The integrated approach of learning tuberculosis transmission within and outside households via random directed graph models

Tuberculosis transmission, Household contact study, Markov Chain Monte Carlo methods, Random directed graph models, Epidemic models, Bayesian inference . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. 2 1 Introduction Tuberculosis (TB) is one of the oldest infectious diseases, yet it remains a major threat to global health.1,2 One obstacle for global TB control is limited knowledge about TB transmission dynamics, which is complicated by long latency, imperfect TB diagnostics as well as the uncertainty about the determinants of progression to active TB disease.3,4 Household contact studies, which enroll newly diagnosed pulmonary TB patients (index cases) and their household contacts, are often used to study TB transmission.5-7 Inferences based on household contact studies seldom consider extra-household transmission,8,9 which contradicts the findings of a growing body of literature that suggests extra-household transmission is even more substantial than household transmission in high burden settings. 10,11 Consequently, such inferences may bias estimates of the general transmission dynamics of TB as well as the nature and strength of the associations between risk factors for latent TB infection (LTBI). 12 For this reason, models that allow concurrent estimation of household and extra-household transmission are important to consider. Few models exist to concurrently account for household and extra-household transmission of disease. The unified probability model (UPM) has been applied to household contact studies of TB for concurrent estimation of household and extra-household transmission of TB.4 Built on a mixed effects modeling structure, it estimates the association between LTBI and exposed host, infector, and environmental risk factors as fixed effects while controlling for household random effects. The UPM additionally has a parameter that characterizes the probability of extra- household transmission so that it can separate the probability of extra-household transmission from the probability of household transmission. Nevertheless, the UPM has three major limitations. First, the extra-household transmission probability may still be non-identifiable due . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 3 to the fact that it is a part of the intercept in the model, though the non-identifiability issue could be ameliorated by controlling for the most powerful predictors of household transmission. Second, the UPM does not easily model risk factors associated with extra-household transmission. Third, the UPM considers the index cases as the only source of household transmission, which is likely not realistic for households with other active TB diseased cases (called secondary TB diseased cases). Another approach for modeling household and extra-household sources of infection is the random directed graph model (RDM). 13 RDMs have proven to be informative in household contact studies for influenza research.14,15 Using Markov Chain Monte Carlo (MCMC), RDM concurrently estimates extra-household and household transmission by modeling and imputing the transmission paths outside and within households. RDMs allow one to incorporate factors associated with extra-household transmission and allow all infected cases to be sources of household transmission. This model has typically been implemented in settings where community controls, i.e., households without an index case, are enrolled, creating important data to inform extra-household transmission paths along with serological data. Household contact studies for TB do not typically enroll community controls and serology data is not relevant for TB. Given community controls play a vital role in distinguishing extra- household transmission from household transmission, RDMs may not be identifiable for household contact studies for TB without modification. Furthermore, RDMs have been developed in the context of influenza which has distinct transmission patterns from TB, so a modified RDM is needed. We have two aims in this paper. First, we develop an RDM that is modified for TB household contact studies. The central task in this aim is to formalize the rules of drawing random directed . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 4 graphs for TB household contact studies. The second aim is to explore strategies to solve the non-identifiability issue for RDMs and TB household contact studies. We present three possible strategies: 1) include extremely predictive covariates in the models of extra-household and household transmission; 2) enroll community controls; or 3) use random weights in the MCMC algorithm where the weights serve as prior information for updating transmission paths outside or within a household. We describe how different weighting schemes can be evaluated based on deviance information criterion (DIC). Our paper is structured as follows: In section 2, we describe our motivating data example, a Brazilian household contact study of TB, as well as the traditional RDM in influenza research. In section 3, we detail a modified RDM for TB household contact studies as well as the weighted MCMC approach for estimation. In section 4, we present the results of two simulation studies as well as the Brazilian household contact study. Section 5 concludes the paper with a discussion of our findings. 2 Background 2.1 Brazilian household contact study Led by the “US-Brazil Research Collaboration on Strain Variation in TB” study group, a research initiative under the US National Institute of Health and a part of the International Collaboration in Infectious Diseases Research program, the Brazilian household contact study targeted the population living in the greater metropolitan area of Vitória, Brazil. This study enrolled 160 households, each with an index case who fulfilled the following criteria: 1) age ≥ 18 years with coughing ≥ 3 weeks; 2) a sputum test result of acid-fast bacilli (AFB) ≥ 2+ with subsequent M. tuberculosis growth in culture; 3) has no less than three household contacts. This . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 5 study excluded index TB cases who had been infected with HIV (or refused to be tested for HIV) or treated for TB disease in the past. A household contact in this study was defined to be an individual of any age that had close contact with the index case for more than 3 months. Demographic and environmental information on all 160 index cases and their 934 household contacts has been collected and LTBI status was determined by tuberculin skin test (TST) reading(s) conducted by staff trained by the National TB Program. The household contacts who had TST readings ≥ 10 mm by 8 weeks after enrollment of the index case were considered LTBI according to TST study criteria. Fifty-five household contacts had active TB disease in addition to the initial index cases during the six-years of follow-up visits. The dates of diagnoses for cases who had active TB disease were recorded. In total, the study enrolled 160 households in four municipalities and had 215 cases that had active TB disease and 545 LTBI cases. The study protocol has been described in detail elsewhere. 