PeteDodd Mathematical modelling approach to estimating TB ...€¦ · Results China All countries LTBI Comparison Discussion Limitations Advantages 11 Beta-binomialdistributions (Method/conclusion)

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

1

Mathematical modelling approach toestimating TB incidencePete Dodd (University of Sheffield)

Tuesday, 31 March 2015

Health Economics & Decision ScienceSchool of Health & Related ResearchUniversity of Sheffield

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

2

Overview

Goal:

Can a simple transmission model be used in a statisticallyrigorous manner to obtain consistent estimates of TB burdenusing:

• notification• prevalence• mortality

data?

Other criteria:

• Must be scalable & automated• Must include age structure• Must fairly account for all uncertainty

Motivation:using different data types requires modelling assumptions!

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

2

Overview

Goal:

Can a simple transmission model be used in a statisticallyrigorous manner to obtain consistent estimates of TB burdenusing:

• notification• prevalence• mortality

data?

Other criteria:

• Must be scalable & automated• Must include age structure• Must fairly account for all uncertainty

Motivation:using different data types requires modelling assumptions!

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

3

Countries considered

country WHO TB incidence(per 100Ky)

population(millions)

Cambodia 400 (366 - 444) 15China 70 (66 - 77) 1,386Indonesia 183 (164 - 207) 250Myanmar 373 (340 - 413) 53Nigeria 338 (194 - 506) 174Pakistan 275 (205 - 357) 182Philippines 292 (261 - 331) 98Thailand 119 (106 - 134) 67Viet Nam 114 (121 - 174) 92

Table: The 9 countries considered, together their WHO estimate of TBincidence for 2013 and their population in 2013.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

4

Prevalence data

country prevalence survey yearsCambodia 2002, 2011China 1990, 2000, 2010Indonesia 2004Myanmar 1994, 2009Nigeria 2012Pakistan 2011Philippines 1997, 2007Thailand 1991, 2012Viet Nam 2007

Table: Years of available prevalence survey data for the 9 countriesconsidered.

• Not available in age-structured n/N form for this work.• Approach to different reporting documented in report.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

5

Mortality data

country iso3 VR data points mortality sourceCambodia 0 CFRChina 22 VRIndonesia 0 CFRMyanmar 0 CFRNigeria 0 CFRPakistan 0 CFRPhilippines 13 VRThailand 15 VRViet Nam 2 VR

Table: Approaches to mortality, and sources in the 2013 GTB report.CFR=approach from CFR; VR=from vital registration data.

• By age (0-14, 15-24, 25-34, 35-44, 45-54, 55-64, 65+),sex and calendar year.

• TB death in HIV -ve individuals.• Used B02 for ICD-9 COD coding; A15-A19 for ICD-10 coding.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

6

Other data

Notifications

• By age (0-14, 15-24, 25-34, 35-44, 45-54, 55-64, 65+),sex and calendar year.

• Available for most years

Demography

• UN ESA Population division modelled population size by5-year age group, sex, and calendar year.

• UN ESA estimated birth rates.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

7

Model overview

It

nt

mt

Xt

notified un-notified

mt

death deathsurvival survival

VR process

ut

ptn,pt

u

pt

survey

1 2

3 4

5

6

8

7

9

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

8

Summary table of parameter values and priors

..

name meaning distributionλ0 initial FOI Γ(0.01, 2.5).1[0.01,0.04]β transmission coefficient Γ(1,6).1[1.5,9]πA primary progression B(2, 20).1[0.075,0.125]πK primary progression (age 0-14)∗ Γ(4.2, 50.4).1[πA, 1]v partial protection B(3, 5).1[0.6,0.9]ϵ endogenous progression Γ(10−3, 5).1[5.10−4, 1.10−2]

CFRu un-notified case fatality B(3, 2).1[0.4,0.6]CFRn notified case fatality B(1, 20).1[0.05,0.1]Tu un-notified disease duration ℓN (log 3, .1).1[1.5, 5.5]Tn notified disease duration ℓN (log 0.5, .4).1[0.1, 1.3]VR probability TB death in register B(3, 1).1[0.01,0.9]CDR final case detection probability B(3, 1).1[0.4,0.9]dCDR rate change in CDR 1[0.01,0.3]

Table: First half represents additional transmission modelparameters; second half are the parameters in current use in WHOestimation processes (with the exception of dCDR). Γ(s, r) denotes aGamma distribution with shape s and rate r; B(a, b) denotes a Betadistribution with shape parameters a, b; ℓN (L,S) denotes a log-normaldistribution with parameters L and S; and 1[a, b] denotes an indicatorfor belonging to the interval [a, b].(∗Not involved in inference.)

