Capturing heterogeneous infectiousness in transmission dynamic models of tuberculosis: a compartmental modelling approach

Infectiousness heterogeneity among individuals with tuberculosis (TB) is substantial and is likely to have a significant impact on the long-term dynamics of TB and the effectiveness of interventions. However, there is a gap in capturing heterogeneous infectiousness and evaluating its impact on the effectiveness of interventions. Informed by observed distribution of secondary infections, we constructed a deterministic model of TB transmission using ordinary differential equations. The model incorporated assumption of heterogeneous infectiousness with three levels of infectivity, namely non-spreaders, low-spreaders and super-spreaders. We evaluated the effectiveness of dynamic transmission untargeted and targeted implementation of an intervention intended to represent active case finding with a point-of-care diagnostic tool. The simulated intervention detected 20% of all TB patients who would otherwise have been missed by the health system during their disease episode and was compared across four epidemiological scenarios. Our model suggested that targeting the active case finding intervention towards super-spreaders was more effective than untargeted intervention in all setting scenarios, with more effectiveness in settings with low case detection and high transmission intensity. For instance, a targeted intervention achieved a 42.2% reduction in TB incidence, while the untargeted intervention achieved only a 20.7% reduction over 20 years, given the same number of people treated. Although the most marked impact on equilibrium TB incidence came from the rate of late reactivation, the proportion of super-spreaders and their relative infectiousness had shown substantial impact. Targeting active case-finding interventions to highly infectious cases likely to be particularly beneficial in settings where case detection is poor. Heterogeneity-related parameters had an equivalent effect to several other parameters that have been established as being very important to TB transmission dynamics.


Abstract
Infectiousness heterogeneity among individuals with tuberculosis (TB) is substantial and is likely to have a significant impact on the long-term dynamics of TB and the effectiveness of interventions. However, there is a gap in capturing heterogeneous infectiousness and evaluating its impact on the effectiveness of interventions.
Informed by observed distribution of secondary infections, we constructed a deterministic model of TB transmission using ordinary differential equations. The model incorporated assumption of heterogeneous infectiousness with three levels of infectivity, namely nonspreaders, low-spreaders and super-spreaders. We evaluated the effectiveness of dynamic transmission untargeted and targeted implementation of an intervention intended to represent active case finding with a point-of-care diagnostic tool. The simulated intervention detected 20% of all TB patients who would otherwise have been missed by the health system during their disease episode and was compared across four epidemiological scenarios.
Our model suggested that targeting the active case finding intervention towards superspreaders was more effective than untargeted intervention in all setting scenarios, with more effectiveness in settings with low case detection and high transmission intensity. For instance, a targeted intervention achieved a 42.2% reduction in TB incidence, while the untargeted intervention achieved only a 20.7% reduction over 20 years, given the same number of people treated. Although the most marked impact on equilibrium TB incidence came from the rate of late reactivation, the proportion of super-spreaders and their relative infectiousness had shown substantial impact.
Targeting active case-finding interventions to highly infectious cases likely to be particularly beneficial in settings where case detection is poor. Heterogeneity-related parameters had an 2015 TB elimination activities, with the ambition of a 95% reduction in TB deaths and a 90% reduction in TB incidence by 2035 by comparison to 2015 rates [2]. However, the natural history of TB remains poorly understood, and consequently, the uncertain potential impact of control interventions limits confidence about the possibility of its elimination. The heterogeneous transmission of Mycobacterium tuberculosis (Mtb) within populations is wellestablished, but its epidemiological impact is poorly understood. Moreover, up to a third of TB cases are not diagnosed, so finding and treating infectious cases is key to achieving TB control, and finding and treating highly infectious people is the key to TB control in high transmission settings [3,4].
We define infectiousness heterogeneity as the variability in the capacity of infectious patients to produce secondary infections, which may be attributable to characteristics of the host, the agent and the environment [5][6][7][8][9]. Heterogeneity of infectious individuals in spreading Mtb infection is well-recognised, with a small group of highly infectious individuals producing a large proportion of secondary infections, while many others produce very few or none [10].
Previous TB genomic epidemiology has identified TB super-spreading events by quantifying heterogeneity in the infectiousness of TB patients through fitting a standard statistical distribution (the negative binomial distribution, NBD) to the distribution of secondary cases produced by each infectious patient (the "offspring distribution") [11]. More recently, data from TB contacts in a low-transmission setting have allowed estimation of the proportion of all TB patients that can be categorised as super-spreaders, finding it to be approximately 10% [9]. In such studies, super-spreaders were typically defined as patients who produced a number of secondary "infections/cases" greater than the 99 th centile of a standard Poisson offspring distribution, with distribution mean equal to the average number of secondary infections per index [9,12]. This heterogeneity is likely to have implications for both the burden of disease and the effectiveness of control interventions. Thus, we propose that it is necessary to capture this heterogeneity when modelling TB transmission dynamics.
Some past compartmental models of TB transmission dynamics have attempted to capture heterogeneity in patients' infectiousness by stratifying the active TB compartment into different levels of infectivity. Amongst the most typical approaches is stratification as either infectious (usually representing pulmonary TB) or non-infectious (extrapulmonary TB) [13][14][15][16][17]. This approach implies that all pulmonary TB patients are equally infectious, while all extrapulmonary patients are entirely non-infectious, and so does not capture the heterogeneity among pulmonary patients. Other TB models have considered sputum smear status as a factor in stratifying patients' levels of infectiousness, considering both smear-negative and smearpositive pulmonary TB to be infectious, with the relative infectiousness of smear-negative patients compared to smear-positive typically set between 15 and 25% [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. While stratifying pulmonary TB patients based on smear status captures an additional clinical attribute that is important in determining infectiousness, even these models still do not capture the full picture of TB patients' infectiousness variation, since several behavioural and demographic factors other than smear status can affect the level of infectiousness [5]. susceptibility on transmission location. The study defined super-spreading defined based on the number of secondary cases instead of secondary infections and suggested that the majority of disease resulted from infection by a small proportion of people with TB or superspreaders [34]. However, as TB disease activation may take long time and also depend on mainly on the characteristics of contact person, the definition of super-spreading used in the study may not show true heterogeneity in Mtb transmission [35]. In the current study, we present a deterministic compartmental Mtb transmission model that incorporates three levels of TB patients' infectivity, namely non-spreaders, low-spreaders and super-spreaders. Using this framework, we analysed how infectiousness heterogeneity affects disease dynamics and evaluated the effectiveness of a hypothetical active case finding intervention and the impact of targeting super-spreaders.