6,16 2.2 Random directed graph model Developed in influenza transmission research, the random directed graph model (RDM) considers the unobserved transmission paths as missing data and imputes them using MCMC.14 Specifically, for each household one can create a random directed graph where its vertices represent the household members and edges represent the possible transmission paths. The directed graph should be consistent with the observed data and follow rules consistent with disease transmission. These rules are: 1) Infected household members, or those with unknown status, can send and receive edge(s) from the other infected household members and/or the community. 2) Infected household members receive at least one edge. 3) Non-infected household members do not receive or send edges. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 6 In general, the posterior distribution for the RDM is given by: /g2172/g4666/g2163, /g2242| /g2207 /g4667/g1503 /g2172 /g4666 /g2207 |/g2163/g4667/g2172/g4666/g2163|/g2242/g4667/g2172/g4666/g2242/g4667 (1) where G denotes the directed graph of the transmission paths for each household, θ represents the transmission model parameters and y is the observed data. P /g4666 y | G /g4667 equals 1 if the underlying directed graph is consistent with the observed data and 0 otherwise. P /g4666 G | θ /g4667 is the likelihood of the directed graph given the transmission model and it has the following generic form: /g2172 /g4666 /g2163 | /g2242 /g4667 /g3404/g4688 /g3537 /g3537 /g4670 /g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667/g4671 /g2204 /g2185/g2191/g4670 /g2778/g33i8/g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667/g4671 /g2778/g2879/g2204 /g2185/g2191 /g2196 /g2185 /g2191/g2880/g2778 /g2159 /g2185/g2880/g2778 /g468i ·/g4688 /g3537/g3537/g3537 /g3427 /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3431 /g2204 /g2190/g2191/g2192 /g3427/g2778 /g33i8 /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3431 /g2778/g2879/g2204 /g2190/g2191/g2192 /g2192 /g1488/g2960 /g2190 /g2196 /g2190 /g2191/g2880/g2778 /g2164 /g2190/g2880/g2778 /g468i. (2) For extra-household transmission, /g1829 refers to the number of communities enrolled in household contact study and there are /g1866 /g3030 participants in the community /g1855 (/g1855 = 1, …, /g1829 ). The probability that an infectee /g1861 is infected by someone outside the household is /g1868 /g3030/g3036/g4666 /g1876 /g3036,/g1876 /g3030/g4667 , which depends on the features of individual /g1861 , /g1876 /g3036 and the features of the community of residence, /g1876 /g3030. For a directed graph, /g1874 /g3030/g3036 equals 1 if an edge from the community /g1855 to the infectee /g1861 is present or 0 if it is absent. The model for extra-household transmission is /g21i8 /g2185/g2191 /g4666/g2206 /g2191 ,/g2206 /g2185 /g4667/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4666 /g33i8 /g2245 /g2201 /g4666 /g2206 /g2191 /g4667 · /g2245 /g2185 /g4666/g2206 /g2185 /g4667/g4667 (3) where /g2019 /g3046/g4666 /g1876 /g3036/g4667 and /g2019 /g3030/g4666 /g1876 /g3030/g4667 characterize the susceptibility of the infectee /g1861 and the general risk of extra-household transmission in the community /g1855 respectively. For household transmission, /g1834 refers to the number of enrolled households and /g1866 /g3035 refers to the number of participants in the household /g1860 (/g1860 = 1, …, /g1834 ). The group of infectious participants in the household /g1860 is denoted as Ω /g3035. The probability that an infectee /g1861 is infected by an infector /g1862 in . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 7 the household /g1860 is /g1868 /g3035/g3036/g3037/g3435 /g1876 /g3035,/g1876 /g3036,/g1876 /g3037/g3439 , which is determined by the features of the household /g1860 , the infectee /g1861 as well as the infector /g1862 . For a directed graph, /g1874 /g3035/g3036/g3037 equals 1 if an edge from the infector /g1862 to the infectee /g1861 in the household /g1860 is present or 0 if it is absent. The model for household transmission is /g21i8 /g2190/g2191/g2192 /g4666/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g4667/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4666 /g33i8 /g2245 /g2190 /g4666 /g2206 /g2190 /g4667 · /g2245 /g2201 /g4666/g2206 /g2191 /g4667· /g2245 /g2165 /g4666/g2206 /g2192 /g4667/g4667 (4) where /g2019 /g3035/g4666 /g1876 /g3035/g4667 , /g2019 /g3046/g4666 /g1876 /g3036/g4667 and /g2019 /g3010/g3435 /g1876 /g3037/g3439 characterize the general risk of household transmission in the household /g1860 , the susceptibility of the infectee /g1861 and the infectivity of the infector /g1862 respectively. 3 Methods 3.1 Random directed graph for TB transmission There are important differences between the transmission dynamics of TB and influenza that impact the construction of the RDM. Unlike influenza, a large portion of individuals with latent TB infection remain in that state indefinitely and are not infectious. Those who progress to disease do so on a time scale of months or even years, compared to the rapid progression of influenza within days or hours of infection. It is generally accepted that LTBI is a noninfectious state, which may not be the case in influenza. The impact of this on the model is that in influenza, all infected cases are considered infectious, while in TB, only the LTBI cases who progress to active TB disease are infectious. To formalize our TB transmission model, we classify household contact study participants into three categories: 1): the active TB diseased (ATB) cases as the participants who are either the index cases or secondary TB diseased cases in household contact study; 2): the latent TB infection (LTBI) cases who have a TST reading ≥ 10 mm recorded by household contact study; and 3): the no TB infection (NTBI) cases whose TST reading(s) is . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 8 constantly below 10 mm. The outcome of our model is infection status which is 1 for the ATB and LTBI cases and 0 for the NTBI cases. An RDM for TB requires the following information: 1) observed infection status for all household members, namely ATB, LTBI, and NTBI; 2) dates of diagnoses for all ATB cases; and 3) covariate information on the individuals and environment that correlates with transmission. The graph is drawn by adhering to three rules, modifying the rules in 2.2 to be specific to TB. These are: 1) Only ATB cases can send edges and they can only send edges to others within the same household. 2) ATB cases can only send edges to other ATB cases with later diagnosis dates and LTBI cases, as an ATB case can only infect those who progress to ATB at a later date and LTBI cases. 3) To produce a graph that is consistent with the observed data, the ATB case(s) with earliest diagnosing date must receive an edge from outside the household. However, we do not consider the edge of the earliest ATB cases for our estimation and results when community controls are not enrolled since including these edges in a household contact study without community controls would upwardly bias the estimates of extra-household transmission force. 4) ATB and LTBI case(s) should receive at least one edge and NTBI case(s) should receive no edges. