Quantifying progression and regression across the spectrum of pulmonary tuberculosis: a data synthesis study

Summary Background Prevalence surveys show a substantial burden of subclinical (asymptomatic but infectious) tuberculosis, from which individuals can progress, regress, or even persist in a chronic disease state. We aimed to quantify these pathways across the spectrum of tuberculosis disease. Methods We created a deterministic framework of untreated tuberculosis disease with progression and regression between three states of pulmonary tuberculosis disease: minimal (non-infectious), subclinical (asymptomatic but infectious), and clinical (symptomatic and infectious). We obtained data from a previous systematic review of prospective and retrospective studies that followed and recorded the disease state of individuals with tuberculosis in a cohort without treatment. These data were considered in a Bayesian framework, enabling quantitative estimation of tuberculosis disease pathways with rates of transition between states and 95% uncertainty intervals (UIs). Findings We included 22 studies with data from 5942 individuals in our analysis. Our model showed that after 5 years, 40% (95% UI 31·3–48·0) of individuals with prevalent subclinical disease at baseline recover and 18% (13·3–24·0) die from tuberculosis, with 14% (9·9–19·2) still having infectious disease, and the remainder with minimal disease at risk of re-progression. Over 5 years, 50% (40·0–59·1) of individuals with subclinical disease at baseline never develop symptoms. For those with clinical disease at baseline, 46% (38·3–52·2) die and 20% (15·2–25·8) recover from tuberculosis, with the remainder being in or transitioning between the three disease states after 5 years. We estimated the 10-year mortality of people with untreated prevalent infectious tuberculosis to be 37% (30·5–45·4). Interpretation For people with subclinical tuberculosis, classic clinical disease is neither an inevitable nor an irreversible outcome. As such, reliance on symptom-based screening means a large proportion of people with infectious disease might never be detected. Funding TB Modelling and Analysis Consortium and European Research Council.


S1
Disease model structure Our model structure focuses on the spectrum of disease, with infection and progression from infection already assumed to have happened. We split the progression into three states, as shown in figure 1: minimal, subclinical, and clinical disease.
Minimal disease is the earliest stage of disease from infection, being non-infectious, but with pathological changes to the lung visible on, originally, chest x-ray but also other forms of chest imaging such as computed tomography (CT). Along with being the first stage of disease after infection, minimal is the final stage before recovery, with regression back to minimal possible within the spectrum framework, and then natural recovery from disease possible.
In the forward progression, the stage after minimal is subclinical. This is an infectious disease state, but without sufficient symptoms to present for screening. In other words, this is an infectious but effectively asymptomatic state. Within the spectrum, there is progression to clinical disease (i.e., development of symptoms) and regression to minimal disease (i.e., becoming non-infectious) from subclinical disease.
Clinical disease, symptomatic and infectious, is the final disease state. As with the other two states, there are two transition possibilities out of clinical disease, regression to subclinical disease (i.e., resolution of symptoms but remaining infectious), and death from TB.
In the model structure, there is possibility to both progress and regress, but in visualising the model, such as figure 1, arrows that point right indicate the disease progressing to a more severe state, and arrows that point left indicate the disease regressing to a less severe state.

S1.1 Alternative structures
We opted to parameterise a three state, linear model structure. There are data to support this choice, and as a sense of progression of disease, it matches with previous conceptualisations. [1][2][3][4][5] However, the data to support this structure could also inform other, more complex, model structures. For example, minimal disease could be split into two or more different states based on symptoms or perceived activity of lesions seen on radiography. However more disease states and transitions dilute the available data which, despite the systematic review that underpins it, is already limited. It is always possible to make a model structure more complex but here we looked for the most useful balance between simplicity, representing the data available, and providing answers to questions being asked. We feel the three state, linear model structure used here provides this balance.

S1.2 Model Assumptions
No model can perfectly describe a system, and as such, assumptions are required to best represent the system for the purposes of the model.