Literature review
Before constructing our model, we systematically reviewed how previous TB models have approached heterogeneous infectiousness [36]. In the review, we found that TB models frequently stratified the active TB compartment according to one or more patient-related factors. We constructed a flexible model with three levels of infectiousness, without restricting to a given factor, and the ability to transition between these levels using empirical measures of heterogeneous infectiousness to parametrise the model. infection, with cases of subsequent active TB disease linked to these contact episodes now extending to March 2017 (see [37] for earlier publication of linkage process). We constructed empirical offspring distributions from the detailed contact tracing data set of the VTP.
Ethical approval was obtained from Monash University, Human Research Ethics Committee (Project Number: 7776) and permission was given by the VTP and DHHS.
Model structure Using ordinary differential equations (ODE), we constructed a deterministic model of Mtb transmission in a hypothetical high TB burden setting. To represent heterogeneous infectiousness and super-spreading in Mtb transmission, we stratified the active TB compartment into three sub-classes with different levels of infectiousness: non-spreaders ( I 0 ), low-spreaders ( I 1 ) and super-spreaders ( I 2 ). We did not use any specific factor to discriminate I 0 , I 1 and I 2 a priori. Instead, we use the offspring data (number of secondary infections per index) and a Bayesian approach to find the most realistic parameterisation to implement heterogeneous infectiousness in our model. Following infection, individuals enter a rapid-sojourn early latent compartment ( L A ), from which they may progress to one of the three active TB compartments ( I 0 , I 1 or I 2 ) at a total rate of ε or enter the low-risk late latent stage ( L B ) at rate κ. From the late latent state they may progress more slowly to disease (total rate ν), also entering the active TB compartments ¿). Re-infection may occur during late latency with a reduced force of infection λ r ( λ r =r × λ ) and, if re-infection occurs, these individuals similarly enter the early latent compartment, progressing to the disease at the same rates as for newly infected persons. Individuals with active disease may either die due to TB (rate μ i ) or background mortality (μ); or spontaneously recover and return to the late latent state (rate γ ); or be detected and treated by the health system (rate δ) and return to the susceptible compartment (S). The resulting statistical distribution was then compared in a Bayesian context with empiric data obtained from our previous analysis of TB contact tracing data in the Australian state of Victoria [9]. A comparison between the distribution of the number of secondary infections per index simulated by our calibrated model with the one observed empirically is presented in The parameter values other than those pertaining to infectiousness introduced above were estimated from a review of relevant evidence, with the relevant sources presented in Table 1.