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 9 An example graph for TB is shown in Figure 1. There are three main assumptions in this paper. First, we assume the diagnosis dates and (unobserved) infection dates have the same temporal ordering, so that we can use them to determine infectors and infectees. Second, we assume households as well as edges are independent, meaning the edges drawn in one household do not influence the edges drawn in another household. Third, extra-household edges come from another (unknown) individual who lives in the same community. 3.2 TB transmission model with powerful predictors In general, the likelihood function of a graph for TB is modified based on the likelihood function (2): /g2172 /g4666 /g2163 | /g2242 /g4667 /g3404/g4688 /g3537 /g3537 /g3427 /g21i8 /g2185/g2191 /g3435/g2206 /g2191 ,/g2206 /g2185 /g4667 /g2188 /g4666 /g2202|/g2242 /g4667 /g343i/g3431 /g2204 /g2185/g2191/g4670 /g2778/g33i8/g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667 /g2188 /g4666 /g2202|/g2242 /g4667/g4671 /g2778/g2879/g2204 /g2185/g2191 /g2191/g1488/g2157 /g2185 /g2159 /g2185/g2880/g2778 /g468i ·/g4688 /g3537/g3537/g3537 /g3427 /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g2188/g3435/g2202 /g2191/g2192 |/g2242/g343i/g3431 /g2204 /g2190/g2191/g2192 /g3427/g2778 /g2192/g1488/g2960 /g2191/g2191/g1488/g2157 /g2190 /g2164 /g2190/g2880/g2778 /g33i8/g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g2188/g3435/g2202 /g2191/g2192 |/g2242/g343i /g3431 /g2778/g2879/g2204 /g2190/g2191/g2192 /g468i ·/g4688 /g3537/g3537 /g4670 /g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667/g4671 /g2204 /g2185/g2191/g4670 /g2778/g33i8/g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667/g4671 /g2778/g2879/g2204 /g2185/g2191 /g2191/g1488/g2171 /g2185 /g2159 /g2185/g2880/g2778 /g468i ·/g4688 /g3537/g3537 /g3537 /g3427 /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3431 /g2204 /g2190/g2191/g2192 /g3427/g2778 /g33i8 /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3431 /g2778/g2879/g2204 /g2190/g2191/g2192 /g2192 /g1488/g2960 /g2190/g2191/g1488/g2171 /g2190 /g2164 /g2190/g2880/g2778 /g468i (5) where /g1827 /g3030 and /g1827 /g3035 refer to the collection of all ATB cases (except the first in each household) in the community /g1855 and in the household /g1860 respectively. Similarly, /g1841 /g3030 and /g1841 /g3035 refers to the collection of all non-ATB cases in the community /g1855 and in the household /g1860 respectively. Ω /g3036 is the collection of infectors that could infect the infectee /g1861 and it is a subset of Ω /g3035. For example, . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 10 Ω /g2870 in Figure 1 is {subject 1, community} and Ω /g3035 is {subject 1, subject 2, community}. We modified the likelihood function for ATB cases specifically: /g1858 is the density function of serial interval and /g1872 /g3036/g3037 is the lag between the dates of diagnoses of infector /g1862 and infectee /g1861 , i.e., /g1872 /g3036/g3037/g3404/g1856 /g3037/g3398/g1856 /g3036. t is a random number drawn from /g1858/g4666/g1872|/g2016/g4667 and represents the serial interval between the infectee /g1861 and an unknown infector outside the household. The reason for such modification is to incorporate information about the serial interval /g1858 and dates of diagnoses /g1856 /g3036,/g1856 /g3037 that are only available for ATB cases into the calculation of the relative likelihood of household transmission to extra-household transmission, with the hope that we can better impute the transmission paths between a pair of ATB cases. In general, when one has enough powerful predictors for imputing transmission paths, the model for extra-household transmission and household transmission follows from (3) and (4): /g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667 /g3404/g2778/g33i8/g1805 /g1824 /g1816 /g3435 /g33i8 /g2235· /g2245 /g2201 /g4666 /g2206 /g2191 /g4667 · /g2245 /g2185 /g4666 /g2206 /g2185 /g4667 /g343i . (6) /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4672 /g33i8 /g2235· /g2245 /g2190 /g4666 /g2206 /g2190 /g4667 · /g2245 /g2201 /g4666 /g2206 /g2191 /g4667 · /g2245 /g2165 /g3435/g2206 /g2192 /g343i/g4673. (7) For the model above, α characterizes the risk of TB infection for the baseline group and all the predictors (i.e., /g1876 /g3036,/g1876 /g3030,/g1876 /g3035,/g1876 /g3037) are dummy variables representing risk factors for TB infection. Therefore, the baseline group in this case refers to the group of people who have the minimum risk of TB infection when the most important risk factors of TB infection have been controlled for in the transmission model. In this case, we think the risk of household transmission should be approximately the same as the risk of extra-household transmission for the baseline group. The model parameterization above can solve the non-identifiability issue by itself, and therefore it is estimable for a household contact study without community controls. However, this is only . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 11 appropriate if most powerful predictors for TB transmission have been included in the model, an assumption that is impossible to test in practice and likely not achieved. 3.3 TB transmission model with insufficient predictors If we do not have enough powerful predictors for describing TB transmission (due to measurement error, poor diagnostics, etc.), the earlier model is no longer appropriate as the baseline risks of TB infection for household and different communities will not be identical. Therefore, we need to reparameterize the transmission model (6) and (7) /g21i8 /g2185,/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667 /g3404/g2778/g33i8/g1805 /g1824 /g1816 /g3435 /g33i8 /g2235 /g2185 · /g2245 /g2201 /g4666 /g2206 /g2191 /g4667 /g343i . (8) /g21i8 /g2190,/g2191, /g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4672 /g33i8 /g2235· /g2245 /g2190 /g4666 /g2206 /g2190 /g4667 · /g2245 /g2201 /g4666 /g2206 /g2191 /g4667 · /g2245 /g2165 /g3435/g2206 /g2192 /g343i/g4673. (9) In this model, /g2009 /g3030 characterizes the baseline risk of extra-household transmission in the community /g1855 (/g1855 = 1, …, /g1829 ) and /g2009 characterizes the baseline risk of household transmission. The above model parameterization is non-identifiable and thus non-estimable for a household contact study without community controls, even after taking dates of diagnoses for ATB cases into account. This is because the majority of infectees are typically LTBI cases and information on the dates of diagnosis is not available, making it challenging to impute transmission paths based on insufficient predictors. Consequently, the household contact study does not provide any direct information on transmission outside of enrolled households, if both community controls and powerful predictors are unavailable.17 To address this issue, there are two strategies: The first strategy is to enroll community controls, which are used to identify extra-household transmission and thus separate household transmission from extra-household transmission. The drawback of this strategy is a large number of community controls may not be available in practice. The second strategy is to modify the . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 12 MCMC algorithm for the household contact study data without community controls by a priori weighting extra-household transmission paths versus household transmission paths (or vice versa), which is detailed next. 3.4 Bayesian inference We introduce a new MCMC algorithm that uses random weights to impute the transmission paths between an ATB case and an LTBI case. This approach allows researchers to incorporate knowledge about the relative strength of extra-household to household transmission when the model structure does not easily permit this information to be included as a proper prior. The weighted MCMC algorithm has the following steps: 1. Choose a lognormal distribution with the mean μ and the standard deviation σ from which to draw the weights. 2. In each iteration, draw a random number /g1872 from the density function of serial interval and a random number /g1870 from the lognormal distribution. 3. Update the parameters with the following probability of acceptance: /g2168/g4666/g2163|/g2242 /g4593 /g4667/g2172/g4666/g2242 /g4593 /g4667/g21ii/g4666/g2242|/g2242 /g4593 /g4667 /g2168/g4666/g2163|/g2242/g4667/g2172/g4666/g2242/g4667/g21ii/g4666/g2242 /g4593 |/g2242/g4667 /g1512/g2778 (10) based on the likelihood function /g1838/g4666/g1833|/g2016/g4667 , the prior /g1842/g4666/g2016/g4667 and the proposal /g1869/g4666/g2016 /g4593|/g2016/g4667 for the current value /g2016 and the candidate /g2016 /g4593 . 4. Update the graph for each infected individual, except the earliest ATB case in each household, with the probability of acceptance for ATB and LTBI defined separately. 4.1. For an ATB recipient /g1861 the acceptance probability is . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 13 /g2168/g4666/g2242|/g2163 /g2191 /g4593 /g4667/g21ii/g4666/g2163 /g2191 |/g2163 /g2191 /g4593 /g4667 /g2168/g4666/g2242|/g2163 /g2191 /g4667/g21ii/g4666/g2163 /g2191 /g4593 |/g2163 /g2191 /g4667 /g1512/g2778 (11) where /g1833 /g3036 is a graph that shows presence and absence of all edges leading to the infectee /g1861 . The new graph /g1833 /g3036 /g4593 is obtained by randomly deleting or adding one path based on /g1833 /g3036 .The proposal ratio is: /g21ii /g4666 /g2163 /g2191 | /g2163 /g2191 /g4593 /g4667 /g21ii /g4666 /g2163 /g2191 /g4593 /g3627/g2163 /g2191 /g4667 /g3404 /g3422 /g2183/g33i8/g2184 /g2184/g33i7/g2778 /g1806/g1815/g1818 /g1801/g1804/g1804/g180i/g1814/g1807 /g1801/g1814 /g1805/g1804/g1807/g1805 /g2184 /g2183/g33i8/g2184/g33i7/g2778 /g1806/g1815/g1818 /g1804/g1805/g1812/g1805/g1820/g180i/g1814/g1807 /g1801/g1814 /g1805/g1804/g1807/g1805 /g1 (12) where /g1853 is the total number of possible edges leading to individual /g1861 and /g1854 is the number of edges in /g1833 /g3036.15 4.2. For a LTBI recipient /g1861 the acceptance probability is /g2205· /g2168/g4666/g2242|/g2163 /g2191 /g4593 /g4667/g21ii/g4666/g2163 /g2191 |/g2163 /g2191 /g4593 /g4667 /g2168/g4666/g2242|/g2163 /g2191 /g4667/g21ii/g4666/g2163 /g2191 /g4593 |/g2163 /g2191 /g4667 /g1512/g2778 (13) where /g1875 is determined as follows based on the weights. If μ < 0, household transmission paths would be penalized for each LTBI infectee by defining the weights by /g2205/g3404/g4688 /g2200 if add a household transmission path /g2200 /g2879/g2778 if delete a household transmission path /g2778 if add/delete an extrahousehold transmission path /g1 (14) If μ > 0, extra-household transmission paths would be penalized for each LTBI infectee by defining the weights by /g2205/g3404/g4688 /g2200 /g2879/g2778 if add an extrahousehold transmission path /g2200 if delete an extrahousehold transmission path /g2778 if add/delete a household transmission path /g1 (15) When either powerful predictors or community controls are available for the household contact study, weights are not needed and the above algorithm can be simplified (the simple MCMC algorithm). In each iteration, . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 14 1. Draw a random number t from the density function of serial interval 2. Update the parameters with the following probability of acceptance: /g2168/g4666/g2163|/g2242 /g4593 /g4667/g2172/g4666/g2242 /g4593 /g4667/g21ii/g4666/g2242|/g2242 /g4593 /g4667 /g2168/g4666/g2163|/g2242/g4667/g2172/g4666/g2242/g4667/g21ii/g4666/g2242 /g4593 |/g2242/g4667 /g1512/g2778 . (16) 3. Update the graph for each infected individual, except the earliest ATB case in each household, with the following probability of acceptance: /g2168/g4666/g2242|/g2163 /g2191 /g4593 /g4667/g21ii/g4666/g2163 /g2191 |/g2163 /g2191 /g4593 /g4667 /g2168/g4666/g2242|/g2163 /g2191 /g4667/g21ii/g4666/g2163 /g2191 /g4593 |/g2163 /g2191 /g4667 /g1512/g2778 . (17) 3.5 DIC and its application When strong prior knowledge about /g1875 is not available, it is recommended that the MCMC algorithm is repeatedly run with multiple distinct weighting schemes, i.e., lognormal distributions with different μ . The idea is to repeatedly draw posterior samples under various weighting schemes ranging from weights strongly favoring household transmission to weights strongly favoring extra-household transmission and select the best-fitting weighting scheme(s) among them. In this section, we discuss how to use DIC to compare multiple samples drawn from the joint posterior with distinct weighting schemes. /g1842/g4666/g1877|/g2016/g4667 can be approximated in simulation by /g2172/g3553 /g4666 /g2207 |/g2242/g4667 /g3404 /g3537 /g4678 ∑ /g2778 /g2207 /g2191/g2193/g2880 /g2207 /g2193|/g2163 /g2193 ,/g2242/g2175 /g2193/g2880/g2778 /g2175 /g467i /g2196 /g2191/g2880/g2778 (18) where S is the number of iterations for simulation and /g1833 /g3038 is the observed graph in each iteration. The DIC is computed based on /g1842/g3552 /g4666/g1877|/g2016/g4667 , and in general we select weighting schemes to avoid under-weighting such that the difference between the DIC of selected weighting scheme and the minimum DIC should be no larger than 5. We also want to avoid over-weighting by selecting . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 15 weighting scheme(s) that should not be more informative (i.e., μ should not be further away from 0) than the one with minimum DIC. Selected weighting scheme(s) should be consistent with one’s knowledge about whether extra-household transmission is stronger than household transmission (or vice versa). We illustrate the application of DIC in the results section. 