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

9

Inference overview

Philosophy

Bayesian approach =⇒ uncertainty in all model parameterssampled, consistent with the data.

• many unobserved states to be summed over• some parameters don't effect fit (nuisance parameters)• some parameters correlated given data

(Don't really care about parameter values)

Details

• Affine invariant MCMC sampler• many chains started near MAP• simple to tune & parellizable• handles correlations well

• Average log-likelihood from 10 runs used for each step• 500 steps with 1,000 chains

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

10

Likelihood approximation

0.000

0.025

0.050

0.075

0.100

−1060 −1050 −1040 −1030 −1020LL

dens

ity

Figure: ℓ− Eℓ ∼ N(0, σ)

|logELik− E log(Lik)| = log(Eeℓ−Eℓ) ≈ σ2

2≲ 1%

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

11

Beta-binomial distributions(Method/conclusion)

0 200 400 600 800 1000

0.000

0.002

0.004

0.006

0.008

0.010

1:1000

dbet

abin

om(1

:100

0, s

ize

= 10

00, t

heta

= 2

00, p

= 0

.7)

Figure: Beta-binomial has 2-levels: p ∼ Beta, n ∼ Binom(N, p)

• Binomial representations of processes like detection aretightly weighted for moderate N

• Unrealistic representation of certainty• Leads to an extremely peaked likelihood, that undervalues

prevalence surveys and under-represents uncertainty

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

12

Demography

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1991 1992 1993 1994 1995

1996 1997 1998 1999 2000

2001 2002 2003 2004 2005

2006 2007 2008 2009 2010

0−45−9

10−1415−1920−2425−2930−3435−3940−4445−4950−5455−5960−6465−6970−7475−7980−8485−8990−9495−99

100−

0−45−9

10−1415−1920−2425−2930−3435−3940−4445−4950−5455−5960−6465−6970−7475−7980−8485−8990−9495−99

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10−1415−1920−2425−2930−3435−3940−4445−4950−5455−5960−6465−6970−7475−7980−8485−8990−9495−99

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0−45−9

10−1415−1920−2425−2930−3435−3940−4445−4950−5455−5960−6465−6970−7475−7980−8485−8990−9495−99

100−

−40000 0 40000 −40000 0 40000 −40000 0 40000 −40000 0 40000 −40000 0 40000Number (thousands)

Age

sex

●●

●●

female

male

China

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

13

Overview of other outputs

●● ● ● ● ● ● ●

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●

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● ● ● ● ●

0

50

100

150

1990 1995 2000 2005 2010year

rate

per

100

,000

per

yea

r

variable

●

●

●

●

●

●

e_inc_100k

e_mort_exc_tbhiv_100k

incidence

mortality

notifications

VR

●● ● ● ● ● ● ●

●

●

● ● ● ●

● ● ● ● ●

0

500,000

1,000,000

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1990 1995 2000 2005 2010year

num

bers

per

yea

r

variable

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●

e_inc_num

e_mort_exc_tbhiv_num

incidence

mortality

notifications

VR

0

1,000,000

2,000,000

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1990 1995 2000 2005 2010year

num

bers variable

e_prev_num

prevalence

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1990 2000 2010

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55−

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65+

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35−

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45−

55

55−

65

65+

0−15

15−

25

25−

35

35−

45

45−

55

55−

65

65+

age

TB

pre

vale

nce

per

100,

000

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

14

Notifications, incidence, mortality

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50

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rate

per

100

,000

per

yea

r

variable

●

●

●

●

●

●

e_inc_100k

e_mort_exc_tbhiv_100k

incidence

mortality

notifications

VR

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

15

Prevalence by age in survey years

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1990 2000 2010

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25−

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35−

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45−

55

55−

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65+

age

TB

pre

vale

nce

per

100,

000

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

16

Prevalence through time

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rate

per

100

,000

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yea

r

variable●

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e_inc_100ke_mort_exc_tbhiv_100kincidencemortalitynotificationsVR