Assumption
Justification Three state disease model Discussed in section S1.1 Death only from clinical disease We assumed it was unlikely that a person would be entirely healthy with no symptoms before dying from the effects of their TB disease. Recovery only from minimal disease Recovery from subclinical disease and clinical disease is not assumed to be instantaneous and instead a regression through less and less severe disease, until reaching minimal disease, at which point recovery is possible No relapse from recovery without reinfection We did not find data on any relapses from natural recovery and would not have been able to separate relapse from reinfection. Therefore, these estimates are based on cohorts until initial recovery or death, discounting any relapse or reinfection. Data where symptoms are stated only at the start, it is assumed they persist and are the same at the end This assumes that if symptoms were important enough to mention at the start of follow-up, they would also be important enough to mention at the end of follow-up if they had changed, as discussions focused on the change in state. We have explored the impact of this assumption in a sensitivity analysis. See section S10.1.4 for more information.

S3 Symptomatic Minimal
Minimal disease was classified as when an individual had radiological changes attributable to TB but negative bacteriology, regardless of symptoms. Although progressing from a potentially symptomatic bacteriologically negative state to an asymptomatic bacteriologically positive seems unlikely, a number of sources suggested that there was no need to consider an alternative progression for symptomatic minimal.
Firstly, there is insufficient data to show an obvious split in progression between symptomatic and asymptomatic bacteriologically negative individuals (see figure 2), the diagnostic standards from the 1940s placed significantly less weight on symptoms if they were not accompanied by a positive sputum, and the prevalence survey in Cambodia in 2002 found that symptoms in culture negative individuals were not associated with future bacteriological positivity. 12,13 We also know that TB symptoms are highly non-specific, and so there is no guarantee that symptoms occurring whilst an individual has minimal disease are actually caused by TB and not by something else.
Therefore, we have considered all bacteriologically negative individuals to be minimal, regardless of symptom status. As explained in the main text, there were two study types included. In table 1 we report the data types for each line of data, in column "Follow-up method". There were 38 data points reported as cumulative follow-up, and 16 reported as single follow-up.

S4.2 Timings
For single follow-up data, if an average follow-up time was given, that is the time used. Otherwise, if a minimum and maximum follow-up time was given, the times have been summed and halved to give the follow-up time used. For cumulative follow-up data, the maximum follow-up time given was used. The times in table 1, column "Months of followup" reflect these choices.

S4.3 Included data
All data that was included in the final data fit is in table 1.
These cohorts were followed for intervals between 1923 and 2004, with studies conducted in North America (6), Europe (7), Asia (7), and one each from South America and Africa. In total there were 5 data points from minimal to subclinical, 14 data points from minimal to clinical, 15 data points from clinical to minimal, 18 data points from minimal to infectious, and 2 data points from infectious to minimal.

S4.4 Excluded data
As can be seen in table 2, some of the data that was originally extracted for the wider review was not eligible for this work. In total 6 cohorts were excluded, from 10 different studies. Most (5) rows of data that were excluded, had an initial state with x-ray negative, and one study was observing a cohort who, although already x-ray positive, were not expected to progress to infectious TB disease. Others were excluded for too much uncertainty within the start and end states, or changes only within states, such as change in x-ray severity, but no change in bacteriological or symptom status. These reasonings are laid out in table 2

S5 Fitting process
The equations to define the model system are: where: • , , and are the states for minimal, subclinical, and clinical respectively • _ , _ , _ , and _ are transitions between the states, where the first letter is the start state and the second letter is the end state • is recovery from minimal disease • is death from clinical disease (there is no other death included in the model) As the data described how a cohort changed over time, and described only one outcome, the fitting process used a model system for each of the transitions and data types, totalling 16 different versions of the model system. Full code is available on GitHub.
We used uniform priors for the four estimated parameters, all with a range of 0 to 12, where 12 would be equivalent to changing state once a month. During the fitting process, potential parameters are trialled within this range. figure 3 shows the different parameter values that were accepted over the 10,000 iterations of the model fit.
These accepted parameters in turn, inform figure 4, which shows the correlation between two parameters. It also shows the overall distribution of the accepted parameters. We see a strong positive correlation between the parameters that control transition between minimal and subclinical; as one transition increases, the also has to increase to prevent there being excess people in one state and too few in another. We also see this with the parameters that control transition between subclinical and clinical.
The rest of the pairings have slightly weaker, negative correlations. This is clearest with the subclinical to minimal and subclinical to clinical pairing, as if one increases, the other has to decrease to make sure that there are still sufficient individuals in subclinical to fit the data.