4 Results 4.1 Simulation 1: Simulation with powerful predictors We simulated TB epidemics for 1500 households with the following dichotomous predictors: age ( 2), sleeping proximity (sleeping in the same room with an ATB case), majority of time at home, severity of disease for ATB cases, high community burden, high socioeconomic status, and proximity to someone with a cough (Table 1). We assumed transmission was fully determined by the predictors above: /g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667 /g3404 /g2778 /g33i8 /g1805/g1824/g1816 /g4672/g33i8/g2235 · /g2245 /g2201 /g2183/g2186/g2203/g2194/g2202 · /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187 · /g2245 /g2201 /g2185/g2197/g2203/g2189/g2190 · /g2755 /g2185 /g2184/g2203/g2200/g2186/g2187 /g2196 · /g2755 /g2185 /g2201/g2187/g2201 /g4673 (19) /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4672 /g33i8 /g2235· /g2245 /g2190 /g2185/g2200/g2197/g2205/g2186 · /g2245 /g2201 /g2183/g2186/g2203/g2194 /g2202 · /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187 · /g2245 /g2201 /g2201/g2194/g2187/g2187/g2198 · /g2245 /g2201 /g2202/g2191/g2195/g2187 · /g2245 /g2165 /g2201/g2187/g2204/g2187/g2200/g2191/g2202 /g2207 /g4673. (20) where /g2019 /g3035 /g3030/g3045/g3042/g3050/g3031 equals /g2019 /g3035 /g3030/g3045/g3042/g3050/g3031/g2880/g2869 for a crowded household (crowd = 1) or 1 for a non-crowded household (crowd = 0). The interpretations for all other parameters are similar. We only selected the households with at least one ATB case for subsequent estimation and, in essence create a “perfect” household contact study where we assumed we were able to trace and enroll all TB patients and their household contacts. The size of the final data was similar to the Brazilian household contact study (average number of TB households was 155). The simulation generated 1000 simulated datasets and for each dataset the simple MCMC algorithm was iterated . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 16 220,000 times with a burn-in of 20,000 and a thinning of 20. Each run took about 2 hours by using the Rcpp package on a laptop (Intel® core i7-8550U [email protected] GHz). The results showed that the simple MCMC algorithm generated good estimates and reliable confidence intervals for all parameters, without using any weights or community controls (Figure 2). The actual coverage rates ranged from 94.3% to 98.8% across the parameters (Table 1). Therefore, for household contact study data (i.e., the households with at least one ATB case; henceforth we refer it as the HHC data) with powerful predictors, we confirm that there was no need to employ community controls or weights and there is no concern regarding the non- identifiability issue. 4.2 Simulation 2: Simulation with insufficient predictors As previously discussed, in practice we likely do not have a set of predictors that are sufficient to estimate the transmission paths of the RDM. In this simulation, we only used three predictors (which have been defined in the first simulation): age (</ ≥ 18), biological sex, and household crowding. We assumed transmission was fully determined by the predictors above: /g21i8 /g2185/g2191 /g4666 /g2206 /g2191 ,/g2206 /g2185 /g4667 /g3404 /g2778 /g33i8 /g1805/g1824/g1816 /g4672/g33i8/g2235 /g2185 · /g2245 /g2201 /g2183/g2186/g2203/g2194/g2202 · /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187 /g4673 (21) /g21i8 /g2190/g2191/g2192 /g3435/g2206 /g2190 ,/g2206 /g2191 ,/g2206 /g2192 /g343i/g3404/g2778/g33i8/g1805 /g1824 /g1816 /g4672 /g33i8 /g2235· /g2245 /g2190 /g2185/g2200/g2197/g2205/g2186 · /g2245 /g2201 /g2183/g2186/g2203/g2194/g2202 · /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187 /g4673. (22) We consider four scenarios depicting varying relative magnitudes of household transmission. 1) Extra-household transmission accounted for approximately 90% of the total infections. 2) Extra-household transmission accounted for 50-60% of the total infections. 3) Household transmission accounted for 50-60% of the total infections. 4) Household transmission accounted for 90% of the total infections. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 17 We simulated 25 datasets for each scenario and for each dataset we ran three different kinds of analyses. First, we used all households, including those with and without ATB case(s), and ran the simple MCMC algorithm on this data (the whole data approach). Second we used the HHC data (only households with at least one ATB case) and ran the simple MCMC algorithm on it. Third, we used the HHC data and separately ran the weighted MCMC algorithm with 10 different weighting schemes. All MCMC algorithms were iterated 220,000 times with a burn-in of 20,000 and a thinning of 20. The final size of the HHC data was similar to the Brazilian household contact study (the average number of TB households was 177, with minimum = 140 and maximum = 214). We chose five of the ten weighting schemes to favor extra-household transmission and the other five to favor household transmission and show how DIC can be used to select the best weights. For the weighting schemes favoring extra-household transmission, we chose the mean of lognormal distribution as -0.25, -0.2, -0.15, -0.1, and -0.05 with standard deviation as 0.05. For the weighting schemes favoring household transmission, the mean of lognormal distribution was -0.02, -0.04, -0.06, -0.08 and -0.1 with standard deviation 0.02. The average DIC of the 10 posterior samples drawn under the weighting schemes in the four simulation scenarios are shown in Figure 3. DIC correctly informs us about whether a weighting scheme should favor extra- household transmission (or household transmission) under the simulation scenario 1, 2 and 4. For example, in scenario 1 where weighting scheme should strongly favor extra-household transmission, the weighting schemes favoring household transmission all had significant higher DIC than the weighting schemes favoring extra-household transmission, suggesting that the weighting scheme(s) favoring household transmission poorly fit to the data. In scenario 3, we still observed that the weighting schemes favoring household transmission have lower DIC than . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 18 the weighting schemes favoring extra-household transmission, although the differences were not statistically significant. Not surprisingly, DIC is more informative under the scenarios 1 and 4 than under the scenarios 2 and 3. However in scenarios 2 and 3 it is not clear how consequential the weights are since household and extra-household transmission are almost equally likely. Next, we compared the parameter estimates based on the HHC data without weights, the whole data without weights, and the HHC data with weights. For the third approach, we selected weighting schemes using DIC. Using weights on the HHC data helped identify the extra- household and household baseline transmission forces and thus reduce bias of the estimates, compared to the estimates based on the HHC data without weights (Figure 4). Using community controls helped identify the baseline extra-household transmission forces and improved the accuracy of estimates compared to the other two approaches. When household transmission is stronger, community controls also automatically leaded to identification of the baseline household transmission force (Tables 4 and 5). However, when extra-household transmission dominates, community controls do not appear sufficient to accurately estimate household baseline transmission (Tables 2 and 3). Interestingly, the estimates of risk of infection for the covariates (age, crowding and biological sex) were comparable for all three estimation approaches (Figure 5). This indicates that the non- identifiability issue only exists for baseline extra-household and household transmission force estimates and does not appear to impact estimation of risk factors of TB transmission model when using RDMs. 4.3 The Brazilian household contact study . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 19 The Brazilian household contact study enrolled households from four municipalities and the following predictors were used in the transmission model: adult ( 2). The first two variables were included in both the extra-household and household transmission model and the last one was used in the household transmission model only. Since Brazil has substantial TB in the community based on prior work, it is reasonable to assume that extra-household transmission is stronger than household transmission. 4,12 To be objective in implementing the weighted MCMC algorithm we explore weights similar to those we used in the simulations (i.e. five favoring extra household and five favoring household transmission with standard deviation fixed at 0.05). All MCMC algorithms were iterated 220,000 times with a burn-in of 20,000 and a thinning of 20. The DIC suggests extra-household transmission is stronger than household transmission. (Figure 6) We then selected three weighting schemes whose μ (mean of lognormal distribution) were -0.2, -0.15, -0.1 since their DIC were not significantly different from the minimum DIC. We excluded the weighting scheme whose μ was -0.25 to avoid overweighting. Subsequently, we used the three corresponding posterior samples as parameter estimates. To enhance interpretability, we simulated TB epidemics based on the parameter estimates and reported the underlying TB transmission pattern as our result.14,15 In total, there were 30,000 simulated TB epidemics and each of them was done for 10,000 households in four municipalities, which were set to have similar features as the Brazilian household contact study. Overall, the relative risk of being infected within the household to being infected outside the household was 0.23 (95% CI, 0.03-0.49), demonstrating that extra-household transmission was the predominant transmission force (Figure 7). Consistent with this, we estimate that 82% (95% CI, 63%-98%) of infections are attributed to extra-household transmisson while only 6% (95% . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 20 CI, 1%-15%) of infections are estimated to be attributed to household transmission. Those proportions were consistent across the four municipalities (Figure 8). We also studied the risk factors for TB transmission (Figure 9) and found that adults had 9% more risk of TB infection than non-adults on average (RR: 1.09, 95% CI: 0.99-1.20). Females had the same of risk of TB infection compared to males (RR: 1.00, 95% CI: 0.91-1.10). Living in a crowded household did not increase the risk of TB infection (RR: 1.01, 95% CI: 0.99-1.04), likely due to the fact that crowding was only associated with household transmission which was minimal in this context. 5 Discussion In this study, we discuss the application of random directed graph model (RDM) to understand TB transmission. We first formalized rules of drawing random directed graph appropriate for TB transmission, and modified the likelihood model by treating ATB cases differently from LTBI and NTBI cases. By doing this, we can incorporate the diagnosing dates of ATB cases to better impute transmission paths among ATB cases as well as the temporal order of TB transmission chains. We further addressed the identification issue of RDM in this context with three different strategies: 1) control for the most powerful predictors in the TB transmission model; 2) enroll community controls in the household contact study; 3) use weights and DIC if the above two strategies become unviable. Having the identification issue addressed, simulation showed that the RDM could successfully estimate the relative risk and importance of household transmission versus extra-household transmission even without community controls. Furthermore, RDM is also a valuable tool for incorporating community controls and would gain power for both identification and estimation if they are available. 5 RDM can consistently generate unbiased estimate of the relative risks of risk factors included in the transmission model, regardless of the . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 21 usage of powerful predictors, community controls or weights. This is attractive compared to mixed model which is known to output biased estimates in the same setting.4 Our work is motivated by TB household contact study through which one seeks to infer household and extra-household transmission and their contributions to the general TB transmission dynamics. Traditionally, TB household contact studies assume TB transmission only happens within the household, which is not consistent with TB literature that suggests TB transmission in some settings is more likely to happen outside the household than in the household. 10,11 Moreover, strain analysis using restriction fragment length polymorphism or spoligotyping of secondary diseased cases suggests that extra-household transmission could account for up to 70% of the total infections.18-22 However most of the models for household contact studies are logistic regression or mixed models assuming transmission only happens within the household. 8 When using such models in a context where extra-household transmission is considerable, risk estimates would be biased especially for risk factors that solely associated with household transmission such as crowding, which may result in misguided public health planning.4,23 Therefore, it is critical to use models allowing concurrent estimation of household and extra-household transmission (such as UPM and RDM) in order to quantify the relative importance of household versus extra-household transmission. There are limitations for our approach. First, our approach is simulation-based, which demands the model to be consistent with the simulation which should be accurate and realistic. In our paper, we assume the simulation is fully determined by controlled covariates in the model. However, this assumption is hardly testable and thus we don’t consider the confounding effect in this paper. Second, the transmission model parameters are not quite interpretable and to gain interpretability we recommend that the relative risk as well as risk estimates are obtained through . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 22 simulation. Third, our approach is computationally intensive compared to other Bayesian approaches given the additional work on simulation and possibly weighting. In summary, RDM is a valuable tool for TB household contact study as it provides additional knowledge on extra-household transmission as well as the roles of extra-household and household transmission in TB transmission dynamics. It worth emphasizing that, the knowledge gained by RDM cannot be picked up by a traditional statistical method, but it is clearly needed for public health programs to better plan their resources and target interventions. Supplementary Material Supplemental material is available online and contains more details on the MCMC algorithm and simulation. The code and data for this paper are available at https://github.com/tenglongli/tb- rdm-method.

We thank David Alland, Padmini Salgame, Jerrold Ellner and Reynaldo Dietze for sharing the data.

