BACKGROUND DOCUMENT 5b

A mathematical modelling approach to estimating TB incidence

Prepared by:

Pete Dodd

Questions for discussion

1. Would a different choice of (relatively un-tuned) inference methodology be likely to

be more efficient, reliable, and/or facilitate model comparison?

2. Is the beta-binomial model of reporting processes the best way to avoid unrealistic

balances between the information content of notification and VR data vs prevalence

data? Is there a sensible way of determining the mixture parameter?

3. What is the best way to make use of numerator/denominator data (e.g. prevalence

survey data) in the presence of design effects and adjustment?

A mathematical modelling approach to

estimating TB incidence

P.J. Dodd∗1

1Health Economics and Decision Science, School of Health &Related Research, University of Sheffield, Sheffield, UK.

February 27, 2015

Abstract

In this report, we present a framework that uses notification, preva-lence and mortality data to inform TB burden estimates in a single rigor-ously defined statistical framework. Temporal correlations are handled byintroducing a simple stochastic compartmental model of TB transmissionthat builds on current modelling assumptions used in WHO estimates,and that includes demographic trends in ages and sexes of populations.Each of the 12 parameters characterising the model is treated as uncertain,and Monte Carlo methods are used to perform inference and prediction ina Bayesian framework for 9 countries with prevalence survey data avail-able. Resulting burden estimates are presented and compared with thoseof WHO. We discuss difficulties and potential further work, and concludethat this approach holds promise as a novel tool for consistent burdenestimation.

∗P.J.Dodd@sheffield.ac.uk

1

CONTENTS CONTENTS

Contents

List of Figures 3

List of Tables 3

1 Introduction 4

2 The data 52.1 Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 TB notifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 TB prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 TB mortality in vital registration statistics . . . . . . . . . . . . 6

3 The model 73.1 Conceptual overview . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Variables and notation . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Specification of submodels . . . . . . . . . . . . . . . . . . . . . . 123.5 Measurement processes . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5.1 P (mt|mtα) - VR statistics . . . . . . . . . . . . . . . . . 123.5.2 P (nt|pnt α) - notifications . . . . . . . . . . . . . . . . . . 123.5.3 P (pt|put pntXtα) - observed prevalence . . . . . . . . . . . 13

3.6 Unobserved states . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6.1 P (mt|pnt put α) - true mortality . . . . . . . . . . . . . . . 133.6.2 P (pxt |Itpxt−1xtα) - true prevalence . . . . . . . . . . . . . 133.6.3 P (It|Xt−1p

nt−1p

ut−1α) - incidence . . . . . . . . . . . . . . 14

3.6.4 P (Xt|Xt−1pnt−1p

ut−1α) - population state dynamics . . . 14

3.6.5 P (p0X0|α) - the initial state . . . . . . . . . . . . . . . . 153.7 Parameters & priors . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.7.1 Parameters in common with WHO approach . . . . . . . 163.7.2 Other parameters . . . . . . . . . . . . . . . . . . . . . . . 16

4 The likelihood and its ingredients 174.1 Age groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Sexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Beta-binomial distributions . . . . . . . . . . . . . . . . . . . . . 19

5 Inference approach 195.1 Unobserved variables . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 MCMC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Results 216.1 The example of China . . . . . . . . . . . . . . . . . . . . . . . . 21

6.1.1 Demographic fits . . . . . . . . . . . . . . . . . . . . . . . 216.1.2 MCMC fits . . . . . . . . . . . . . . . . . . . . . . . . . . 226.1.3 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Summary of estimates . . . . . . . . . . . . . . . . . . . . . . . . 27

2

LIST OF FIGURES LIST OF TABLES

6.3 LTBI estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4 Comparison with WHO outputs . . . . . . . . . . . . . . . . . . . 29

7 Discussion 30

Acknowledgments 33

Abbreviations 34

References 35

A Country demographic fits 37

B Country TB fits 47

C Country MCMC sample pair plots 57

D Country MCMC chains 67

List of Figures

1 Schematic model structure. . . . . . . . . . . . . . . . . . . . . . 112 Stochasticity of log-likelihood . . . . . . . . . . . . . . . . . . . . 203 Demographic fit for China . . . . . . . . . . . . . . . . . . . . . . 214 ACF plots for China . . . . . . . . . . . . . . . . . . . . . . . . . 225 MCMC chains for China . . . . . . . . . . . . . . . . . . . . . . . 236 Triangle plot for China . . . . . . . . . . . . . . . . . . . . . . . . 247 Predicted rates for China . . . . . . . . . . . . . . . . . . . . . . 258 Predicted prevalence for China. . . . . . . . . . . . . . . . . . . . 269 Predicted prevalence by age for China. . . . . . . . . . . . . . . . 2710 Model & WHO estimates compared. . . . . . . . . . . . . . . . . 30

List of Tables

1 Countries considered . . . . . . . . . . . . . . . . . . . . . . . . . 52 Prevalence surveys . . . . . . . . . . . . . . . . . . . . . . . . . . 63 WHO mortality approach . . . . . . . . . . . . . . . . . . . . . . 74 Notations & meaning. . . . . . . . . . . . . . . . . . . . . . . . . 105 Parameters & priors. . . . . . . . . . . . . . . . . . . . . . . . . . 176 Esimates summary . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Estimates summary (numbers). . . . . . . . . . . . . . . . . . . . 288 LTBI estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3

1 INTRODUCTION

1 Introduction

The incidence of tuberculosis (TB) disease is the preferred metric of TB bur-den, partially as it untangles differences in the duration as prevalent cases thatmay depend on behavioural factors, health system factors and HIV prevalence.Unfortunately, the incidence of TB disease is impractical to measure directly ona national level, as this would require following up very large numbers of indi-viduals. However, the vast majority of countries report TB notifications, whichare collated by the World Health Organisation (WHO). While underdiagnosisand underreporting mean that these underestimate incidence, notification datado provide very useful information for estimating TB incidence.

Although different methods are used dependent on certain conditions, for themajority of TB the WHO incidence estimate is based on inflating the reportednotification rate by an uncertain factor (the inverse of the case detection ratio,CDR) that represents knowledge about the shortfall [1]. Information about thevalue of the CDR has historically relied on quantified expert opinion, but isincreasingly based on capture-recapture modelling [2].

WHO estimates of mortality due to TB are based on data from vital reg-istration (VR) systems, where they exist (adjusting for imperfect coverage andmisclassification of deaths), and on uncertain knowledge of case fatality rates(CFRs) in notified and non-notified TB where VR systems are not present.

In recent years, an increasing number of high-quality national prevalencesurveys have been completed [1]. These provide direct information on the TBprevalence in a country, and are much less affected by bias then notification orVR data. Gaining information on TB incidence from prevalence data requiresa model of some kind that can describe the typical duration of prevalent TBcases. This is indeed the approach used by WHO to estimate TB prevalencefrom incidence in countries without prevalence surveys, and a reverse of thisprocedure is used in countries with direct prevalence data.

Using separate approaches to mortality, incidence and prevalence means thatthey are not guaranteed to be consistent with one another. Moreover, it meansthat mortality data, e.g., is not able to inform on the likely incidence. Thisargues for a unified statistical framework - a model that can account for noti-fication, mortality, and prevalence data and the likely relations between them,and account for the different degrees of information each data source provides.In order to relate prevalence to incidence, one needs a model that least char-acterises the typical survivorship of prevalent TB after incidence, and to makecontact with mortality data, one also needs assumptions about the case fatal-ity ratios in various cases. As mentioned above, some of the approaches usedby WHO include assumptions of this kind. It is also natural to believe thatincidence, prevalence and mortality in one year are partially informative aboutincidence, prevalence and mortality in the next year. One approach to usinginformation from different years is to include a statistical model for the relation-ship between these quantities in one year and other years. Another approach isto explicitly model the transmission process, so that incidence in a given yearis dependent on a history of infection, driven by prevalence.