S5.1 Weighting
When calculating the likelihood, larger studies were weighted by the original cohort size to reflect the increased confidence that such studies provide. Thus, larger cohorts have a heavier weighting and constrain the model more.
In order to prevent a single study with multiple observations being over-represented in the fit of a transition, we down-weighted both the sample size and the number of people transitioning by the number of repeated observations. This maintains the observed proportion to transition whilst reducing confidence and thus importance given to each individual data point within the study. This is shown in table 1, with the number of repeats and the re-calculated effective cohort size and number transitioning. This ensures that the proportion remains the same, but less weight is given to each individual data point.

S5.2 Duration of disease
Tiemersma et al use an assumption of exponential duration of disease to quote an "average" duration of disease as three years and calculates this from the incident cases occurring between each survey. 6,33 They state that a of 0.3 fits the cumulative distribution for the number of observed cases and thus 1 = 3.33 years is the average duration given by the data, but that missed cases mean that is likely an over-estimate and so 3 years is the average duration of disease . What they are then quoting as average is the mean, so the median duration can be given by (2) . Taking = 0.33 so that mean duration is 3, gives a median duration of 2.1 years.
The numbers quoted are incident cases between each survey, so we can use the duration of disease looking at a cohort that starts in subclinical disease. Therefore, we need to find that at 2 years, 50% of the cohort that started in subclinical, is either still subclinical or is clinical.
To use this as a fitting point, we want to look at time 2 years, and see how close to 50% the number of people in subclinical + clinical from the subclinical cohort is.
An exponential function can be written as = ( ) where is prevalence and is time. We know that at = 0, = 1 so = 1 and the equation simplifies to = .
We want to look at 2 years, find the prevalence, and then from that, calculate the time at which the prevalence would be 0.5. So, to calculate b, we set = 0.5 and = .
So, the full equation, at the fitting point of = 2 and rearranging for gives: This means, from fitting at a single time point, we can estimate the median duration of disease using the assumption of an exponential distribution of duration.

S5.3 Prevalence ratios
Prevalence surveys have found that approximately 50% of people with bacteriologically positive disease do not report experiencing symptoms,. 52 Whilst harder to ascertain, estimates of the proportion of people with bacteriologically negative disease range from two to three times the number of people with infectious disease. Both these have been included in the model fit as data points, calculated from the steady states of the system equations.
The subclinical to clinical ratio is calculated: For simplicity, the parameters representing transitions have been simplified to single letters: and to create a non-zero steady state, an unknown is the flow of new disease.
The system of equations then becomes: Assuming a steady state and using the equation for ̇ we can get an equation for M in terms of S and C: Substituting C in terms of S: Then to calculate + : The prior value for the ratio of subclinical to clinical disease is taken from a systematic review of prevalence surveys that found approximately 50% of people with bacteriologically positive disease did not screen positive on symptoms. 13,52 This is from populations with treatment, however we do not have an equivalent source in absence of treatment. However, to compensate for this, we allowed wide priors for the model to settle on the best value given the data. For the purposes of fitting, we assume this prior value is what is found in a steady state situation.
There is not a similar review to inform the ratio between minimal and infectious. We have used results from a post analysis of the 2016 Kenyan prevalence survey to provide an estimate for the prior, that is used with the same assumptions as the subclinical to clinical ratio. 53 Whilst this cannot be perfect, it again provides a prior range for the calibration to consider. This was paired with a wide uncertainty for the model to consider. In the survey, 10% of those screened with CXR were considered to have TB abnormalities, but on expert review, only 60% were truly considered abnormal. On Xpert testing, 90% of all originally screened as TB were negative, with 10% positive (which would give a 9:1 ratio). However, taking into account that only 60% were considered TB on expert review, that brings the ratio down to 5:1. Our own priors felt that this number was still too high, so we halved the ratio (2.5:1). This estimate is also consistent with a repeated prevalence survey in Cambodia. 13

