Understanding the incremental value of novel diagnostic tests for tuberculosis

Transcription

1 Understanding the incremental value of novel diagnostic tests for tuberculosis Nimalan Arinaminpathy & David Dowdy Supplementary Information Supplementary Methods Details of the transmission model We use a simple transmission model, represented by a system of coupled ordinary differential equations, to capture the different stages of infection and careseeking, as well as relative infectiousness and durations in each of these stages. We denote: U for the proportion of the population that is uninfected; L 0 for the proportion of the population that harbors latent infection; L 1 for latent infections with imminent breakdown to active disease (on average within the next two years); for cases of active disease prior to care-seeking; for cases of active disease that have started care-seeking, but with mild, nonspecific symptoms, and I 2 for cases of active disease, seeking care with more prolonged and TB-specific symptoms. Together,, I 2, and I 3 represent the different stages of diagnosis explored in the main text. Further, we allow for a segment of the population having relatively low access to care, labeling this segment with a prime (for example, writing U for uninfected individuals in the general population and U for corresponding individuals with poor access ). Representing time derivatives with a dot, governing equations are thus: 1

2 U = λu + ( 1 p low )b L 0 = ( 1 p fast )λu ( r 0 + µ )L 0 + d 0 + d 1 + d 2 I 2 L 1 = p fast λu + r 0 L 0 ( r 1 + µ )L 1 = r 1 L 1 ( γ 0 + c + d 0 + µ TB ) = γ 0 ( γ 1 + c + d 1 + µ TB ) I 2 = γ 1 ( c + d 2 + µ TB )I 2 ( ) + c( + + I 2 ) U = λ U + p low b L 0 = ( 1 p fast )λ U ( r 0 + µ ) L 0 + d 0 + d 1 + d 2 I 2 L 1 = p fast λ U + r 0 L 0 r 1 + µ ( ) L 1 ( ) ( d 1 + µ TB ) ( d 2 + µ TB ) I 2 I 0 = r 1 L 1 γ 0 + c + d 0 + µ TB I 1 = γ 0 I 2 = γ 1 γ 1 + c + c + ( ) + c ( + + I 2 ) Here, the risk of infection λ is given by: λ = β I + I + β I + I + β (I + I ). The terms d, d, d are the rates of diagnosis and cure; at baseline (representing the situation with no novel diagnostic tests), we assume that essentially all TB diagnosis happens at stage I 2, thus taking d = d = d = d = 0, and d = kd, where k is an access disparity parameter, representing the degree to which low-access to care is diminished with respect to regular access: for the sake of illustration, we assume in the base case that k = 0.5. Otherwise, we assume natural history parameters (e.g., development of active disease, spontaneous resolution) to be the same for both access groups. The birth rate b is chosen to balance the overall death rate in the population, thus allowing an equilibrium population. Remaining parameter definitions are given in Supplementary Table S1: for the purpose of illustration, assumptions were made for parameters such as p low, the fraction of the population that has poor access to care. We present parameter values in terms of mean times spent in each compartment. An assumption implicit in any compartmental formulation such as the one presented here, is that of exponentially distributed durations in each compartment; previous work has highlighted the potential effects of alternative distributions. 1 For example, delay times in patient care-seeking and diagnosis of TB are unlikely to be exponentially distributed. 2, 3 This represents an important limitation of our model, which is not intended to predict the incremental value of any specific test in any specific setting. Nonetheless, the present approach offers a simple and transparent framework for capturing the essential structure being explored in this work. 2

3 To implement the different scenarios in the main text (late/early diagnostic gaps, and high access disparity scenarios), with the relative sizes of β, β, β being specified under different scenarios, we constrain the system such that only one of these parameters (say β ) remains free to be determined. Under each scenario, we calibrate β to yield an annual incidence of 150 per 100,000, consistent with estimates for the global incidence of TB in (noting that at equilibrium, the annual incidence of active disease is given by r L + r L - i.e. a sum of influx terms into, integrated over a unit time interval). Taking the state values at equilibrium, we then calculate the following quantities. Transmission load As described in the main text, the transmission load associated with any given care-seeking stage (, or I 2 ) is the annual number of infections arising from all TB cases in that stage. Here and in what follows, it is helpful first to define the average number ρ of infections per stage, and per case. At equilibrium, this is given by a ratio of the associated transmissibility (β terms) and durations in each stage, multiplied by the number of uninfected individuals at equilibrium. Specifically, the values associated with, and I 2 are, respectively: β 0 ρ 0 = γ 0 + c + d 0 + µ TB β 1 ρ 1 = γ 1 + c + d 1 + µ TB β 2 ρ 2 = c + d 2 + µ TB ( U +U ') ( U +U ') ( U +U ') and likewise for those associated with I 0, I 1 and I 2. At the population level, the transmission loads TL associated with each stage are simply given by: TL 0 = ρ 0, TL 1 = ρ 1, TL 2 = ρ 2 I 2 As illustrated in Fig. 3, to find the numbers of infections averted by diagnosing a single case in a given stage, we incorporate not only the transmission load in that stage, but in all subsequent stages as well. Estimating the incremental value We aim to compare different types of new diagnostic test, assuming these to be implemented in addition to the current standard of care (smear plus clinical judgment and any other ancillary tests [e.g., chest X-ray] that might currently be performed, at stage I 2 ). In particular, we calculate the incremental number of transmission events that would be averted by an algorithm containing a new diagnostic test, divided by the incremental 3

