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Research Methods in Healthcare Epidemiology and Antimicrobial Stewardship—Mathematical Modeling

Part of: Research and Methodology in Infection Control

Published online by Cambridge University Press:  08 August 2016

Sean L. Barnes ,
Parastu Kasaie ,
Deverick J. Anderson  and
Michael Rubin
Sean L. Barnes*
Affiliation:
Department of Decision, Operations & Information Technologies, Robert H. Smith School of Business, University of Maryland, College Park, Maryland
Parastu Kasaie
Affiliation:
Department of Health, Behavior and Society, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, Maryland
Deverick J. Anderson
Affiliation:
Department of Medicine, Duke University School of Medicine, Durham, North Carolina
Michael Rubin
Affiliation:
Department of Internal Medicine, University of Utah School of Medicine, Salt Lake City, Utah
*
Address correspondence to Sean L. Barnes, PhD, Department of Decision, Operations & Information Technologies, Robert H. Smith School of Business, University of Maryland, 4352 Van Munching Hall, 7699 Mowatt Ln, College Park, MD 20742-1815 (sbarnes@rhsmith.umd.edu).
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Abstract

Mathematical modeling is a valuable methodology used to study healthcare epidemiology and antimicrobial stewardship, particularly when more traditional study approaches are infeasible, unethical, costly, or time consuming. We focus on 2 of the most common types of mathematical modeling, namely compartmental modeling and agent-based modeling, which provide important advantages—such as shorter developmental timelines and opportunities for extensive experimentation—over observational and experimental approaches. We summarize these advantages and disadvantages via specific examples and highlight recent advances in the methodology. A checklist is provided to serve as a guideline in the development of mathematical models in healthcare epidemiology and antimicrobial stewardship.

Infect Control Hosp Epidemiol 2016;1–7

Type
SHEA White Papers
Information
Infection Control & Hospital Epidemiology , Volume 37 , Issue 11 , November 2016 , pp. 1265 - 1271
Copyright
© 2016 by The Society for Healthcare Epidemiology of America. All rights reserved 

Mathematical models are abstract representations of real-world systems and can serve as tools to inform clinical decision-making. They can support a variety of efforts in healthcare epidemiology and antibiotic stewardship (HE&AS), including guiding data collection and empirical analysis, testing explanatory hypotheses about mechanisms driving observed real-world patterns, and informing policy as well as intervention design and evaluation. Mathematical models are classified on the basis of several criteria, including whether they are used to model system behavior that is static or dynamic, stochastic or deterministic, and discrete or continuous. In this review, we provide an overview of 2 of the more common types of mathematical models used to understand and describe infectious diseases: compartmental models and agent-based (AB) simulation models. More detailed information about mathematical modeling (as applied to HE&AS) is available in several key references.Reference Bonten, Austin and Lipsitch 1 – Reference Riley 10

ADVANTAGES AND DISADVANTAGES

Mathematical modeling provides several advantages over observational and experimental approaches (Table 1).Reference North, Collier and Vos 11 – 13 For instance, models can be used to gain insight when experimenting with the real system is too difficult, time consuming, expensive, or unethical. In addition, mathematical models can help to evaluate the external validity of traditional studies and explore scenarios beyond observed settings. By contrast, translating insight generated from modeling analyses into practice is difficult because practitioners often rely on more traditional experimental methods (eg, randomized controlled trials) to inform decision-making. Modeling teams also require a distinct and typically multidisciplinary set of individuals to be productive; in addition to clinicians, mathematicians, and programmers, modeling groups often include statisticians, epidemiologists, and data analysts, among others, which can raise the barrier to entry for research in this domain.

TABLE 1 Advantages, Disadvantages, and Potential Pitfalls of Using Mathematical Modeling in Healthcare Epidemiology and Antimicrobial Stewardship Research

NOTE. AB, agent-based.

