Mathematical Modeling of "Chronic" Infectious Diseases: Unpacking the Black Box

Abstract

Background: Mathematical models are increasingly used to understand the dynamics of infectious diseases, including "chronic" infections with long generation times. Such models include features that are obscure to most clinicians and decision-makers.

Methods: Using a model of a hypothetical active case-finding intervention for tuberculosis in India as an example, we illustrate the effects on model results of different choices for model structure, input parameters, and calibration process.

Results: Using the same underlying data, different transmission models produced different estimates of the projected intervention impact on tuberculosis incidence by 2030 with different corresponding uncertainty ranges. We illustrate the reasons for these differences and present a simple guide for clinicians and decision-makers to evaluate models of infectious diseases.

Conclusions: Mathematical models of chronic infectious diseases must be understood to properly inform policy decisions. Improved communication between modelers and consumers is critical if model results are to improve the health of populations.

Keywords: Bayesian analysis; HIV; hepatitis C; theoretical models; tuberculosis.

Figure 1.

The effect of model structure on results. Each panel presents a model structure adjacent to a graphical representation of the fit of that model. The dark squares indicate the World Health Organization incidence estimates (the same for all models). The light circles and ribbons indicate model estimates of tuberculosis (TB) incidence and 95% uncertainty intervals in the absence of the intervention. The dark triangles and ribbons indicate model estimates of TB incidence and 95% uncertainty intervals in the presence of the intervention. The white inset indicates the estimate and 95% uncertainty interval of our primary outcome: the change in 2030 TB incidence with vs without the intervention. In addition to the structural elements depicted in the figures, each model also assumes that the transmission rates of tuberculosis decline linearly from 2002 to 2030. Latent TB infection (LTBI) < 5 years denotes a latent infection acquired within the past 5 years. LTBI > 5 years denotes a latent infection acquired more than 5 years earlier.

Figure 2.

The effect of prior distributions on results. Panels A, B, and C present graphical representations of the model fit under different formulations of prior distributions. Panel D presents a box-and-whiskers plot for the primary outcome under each set of priors: the dark horizontal line indicates the estimate, the shaded box indicates the 50% uncertainty interval, and the whiskers indicate the 95% uncertainty interval. ^aPrior distributions with central tendencies are either log-normal distributions when the parameter is a rate (with possible values 0 to infinity) or logit-normal distributions when the parameter is a proportion (with possible values 0 to 1). The standard deviations of these distributions are chosen so that the sampling range corresponds to a 95% confidence interval. TB, tuberculosis.

Figure 3.

The effect of likelihood on results. Panel A compares the use of 2 distributions—normal and uniform—in the likelihood at years 2000 and 2015.The light circles represent a sample simulation, while the dark squares represent the World Health Organization (WHO) estimates. The curves on the right represent the distributions for those years, and the value of the likelihood is the vertical distance to the intersection of the curve and the WHO estimate, indicated by the green circle. The normal likelihood gives greater weight when the simulation estimate is closer to the WHO estimate, while the uniform likelihood gives equal weight to any estimate that falls within the WHO range. The next 4 panels present graphical representations of the model fit using a likelihood made of (B) normal distributions at 2000 and 2015, (C) uniform distributions on each of the 16 years from 2000 to 2015, (D) independent normal distributions on the 16 years from 2000 to 2015, and (E) a normal likelihood for the 16 years from 2000 to 2015, with a correlation coefficient of 0.5 between the errors in any 2 years. Panel F shows a box-and-whiskers plot for the primary outcome under each formulation of the likelihood: the dark horizontal line indicates the estimate, the shaded box indicates the 50% uncertainty interval, and the whiskers indicate the 95% uncertainty interval. TB, tuberculosis.

Figure 4.

The effect of calibration targets on results. Panels A and D display the ability to reproduce incidence and mortality trends, respectively, of a model calibrated to incidence alone (using a correlated error likelihood). Panels B and E display the ability to reproduce the incidence and mortality, respectively, of a model calibrated to both incidence and mortality. Panels C and F show the changes by 2030 in incidence and mortality, respectively, if the intervention is undertaken under each of the 2 calibration procedures: the dark horizontal line indicates the estimate, the shaded box indicates the 50% uncertainty interval, and the whiskers indicate the 95% uncertainty interval. TB, tuberculosis.

Figure 5.

Sensitivity analyses. Panel A shows a comparison of the primary outcome (relative reduction in 2030 tuberculosis [TB] incidence if the intervention is undertaken) in high vs low quintiles of each input parameter. For each parameter, the light circle and bar show the results (estimate and 95% uncertainty interval) if we restrict our analysis to those simulations where the parameter value is among the highest 20% of all sampled values for that parameter; the dark square and bar show the results for simulations where the parameter value is in the lowest 20%. Panel B shows partial rank correlation coefficients, the correlation between the rank of a parameter and the rank of the outcome adjusted for all other parameters [33]. A value of 1 would indicate perfect square: the simulation with the greatest value of the parameter having the greatest value of the outcome, the simulation with the second-highest parameter having the second-greatest outcome, etc. The 6 most influential parameters for each analysis are shown.