Sensitivity analysis
We first performed a one-way sensitivity analysis on each parameter to understand the sensitivity of equilibrium incidence to parameter ranges in the absence of interventions. We

Baseline sensitivity analyses
The sensitivity analyses show that the most marked impact on equilibrium incidence arose from the rate of late TB progression followed by the CDR (Fig 3). The proportion of superspreaders among persons with infectious TB and the relative infectiousness of superspreaders compared to low-spreaders also had a substantial impact. Supplementary Fig 2 presents a multidimensional sensitivity analysis, using the LHS method to sample 1000 parameter sets, which was consistent with the results of the one-way sensitivity analysis.

Impact of targeted interventions
Targeting the active case finding intervention towards super-spreaders was more effective than mass intervention in all setting scenarios. However, its effectiveness was particularly marked in settings with low case detection, regardless of transmission rates. For example, in the first scenario of both high transmission and high CDR, a 20% untargeted active case finding intervention led to a 22.8% reduction in incidence over 20 years, while with an 80% targeting an equal active case finding intervention resulted in a 35.9% reduction in TB incidence over the same period. However, in the scenario of high transmission but low CDR, the targeted intervention is much more effective (achieving a 42.2% reduction) than the untargeted intervention, which only reduced incidence by 22.8% over 20 years, considering a 20% active case finding and an 80% level of targeting. Interventions in settings of low transmission and high case detection rates had a relatively minor impact on the burden (Supplementary Fig. 3).
We evaluated the impact of intensifying active case finding (q) from zero to 40% detection and varying the proportion of this intervention targeting the super-spreaders (d). As shown in  Active case finding has been described as "turning off the tap" in TB control intervention strategies since it represents a method of identifying individuals with TB and promptly initiating treatment to avoid further onward transmissions [50]. Previous TB modelling has suggested that active case finding with highly sensitive diagnostic tools can reduce delay to treatment and so significantly reduce TB transmission [51]. are consistent with what has previously been assumed. Nevertheless, the exact values of these quantities remain uncertain, and future research to refine these quantities may modify this conclusion.
The current model structure enhances flexibility around the assumption of infectiousness heterogeneity that allowed for reflection of a broad range of factors that can alter active TB cases infectivity level [5,58]. Thus, the model is able to incorporate many previous compartmental modelling structural approaches, including those that stratified active TB cases' infectivity into two levels, as non-infectious and infectious [13][14][15][16][17], or model structures that incorporate three levels of infectivity, as non-infectious, smear-negative infectious and smear-positive infectious [30,31,59]. However, in the application of this model to a particular epidemiological setting, data to estimate the proportion of superspreaders is essential, since this information has a particularly marked impact on both baseline disease burden and intervention effectiveness.
We used empirical data to parameterise our model, but a noteworthy limitation is that these results may not be generalizable to other settings. However, we are not aware of any past work that has used empiric data to inform model parameters on the proportions of infectious TB and super-spreaders at all. As with any model-based analyses, our study has limitations that arise from its assumptions. In addition to those introduced above, our model is not intended to represent a specific setting, and as such does not incorporate stratification by age,

Conclusions
The approaches we used to inform model parameters related to infectiousness heterogeneity from the observed distribution of secondary infections can be useful for future modelling studies of TB, in particular, and other infectious diseases, in general. The principles of implementation could be used in future TB modelling studies that represent specific epidemiological settings, especially when TB contact investigation data are available to estimate the proportion of super-spreaders. In the usage of advanced point-of-care technologies, targeting active case-finding interventions to super-spreaders is likely to be especially beneficial in low CDR settings.

Ethics approval and consent to participate
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