeling for epidemics Marian Farah Biostatistics Cambridge • Transmission model : Outline � � � Introduction getting infected latent period infectious period � � � S ( t ) E ( t ) I ( t ) R ( t ) − → − → − → Methods Results Discussion • Transmission depends on the virulence , the mixing patterns in the population , and the transition rates among the S , E , I , and R states. • Transmission dynamics are typically described by a system of differential equations .

Dynamic Bayesian Birrell et al. (2011) H1N1 model modeling for epidemics η 1 ,η 2 η 3 η 4 Marian Farah − → E ( t ) − → I ( t ) − → R ( t ) S ( t ) Biostatistics Cambridge Outline Introduction Methods Results Discussion

Dynamic Bayesian Birrell et al. (2011) H1N1 model modeling for epidemics η 1 ,η 2 η 3 η 4 Marian Farah − → E ( t ) − → I ( t ) − → R ( t ) S ( t ) Biostatistics Cambridge η 5 ↓ incubation Outline Expected # of symptomatic individuals Introduction η 6 ↓ propensity to consult doctor Methods Expected # of doctor consultations Results ↓ delay in reporting Discussion Expected # of reported cases, µ ( η , t )

Dynamic Bayesian Birrell et al. (2011) H1N1 model modeling for epidemics η 1 ,η 2 η 3 η 4 Marian Farah − → E ( t ) − → I ( t ) − → R ( t ) S ( t ) Biostatistics Cambridge η 5 ↓ incubation Outline Expected # of symptomatic individuals Introduction η 6 ↓ propensity to consult doctor Methods Expected # of doctor consultations Results ↓ delay in reporting Discussion Expected # of reported cases, µ ( η , t ) • η = ( η 1 , . . . , η 6 ): underlying parameters of the epidemic. • Proportion of symptomatic cases, propensity to consult, exponential growth rate, expected infectious period, a measure of the initial number of infected individuals, population interaction parameters.

Dynamic Bayesian Computational challenge modeling for epidemics Marian Farah Biostatistics Cambridge Outline • The likelihood of reported data, z ( t ), t = 1 , . . . , T , Introduction depends on µ . Methods Results T � � • p ( η | z { 1: T } , µ ) ∝ � z ( t ); µ ( η , t ) × p ( η ) p Discussion t =1 • µ ( η, t ) must be computed at every MCMC iteration. • µ is computationally expensive .

Dynamic Bayesian Computational challenge modeling for epidemics Marian Farah Biostatistics Cambridge Outline • The likelihood of reported data, z ( t ), t = 1 , . . . , T , Introduction depends on µ . Methods Results T � � • p ( η | z { 1: T } , µ ) ∝ � z ( t ); µ ( η , t ) × p ( η ) p Discussion t =1 • µ ( η, t ) must be computed at every MCMC iteration. • µ is computationally expensive . • What about an efficient estimate?

Dynamic Bayesian Computer simulator modeling for epidemics Marian Farah Biostatistics specify inputs η = X run code outputs Cambridge  x 1 , 1 x 1 , 6  µ ( x 1 , 1) , . . . , µ ( x 1 , T ) . . . Outline Birrell x 2 , 1 x 2 , 6 µ ( x 2 , 1) , . . . , µ ( x 2 , T ) . . . Introduction   et al.  → → X =  . . .  . . . . .   Methods . . . µ ( x 2 , t ) .  (2011) x n , 1 x n , 6 µ ( x n , 1) , . . . , µ ( x n , T ) Results . . . Discussion

Dynamic Bayesian Computer simulator modeling for epidemics Marian Farah Biostatistics specify inputs η = X run code outputs Cambridge  x 1 , 1 x 1 , 6  µ ( x 1 , 1) , . . . , µ ( x 1 , T ) . . . Outline Birrell x 2 , 1 x 2 , 6 µ ( x 2 , 1) , . . . , µ ( x 2 , T ) . . . Introduction   et al.  → → X =  . . .  . . . . .   Methods . . . µ ( x 2 , t ) .  (2011) x n , 1 x n , 6 µ ( x n , 1) , . . . , µ ( x n , T ) Results . . . Discussion 4 12x 10 12 10 10 8 8 log µ ( η ,t) µ ( η ,t) 6 6 4 4 2 2 0 0 0 50 100 150 200 250 0 50 100 150 200 250 time time