Compartmental Models

Compartmental modeling is a widely used methodology for simulating the behavior of complex systems, typically characterized by dynamic, tightly coupled, and nonlinear behavior that is difficult to characterize using other methods.Reference Sterman 14 The most common form of a compartmental model leverages a system of coupled ordinary differential equations to model the dynamics of one or more quantities of interest over time. As applied to healthcare epidemiology, compartmental models are most often used to model how proportions of susceptible, infected, and recovered individuals evolve over time in a healthcare or community setting (Figure 1).

FIGURE 1 Key feature summary of compartmental and agent-based transmission models. Compartmental modeling is a top-down approach that models the evolution of stocks of susceptible, infected, and recovered individuals over time and flows of individuals between the compartments. Note that β is the transmission rate and γ is the recovery rate. Agent-based modeling is a bottom-up approach that explicitly simulates interactions between potentially heterogeneous agents (eg, patients, healthcare workers) in a defined environment, which serves as the mechanism for transmission. Individual agent states are aggregated to monitor stocks of susceptible, infected, and recovered individuals over time. Both modeling approaches can generate the dynamics shown in the figure to the right; however, stochastic variations are not shown, which are present in some systems dynamics models and most agent-based models.

Compartmental models are often specified via a set of analytical equations that are computationally inexpensive to simulate and therefore are easily scaled to different-sized systems. These equations can be used to explicitly derive important relationships between model parameters. The most notable result from this type of analysis is the basic reproduction number (R0), which represents the expected number of secondary cases per primary case in an entirely susceptible population. These types of results can provide significant insight into system dynamics because the results are generalizable across all parameter values and do not have to be explicitly observed or simulated.

By contrast, these models often oversimplify transmission dynamics in favor of analytic tractability (ie, the ability to solve and simulate the system of equations) and are somewhat limited in their ability to capture heterogeneity (ie, individual characteristics and behavior). For example, compartmental models often assume the principle of mass action—through which all individuals in the population are equally likely to interact with each other.Reference Rahmandad and Sterman 15 Also, most experimentation with compartmental models is conducted in the form of sensitivity analysis, through which model parameters are varied and the effects on primary outcomes are observed. Although useful, this approach can sometimes have limited impact if the experimentation does not reflect realistic scenarios.

AB Models

AB modeling provides a more explicit representation for studying complex, dynamic systemsReference An 16 – Reference Zorzenon dos Santos and Coutinho 19 and serves as a virtual laboratory for exploring new approaches to infection control. AB models consist of a set of “agents” that encapsulate the behaviors of various entities that constitute the system of interest. Agents can interact with each other and/or the environment on the basis of a set of simple rules that lead to a series of population-level patterns or outcomes (Figure 1), many of which may be unexpected (a concept known as emergence). In doing so, these models can be used to estimate the risks of disease and the effects of interventions at the individual level.

AB models offer significant flexibility for modeling dynamics at the individual level that will generate behaviors at the population level. In the context of infectious disease modeling, these dynamics include the definition of contact networks, mobility patterns, and characteristics relevant to disease progression and treatment outcomes at the individual level that govern disease transmission at the population level. Moreover, the multilevel nature of such models enables explicit definition of various interventions at the patient, facility, and community levels (eg, contesting TB contact-tracing with an active-case finding campaignReference Kasaie, Andrews, Kelton and Dowdy 20 ) and provides a powerful experimental platform to study the system’s behavior and predict future trends. Unlike many (deterministic) compartmental models, AB models explicitly capture system uncertainty via the direct application of random variables, thus providing a more realistic framework for representing stochastic (ie, random) behaviors and estimating risks associated with potential interventions.

