Mathematical Modelling and Tuberculosis: Advances in Diagnostics and Novel Therapies

Abstract

As novel diagnostics, therapies, and algorithms are developed to improve case finding, diagnosis, and clinical management of patients with TB, policymakers must make difficult decisions and choose among multiple new technologies while operating under heavy resource constrained settings. Mathematical modelling can provide helpful insight by describing the types of interventions likely to maximize impact on the population level and highlighting those gaps in our current knowledge that are most important for making such assessments. This review discusses the major contributions of TB transmission models in general, namely, the ability to improve our understanding of the epidemiology of TB. We focus particularly on those elements that are important to appropriately understand the role of TB diagnosis and treatment (i.e., what elements of better diagnosis or treatment are likely to have greatest population-level impact) and yet remain poorly understood at present. It is essential for modellers, decision-makers, and epidemiologists alike to recognize these outstanding gaps in knowledge and understand their potential influence on model projections that may guide critical policy choices (e.g., investment and scale-up decisions).

Figure 1

A simple epidemiological model of TB. Uninfected individuals that are exposed to TB can become infected with TB, which can result in either a long-standing infection that is asymptomatic and noninfectious (latent TB) or progress at some point (“reactivation”) to a condition that is infectious and generally symptomatic (active TB). Detection and effective treatment can cure active TB. For simplicity, some other important features of natural history of TB are not shown here (but are generally included in compartmental models of TB), including reinfection, spontaneous resolution (“self-cure”), and mortality.

Figure 2

(a) The rate of bacterial shedding over the duration of infection may be a constant function (black line) or change over time (blue line). (b) The cumulative number of contacts exposed to TB over the duration of infection may increase linearly over time (black line) or may plateau as contact pool becomes saturated or patient is too ill to circulate in the community. The potential impact of a novel intervention may depend on this assumption; given a linear increase, earlier intervention (t ₁) would be likely preferred. While given the second curve with only a small increase and plateau, the impact between intervening at t ₁ and t ₂ might not be as great; therefore other factors including cost-effectiveness may come into play. (c) The cumulative number of secondary cases resulting from one index case in relation to the number of cumulative infected contacts: there are factors associated with bacterial virulence and host susceptibility that impact the rate of progression from infection to disease. This rate may be steeper among immunosuppressed contacts, for example, (dotted line) compared with immunocompetent contact (solid line). (d) The effective reproductive number (R _e) is the number of secondary cases generated over a given time period. Bacterial shedding, contact mixing pattern, and bacterial and host susceptibility all contribute to the overall rate of secondary cases generated over time; depending on what assumptions are made these rates could be thought to stay constant over time or vary, perhaps tapering off over the duration of infection. By reducing the time to diagnosis and treatment initiation we hope to reduce the number of secondary cases but the amount of impact depends on assumptions around the shape of the curve over time. The hashed area represents the secondary cases generated from one index case while the shaded area represents the potential reduction in secondary cases given an intervention at t ₂. Figure adapted with permission from Dowdy et al. [7].

Figure 3

A simple epidemiological model of drug resistant (DR-)TB. This model divides the transmission cycle of TB into two arms: transmission of DS-TB and DR-TB (which is shown in red). For simplicity and comparability, the transmission cycle of DR-TB is structurally similar to DS-TB. The difference between DS-TB and DR-TB can be characterized by difference in rates of transition between different compartments. (E.g., if the transmission fitness of DR-TB is less than that of DS-TB, the rates of new infections of DR-TB are lower compared to DS-TB.) The acquisition of drug resistance during treatment resulting from de novo mutations is a primary way in which drug resistance enters the population. Subsequently, drug resistance can spread via transmission events. Increasing the rate at which DR-TB is successfully diagnosed and treated (e.g., through drug susceptibility testing and regimen modification) can be modeled as an increase in the flow from compartment “Active DR-TB” back to “Latent DR-TB” (or, in an alternative formulation, back to uninfected).

Figure 4

Proliferation of drug resistance following the launch of new first-line drug regimen. (a) The effective reproductive ratio of DR-TB (R _e,dr) is the expected number of secondary cases of active, resistant TB resulting from a single case of DR-TB (shown as the grey shaded area). An increase in the relative transmission fitness of DR-TB (e.g., due to compensatory mutations; shown by the blue arrow) increases R _e,dr (shown by the blue hatched area). Shortening the average duration of DR-TB infections (e.g., by deployment of DST, and effective second-line treatment; shown by the red arrow) decreases R _e,dr (shown by the red hatched area). However, the rate of acquisition of drug resistance (e.g., due to de novo mutations against drugs in the treatment regimen) does not factor in the calculation of R _e,dr (b, c, and d). The trajectories of the prevalence of DR-TB just following the launch of a hypothetical new drug regimen are affected by both the acquisition rates and the R _e,dr of DR-TB, but their effects will be more pronounced at different time periods. Acquisition-driven drug resistance is expected to be more frequent in the first 5 years (pink area), while transmission-driven TB relatively later (blue area). (b) For two hypothetical DR-TB strains with similar R _e,dr, but different acquisition rates, we may observe difference in their prevalence in the short term, but over time they are expected to result in similar levels of resistance. (c) In contrast, for strains with similar acquisition rates, but different R _e,dr, we may not observe significant difference in their prevalence in the short term, but the levels of drug resistance can diverge significantly. Factors that affect R _e,dr will affect the trajectories of DR-TB prevalence—for example, deployment of DST that achieve reduction in average duration of infection (red arrow) can reduce prevalence of DR-TB over longer term. (d) DR-TB strain with larger acquisition rate and smaller R _e,dr is expected to be more prevalent over the short term compared to a strain with lower acquisition rate and higher R _e,dr, but the prevalence of DR-TB is flipped between two hypothetical strains over longer term. Hence, short term prevalence of DR-TB alone may not be a reliable predictor of the prevalence over longer term. Figures are only illustrative and not drawn to scale.