In this report, we pursue the latter possibility. We consider a simple trans-mission model framework that includes the current WHO models of durationand case fatality, and minimally extends these to account for latent infection and

4

2 THE DATA

the influence of the force-of-infection on incidence. We include age and sex, andaccount for the effect of demography and demographic change on the prevalenceof infection and the pattern of disease. In doing this we have to make additionalassumptions about the natural history, but have tried to keep the model assimple as possible, for reasons of transparency, computational burden, and tominimise the number of parameters introduced.

We have taken a Bayesian approach to the calibration of this model and gen-eration of estimates from it. All of the parameters are assumed to be uncertain,and Monte Carlo methods are used to sample these parameters conditioned onthe data for each country. In this way, some parameters are restricted to beconsistent with the data, and others have their ranges more freely explored. Theuncertainty given the data is then propagated into predictions about quantitiessuch as TB incidence. The same inference and prediction approach is used inan automated fashion for all the countries we consider.

The rest of this document is structured as follows. In Section 2, we go onto discuss the data sources and their structure, before describing the model inSection 3. Section 3.1 is a conceptual overview of the model that describesand motivates the features included. Section 3.7 discusses the parameterisationof the model. The construction of the likelihood and inference approach arediscussed in Sections 4 and 5, respectively, before presenting the results forChina in Section 6. (Results for the other countries considered are presentedin the Appendices.) The model estimates are compared with those of WHO inSection 6.4 before a concluding discussion in Section 7.

2 The data

2.1 Countries

Although data were collected, cleaned and collated in a standardised fashion forall countries, the remit of this report was to consider the 9 countries listed inTable 1.

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 1: Countries considered, together their WHO estimate of TB incidencefor 2013 and their population in 2013.

Demographic data on birth rates by year and sex, and estimated population

5

2.2 TB notifications 2 THE DATA

counts by year, sex and 5-year age category were taken from UN Economic andSocial Affairs Population Division (http://esa.un.org/wpp/).

2.2 TB notifications

TB notifications reported to WHO by year, sex and age category (in this case:0-14, 15-24, 25-34, 35-44, 45-54, 55-64, 65+ years of age) were downloadedfrom http://www.who.int/tb/country/data/download/en/. WHO estimatesof TB burden used below in comparisons, were also taken from this source.

2.3 TB prevalence

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

Table 2: Prevalence surveys. Years of available prevalence survey data for the9 countries considered.

All of the 9 countries had data from at least one prevalence survey available (seeTable 2). The data were aggregated over age. Some prevalence surveys usedsmear, and others used bacteriological confirmation; some reported adjustedprevalence (i.e. standardised, often with cluster-sampling design effects), othersonly crude prevalence. We used the following approach to standardising thisdata into the effective numerator/denominator data required by the likelihood(under a normal approximation to a simple binomial sample) : if adjusted esti-mates were present, this point-estimate was used together with the upper- andlower-confidence intervals to generate a numerator/denominator; else if adjustedestimates were absent, crude bacteriological confirmed point-estimates and con-fidence intervals were used; else if confidence intervals were not available forcrude bacteriologically confirmed estimates, the point-estimate and number ofparticipants were used; else if only estimates based on smear were available(rare), the point-estimate divided by 0.7 (to account for the sensitivity of smearmicroscopy) and the number of participants were used to an generate effectivenumerator and denominator.

2.4 TB mortality in vital registration statistics

The vital registration data are available at: http://www.who.int/healthinfo/statistics/mortality_rawdata/en/, and included some years in the 9 coun-tries we consider (see Table 3). Different ICD coding for cause of death (COD)

6

3 THE MODEL

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

were used in different years. The data were stratified by age (0-14, 15-24, 25-34,35-44, 45-54, 55-64, 65+ years of age) and sex and calendar year. For ICD-9COD coding, we used code B02 as representing a TB death; for ICD-10 coding,we used codes A15 - A19.

3 The model

3.1 Conceptual overview

The model structure is driven by the need to make contact with notification,mortality and prevalence data. The aim was to parallel modelling assumptionsthat are already made in some WHO analyses [1], and introduce as few addi-tional states as possible. It was also desirable to introduce as few additionalparameters as possible: each parameter was treated as imperfectly known and socontributed more uncertainty; and sampling procedures tend to perform morepoorly as the number of parameters increases. Finally, the model had to befast to run as exploring parameter space requires many runs, but easy to workwith and analyse output (choice of implementation is discussed in Section 3.2).Finally, as the model is required to produce estimates of TB cases by age andsex, this structure was included in such a way as to match data on demographictrends.

In order to match mortality data, we needed both a true rate of mortality,and a process that captures the fact that not all TB deaths (2 in Figure 1) arerecorded in vital registers (the data source). The imperfect reporting processis shown by arrow 1 in Figure 1, and described in detail in Section 3.5.1. Truemortality results as an endpoint for both notified and non-notified TB cases.This is in common with the WHO model for TB mortality based on character-ising case fatality ratios (CFRs), in countries without VR systems [1]. In themodel, this means that a certain proportion of those notified die. Applying adifferent CDR to those who are not notified requires the notion of ‘non-notifiedresolution’, i.e. ceasing to be a prevalent TB case either due to death, or self-cure, without notification. The notified cases in the model correspond logicallyto the notifications reported to WHO.

In order to relate to prevalence data, the model must include a prevalence

7

3.1 Conceptual overview 3 THE MODEL

state that individuals pass through before notification or non-notified resolution.We model the survey process, based on this prevalence, i.e. the number of casesone would expect to find with a simple random sample of a given size. However,there are choices about how to model the split between notified and non-notifiedoutcomes.

One could include one class of prevalent TB, and assume that a certain pro-portion of this pool are notified or non-notified when they cease being prevalent.However, the overall rate of ceasing to be prevalent must then be an average ofthe rates associated with the timescales for notified and non-notified cases. Thisis a problem if, e.g. the timescale for notified TB is small (as may be the casewhen exploring parameter space) because it would lead to a large overall rateof ceasing to be prevalent, and correspondingly, a small prevalence; regardlessof the timescale for non-notified cases. A more satisfactory possibility is to splitthe prevalent class into two: prevalent cases who will be notified, and prevalentcases who will not be notified. A small timescale for notified TB cases willthen not artificially deplete the overall prevalence since a different rate appliesto the pool of prevalent cases who will not be notified. The rates of leavingthe prevalent pool determine how much of this pool at one time contributes toprevalence at the next.

The probability that an incident case entered the pool of prevalent cases thatwill be notified (as opposed to the pool of cases that will not be notified) waslabelled CDR but is not directly related to the epidemiological case detectionratio (ratio of notified over incident cases in a year). The example of Chinasuggests it does not make sense to treat case detection as a static quantity.There, the prevalence to notification ratio (which is a measure of case detection)changed substantially between prevalence surveys, and notifications increasedsharply over a period where three prevalence surveys showed a declining trendin TB prevalence. We therefore allowed our parameter CDR to depend oncalendar time, introducing a parameter allowing it to decrease going backwardsin time from the present.

Up to here, no model of transmission has been introduced. The transmissionmodel serves to determine incidence at a given point in time in terms of the his-tory of the states discussed above. The model consists of a linear relationshipbetween the prevalence and the M.tb force-of-infection. The force-of-infectionthrough time determines the proportions of each age-group in the base popula-tion that are latently infected with M.tb (right-most dashed arrow 7 in Figure 1).The incidence is then determined as a constant rate of endogenous reactivationfrom those latently infected, and a second contribution proportional to the cur-rent force-of-infection, representing progression following primary infection andreinfection (against which latent infection confers partial protection). Theserates are implemented in a way that is standard in the TB modelling literature.