The realism generated by these models, however, is offset by the additional complexity in development and analysis, as well as computational demands for experimentation (particularly for large-scale models). Traditional programming languages used to develop AB models demand considerable computer programming expertise and can be time consuming. A standard curriculum for teaching AB simulation rarely exists.Reference Macal and North 21 Furthermore, the complex structure and parameter-rich nature of AB models pose several challenges for calibration, validation, and sensitivity analysis.Reference Kasaie and Kelton 22

PITFALLS AND TIPS

Some pitfalls in mathematical modeling can be limited through use of best practices (Tables 1 and 2). First, model detail should match the complexity of the problem being studied. Compartmental models often oversimplify model dynamics, which can lead to models that do not capture the behavior of the system sufficiently well. By contrast, AB models are often overly complex relative to the problem being modeled. Excessively detailed models can lead to unreasonable data and computational demands that may limit the scope of analysis and, in turn, serve as a barrier to translating findings into practice. The best practice of modeling is seeking the simplest approach that enables answering the question of interest, or in other words, “make things as simple as possible but no simpler” (attributed to Albert Einstein). Along these lines, modelers and practitioners must work closely together to ensure that model complexity is appropriate given the problem under study.

TABLE 2 Checklist of Key Considerations When Developing a Mathematical Model for Healthcare Epidemiology and Antimicrobial Stewardship Research

NOTE. AB, agent-based.

Another key pitfall of mathematical modeling involves validation and calibration of the model, because these processes improve the model’s ability to produce results similar to the actual system. Many modeling studies are limited in this regard, performing only cursory (and often subjective) checks that 1 or 2 model outputs are similar to observed measures of the same. Calibration is especially challenging for large-scale AB models that employ many parameters, and these models often require a multilevel statistical validation focused on both individual- and population-level outcomes. Modeling human behavior and the interactions between agents is a difficult challenge; thus, efforts to validate modeled behavior with social theories are often neglected. Finally, many models are often calibrated to reproduce the behavior of a single site, rather than to approximate more representative behavior across multiple sites. We recommend that future modeling studies dedicate more robust efforts to this process and execute proper hypothesis testing to demonstrate that model outputs are representative of observed performance measures.

MATHEMATICAL MODELING IN HEALTHCARE EPIDEMIOLOGY AND ANTIBIOTIC STEWARDSHIP—REVIEW OF SPECIFIC EXAMPLES

Many applications of mathematical modeling in HE&AS research consist of 2 primary objectives:

  1. 1. To simulate the dynamics of patient-to-patient transmission of resistant or susceptible organisms in a healthcare or community setting

  2. 2. To evaluate the effect of model-based parameters or interventions on transmission of, and infection with, clinically relevant organisms

Deterministic compartmental models—for which model dynamics are entirely predictable for a given set of model equations and initial conditions—have been adapted to many applications in HE&AS research since their inception.Reference Austin and Anderson 23 – Reference Sebille, Chevret and Valleron 28 D’Agata and colleaguesReference D’Agata, Horn, Ruan, Webb and Wares 26 improved on earlier transmission models of the type using proportions of susceptible, infected, and recovered individuals by including additional compartments for patients on the basis of whether they were receiving antibiotics, which can affect the likelihood of acquisition or transmission. However, the inclusion of these additional compartments—although more realistic—also led to complex model equations with many unknown parameters.

A major limitation of many of the aforementioned studies is that they do not account for the variability often observed in the prevalence of colonized and/or infected patients and healthcare workers over time. More recent compartmental models have incorporated stochastic dynamics, which better account for this behavior, particularly in smaller systems such as intensive care units.Reference Austin, Bonten, Weinstein, Slaughter and Anderson 29 – Reference Bootsma, Diekmann and Bonten 35 For example, Bootsma and colleaguesReference Bootsma, Diekmann and Bonten 35 developed such a model across 3 hospitals to evaluate several infection control interventions with many parameters informed by a large, tertiary care medical center in the Netherlands. This was one of the first such compartmental models to attempt to capture the effect of superspreaders and patients with high risk of acquisition, 2 characteristics more naturally suited to AB modeling.