Dynamic Bayesian Calibration and Emulation modeling for epidemics Marian Farah Biostatistics Cambridge • Calibration : (e.g., Higdon et al., 2004) Posterior inference for η through the simulator, µ , and Outline “field” observed data z ( t ), Introduction Methods Observed = Reality + Error Results Observed = Simulator + bias + Error Discussion z ↑ µ ↑ b ↑ T � � � • p ( η , b | z { 1: T } , µ ) ∝ p z ( t ); µ ( η , t ) + b × p ( η ) p ( b ) t =1 • Emulation : (e.g., Kennedy and O’Hagan, 2000) Estimating a slow computer simulator output, µ , using fast statistical model (an emulator ), say ˆ µ .

Dynamic Bayesian Calibration and Emulation modeling for epidemics Marian Farah Biostatistics Cambridge Outline Introduction • Idea : (e.g., Bayarri et al., 2007a) Methods Replace the slow simulator output, µ , with the fast Results emulator estimation, ˆ µ , and obtain posterior inference for Discussion η through T � � • p ( η , b | z { 1: T } , ˆ µ ) ∝ � p z ( t ); ˆ µ ( η , t ) + b × p ( η ) p ( b ) t =1

Dynamic Bayesian modeling for epidemics Marian Farah Biostatistics Cambridge Outline Introduction Emulation and calibration of Methods Results dynamic models Discussion

Dynamic Bayesian Emulation review modeling for epidemics Marian Farah Biostatistics Cambridge • A deterministic computer simulator is a function f ( · ) Outline that maps input x to a unique output y = f ( x ). Introduction Methods • The function f ( · ) is treated as unknown and given a prior . Results Discussion • Likelihood : data are runs of the simulator , given a design over the input space , e.g., Latin Hypercube. • Emulator : the posterior (predictive) distribution of f ( · ).

Dynamic Bayesian The Gaussian process modeling for epidemics Marian Farah Biostatistics y ( x ) ∼ GP ( m ( x ) , v c ( x , x ′ ) ) Cambridge m ( · ), v , and c ( · , · ) are the mean, variance, & correlation Outline function (e.g., Neal 1998; Rasmussen & Williams 2006). Introduction Methods 6 Results 5 Discussion 4 3 2 1 0 y −1 −2 −3 −4 −5 −3 −2 −1 0 1 2 3 x

Dynamic Bayesian Toy example modeling for epidemics Marian Farah Biostatistics Cambridge 10 Outline Introduction 8 Methods 20 output function Results 6 ouput Discussion f ( x ) = x + 3 sin ( x / 2) 4 2 0 −2 0 2 4 6 8 input

Dynamic Bayesian Toy example modeling for epidemics Marian Farah Biostatistics Cambridge 10 Outline Introduction 8 Methods output function Results 6 simulator data ouput Discussion prior realizations 4 2 0 −2 0 2 4 6 8 input

Dynamic Bayesian Toy example modeling for epidemics Marian Farah Biostatistics Cambridge 10 Outline Introduction 8 Methods output function simulator data Results 6 prior realizations ouput Discussion 95% posterior region 4 2 0 −2 0 2 4 6 8 input

Dynamic Bayesian Toy example modeling for epidemics Marian Farah Biostatistics Cambridge 10 Outline Introduction 8 output function Methods simulator data Results 6 prior realizations ouput Discussion 95% posterior region 4 2 0 −2 0 2 4 6 8 input

Dynamic Bayesian Calibration review modeling for epidemics Marian Farah Biostatistics Cambridge • Simulator: Specify x → f ( x ). Outline • For x = η , f ( η ) simulates a physical system. Introduction Methods • η is uncertain . Results Discussion • Calibration : solving the inverse-problem , i.e., η | z , f ( · ). • If f ( · ) is computationally expensive, it is emulated. • Priors for η and f ( · ). • Likelihood : data come from field observations and simulator runs.

Dynamic Bayesian Toy example modeling for epidemics Marian Farah Biostatistics Cambridge 10 Outline Introduction 8 Methods output function Results 6 8 simulator data ouput Discussion 6 field data 4 2 0 −2 0 2 4 6 8 input