The base population therefore records the proportion of each age group whoare latently infected. This is affected by birth rates. The birth rates are takenfrom UN ESA data (see Section 2.1), and the mortality rates depend on calendartime and age and sex in such a way as to closely match the overall populationtrends. The population counts in this base population are not adjusted toaccount for incidence, as this is taken to be negligible. Similarly, TB mortalityis not used to advance the base population as it is already included in the totalmortality. Thus, the base population advances in a quasi-independent fashion,with only the proportion of each sub-category affected by the force-of-infection.

8

3.2 Implementation 3 THE MODEL

The model needs to be stochastic so that there is a well-defined probability ofthe data taking a particular (integer) value. However, the dynamics of the basepopulation are deterministic conditional on the force-of-infection, but becomestochastic due to the stochasticity of the force-of-infection. Because the dataare all available at yearly intervals, the model is designed to update with yearlyintervals. Full details of the model are provided in the rest of this section.

3.2 Implementation

It was intended to make the model as easy as possible to install and use, suggest-ing an established high-level language with a packaging mechanism. The R [3]language and environment for a statistical computing is a natural choice: itallows packages containing data, has strong data manipulation, statistical anal-ysis and graphics capability. It also allows portions of packages to be written incompiled languages to improve computational speed, without users needing tobe aware of this. We have implemented the underlying model in fortran 90 [4],which is suited to array manipulations, resulting in millisecond execution timefor a run of 20 years.

3.3 Variables and notation

Table 4 summarizes the quantities involved in describing the model, and theirmeaning. A t subscript denotes the value of some quantity in year t; the absenceof a t subscript means either that the variable is not dependent on time. Somevariables are observed, in the sense that they are directly related to measure-ments that can be made. Individually, these are distinguished as having hats,and are collectively denoted yt.

Other variables are not observed. They may be directly related to TB burdenin a given year (e.g. : It, the number of new TB cases in year t, mt the numberof TB deaths in year t) and involved in determining observable quantities andthe dynamics. Other unobserved quantities are part of the ‘base’ population, Xt

(see Section 3.1 and Figure 1), which divides the populationinto non-overlappingcompartments according M.tb-infection status, age and sex.

All variables measuring population states have age and sex structure, andare to be considered as arrays. We normally suppress indices related to thisstructure for clarity.

Finally, there are quantities that do not fall into these groupings - the col-lection of all observed and unobserved quantities (Zt); the data (D); the modelparameters (α).

9

3.3 Variables and notation 3 THE MODEL

type name meaning

ob

serv

ed

mt number of TB deaths in VRnt TB notificationspt prevalent TB cases found in surveyyt {mt, nt, pt}, all observed variables

un

obse

rved T

B

mt true TB deathsIt TB incidenceλt force-of-infectionut resolutions of non-notified TBpnt prevalent TB that will be notifiedput prevalent TB that will not be notified

bas

e

Ut M.tb uninfectedLt M.tb infectedNt Ut + Lt, total base populationXt {Ut, Lt}, base population stateZt all unobserved and observed variablesDt {Dm

t , Dnt , D

pt }, all data

(mortality, notifications, prevalence)α all model parameters

Table 4: Notations & meaning. Quantities used in describing the model. Ob-served quantities are measured directly in some way; unobserved quantities arenot. A subscript t denotes the value of some quantity in year t. All variablesexcept force-of-infecion (λt) and model parameters (α) are arrays representingsex- and age-structure.

10

3.3 Variables and notation 3 THE MODEL

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

Figure 1: Schematic model structure. n=notifications; m=mortality;p=prevalence (pu for prevalence that will not be notified; pn for preva-lence that will be notified); I=incidence; X=underlying population state.Diamonds are rates and boxes are counts. Observed quantities have a redborder. Labeled arrows are processes referred to in text. This diagram repeatsfor each time, with states for prevalence and the underlying population stateat time t influencing those at time t+ 1.

11

3.4 Specification of submodels 3 THE MODEL

3.4 Specification of submodels

• P (mt|mtα) - a model of recorded mortality in VR statistics given the truemortality mt,arrow 1 in Fig 1, specified in section 3.5.1

• P (nt|pnt α) - a model of notifications given the numbers prevalent,arrow 3 in Fig 1, specified, in section 3.5.2

• P (pt|pnt putXtα) - a model of observed prevalence given true prevalence,arrows 5 in Fig 1, specified in section 3.5.3

• P (mt|pnt put α) - a model of true mortality given the true prevalence,arrow 2 in Fig 1, specified in section 3.6.1

• P (ut|put α) - a model of un-notified prevalence cessation,arrow 4 in Fig 1, specified, in section 3.5.2

• P (pxt |Itpxt−1xtα) - a model of prevalence (x ∈ {n, u}) given incidence andlast year’s prevalence,arrow 6 in Fig 1, specified in section 3.6.2

• P (It|Xt−1pnt−1p

ut−1α) - model of incidence given the prevalence & popula-

tion state,arrow 8 in Fig 1, specified in section 3.6.3

• P (Xt|Xt−1pnt−1p

ut−1α) - model of population state dynamics given preva-

lence,arrow 9 in Fig 1, specified in section 3.6.4. This depends on the model forthe force-of-infecion (arrow 7 in Fig 1), also discussed in section 3.6.4.

• P (p0X0|α) - a model of the initial state,not shown in Figures, specified in section 3.6.5

3.5 Measurement processes

3.5.1 P (mt|mtα) - VR statistics

It is assumed that there is an independent probability V Rt that each death dueto TB is recorded in the register, but that this is a random effect giving rise toa beta-binomial distribution with dispersion parameter θ = 10 (see Sections 4.3and 7 for discussion), so that:

P (mt|mtα) = BetaBin(mt|mt, V Rt, θ) (1)

We take V Rt = V R to be a constant.This is process 1 in Figure 1.

3.5.2 P (nt|pnt α) - notifications

It is assumed that all those in the pool, pnt , of prevalent TB that will be notifiedare subject to a hazard of notification equal to 1/Tn (i.e. have exponentially dis-

12

3.6 Unobserved states 3 THE MODEL

tributed times to notification with mean Tn), and that all those in the pool, put ,of prevalent TB that will not be notified are subject to a hazard of resolutionequal to 1/Tu (i.e. have exponentially distributed times to death-or-self-curewith mean Tu). This means the probability of notification (non-notified reso-lution) after a year is 1 − e−1/Tn (or 1 − e−1/Tu , respectively). The numbersof notifications, nt and non-notified resolutions, ut are therefore binomially dis-tributed with these probabilities:

P (nt|pnt α) = Bin(nt|pnt , (1− e−1/Tn)

)(2)

P (ut|put α) = Bin(ut|put , (1− e−1/Tu)

). (3)

These are processes 3 & 4 in Figure 1.

3.5.3 P (pt|put pntXtα) - observed prevalence

If a prevalence survey in year t is conducted as a simple random sample withsample size Ns

t , then we have a simple binomial form for the observation model:

P (pt|put pntXtα) = Bin (pt|Nst , (p

ut + pnt )/Nt) . (4)

This can be understood as a product over the terms stratified by age categoryand sex. In fact, since numerator and denominator data were not available forprevalence surveys stratified by age and sex, the version used in this modellingemployed total prevalence in those over 15 years of age, and Equation 4 can beread with put + pnt representing the total numbers of TB cases among those over15 years of age, and Nt the total population aged over 15 years.

This is process 5 in Figure 1.

3.6 Unobserved states

3.6.1 P (mt|pnt put α) - true mortality

The mortality in a given year, is made up of deaths from those who are notified(i.e. among nt), which happen with probability CFRn; and deaths among thosewho resolve their TB without notification (i.e. among ut), which happen witha larger probability CFRu. The true mortality in year t is therefore a sum oftwo binomially distributed random variables:

mt = mut +mn

t (5)

P (mut |ut, α) = Bin(mu

t |ut, CFRu) (6)

P (mnt |nt, α) = Bin(mn

t |nt, CFRn). (7)

This is process 2 in Figure 1.