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DNA fingerprinting of Mycobacterium tuberculosis isolates from epidemiologically linked case pairs. Emerging Infectious Diseases. 2008;8(11):1124. 20. Brooks-Pollock E, Becerra MC, Goldstein E, Cohen T, Murray MB. Epidemiologic inference from the distribution of tuberculosis cases in households in Lima, Peru. Journal of Infectious Diseases. 2011;203(11):1582–1589. 21. Cohen T, Murray M, Abubakar I, Zhang Z, Sloutsky A, Arteaga F, Chalco K, Franke MF, Becerra MC. Multiple introductions of multidrug-resistant tuberculosis into households, Lima, Peru. Emerging Infectious Diseases. 2011;17(6):969–975. 22. Glynn JR, Guerra-Assunção JA, Houben RM, Sichali L, Mzembe T, Mwaungulu LK, Mwaungulu JN, McNerney R, Khan P, Parkhill J, Crampin AC. Whole genome sequencing shows a low proportion of tuberculosis disease is attributable to known close contacts in rural Malawi. PloS One. 2015;10(7) e0132840. 23. Yuen CM, Amanullah F, Dharmadhikari A, Nardell EA, Seddon JA, Vasilyeva I, Zhao Y, Keshavjee S, Becerra MC. Turning off the tap: stopping tuberculosis transmission through active case-finding and prompt effective treatment. The Lancet. 2015;386(10010):2334–2343. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 25 Fig ure 1 : Examp le of a p lausib le d irected g rap h f or a househo ld with 2 active TB cases (subject 1 and subject 2), 1 LT BI case (subject 3) an d 1 non -in fected (N TBI) case (subject 4). A sol id a rr ow rep resents the presence of a pot ential transmissio n path and a dashed a rr ow rep resents the absence of a p otential tra nsmission path. Su bject 1 is the ATB ca se with earliest d iag nosing date (July 1 st , 201 5) an d he/she sho uld on ly rece ive an edg e fr om outside the ho usehol d. Th is edg e i s not consi dered when community c ontro ls are unavailab le. Subject 2 is a n ATB case b ut later than subject 1 , so he/she co ul d receive an edg e fr om subject 1 and /or fr om o utside the h ousehol d ( in t his g raph , he/she receives both edg es). Subject 3 i s an LTBI case and sh oul d at least receive one edg e ( in th is g raph , the edg e is f r om outside the ho usehold), but he/she ca nnot se nd edg es to someone e lse. Subject 4 is not infected , so he/she does not recei ve any edg es. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 26 Figure 2 : Sim ulati on results for the mode l with po werful pred ictors. The results were summar ized o ver 100 0 simu lated datasets which were sim ila r to Brazi lian ho usehold contact study. The solid l ines rep resent the average 95% C.I. for all mode l parameters, and the bla ck dots rep resent the mean of me dians in the posterio r sample of parameters. The red cr osses are the simulated values of the pa rameters. The ho rizo ntal dashed line co rrespon ds to the value 1, which is the threshol d o f whether a predict or signa ls higher risk o f TB i nfectio n. It is cl ear that fo r the mo del w ith power ful pre dicto rs th e MCMC algortihm (witho ut weights) can lead to good estimates and goo d i de ntifiab il ity, even witho ut any comm unity co ntro ls. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 27 Figure 3 : The pl ot of average DIC for 1 1 poster ior samples, 10 of whic h were d rawn u nder the weighting schemes and one based on the sim ple MCMC algor ithm (shown at weight = 1). The red dashed l ine represents the value whi ch eq uals the minimum DIC + 5, which is the th reshold fo r dec i ding whether a weighting scheme is signi ficant ly d ifferent fr om the mi nimum DIC. In this case, we only consi der the weighting schemes that have DIC lower than this th reshold. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 28 Figure 4 : Compar ing the basel ine tra nsmission f or ce estimates of househo ld an d the c ommun ities across the fou r simu latio n scenarios. The li ne rep resents the average 95% CI for 25 datasets; The black symbo ls represent mea n of me dian estimates among 25 datasets, and the red cross rep resent the unde rlying simu latio n values. ‘HHC ’ i n the f igur e above refers to the approac h that we used the H HC data witho ut wei ghts. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 29 Figure 5 : Compar ing the risk facto r estimates acro ss the four simu latio n scenar ios. The li ne rep resents the average 95% CI for 25 datasets; The b lack symbols re present mean o f median estimates among 25 datasets, and the re d cross rep resent the under lying simulat ion values. ‘HHC’ in the figu re a bove refers to the app roach that we used the HHC data without weights. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 30 Figure 6 : The pl ot of DIC for the Brazi lia n ho usehol d contact study. We ploted DIC for 11 p osterio r samples, 1 0 of whic h were drawn un der the weighting schemes and o ne based on t he simple unweighted MCMC algorithm (x value = 1 ). The red dashed line represents the value wh ich eq uals mi nimum DIC + 5, whi ch is the thresho ld f or deci ding wheth er a weighting scheme is significant ly di fferent from the one with mi nimum DIC (represented by the square dot). In th is case, we chose the three weighting schemes that represented by the squar e and tr iangle d ots as our candi dates. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 31 Figure 7 : The vio lin pl ot of relat ive risk of ho useho ld versus extra-househo ld transmission ac ross the fou r mun ici pal ities. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 32 Figure 8 : The vio lin pl ot of the p rop orti ons of TB i nfection attrib uted to extra-h ouseho ld an d ho usehold transmission across the four mun icipa lit ies. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 33 Figure 9: The vio lin pl ot of the re lative risk o f T B i n fection for the r isk factors inc luded in TB t ransmission mo del. The ho rizonta l dashed li ne co rrespo nds to RR = 1 , whi ch means a risk facto r d oes not have any effe ct in TB t ransmi ssion. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 34 Table 1: Simulation results for model with powerful predictors. Baseline and relative susceptibility parameters Simulation value Mean of Median Number of times 95% C.I covers the simulation value in 1000 simulations α : baseline risk 0.03 0.03 965 /g2245/g2201 /g2183/g2186/g2203/g2194/g2202/g2880/g2778 : adult vs. child 1.2 1.21 978 /g2245/g2201 /g2188/g2187/g2195/g2183/g2194/g2187/g2880/g2778: female vs. male 1 1 988 /g2245/g2190 /g2185/g2200/g2197/g2205/g2186/g2880/g2778: crowded vs. uncrowded household 1.5 1.5 952 /g2245/g2201 /g2201/g2194/g2187/g2187/g2198/g2880/g2778: slept with vs. not slept with TB patient 5.5 5.02 943 /g2245/g2201 /g2202/g2191/g2195/g2187/g2880/g2778: spent more time vs. less time in household 2 1.92 956 /g2245/g2165 /g2201/g2187/g2204/g2187/g2200/g2191/g2202/g2207/g2880/g2778: Infector with high severity vs. low severity 2.8 2.62 966 /g2755/g2185 /g2184/g2203/g2200/g2186/g2187/g2196/g2880/g2778: lived in high vs. low burden community 3.5 3.26 970 /g2755/g2185 /g2201/g2187/g2201/g2880/g2778 : lived in higher vs. lower socio- economical communty 3 2.86 977 /g2245/g2201 /g2185/g2197/g2203/g2189/g2190/g2880/g2778: experiencing cough vs. no such experience in community 2.5 2.44 971 . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 35 Table 2: Simulation result for scenario 1: extra-household transmission is predominant. The