S5.4 True Minimals
In the systematic review preceding this work, x-ray positive, bacteriologically negative disease was analysed based on reporting of the presumed activity (whether the x-rays were classified as active or inactive). For modelling purposes, there was insufficient data to split groups starting in minimal disease beyond the symptoms at the end and the follow-up collection type, so the distinction between active and inactive x-rays has not been included. However, determining who truly has TB when the only test is an x-ray is difficult. To take this into account, we have used tuberculin skin test (TST) as a proxy for determining whether a positive x-ray is a result of TB infection progressing to disease, and so we can estimate the proportion of people classed as minimal that are actually minimal. These papers were not selected systematically but span a range of time and location. Table 3 shows each of the studies, the number of people who were found to be x-ray positive in the study, and then the number of those who were also TST negative. Applying a meta-analysis to this, we find that the fixed effects result is 20% of x-ray positives are TST negative and so unlikely to be caused by TB, and the random effects suggests 28%, as can be seen in figure 5. Therefore, throughout this paper we have assumed that 25% of all x-ray positives are non-TB, and thus reduced every cohort that starts in minimal accordingly. To check this assumption, we have tested this with 0% and 40% of x-ray positives being non-TB, as can be seen in table 6.

S5.5 Likelihood Calculation
The likelihood function can be described as follows: Where: • is the study identifier • is the timepoint of the reported observation • , is the observed number of transitions for study k at time point t • is the number of reported observations included for the study • is the cohort size for study k • is the model-predicted proportion transitioned • is the ratio between disease states (with j identifying which disease ratio) • is the model-predicted ratio between disease states • is the expected duration of infectious disease • is the model-predicted duration of infectious disease S6 Minimal disease

S8 Duration of symptoms
Duration of symptoms can be split into three categories: duration before death, duration before regression to subclinical, and when applicable, duration before treatment. These three have not shown a significant difference in our analysis, but of note is the highly skewed distribution that we observe. Of those who become clinical, the minimum time spent clinical is one month (as that is the time step in the model), but a small proportion of individuals have persistent symptoms for a long time

S9 Additional results
Here we consider the median duration of disease and the proportion of people in each state at a given time. We can see that including treatment decreases the duration of disease and decreases the proportion clinical. When including minimal disease in the duration, we see that duration increases significantly, showing the importance of considering all those who are at risk of progressing to infectious disease. In the following figures, the top row is the number of people in all disease states (minimal, subclinical, and clinical) over time, and the bottom row is the number of people in infectious disease states (subclinical and clinical).

S10 Sensitivity Analyses
We have run sensitivity analyses on different areas of the main analysis. For the purposes of comparison, we have included the median parameter estimates, and key outputs; median duration of disease, percentage of people clinical, undulating, subclinical, and minimal after 5 years, and the number of people who have died from TB over 10 years.

S10.1.1 Bootstrap studies
We ran the fit process removing the data from one study at a time. Table 4 shows the key outputs, showing that no one study is driving the fit, with similar results when each study is removed. In the main analysis, where studies mentioned symptoms at the start but not at follow-up, we assumed that the initial symptom status persisted. This allowed study end-points to be classified either as subclinical or clinical where they would otherwise be classified as infectious. Here we remove that assumption and treat all studies where we implemented the assumption instead as minimal to infectious. The method of parameter choice for the simulation was one of three. In the main analysis, each step of the model, for each individual, the relevant parameters were chosen randomly from the posterior distribution. For the other two alternatives, we randomly sampled the parameters at the start of the simulation for each individual and fixed them for the whole run, and the other used the median parameters for each person. Table 8 shows that there is very little difference between either method overall.

S10.2.2 Treatment
Treatment was added to the model to simulate a case detection rate for a care pathway initiated by self-reported symptoms. When considered in the main analysis, the case detection rate was implemented at 70%. In table 9 we compare the difference between case detection rates at 50%, 70%, and 90%.

S10.2.3 Trajectories
The trajectories are based on two variables, the proportion of time in a single state, and the number of state changes both over the previous 12 months. The main analysis defines transitioning as less than nine months in a single state or 3 or more changes in state. In table 10 we compare the definition of transitioning as less than 8 months or less than 10 months, whilst keeping the number of state changes fixed at 3 or more. In table 11 we compare the definition of transitioning with less than 9 months fixed, and the state changes as either 2 or more, or 4 or more.