3.6.2 P (pxt |Itpxt−1xtα) - true prevalence

The function CDRt is not a case detection ratio per se, but the mean probabilitythat an incident case in year t enters the pool of prevalent TB that will be

13

3.6 Unobserved states 3 THE MODEL

notified, pnt , rather than the pool of prevalent cases that will not be notified, put .We allow for this parameter to change through time, linearly on a logit scale(which keeps CDRt ∈ [0, 1]) up to a maximum of CDR in 2013, with a rate ofdCDR (Equation 8).

The number of new TB cases that will be notified in each age and sex cat-egory is determined as a Beta-binomial sample from the newly incident caseswith mean CDRt and dispersion θ = 10 (see Section 4.3 for discussion): Equa-tion 9. The remaining new incident cases, enter the pool of prevalent cases whowill not be notified (put ): Equation 10. The pools of prevalent TB destined fornotification, and destined to resolve without notification, then update account-ing for notifications and non-notified resolutions (Equation 11 and equation 12,respectively).

logit CDRt = logit CDR− dCDR×max(T − t, 0) (8)

P (∆pnt |It, CDRt, α) = BetaBin(∆pnt |It, CDRt, θ) (9)

∆put = It −∆pnt (10)

pnt = pnt−1 + ∆pnt − nt (11)

put = put−1 + ∆put − ut (12)

This is process 6 in Figure 1.

3.6.3 P (It|Xt−1pnt−1p

ut−1α) - incidence

The incidence is a Poisson process with a rate that composed of two parts: onedue to endogenous activation of latent infection (i.e. εLt−1), and another due toprimary progression following (re-)infection (i.e. (Ut−1 + (1 − v)Lt−1).π.λt−1),so that:

P (It|Xt−1pnt−1p

ut−1α) = Po(It|(Ut−1 + (1− v)Lt−1).π.λt−1 + ε.Lt−1) (13)

where the force-of-infection is take constant across age/sex-categories as

λt = βptNt

(14)

where pt is the total number of prevalent TB cases and Nt is the total popu-lation size, both at time t, and β is the transmission coefficient. The secondcontribution to the rate, Ut−1.π.λt−1 represents the contribution from primaryprogression following initial infection, and (1 − v)Lt−1.π.λt−1 represents rapidprogression following successful reinfection, which is supposed to occur with alower hazard due to partial protection from latent M.tb infection (i.e. v).

The primary progression rate π depends on age, taking the value πK in agecategories for those under 15 years of age, and πA in all other age categories.

This is process 8 in Figure 1, with the influence of force-of-infection repre-sented by the leftmost dashed arrow labelled 7.

3.6.4 P (Xt|Xt−1pnt−1p

ut−1α) - population state dynamics

The dynamics of the base population Xt are determinisitic when conditionedon the force-of-infection:

14

3.6 Unobserved states 3 THE MODEL

P (Xt|Xt−1pnt−1p

ut−1α) = δ(Xt, Ft(Xt−1p

nt−1p

ut−1α)) (15)

where Ft is a function that enacts an advance of a discretized version of thefollowing ordinary differential equations:

U ′ = b− λ.U − µ.U (16a)

L′ = λ.L− µ.L (16b)

where U represents TB uninfected individuals, L represents individuals withLTBI. However, our dynamics also include ageing and age dependent mortalityto match demographics,

In fact our state space is more elaborate because we have age and sex. Wewill write

Ut(a, s), Lt(a, s) (17)

for the a-th 5-year age category (a = 1, . . . , 21), and s-th sex category (s = 1, 2for male or female) of U or L respectively at year t.

With the convention that U and L at 0 indices have value 0, the terms in γrepresent aging between categories, and integrating the ODEs over 1 year (underthe assumption that the exogenous time-dependent parameters and the force-of-infection, λt, remain fixed), we obtain the approximate annual transition:

Ut+1(a, s) = (1− e−γ−λt).Ut(a, s) +γ

γ + λt.(1− e−γ−λt).Ut(a− 1, s)

+ δa.bts − µtas.Ut(a, s) (18a)

Lt+1(a, s) =λt

γ + λt.(1− e−γ−λt).Ut(a− 1, s) + e−γLt(a, s)

+ (1− e−γ).Lt(a− 1, s)− µtas.Lt(a, s) (18b)

The value of γ is determined as the inverse of the the age-bin widths (i.e.γ = 1/5). The values of bts, µtas are obtained deterministically from the UNESA data (Section 2.1).

This is process 9 in Figure 1, with the influence of force-of-infection repre-sented by the rightmost dashed arrow labelled 7.

3.6.5 P (p0X0|α) - the initial state

We choose divide the underlying population into uninfected and latently infectedstates, as if they had been subject to a constant force-of-infection, λ0, i.e.

U(a, s) = P (a, s).e−λ0Aa (19)

L(a, s) = P (a, s).(1− e−λ0Aa

)(20)

where Aa are the mid-points of the age rages for each category. This impliesan initial incidence, as in Section 3.6.3. The initial overall prevalence is isdetermined from λ0 using Equation 14 and distributed among age categories

15

3.7 Parameters & priors 3 THE MODEL

to match the relative pattern seen in the initial incidence. The initial split ofprevalence between those destined for notification (pn) and those destined notto be notified (pu) is at random with probability determined by the initial valueof the parameter CDR weighted by the relative yearly probability of remainingprevalent for each category, i.e. a probability

CDR.e−1/Tn

CDR.e−1/Tn + (1− CDR).e−1/Tu (21)

of being in the pn category.This is not shown in Figure 1.

3.7 Parameters & priors

The parameters used in the model are summarised in Table 5. All parametersare characterised by distributions to capture uncertainty, and are additionallyrestricted to ranges for numerical reasons. These characterisations are relativelyad hoc.

3.7.1 Parameters in common with WHO approach

In common with some of the WHO approaches, we make assumptions aboutthe case fatality ratios among cases that have been notified and cases which arenon-notified (CFRn and CFRu, respectively). We also assume that prevalentcases resulting in notification have a shorter typical duration than those whichresolve without notification (Tu and Tn, respectively). We also assume that notall TB deaths show up in VR data (parameter V R).

We introduce a parameter labelled CDR, which is not the case detectionratio (see Section 3.7.1), but essentially controls it. WHO estimation procedurescan use a different case detection ratio each year; we introduce a single extraparameter (dCDR) controlling the time trend in case detection and reporting.

These parameters are described in the lower half of Table 5.

3.7.2 Other parameters

In addition, the transmission modelling component introduces extra parametersdescribing the natural history and epidemiology of TB (top half of Table 5).These are used in a way standard in the TB modelling literature.

We have a probability of primary progression following TB infection (πAin adults; πK in children under 15). πK is discussed more in Section 4.1.1,and is not used in the inference. If already infected, individuals have a partialprotection, v, against reinfection and progression. Latently infected individualsdevelop active TB via endogenous activation to TB disease at rate ε.

We introduce a transmission coefficient, β, which serves as a proportionalitybetween the prevalence and force-of-infection. Finally, we characterise the initialstate by a single parameter, λ0: the initial force-of-infection (see Section 3.6.5).