AB models have been applied to the study of disease spread in a variety of settings, ranging from specific care unitsReference Hotchkiss, Strike, Simonson, Broccard and Crooke 18 , Reference Forrester and Pettitt 36 to emergency departments,Reference Laskowski, Demianyk, Witt, Mukhi, Friesen and McLeod 37 hospitals,Reference Barnes, Golden and Wasil 38 – Reference Meng, Davies, Hardy and Hawkey 40 nursing homes,Reference Jaramillo, Taboada, Epelde, Rexachs and Luque 41 , Reference Lee, Singh and Bartsch 42 and communities.Reference Macal, North and Collier 43 AB models enable better incorporation of population heterogeneity with regard to population demographic characteristics, contact networks, and mobility patterns.Reference van Kleef, Robotham, Jit, Deeny and Edmunds 6 For example, Macal et alReference Macal, North and Collier 43 modeled community-based MRSA transmission in a synthetic population based on Chicago, Illinois, which included distinct representations of individual demographic characteristics (eg, age, gender, race) and activity patterns informed by national survey data. This model produced an accurate estimate of community-associated MRSA incident rates from 2000 through 2010, but also required a broad set of assumptions, extensive data from disjoint sources, and extensive development and calibration.

Some studies actually employ both methods and compare them directly.Reference Rahmandad and Sterman 15 , Reference D’Agata, Magal, Olivier, Ruan and Webb 27 For example, Rahmandad and StermanReference Rahmandad and Sterman 15 construct susceptible-exposed-infected-recovered models using compartmental and AB approaches over a variety of contact network structures. They highlight many of the aforementioned advantages and disadvantages of each approach but also stress the importance of finding a balance between the 2 paradigms, stating that results can be indistinguishable for larger and more homogenous populations. Sensitivity analysis is also critical, but extensive analysis in this dimension is difficult for computationally intense AB models.

Many compartmental and AB transmission models are accompanied by systematic analysis of one or more model-based parameters or potential infection control interventions. Hand hygiene compliance is by far the most studied interventionReference Hotchkiss, Strike, Simonson, Broccard and Crooke 18 , Reference Beggs, Shepherd and Kerr 25 , Reference D’Agata, Horn, Ruan, Webb and Wares 26 , Reference Sebille, Chevret and Valleron 28 – Reference Cooper, Medley and Scott 30 , Reference Grundmann, Hori, Winter, Tami and Austin 32 , Reference McBryde, Pettitt and McElwain 33 , Reference Barnes, Golden and Wasil 38 , Reference Rubin, Jones and Leecaster 44 – Reference Montville, Chen and Schaffner 46 ; however, many studies have also used mathematical models to investigate the potential benefits of cohorting,Reference Beggs, Noakes, Shepherd, Kerr, Sleigh and Banfield 24 , Reference Austin, Bonten, Weinstein, Slaughter and Anderson 29 , Reference Grundmann, Hori, Winter, Tami and Austin 32 , Reference McBryde, Pettitt and McElwain 33 , Reference Barnes, Golden and Wasil 38 , Reference Barnes, Golden, Wasil, Furuno and Harris 45 active surveillance and diagnostic testing (ie, to identify colonized but asymptomatic patients),Reference D’Agata, Horn, Ruan, Webb and Wares 26 , Reference Austin, Bonten, Weinstein, Slaughter and Anderson 29 , Reference Cooper, Medley and Scott 30 , Reference Robotham, Jenkins and Medley 34 , Reference Bootsma, Diekmann and Bonten 35 , Reference Barnes, Golden and Wasil 38 , Reference Eubank, Guclu and Kumar 47 contact precautions and isolation,Reference D’Agata, Horn, Ruan, Webb and Wares 26 , Reference Cooper, Medley and Stone 31 , Reference Bootsma, Diekmann and Bonten 35 , Reference Barnes, Golden and Wasil 38 decolonization,Reference Barnes, Golden and Wasil 38 , Reference Barnes, Harris, Golden, Wasil and Furuno 48 , Reference Hetem, Bootsma and Bonten 49 and environmental cleaning.Reference Barnes, Morgan, Harris, Carling and Thom 39 , Reference Rubin, Jones and Leecaster 44 In addition, authors have used these models to investigate the impact of several key model-based parameters, such as admission prevalence, unit/ward size, pathogen transmissibility, contact rates, and length of stay.Reference D’Agata, Magal, Olivier, Ruan and Webb 27 , Reference Cooper, Medley and Scott 30 , Reference McBryde, Pettitt and McElwain 33 , Reference Rubin, Jones and Leecaster 44