4 resource outlay that would be required to add the novel test to the existing standard-of-care algorithm. Here, resources could be interpreted, for example, as the number of individuals that must be incrementally tested, or (more realistically) that number multiplied by a resource factor that accounts for the cost, personnel and effort involved in screening each person. We assume that the standard of care involves diagnosis of individuals in the stage I 2 after an average duration of one month in this stage. As described in the text, we calculate the maximum transmission load avertible by augmenting this standard of care with a novel diagnostic test. In other words, we assume that a novel diagnostic test could be applied sufficiently early in the disease course, and with perfect sensitivity, as to avert the entire transmission load arising from a given patient stage. Additional work to evaluate actual tests being developed would then need to assess the proportion of this maximum impact that would actually be achievable by a given test in reality, and would reduce the estimate of the averted transmission proportionally. For a new diagnostic test applied at a given stage i in the general population, we therefore assume that an incremental amount of resource ΔR(i) is needed to avert this maximum transmission load from a single individual. Correspondingly, we denote this as ΔR (i) in the poor-access population. Thus, each case diagnosed in the general population represents a maximum potential impact of an additional ΔT i = ρ(i) cases averted, with ρ i as defined as above, and correspondingly for the poor-access population. To find the maximum incremental value, we divide the maximum incremental impact by the incremental resource requirement. Thus in equation (1) in the main text, for an intervention targeted at stage i, T 1 T 0 is simply ΔT(i), and R 0 R 1 is the same as ΔR i. We additionally benchmark our analysis against a value of 1 for the smear-replacement test as implemented in the general population (labeled with subscript 2 in the main text). Supplementary Table S2 shows the numbers used for ΔR(i), under two different examples: one where resource requirements are driven purely by the number needed to screen to diagnose one incremental case, and the second a more cost-like example where we additionally account for the different amount of effort that may be involved per patient in each care-seeking stage. 4

5 Supplementary Table S1: Parameters assumed in a simplified model of TB diagnosis and transmission Parameter Symbol Value Comments/References Rate of progression, L0 to L1 r Rate of breakdown, L1 to I0 r 0.5 Assume breakdown on average in 2 years Proportion of infections being 'fast' progressors p #$ Proportion of cases having poor access to care Progression rates between careseeking stages p # 0.1 Illustrative assumption From to γ 2 Corresponding to an assumed 6- month duration From to I 2 γ 4 Corresponding to an assumed 3- month duration Existing rate of diagnosis Regular access d 12 Assuming 1 month duration in I2 before diagnosis Poor access d 12k Taking k = 0.5 for illustration Per- capita mortality rate Non- TB μ 1/66 To yield TB- free life expectancy of 66 years TB cases μ 1/6 Assuming 50% mortality in 3 years for untreated TB (alongside self cure rate) 7 Per- capita self- cure rate c 1/6 Assuming 50% self- cure in 3 years for untreated TB (alongside mortality rate) 7 5

6 Supplementary Table S2: Components used in estimating ΔR(i), the resource required to avert the transmission load from a single case. Measure Number needed to screen to diagnose one case (NNS) Effort needed to test one suspect (ET) Biomarker at stage L 1 (GP) Cough triage at stage (GP) Smear substitute at stage I 2 (GP) Point- of- care at stage I 2 (PA) 1/L GP denotes the general population, and PA denotes the poor-access population. In the simplest illustrative case (Figure 4), we assume that resource requirements are driven only by the number of people needed to test to identify an additional case of active TB, thus taking ΔR(i) = NNS(i). In a more realistic example (Figure 5), we further allow for a costlike resource measure, allowing different levels of effort or cost in reaching patients from different stages. In this case, using the values in the table, we take ΔR i = NNS i ET(i). 6

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