16

4 THE LIKELIHOOD AND ITS INGREDIENTS

name meaning distribution sourceλ0 initial FOI‡ Γ(0.01, 2.5).1[0.01, 0.04] [5]β transmission coefficient Γ(1, 6).1[1.5, 9] [6]πA primary progression B(2, 20).1[0.075, 0.125] [7]πK primary progression (age 0-14)∗ Γ(4.2, 50.4).1[πA,∞] [6]v partial protection B(3, 5).1[0.6, 0.9] [8]ε endogenous progression Γ(10−3, 5).1[5.10−4, 1.10−2] [7, 9]

CFRu un-notified case fatality B(3, 2).1[0.4, 0.6] [10]CFRn notified case fatality B(1, 20).1[0.05, 0.1] [10]Tu un-notified disease duration `N (log 3, .1).1[1.5, 5.5] [11]Tn notified disease duration `N (log 0.5, .4).1[0.1, 1.3] assumedV R probability TB death in register B(3, 1).1[0.01, 0.9] assumedCDR final case detection probability† B(3, 1).1[0.4, 0.9] [1]dCDR rate change in CDR† 1[0.01, 0.3] assumed

Table 5: Summary table of parameter values and priors. First half representsadditional transmission model parameters; second half are the parameters incurrent use in WHO estimation processes (with the exception of dCDR). ∗Notinvolved in inference - see text Section 4.1.1. †As discussed in Section 3.7.1,this is not the usual case-detection ratio, but controls the chances of detection.‡FOI = force-of-infection. Γ(s, r) denotes a Gamma distribution with shapes and rate r; B(a, b) denotes a Beta distribution with shape parameters a, b;`N (L, S) denotes a log-normal distribution with parameters L and S; and 1[a, b]denotes an indicator for belonging to the interval [a, b].

4 The likelihood and its ingredients

General structure of the likelihood, given in terms of the probability of the datayear t, D, given the parameters, α

P (D|α) = P (m = Dm, n = Dn, p = Dp|α) =∑z0:T

(T∏t=1

P (mt = Dmt |mtα)P (nt = Dn

t |pnt α)P (pt = Dpt |put pntXtα)

)P (Z0:T = z0:T |α)

(22)

where Zt is the complete model state in year t, and the Markov property ofthe dynamics means

P (Z0:T |α) = P (ZT |ZT−1α) . . . P (Z1|Z0α)× P (Z0|α) (23)

In full, therefore

P (D|α) =∏t

∑zt

P obs(yt = Dt|ztα)P dyn(zt|zt−1α)P (z0|α) (24)

Using y as a shorthand for observed variables, and where

17

4.1 Age groups 4 THE LIKELIHOOD AND ITS INGREDIENTS

P obs(yt = Dt|ztα) = P (mt = Dmt |mtα)P (nt = Dn

t |pnt α)P (pt = Dpt |put pntXtα)

(25)which are each specified in Section 3.5, P dyn(zt|zt−1α) is implicit in the dynam-ical rules specified in Section 3.6, and P (z0|α) is the initial state specified inSection 3.6.5.

The log-likelihood LL(α|D) is therefore

LL(α|D) ∝∑t

log

(∑zt

P obs(yt = Dt|ztα)P dyn(zt|zt−1α)P (z0|α)

)+ logPrior(α) (26)

4.1 Age groups

In general, the observation part of the likelihood, Equation 25 is a shorthandfor a product over terms specific to the data in each age group:

P obs(yt = Dt|ztα) =∏a

P obs(yat = Dat |ztα). (27)

The notification data and mortality data are both treated in this way. Theprevalence data were only available aggregated over age, and so the likelihoodwas based on the aggregated variables.

4.1.1 Children

Because of the high potential for bias in notification and vital registration datain children (those aged under 15 years), and the fact that prevalence surveysusually exclude this age-group, we only base the likelihood on data and modeloutput in those aged 15 years and older. We use a prior for the primary acti-vation rate in children (πK) based on a fit to simulated data from a previouslydeveloped natural history model of TB in children [6], assuming a uniform agedistribution and averaging over BCG vaccination coverages between 50% and100% and latitudes between 0◦ and 50◦. Because the primary activation ratein adults (πA) is inferred and varies between countries, we additionally imposethe restriction that πK ≥ π in sampling values for burden predictions.

4.2 Sexes

There may be social and biological reasons for differences in TB incidencebetween the sexes and potentially further sex differences differences influenc-ing care-seeking intent and barriers [12–15]. However, modelling the potentialcauses for sex differences in the data would introduce several additional pa-rameters, which the data not be likely to have the power inform. While themodel includes both age and sex explicitly, and the notification data and mor-tality data are disaggregated by age and sex, for the purposes of the likelihoodwe therefore aggregate over both sexes. As discussed above (Section 4.1), theprevalence data are aggregated over age and sex.

18

4.3 Beta-binomial distributions 5 INFERENCE APPROACH

4.3 Beta-binomial distributions

For the models describing the whether a TB death appears in the VR data(Section 3.5.1) and whether an incident TB case will eventually be reportedor not (Section 3.6.2), we have chosen Beta-binomial distributions rather thanbinomial distributions. The Beta-binomial distribution is a broader distribu-tion than the binomial, and can be thought of as a hierarchical model where abinomial sample is taken once the binomial probability is drawn from a Betadistribution. We parametrize the Beta-binomial with the mean Beta probabil-ity, p, and a parameter θ such that θ → ∞ recovers the binomial distribution.In terms of the usual Beta distribution parameters, a and b:

p =a

a+ b(28)

θ = a+ b (29)

In our work below, we choose θ = 10.This modelling choice is discussed further in Section 7.

5 Inference approach

5.1 Unobserved variables

Equation 26 implies that we should be computing ELik, a numerical approx-imation to the expectation of the full data likelihood Lik (i.e. augmentedwith the values of all unobserved variables) over all possible model trajecto-ries for a given set of parameters. Unfortunately, averaging Lik rather than` = log(Lik) is numerically intractable, due to its tiny value, and in general,E log(Lik) 6= log(ELik) (by Jensens’s inequality, log(ELik) ≤ E log(Lik)).

In principle, we can use any non-negative unbiased Monte Carlo estimator ofELik, and a simple example of this is the value of Lik for a single run [16,17]. Ithas been found empirically however, that noisier estimators can result in poorermixing.

It is an identity that:

|logELik − E log(Lik)| = log(Ee`−E`

)(30)

If `−E` ∼ N(0, σ), and Figure 2 suggests this is a reasonable approximation,then e`−E` on the right hand side of Equation 30 log-normally distributed, andthe right hand side of Equation 30 can be evaluated as σ2/2. Since σ2/2 ≈ 5typically for our log-likelihood, and ` ∼ −1, 000, the proportional error is < 1%.Given this, it seemed an acceptable approximation to interchange the expec-tation and logarithm, and work with the log-expected likelihood to improveconvergence.

19

5.2 MCMC algorithm 5 INFERENCE APPROACH

0.000

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−1060 −1050 −1040 −1030 −1020LL

dens

ity

Figure 2: Stochasticity of log-likelihood. This plot shows a histogram of 104

simulated values of the log-likelihood with fixed parameters. The red curve is anormal distribution with matching mean and variance.

5.2 MCMC algorithm

In our case, we have a moderate number of parameters (12), some of which arelikely to be important in fitting to the data, and others of which are likely to benuisance parameters that increase uncertainty without influencing fit substan-tially. In our model, as in other infectious disease models, we can expect thecorrelations to appear in the joint distributions of parameters. Furthermore,we want our inference to run in as automated a fashion as possible, bearing inmind many countries could be of interest.

Affine-invariant ensemble sampling (introduced in [18], and discussed peda-gogically in [19]) is a good choice for this situation. It uses a large number ofMCMC chains simultaneously, from which a proposal distribution is constructedthat maintains affine-invariance. This means that correlations are ‘invisible’ tothe algorithm, and it deals particularly well with correlated variables. Thealgorithm requires minimal tuning, and is easily parallelizable.

We follow the advice in [19], and first determine a maximum a posteriori(MAP) parameter value by using a Nelder-Mead simplex algorithm. We thenstart 1000 chains in randomly perturbed locations within 1% of this parametervalue and run them for 500 iterations. We use a burn-in of d 23 × 500e iterations.

