Integrating mathematical models with experimental data to investigate the within-host dynamics of bacterial infections

Abstract

Bacterial infections still constitute a major cause of mortality and morbidity worldwide. The unavailability of therapeutics, antimicrobial resistance and the chronicity of infections due to incomplete clearance contribute to this phenomenon. Despite the progress in antimicrobial and vaccine development, knowledge about the effect that therapeutics have on the host-bacteria interactions remains incomplete. Insights into the characteristics of bacterial colonization and migration between tissues and the relationship between replication and host- or therapeutically induced killing can enable efficient design of treatment approaches. Recently, innovative experimental techniques have generated data enabling the qualitative characterization of aspects of bacterial dynamics. Here, we argue that mathematical modeling as an adjunct to experimental data can enrich the biological insight that these data provide. However, due to limited interdisciplinary training, efforts to combine the two remain limited. To promote this dialogue, we provide a categorization of modeling approaches highlighting their relationship to data generated by a range of experimental techniques in the area of in vivo bacterial dynamics. We outline common biological themes explored using mathematical models with case studies across all pathogen classes. Finally, this review advocates multidisciplinary integration to improve our mechanistic understanding of bacterial infections and guide the use of existing or new therapies.

Keywords: host–pathogen interactions; mathematical biology; mechanistic model; parameter inference; within-host dynamics.

Figure 1.

Schematic representation of the relationships between different mathematical modeling techniques. Mathematical models can be divided into mechanistic and empirical on account of whether their parameters represent biological processes or simply characterize relations between variables in the data. While empirical models are necessarily data-dependent, mechanistic models can either be system-specific and fitted to data or, generic, explorative and not related to experimental data (theoretical). Additionally, depending on the biological question addressed, mechanistic models can be deterministic when only the average behavior of the system is of interest, or stochastic when unexplained variation in the behavior of the system matters too. Mechanistic data-driven models can serve different purposes: they can either be solved forward in time to make a forecast (prospective analysis) or can be solved backwards in time to perform parameter inference and model selection (retrospective analysis).

Figure 2.

An example of a flow diagram as a schematic representation of mechanistic models in microbiology (adapted from Kaiser et al. 2013). Boxes represent the variables of the system, in this case the number of bacteria in the caecum and caecal lymph node respectively. Mathematically these variables are shortened for convenience as NC and NL. The processes that change the state of the system are represented by arrows and, here, correspond to bacterial migration, replication and clearance. The rates at which these processes take place are quantified by parameters, in this case, μL, rL and cL corresponding to the rates of migration, replication and clearance, respectively.

Figure 3.

Model selection in a virtual study using tagged strains. Mice are infected with an equiproportionate mix of 10 wild-type isogenic strains (WITS) at t0. At t1, mice are sacrificed and bacterial copies per WITS are enumerated in their blood, liver and spleen. Color-filled circles represent present strains, while unfilled circles represent absent strains. The joint distribution of bacteria per WITS in the 3 tissues uniquely describes the state of the system at t1. Estimated bacterial distributions A and B are obtained at t1 for competing models A and B. Each estimated distribution is compared to the observed distribution and their difference summarized by a divergence measure. Model A yields the smallest difference and, thus, provides a better fit to the data.

Figure 4.

Coupling mathematical modeling with TIMER technique to distinguish between true variation in bacterial division rates and variation due to observational process (adapted from Claudi et al. 2014). The observed distribution of fluorescence intensity in the bacterial population appears continuous. Using stochastic models it is possible to quantify the variation expected from different identified sources of noise and compare it to the variation in the experimental data. If the aggregate variation from all sources of noise can account for the variation in the data, it is not necessary to implicate models of higher complexity, such as multiple bacterial subpopulations with distinct division rates.