On the larger scale, few studies have analyzed interfacility or regional effects of transmission.Reference Lee, Singh and Bartsch 42 , Reference Barnes, Harris, Golden, Wasil and Furuno 48 , Reference Lee, Wong and Bartsch 50 – Reference Lee, McGlone and Wong 55 As a representative example, Lee et alReference Lee, McGlone and Wong 55 developed an AB model of MRSA transmission across 20 acute care hospitals in Orange County, California. When discharged patients returned to the community, they could be readmitted to any hospital in the region, thus providing a pathway for MRSA to spread from one hospital to another. This example illustrates the ability of AB models to capture transmission dynamics at individual, facility, and regional levels. These types of studies demonstrate the value of mathematical modeling as a tool for informing national or international efforts to control the spread of infectious pathogens. In addition, many of these studies leverage hybrid modeling approaches that combine compartmental and AB approaches in such a way as to exploit the advantages of each technique.Reference Bobashev, Goedecke, Feng and Epstein 56 – Reference Yu, Wang, McGowan, Vaidyanathan and Younger 58

MAJOR TAKE-HOME POINTS

Most studies using mathematical modeling in HE&AS consist of the application of either compartmental or AB modeling to simulate transmission and evaluate potential interventions. Successful modeling efforts often leverage close collaboration between modeling and practitioner expertise and include thorough validation and calibration prior to experimentation and analysis. In addition, striking the balance between model simplicity and complexity is an important consideration. Regardless of the specific modeling methodology, mathematical models should focus on providing actionable decision support to healthcare epidemiologists and antibiotic stewards. To achieve this objective, and in light of the limitations and pitfalls summarized above, we propose a set of best practices for developing mathematical models (Table 2).

CONCLUSIONS

Mathematical modeling is a valuable methodology for providing insight into relevant problems in HE&AS. It is a complementary approach, one that provides a unique perspective relative to more traditional methods, such as randomized controlled trials. In many cases, mathematical models can be used to validate or mediate results found by observational or experimental methods; in other cases, mathematical models may provide insight that cannot be obtained via these more traditional approaches. In that sense, models are not intended to replace evidence produced by real-world clinical studies; instead, they can be used to explore a range of scenarios and interventions and to generate subsequent hypotheses that can provide clues to researchers and decision-makers faced with seemingly unlimited HE&AS intervention strategies but limited resources to implement and test them. With improved rigor and efforts to better leverage improvements in computational resources, mathematical modeling should play a significant role in future studies in this field.

ACKNOWLEDGMENTS

Financial support. None reported.

Potential conflicts of interest. All authors report no conflicts of interest relevant to this article.

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Figure 0

TABLE 1 Advantages, Disadvantages, and Potential Pitfalls of Using Mathematical Modeling in Healthcare Epidemiology and Antimicrobial Stewardship Research

Figure 1

FIGURE 1 Key feature summary of compartmental and agent-based transmission models. Compartmental modeling is a top-down approach that models the evolution of stocks of susceptible, infected, and recovered individuals over time and flows of individuals between the compartments. Note that β is the transmission rate and γ is the recovery rate. Agent-based modeling is a bottom-up approach that explicitly simulates interactions between potentially heterogeneous agents (eg, patients, healthcare workers) in a defined environment, which serves as the mechanism for transmission. Individual agent states are aggregated to monitor stocks of susceptible, infected, and recovered individuals over time. Both modeling approaches can generate the dynamics shown in the figure to the right; however, stochastic variations are not shown, which are present in some systems dynamics models and most agent-based models.

Figure 2

TABLE 2 Checklist of Key Considerations When Developing a Mathematical Model for Healthcare Epidemiology and Antimicrobial Stewardship Research