5.3 Predictions

Predictions are made by sampling from the post-burnin chains (this includingthe correlations between parameters, and the uncertainty in these parametersgiven the data) and running the model for each sample (thus including a furthercontribution to uncertainty due to the stochastic nature of the model). Typ-ically, we used 100 samples and reported the median and 2.5-th and 97.5-thpercentiles as uncertainty bounds.

20

6 RESULTS

6 Results

6.1 The example of China

In this section, we describe model outputs and estimates using China as anexample from the 9 countries we considered. We chose China because it has thelargest population. The analogous results are for all countries in the Appendices,but we make some comparative marks as we consider each output.

6.1.1 Demographic fits

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Figure 3: Model comparison of demography for China. The population in 5-year age categories is shown for men and women as bars; the dots representmodel output.

Figure 3 compares the population of China by 5-year age category and sex (bars)and the corresponding model values (dots). The model was initialised in 1990and run through to 2012. The overall agreement is good; although a disparity

21

6.1 The example of China 6 RESULTS

creeps in in the 10-14 year old age category. The reasons for this are unclear;however, the other countries display better fits.

6.1.2 MCMC fits

The autocorrelation function (ACF) plots for the first 20 walkers for China areshown in Figure 4. These are typical of other countries, with the ACF decayingwith a typical timescale of ∼ 100 iterations. This motivated the choice of 500steps for the MCMC chains.

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The MCMC chains for the 1000 walkers started from perturbed positionsaround the maximum a posteriori estimate found by a Nelder-Mead simplexalgorithm is shown in Figure 5. The acceptance probability for the samples afterburnin period (i.e. the last b500/3c) was 3.5%. This is below the acceptancerecommended in [19], and was typical of the acceptance rates for the othercountries. Parameters that do not appear to be much informed by the data (i.e.CDRd, CFRu and prot) do appear to have the ranges well-explored.

Whether the chains explore the case fatality rates is a useful diagnostic ofthe inference procedure. This is not the case for Indonesia, Nigeria, Pakistan,and the Philippines.

22

6.1 The example of China 6 RESULTS

Figure 5: MCMC chains for China. 1000 walkers; 500 steps; 10 runs used toaverage each evaluation of the log-likelihood.

Figure 6 shows information about the bivariate marginal distributions bypairs of parameters. It represents a scatter plot of the post-burnin samples(i.e. b500/3c × 1000 = 166, 000). The data have been hexagonally binned andcolor-mapped to indicate the 2d density, with loess trend lines added in red;the diagonal shows histograms of the 1d marginals. The upper-left section ofthe plot shows the correlations between pairs of parameters. It is clear thatmany pairs of parameters are highly correlated, the strongest 5 by correlationmagnitude being: CDR & dCDR (0.91); foi0 & beta (0.80); beta & eps (-0.75); foi0 & eps (-0.71); foi0 & Tu (-0.62). Other variables, notably CDRu,CDRd and prot exhibited little correlation with other variables. These pat-terns of correlation were consistent across countries, and normally representedeasily-undestood competition between model effects (e.g. a smaller endogenousactivation rate, eps, requiring a higher intitial force-of-infection, foi0, or trans-mission parameter, beta, to fit the data).

23

6.1 The example of China 6 RESULTS

Figure 6: Triangle plot of MCMC samples for China. This shows the data from166,000 joint samples from the MCMC chains after burning. The scatter plotsare hexagonally binned and a loess trend line (red) is overlain. The diagonalplots are frequency historgrams and give parameter names. Above the diagnoaldisplays the correlations between the relevant parameters’ samples.

6.1.3 Predictions

Predictions (i.e. estimates of disease burden measures) are produced by jointlysampling 100 parameter sets from the burned-in posterior samples, and usingthem to run the model. This accounts for the uncertainty of the parametervalues (conditioned on the data), the correlation between parameters inducedby the data, and the stochasticity of the process that generates the data. 95%uncertainty bounds are computed as the 2.5-th and 97.5-th percentile aroundthe median values.

Figure 7 compares the model timeseries for rates of incidence, mortalityand notification with the WHO estimates, and notification data. The WHOestimates are shown as fainter points joined by lines, with error bars displayingthe 95% uncertainty bounds. The model predictions for incidence, notificationsand mortality are shown as solid lines, with grey bands displaying the 95%

24

6.1 The example of China 6 RESULTS

uncertainty intervals derived as above. The notification data (and for othercountries, the VR data where they exist) are represented by points. The dashedline shows the median mortality multiplied by V R - the probability of a TBdeath appearing the the mortality statistics - and is therefore comparable withmortality rates from the vital register.

In the case of China, the recent incidence agrees closely with the WHOestimate, but the model mortality rate is substantially higher. Agreement isbetter in more recent times, which is a general feature.

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Figure 8 represents estimates of the total number of prevalent TB casesthrough time. The model estimates are shown as a solid line with grey banddisplaying the 95’% uncertainty range. The WHO estimates are shown as pointsand lines with error bars for 95% uncertainty ranges. For China, the agreementbetween the WHO and model estimates is unusually close.

25

6.1 The example of China 6 RESULTS

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Figure 8: Predicted prevalence for China. Solid line represents model medianand grey band the 95% uncertainty ranges. The fainter points and lines with95% uncertainty shown by error bars, represents the WHO estimates.

Figure 9 shows the model’s age-specific TB prevalence (red) in the yearswhen there were prevalence surveys in China (black). The red points and linesrepresent the median model predictions of age-specific prevalence, with the errorbars representing 95% uncertainty bounds. The horizontal dashed red line rep-resent the population-weighted mean TB prevalence across all ages ≥ 15 years.The black points and error bars represent the point estimate of TB prevalencein adults together with 95% confidence intervals. N.B. this is not relevant toany age-group on the x-axis.

26

6.2 Summary of estimates 6 RESULTS

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Figure 9: Predicted prevalence by age for China. Age-specific model predictionsand 95% uncertainty ranges in red; prevalence survey results for adults, togetherwith 95% confidence intervals in black. The dashed red line is the population-weighted average model prevalence in those older than 15 years.

6.2 Summary of estimates

Model estimates of incidence, mortality and prevalence for 2013 are shown inTable 6 & Table 7, respectively. These also give Bayesian 95% prediction inter-vals for each of these quantities. That is, the uncertainty intervals are based on100 samples from the posterior distribution, and include both the uncertaintyin the underlying model parameters after inference, and the stochasticity of themodel.

27

6.3 LTBI estimates 6 RESULTS

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 6: Summary table of estimates. Incidence, mortality and prevalence areshown, together with 95% credible intervals in brackets, for each country. For2013.

country incident cases number of deaths prevalent cases

Cambodia 32,884 (27,842 - 39,720) 7,101 (5,244 - 10,661) 71,366 (53,940 - 89,824)China 930,237 (817,994 - 1,081,358) 194,254 (139,014 - 250,752) 1,484,199 (1,252,007 - 1,765,340)Indonesia 306,392 (270,687 - 373,420) 42,043 (31,551 - 63,447) 497,104 (412,600 - 618,263)Myanmar 61,874 (47,020 - 84,139) 12,044 (7,325 - 19,339) 107,611 (77,349 - 164,193)Nigeria 158,604 (118,069 - 238,183) 52,551 (37,128 - 86,191) 513,074 (344,472 - 779,934)Pakistan 251,045 (163,631 - 320,801) 78,532 (40,112 - 103,963) 578,359 (349,697 - 764,975)Philippines 353,916 (310,138 - 430,682) 109,761 (79,517 - 144,297) 551,960 (488,029 - 651,400)Thailand 56,210 (47,946 - 65,779) 17,696 (12,222 - 23,350) 99,460 (81,323 - 123,270)Viet Nam 49,729 (45,004 - 56,351) 6,101 (4,551 - 9,797) 67,386 (46,021 - 88,641)

Table 7: Summary table of estimates. Incidence, mortality and prevalence areshown as absolute numbers (per year), together with 95% credible intervals inbrackets, for each country. For 2013.

