Mathematical Modeling of the Transmission Dynamics of Clostridium difficile Infection and Colonization in Healthcare Settings: A Systematic Review

Background We conducted a systematic review of mathematical models of transmission dynamic of Clostridium difficile infection (CDI) in healthcare settings, to provide an overview of existing models and their assessment of different CDI control strategies. Methods We searched MEDLINE, EMBASE and Web of Science up to February 3, 2016 for transmission-dynamic models of Clostridium difficile in healthcare settings. The models were compared based on their natural history representation of Clostridium difficile, which could include health states (S-E-A-I-R-D: Susceptible-Exposed-Asymptomatic-Infectious-Resistant-Deceased) and the possibility to include healthcare workers and visitors (vectors of transmission). Effectiveness of interventions was compared using the relative reduction (compared to no intervention or current practice) in outcomes such as incidence of colonization, CDI, CDI recurrence, CDI mortality, and length of stay. Results Nine studies describing six different models met the inclusion criteria. Over time, the models have generally increased in complexity in terms of natural history and transmission dynamics and number/complexity of interventions/bundles of interventions examined. The models were categorized into four groups with respect to their natural history representation: S-A-I-R, S-E-A-I, S-A-I, and S-E-A-I-R-D. Seven studies examined the impact of CDI control strategies. Interventions aimed at controlling the transmission, lowering CDI vulnerability and reducing the risk of recurrence/mortality were predicted to reduce CDI incidence by 3–49%, 5–43% and 5–29%, respectively. Bundles of interventions were predicted to reduce CDI incidence by 14–84%. Conclusions Although CDI is a major public health problem, there are very few published transmission-dynamic models of Clostridium difficile. Published models vary substantially in the interventions examined, the outcome measures used and the representation of the natural history of Clostridium difficile, which make it difficult to synthesize results and provide a clear picture of optimal intervention strategies. Future modeling efforts should pay specific attention to calibration, structural uncertainties, and transparent reporting practices.

Appendix. Assessment of the quality of reporting of the transmission-dynamic mathematical models We have developed a questionnaire-based grid to assess the quality of reporting of the transmission dynamics models of Clostridium difficile in healthcare settings. Our grid is primarily intended to assess within each study, the abilities of authors to efficiently describe their model and to report the ensuing findings appropriately. We based our criteria in accordance with various existing modeling practice guidelines to inform policy decisions. These guidelines come from various fields of expertise such as modeling in environmental regulation [1,2] and healthcare modeling to inform medical decisions and the management of healthcare resources [3,4,5,6,7,8], which specifically included modeling of the transmission dynamics [9] and model-based economic evaluations (decision-analytic models) [10,11,12,13,14,15].
Questionnaire-based grid to assess the quality of reporting of dynamic models The assessment of the quality of reporting consists into 35 questions which are divided in seven sections containing each five questions. The sections are related to the following elements:

1) research question;
2) natural history representation and the transmission dynamics; 3) parameter estimates and data sources; 4) modeling approaches and mathematical methods; 5) model outcomes; 6) uncertainty analyses and sensitivity analyses; 7) validation and the quality of documentation.

Rating scale definition for the quality of reporting criteria
For the evaluation of each question, we used a four-point rating scale defined as follows: NOT APPLICABLE (N/A): The question does not apply to the study. LOW (L): The question has not been addressed by the study.

MEDIUM (M):
The question has been partially addressed by the study.

HIGH (H):
The question has been fully addressed by the study.