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500,000

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bers

per

yea

r variable●

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e_inc_nume_mort_exc_tbhiv_numincidencemortalitynotificationsVR

0

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2,000,000

3,000,000

1990 1995 2000 2005 2010year

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bers variable

e_prev_numprevalence

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1990 2000 2010

0

100

200

300

400

0−15

15−2

5

25−3

5

35−4

5

45−5

5

55−6

5

65+

0−15

15−2

5

25−3

5

35−4

5

45−5

5

55−6

5

65+

0−15

15−2

5

25−3

5

35−4

5

45−5

5

55−6

5

65+

age

TB p

reva

lenc

e pe

r 100

,000

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

17

MCMC chains

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

18

Correlations in parameter samples

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

19

Summary table of estimates

..

country incidence per 100K/y mortality per 100K/y prevalence per 100KCambodia 241 (204 - 292) 52 (38 - 78) 525 (397 - 660)China 74 (65 - 86) 15 (11 - 19) 118 (99 - 140)Indonesia 125 (110 - 152) 17 (12 - 25) 203 (168 - 252)Myanmar 117 (89 - 159) 22 (13 - 36) 204 (146 - 311)Nigeria 91 (68 - 137) 30 (21 - 49) 296 (198 - 449)Pakistan 140 (91 - 179) 43 (22 - 58) 322 (195 - 426)Philippines 362 (317 - 441) 112 (81 - 147) 565 (500 - 667)Thailand 88 (75 - 103) 27 (19 - 36) 157 (128 - 194)Viet Nam 56 (51 - 64) 6 (5 - 11) 76 (52 - 101)

Table: Incidence, mortality and prevalence are shown, together with95% credible intervals in brackets, for each country. For 2013.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

20

LTBI estimates

country infections %Cambodia 3,793,000 25China 263,233,000 19Indonesia 35,772,000 14Myanmar 8,829,000 17Nigeria 30,392,000 18Pakistan 41,266,000 23Philippines 31,043,000 32Thailand 13,394,000 20Viet Nam 15,356,000 17

Table: Numbers of individuals latently infected with M.tb according tothe model (to the nearest thousand), and the percentage of thepopulation that this represents. For 2013.

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

21

Comparison with WHO estimates

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Cambodia

Cambodia

China

China

China Indonesia

Indonesia

Indonesia

Myanmar

Myanmar

Myanmar

Nigeria

Nigeria

Nigeria

Pakistan

Pakistan

Pakistan

Philippines

Philippines

Philippines

Thailand

Thailand

Thailand

Viet Nam

Viet Nam

Viet Nam

0

200

400

600

0 200 400 600WHO per 100K capita estimate

mod

el p

er 1

00K

cap

ita e

stim

ate

variable

●a

●a

●a

incidence

mortality

prevalence

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

22

Limitations

Cons

• Inference was suboptimal• Priors used were rather ad hoc• No HIV/ART• Sex disaggregation not used• Beta-binomial choice• Difficulties defining appropriate n/N• Only single model structure considered

Pete Dodd

Introduction

Data

Model

Structure

Inference

Results

China

All countries

LTBI

Comparison

Discussion

Limitations

Advantages

23

Advantages

Pros

• Parsimonious, well-defined, consistent, automated and fast• Makes statistically rigorous use of notification, prevalence

and mortality data• Propagates uncertainty• Can be extended to consider other evidence

(e.g. LTBI, capture-recapture data)• Other models giving It → It+1 could be used, compared,

averaged• Under-15 age-groups could be subdivided and refined• Predicted outputs can be age-disaggregated• Structured as R package - model together with cleaned

data. Press-and-go.

PeteDodd Mathematical modelling approach to estimating TB ...€¦ · Results China All countries LTBI Comparison Discussion Limitations Advantages 11 Beta-binomialdistributions (Method/conclusion)

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