6.3 LTBI estimates

The model produces estimates for the number of people lately infected withM.tb. These are typically less uncertain than quantities with smaller numbers(prevalence, incidence, mortality). Although latent infection was not a focus ofthis work, we give estimates of numbers infected (to the nearest thousand), andthe crude population prevalence as a percentage in Table 8.

28

6.4 Comparison with WHO outputs 6 RESULTS

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 8: LTBI estimates. Numbers of individuals latently infected with M.tbaccording to the model (to the nearest thousand), and the percentage of thepopulation that this represents. For 2013.

6.4 Comparison with WHO outputs

Figure 10 is a scatter plot of model estimate against WHO estimate for mor-tality, incidence and prevalence (all per capita). The error bars show the 95%uncertainty ranges for the model and for the WHO estimates. The dashed linerepresents equality between the two estimates. While there is correlation, theestimates of this model tend to be lower than the WHO estimates.

29

7 DISCUSSION

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Figure 10: Model & WHO estimates compared. Comparison between modeland WHO per capita estimates of incidence, mortality, and prevalence. 95%uncertainty ranges are plotted for both model and WHO estimates are shownas error bars. The dashed black line represents equality between model andWHO estimates.

7 Discussion

In this report we have minimally extended existing models of TB mortalityand disease duration to a transmission model including age and sex structure,and used a Bayesian approach to simultaneously calibrate this to data on no-tifications, TB mortality and data from prevalence surveys. The fitting andpredictions were done in an automated fashion for the 9 countries we consid-ered. The model predictions of TB incidence, prevalence and mortality werecorrelated with WHO estimates, but tended to be slightly lower.

A substantial limitation of the current framework is that it does not includeinformation on HIV and ART. Neglecting HIV infection is not a large approxi-mation for the countries we considered, with the exception of Nigeria. There, itis notable that while the TB prevalence estimates are consistent those of WHO,

30

7 DISCUSSION

the TB incidence estimates are much lower - a consequence of neglecting HIV,which raises the ratio between incidence and prevalence as the mean durationof prevalent cases decreases. It would, however, be relatively easy to modify theexisting framework to take information on HIV incidence, mortality and ARTinitiation rates provided by an accepted method of generating these estimates(e.g. Spectrum [20]). Additional parameters would be needed to describe theprogression of HIV and the relative incidence of TB in people living with HIV,with and without therapy, and the case-fatality ratios and durations of HIV-TBdisease. Submodels with these parameters are already used in WHO estimationprocesses related to HIV-TB.

Performing inference in an automated way for 9 countries, with 12 param-eters and a stochastic estimator of the likelihood proved challenging. Our ac-ceptance rates were low, and the sampling was clearly suboptimal for some ofthe countries. Other inference options should be explored, e.g. particle MCMCmethods [21] may allow for more efficient likelihood averaging and achieve betterconvergence.

Our choice of priors for parameters was relatively ad hoc, with some rangesfor parameters chosen for numerical convenience without exploring the effectsof changing these ranges. Several of these parameters have systematic reviewsavailable to inform them, and a more formal representation of the data foundby these reviews as distributions may be possible. However, we found the priorscontributed relatively little compared to likelihood within the ranges considered.

We used the same prior for the parameters V R and CDR (essentially, prob-abilities of TB death and case reporting) across all countries. WHO have amethod for constructing priors for V R based on assessments of a country’s cov-erage and correctness of VR data [1], and it may be more appropriate to usethese priors on a country by country basis. Because our CDR parameter isnot identical to the case detection ratio, and since we control case detection viathe value of CDR in 2013 and its trajectory going back to earlier times (viadCDR), it is not clear how one would incorporate prior information on the casedetection ratio from a given year. However, the likelihood could conceivably beextended to include measurement processes around capture-recapture studies ingiven years, and this source of quantitative information on the case detectionratio included in the general approach.

Our likelihood included numerator and denominator data from prevalencesurveys aggregated over age, as this was the data available. Prevalence surveystypically provide such data disaggregated by age and sex, and event though un-derpowered for calculation of age-stratified rates, it would likely be preferable toinclude the data in age-stratified form. Furthermore, some prevalence surveysdistinguish between TB cases on treatment and TB cases not on treatment. Ashas been suggested in [1] and elsewhere, this distinction should provide informa-tion on the case detection process. This framework could be slightly modifiedto include a treatment category to accept prevalence survey data further dis-aggregated by treatment status. Usually, prevalence surveys involve clusteredsampling and adjustment for population characteristics. If design effects arelarge, it is not obvious how best to include this information in a likelihood.

We considered only a single model structure. The changing prevalence : no-tification ratios in countries such as China motivated introduction of a singleparameter to allow the parameter controlling case detection (CDR) to changethrough calendar time; however, we did not formally evaluate whether introduc-

31

7 DISCUSSION

ing this extra parameter was justified by improvement in likelihood. Similarly,we used only a single parameter (foi0) to characterize our initial state, andthis led some implausible initial values in order to fit trends. Introducing moreparameters to control the initial state may have ameliorated this issue. Moregenerally, given that very different transmission model variants - or indeed sta-tistical models - could be used to model the influence of history on currentincidence, model comparison techniques should be considered to evaluate differ-ent approaches. Model averaging could also be considered.

Using simple transmission models to link incidence at different time points istypically less flexible than using a statistical model. It is interesting that in thefield of HIV estimation, Spectrum originally used a dynamical model to generateincidence [22], but later changed to a statistical model; presumably becausevarious real-world complexities not included in the simple dynamical model ledto more elaborate incidence curves than it was able to reproduce. In the case, ofTB there may well be population heterogeneities which are not included in ourtransmission model and which become more relevant in some contexts (e.g. lowprevalence settings). But there are also potential advantages of a transmissionapproach in this context. E.g. transmission models naturally make predictionsabout age patterns of infection and disease. Here, we took a simplified approachto children, but finer age-categories could be introduced for young children,and knowledge of the natural history in these age groups leveraged to produceestimates of TB incidence in these age groups that do not depend (or whichonly partially depend) on notifications in these age groups. In [6], this was doneessentially using prior information about the transmission parameter to relateprevalence to infection risk. Incorporating a more detailed paediatric naturalhistory as a submodel of this framework would allow conditioning these priorson more local data.

The fact that latent M.tb infection is included in this model means thattuberculin or interferon gamma release (IGRA) surveys are a further potentialsource of data that could be used to inform on TB burden. The reliabilityof LTBI measures for informing burden estimates has been questioned [22],however, data from representative LTBI surveys could easily be incorporated inthe likelihood along side other data in a way that accounted for an uncertaintest sensitivity and specificity. Simulation studies could be used to determinethe improvements in estimate precision due to inclusion of LTBI surveys, andpotentially to compare their cost-effectiveness (in this sense) with that of TBprevalence surveys.

The sex and age structure included in this model means that it could po-tentially be used to investigate alternative hypotheses around patterns sex andage patterns observed in notification and prevalence data. Data pertaining toLTBI infection status could further feed into this by helping to differentiate be-tween differences in exposure and infection, and differences in progression andreporting.