are described in the following format: mean of median (the number of times that CI covers the simulation value in 25 datasets). For example, 0.16 (3) means the average median estimate was 0.16 for 25 datasets and confidence interval covered the simulation value 3 times for 25 datasets. Baseline and relative susceptibility parameters Simulation value Estimates based on the whole data without weights Estimates based on the household contact study without weights Estimates based on the household contact study with weights α : baseline household 0.05 0.16 (3) 0.27 (0) 0.07 (24) /g2235 /g2185/g2778 : baseline for the first community 0.99 0.97 (24) 0.76 (13) 0.98 (25) /g2235 /g2185/g2779 : baseline for the second community 1.07 1.04 (25) 0.82 (11) 1.05 (25) /g2235 /g2185/g2780 : baseline for the third community 0.83 0.83 (22) 0.63 (14) 0.84 (21) /g2235 /g2185/g2781 : baseline for the fourth community 1.26 1.23 (24) 1.02 (13) 1.24 (23) /g2245 /g2201 /g2185/g2200/g2197/g2205/g2186/g2880/g2778 : crowded vs. non-crowded household 1.2 1.07 (23) 1.04 (22) 0.97 (24) /g2245 /g2201 /g2183/g2186/g2203/g2194 /g2202 /g2880/g2778 : adult vs. child 1.2 1.2 (25) 1.2 (25) 1.2 (25) /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187/g2880/g2778 : female vs. male 1 0.99 (21) 0.98 (22) 0.98 (23) . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 36 Table 3: Simulation result for scenario 2: extra-household transmission is stronger. The results are described in the following format: mean of median (the number of times that CI covers the simulation value in 25 datasets). Baseline and relative susceptibility parameters Simulation value Estimates based on the whole data Estimates based on the household contact study without weights Estimates based on the household contact study with weights α : baseline household 0.21 0.29 (10) 0.36 (4) 0.15 (24) /g2235 /g2185/g2778 : baseline for the first community 0.83 0.83 (23) 0.73 (24) 0.99 (21) /g2235 /g2185/g2779 : baseline for the second community 0.91 0.90 (23) 0.79 (24) 1.05 (24) /g2235 /g2185/g2780 : baseline for the third community 0.67 0.68 (25) 0.56 (21) 0.81 (20) /g2235 /g2185/g2781 : baseline for the fourth community 1.08 1.09 (23) 1 (24) 1.26 (21) /g2245 /g2201 /g2185/g2200/g2197/g2205/g2186/g2880/g2778 : crowded vs. non-crowded household 1.2 1.06 (22) 1.06 (22) 1.06 (25) /g2245 /g2201 /g2183/g2186/g2203/g2194 /g2202 /g2880/g2778 : adult vs. child 1.2 1.2 (25) 1.18 (24) 1.17 (24) /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187/g2880/g2778 : female vs. male 1 0.99 (24) 0.96 (24) 0.96 (24) . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 37 Table 4: Simulation result for scenario 3: household transmission is stronger. The results are described in the following format: mean of median (the number of times that CI covers the simulation value in 25 datasets). Baseline and relative susceptibility parameters Simulation value Estimates based on the whole data Estimates based on the household contact study without weights Estimates based on the household contact study with weights α : baseline household 0.8 0.83 (23) 0.62 (12) 0.98 (15) /g2235 /g2185/g2778 : baseline for the first community 0.3 0.31 (22) 0.67 (10) 0.21 (19) /g2235 /g2185/g2779 : baseline for the second community 0.38 0.38 (25) 0.76 (7) 0.24 (20) /g2235 /g2185/g2780 : baseline for the third community 0.2 0.21 (20) 0.55 (13) 0.11 (24) /g2235 /g2185/g2781 : baseline for the fourth community 0.5 0.51 (23) 0.84 (10) 0.28 (17) /g2245 /g2201 /g2185/g2200/g2197/g2205/g2186/g2880/g2778 : crowded vs. non-crowded household 1.2 1.14 (23) 1.16 (24) 1.13 (24) /g2245 /g2201 /g2183/g2186/g2203/g2194 /g2202 /g2880/g2778 : adult vs. child 1.2 1.18 (23) 1.15 (23) 1.16 (23) /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187/g2880/g2778 : female vs. male 1 1 (24) 1 (23) 1 (23) . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 38 Table 5: Simulation result for scenario 4: household transmission is predominant. The results are described in the following format: mean of median (the number of times that CI covers the simulation value in 25 datasets). Baseline and relative susceptibility parameters Simulation value Estimates based on the whole data Estimates based on the household contact study without weights Estimates based on the household contact study with weights α : baseline household 1.2 1.24 (22) 0.64 (4) 1.23 (20) /g2235 /g2185/g2778 : baseline for the first community 0.1 0.1 (24) 1.06 (1) 0.26 (22) /g2235 /g2185/g2779 : baseline for the second community 0.15 0.15 (24) 0.97 (4) 0.2 (21) /g2235 /g2185/g2780 : baseline for the third community 0.05 0.05 (24) 1.1 (2) 0.29 (22) /g2235 /g2185/g2781 : baseline for the fourth community 0.2 0.21 (20) 1.15 (4) 0.25 (20) /g2245 /g2201 /g2185/g2200/g2197/g2205/g2186/g2880/g2778 : crowded vs. non-crowded household 1.2 1.15 (24) 1.15 (25) 1.15 (24) /g2245 /g2201 /g2183/g2186/g2203/g2194 /g2202 /g2880/g2778 : adult vs. child 1.2 1.17 (22) 1.1 (21) 1.11 (23) /g2245 /g2201 /g2188/g2187/g2195/g2183/g2194/g2187/g2880/g2778 : female vs. male 1 1.01 (24) 1 (25) 1 (25) . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0 2 4 6 8 Adult Baseline Burden Cough Crowd Female SES Severity Sleep Time Parameter Estimates . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1424 1426 1428 1430 1432 1434 1436 Scenario 1: Predominant Extra−Household The average weight of household transmission relative to extra−household transmission DIC 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1245 1246 1247 1248 1249 1250 1251 1252 Scenario 2: Stronger Extra−Household The average weight of household transmission relative to extra−household transmission DIC 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1154 1155 1156 1157 1158 Scenario 3: Stronger Household The average weight of household transmission relative to extra−household transmission DIC 0.80 0.85 0.90 0.95 1.00 1.05 1.10 839 840 841 842 843 844 845 Scenario 4: Predominant Household The average weight of household transmission relative to extra−household transmission DIC . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint Stronger Extra−Household Transmission Stronger Household Transmission Predominant Extra−Household Transmission Predominant Household Transmission Community1 Community2 Community3 Community4 Household Community1 Community2 Community3 Community4 Household 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 Parameter Estimates Mean of Median HHC HHC with weights Whole . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint Stronger Extra−Household Transmission Stronger Household Transmission Predominant Extra−Household Transmission Predominant Household Transmission Adult Crowd Female Adult Crowd Female 1 2 3 1 2 3 Parameter Estimates Mean of Median HHC HHC with weights Whole . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0.8 0.9 1.0 1.1 1.2 1.3 1160 1170 1180 1190 The average weight of household transmission relative to extra−household transmission DIC . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Overall Community1 Community2 Community3 Community4 Relative risk of household transmission versus extra−household transmission . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Extra−Household1 Household1Extra−Household2 Household2Extra−Household3 Household3Extra−Household4 Household4 The proportion of infection attributed to . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Adults vs. Children Female vs. MaleCrowded vs. Non−crowded Relative Risk of TB infection . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted August 2, 2020. ; https://doi.org/10.1101/2020.07.30.20165506doi: medRxiv preprint