Description of the quality of reporting criteria
Below, we provide a brief explanation about the seven sections used in our assessment of the quality of reporting. The questionnaire can be applied to evaluate the quality of reporting of a general dynamic model applied to infectious disease. The seven different sections, used in our assessment, are defined as follows: 1. Research question: the study must determine clearly an objective or a problem being addressed [11,12,2,16,7,14,13,8]. In addition, the study must provide information on the target population [11,7,14,13,15,8] and the healthcare settings being modeled [11,7,14,15]. A time frame should also be provided [13,14] in order to contextualize the study within the previous research or to understand the time interval represented by the model. Finally, the study should define the outcomes of interest and indicate their relevance to the research question being addressed [3,11,12,14,13,7,15,8].
2. Natural history representation and the transmission dynamic: the natural history of the disease must be fully specified [3,11,12] by using a description of each health state and by showing each transition between them. Moreover, the study should provide a rationale [3,11,12,14,13,8] about the structure chosen to represent the natural history. The study must define the type of unit of representation (aggregate population or individuals) which is used to model the target population [7,8] and should report the inclusion of population heterogeneities [3,11,14,13,8]. The modeling of heterogeneities can be done by using additional stratifications in the natural history representation of the disease or by the use of an agent-based model which can track individuals with their specific characteristics. Finally, the transmission/exposure pathways (direct/indirect) of the pathogen that are included in the natural history must be clearly reported or be easily identifiable either by a textual description or by conceptual/logic diagrams.
3. Parameter estimates and data sources: every parameter used in the model should be reported (by a point estimate or by a distribution) [9,14,13,8] with their appropriate units. The units should be consistent with dimensions of physical quantities that these parameters are intended to represent (e.g., a rate should have a dimension of 1/(unit of time) or a probability should be dimensionless). Each study should describe the methodologies and data sources used to identify the parameter estimates. These methods can include: expert elicitation [3,11,12,4,13]; literature reviews [3,11,4]; data analyses [11,12,2,4,13,8]; calibration methods [2,4]. All sources of data must be reported using citations.

4.
Modeling approaches and mathematical methods: the study should provide justification for the modeling approach (e.g., aggregate model vs. individual/agent-based model, deterministic vs. stochastic) [11,12,4,5,14,13,15,8] and should report the mathematical methods (e.g., set of differential equations, Markov chains, Monte Carlo methods, logic rules) used for the transmission dynamics (i.e., progression of the disease over time) [9]. All the interactions modeled between the target population and other model components must be reported [8] by specifying where they occur in the model architecture. For the main findings of the study, the time horizon, the cycle length and the number of simulations done should be reported [3,11,12,7,14,13,8]. Also, the information about the simulation software and programming language should be reported [11,16,4].

5.
Model outcomes: the study must provide a clear definition for each outcome of interest [14] and should report clearly their baseline values (before any intervention). For the principal findings of the study, the outcomes must be reported numerically or should be presented in such a way that allows data extraction. Furthermore, the units used for the outcome measurements should be clearly defined in such a way that leaves no space for ambiguity or for misinterpretation (i.e., by using clear definitions of numerators/denominators and time horizon). Finally, the study should report a graphical representation of the transmission dynamics over time (e.g., incidence and/or prevalence of infection/disease) [9].
6. Uncertainty analyses and sensitivity analyses: the study should assess and then reported any kind of uncertainty analysis (to assess the effect of a lack of knowledge or the inherent variability in the model) or sensitivity analysis (a measure of the effect of a change in model outputs by varying the input values) that were conducted. The different types of uncertainty include: stochastic uncertainty (variability in outcomes) [17,5,13], parametric uncertainty (variability in the model inputs (parameters-specific)) [11,17,12,16,4,5,14,13,8], structural uncertainty (variability in the model choice of representation for the biological properties of the disease (natural history)) [11,12,16,5,9,7,14,13,8] [16,6,15,8]. Some graphical descriptions of model concepts should be given [8]. Finally, the study should not contain apparent errors, inconsistencies or missing information.

Results
The nine studies [18,19,20,21,22,23,24,25,26] included in our review were assessed independently by two reviewers for their quality of reporting and we reported it in detail in  [18] was evaluated differently because the study consisted in an "update article" for which the primary intent was to show the usefulness of dynamic modeling for C. difficile infection in order to study the sources of transmission in a hospital setting. In this case, the authors emphasize more on the rationales related to the use of a model of transmission dynamics rather than to report exhaustively their model in all details. Consequently, many questions were not applicable (N/A) for this study.
The percentage of a perfect agreement between the two reviewers reached 76% overall (excluding the items not applicable (N/A)). The few discrepancies between the two reviewer scores that were different by more than one adjacent point were assessed again and, then resolved by consensus or by using a third reviewer. The discrepancies were reported in Table S2.1 using intermediary scales (see Table S2.2) which represents the scores given by the two reviewers.