We introduced Beta-binomial submodels (with extra variability over bino-mial samples) for processes related to whether TB cases became notified, orwhether TB deaths were recorded in VR data. Applying exactly the same prob-ability to many individual trials seemed a mis-representation of reality for theseprocesses, and had particular consequences for the likelihood. Assuming bino-mial processes resulted in likelihoods that were entirely driven by notificationand mortality data, and which essentially ignored prevalence data. Introduc-

32

7 DISCUSSION

ing extra variability through multi-level modelling of these processes, resultedin broader, more balanced likelihoods, and the data from prevalence surveyswas able to influence the inference. Predictions were also more uncertain, otherthings being equal. We did not include the parameter, θ, controlling the de-gree of dispersion in the Beta-binomial in our inference as we were concerned itwould adversely effect its performance, but set θ = 10. Although it is unlikelythat the data can inform it, it would be preferable to consider incorporating θas a nuisance parameter, or at least to systematically assess sensitivity of anal-yses to changing its value. Informally, we found that the value of θ was notinfluential, once reduced below O(103) (not shown). This topic is important,as it influences the relative contribution of prevalence surveys vs notificationdata to estimates of TB burden, and these considerations will apply in someform to any model (transmission-based or statistical) that is informed by bothnotification and prevalence data.

Although introducing a transmission modelling element has introduced ad-ditional parameters, they are relatively few, are epidemiologically meaningful,and have been treated as uncertain quantities with their uncertainty propa-gated into all predictions. Studies could be carried out to improve the precisionof some of these quantities, and uncertainty analyses of within this frameworkcould be used to predict the improvement in the precision of model estimatesresulting from specific studies and prioritise their usefulness.

In conclusion, we believe bringing together the current assumptions aroundcase-fatality and duration of prevalence into a single framework that can makecontact with notification, prevalence and mortality data (as well as potentiallydata from capture-recapture studies) shows promise. A variety of statisticaland transmission models to relate the incidence at one time to other timesshould be formally assessed against each other. Transmission models have thepotential to leverage additional population data, including LTBI surveys, anduse information on natural history to extrapolate reasonably to groups wheredirect data may be particularly unreliable (e.g. children). Availability as apackage for the R statistical framework would facilitate installation and use,and encourage additional contributions and analyses.

Acknowledgments

The author would like to acknowledge useful conversations and correspondencewith Andrew Azman, Carel Pretorius and Philippe Glaziou in the preparationof this work.

33

7 DISCUSSION

Abbreviations

ART antiretroviral therapyCDR case detection ratioCFR case fatality ratioCOD cause of deathFOI force-of-infectionGTB Global tuberculosis programme, WHOHIV human immunodeficiency virusICD international classification of diseasesIGRA interferon gamma release assayLTBI latent tuberculosis infectionMAP maximum a posterioriMCMC Markov chain Monte CarloM.tb Mycobacterium tuberculosisODE ordinary differential equationTB tuberculosisVR vital registrationWHO World Health Organisation

34

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36

A COUNTRY DEMOGRAPHIC FITS

A Country demographic fits

The following pyramid plots show for each country the UN ESA populationstructure (green bars, right side for men [> 0]; red bars, left side [< 0] forwomen) in 5 year age groups, and the corresponding model prediction as ablack point. The model was initialised in 1990 and run to 2010 and the datafor years 1991 to 2010 inclusive are displayed.

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Figure 12: China

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Figure 13: Indonesia

40

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Figure 14: Myanmar

41

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Figure 15: Nigeria

42

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Figure 16: Pakistan

43

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Figure 17: Philippines

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Figure 18: Thailand

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Figure 19: Viet Nam

46

B COUNTRY TB FITS

B Country TB fits

Each of the subsequent plots follows the same four panel layout:Top left: A plot of the per capita incidence, mortality and notification rates

through time. Mortality and notification data are shown with circular points.Model mortality, notifications and estimates are shown as solid lines for medians,with grey bands representing 95% uncertainty intervals. A dashed line showsthe median true mortality scaled by the probability of TB deaths appearing inVR data, and is therefore comparable with mortality data. WHO mortality andincidence estimates (e inc 100k and e mort exc tbhiv 100k) are shown asfaint lines joining points, together with error bars representing 95% uncertaintyintervals.

Bottom left: shows the absolute prevalence through time. Median modelpredictions are shown as a solid line with a grey band representing 95% uncer-tainty intervals; WHO prevalence estimates (e prev num) are shown as faintlines joining points, together with error bars representing 95% uncertainty in-tervals.

Top right: is the same as the top left panel, but rates are absolute rather thanper capita. The WHO incidence and mortality estimates are labeled e inc numand e mort exc tbhiv num respectively.

Bottom right: consists of as many sub-panels as there are prevalence sur-veys, with their years in the panel title. The measured TB prevalence in adultsis shown as a black point with error bars representing the 95% confidence in-terval. Red points and lines show the median model predictions for per capitaTB prevalence in age group, with error bars representing the 95% uncertaintyinterval. The horizontal red dashed line shows the model population weightedmean TB prevalence, over all ages ≥ 15.

47

B COUNTRY TB FITS

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Figure 20: Cambodia

48

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Figure 21: China

49

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Figure 22: Indonesia

50

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Figure 23: Myanmar

51

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Figure 24: Nigeria

52

B COUNTRY TB FITS

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Figure 25: Pakistan

53

B COUNTRY TB FITS

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Figure 26: Philippines

54

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65+

age

TB prevalence per 100,000

Figure 27: Thailand

55

B COUNTRY TB FITS

●●

●●

●●

●●

●●

●●

●●

●●

●●

0

200

400

600

19901995

20002005

2010year

rate per 100,000 per year

variable

●●●●●●

e_inc_100k

e_mort_exc_tbhiv_100k

incidence

mortality

notifications

VR

●●

●●

●●

●●

●●

●●

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●●

●●

0

100,000

200,000

300,000

400,000

19901995

20002005

2010year

numbers per year

variable

●●●●●●

e_inc_num

e_mort_exc_tbhiv_num

incidence

mortality

notifications

VR

0

200,000

400,000

600,000

19901995

20002005

2010year

numbers

variable

e_prev_num

prevalence

●

●

●

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●

●

●

2007

0

100

200

300

400

0−15

15−25

25−35

35−45

45−55

55−65

65+

age

TB prevalence per 100,000

Figure 28: Viet Nam

56

C COUNTRY MCMC SAMPLE PAIR PLOTS

C Country MCMC sample pair plots

Each of the following graphs visualises the post-burnin joint sample from theMCMC routine for a country. The data presented are a sample of size 2 ×b500/3c × 1000, each point being based on 10 model runs. The sub-diagonalplots are hexagonally binned scatter plots, with lighter colours representingmore points, and red lines representing trend lines from a loess smoother. Thediagonal plots contain histograms estimating the marginal density of each pa-rameter, and the parameter names in red. The super-diagonal panels show thecorrelation between pairs of parameters corresponding to the reflection in thediagonal.

57

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 29: Cambodia

58

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 30: China

59

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 31: Indonesia

60

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 32: Myanmar

61

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 33: Nigeria

62

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 34: Pakistan

63

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 35: Philippines

64

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 36: Thailand

65

C COUNTRY MCMC SAMPLE PAIR PLOTS

Figure 37: Viet Nam

66

D COUNTRY MCMC CHAINS

D Country MCMC chains

The following plots show for the MCMC chains for each country. There are 1000chains started from a MAP-estimate, and run for 500 steps with the averageover 10 model runs used in each likelihood evaluation.

67

D COUNTRY MCMC CHAINS

Figure 38: Cambodia

68

D COUNTRY MCMC CHAINS

Figure 39: China

69

D COUNTRY MCMC CHAINS

Figure 40: Indonesia

70

D COUNTRY MCMC CHAINS

Figure 41: Myanmar

71

D COUNTRY MCMC CHAINS

Figure 42: Nigeria

72

D COUNTRY MCMC CHAINS

Figure 43: Pakistan

73

D COUNTRY MCMC CHAINS

Figure 44: Philippines

74

D COUNTRY MCMC CHAINS

Figure 45: Thailand

75

D COUNTRY MCMC CHAINS

Figure 46: Viet Nam

76