Effectiveness of Mathematical and Simulation Models for Improving Quality of Care in Emergency Departments: A Systematic Literature Review

Abstract

Objectives

The purpose of this systematic literature review is to critically evaluate the use of mathematical and simulation models within emergency departments (EDs) and assess their potential to improve the quality of care. This review emphasizes the critical need for quality enhancement in health care systems, with a specific focus on EDs.

Methods

This review incorporates studies that have investigated the quality of care provided in ED settings, employing assorted mathematical and simulation models for adult populations. Based on the selected studies, a narrative approach was used to synthesize the findings, focusing on outcome classification, simulation, and modelling. There are six outcome dimensions: safety, effectiveness, patient-centeredness, timeliness, efficiency, and equity.

Results

This review analyzed 112 studies, uncovering a distinct focus on a set of key performance measures within ED operations, accounting for 222 instances across these studies. Measures assessing timeliness were most frequent, occurring 111 times, indicative of a strong emphasis on operational efficiency aspects such as waiting times and patient flow. A total of 75 examinations were conducted on efficiency-related measures, with a specific focus on identifying and addressing operational bottlenecks and optimizing resource utilization. On the other hand, safety, patient-centeredness, and effectiveness were not as commonly represented, with only 3, 4, and 29 instances, respectively.

Conclusion

This review highlights the considerable potential of mathematical and simulation models to enhance ED operations, particularly regarding timeliness and efficiency. However, aspects such as patient safety, effectiveness, and patient-centeredness were underrepresented, while equity was absent across the studies, indicating a clear need for further research. These findings emphasize the importance of adopting a more thorough approach to evaluating and improving the quality of emergency care. Future research should also concentrate on refining data management practices, incorporating observational studies, and exploring various simulation tools to develop a more balanced and inclusive understanding of these models' applications.

Background and Significance

The Institute of Medicine (IOM) asserts that health care organizations rely on the quality of care and services to attain desired health outcomes. 1 In health care, quality improvement (QI) involves structural processes for delivering evidence-based care and is the World Health Organization's (WHO) priority. 2

According to Donabedian, quality is defined by effectiveness, efficiency, efficacy, acceptability, equity, and legitimacy. These are closely aligned with the six dimensions recommended by the WHO and the IOM. 1 3 4 Within the National Health Service (NHS) in the United Kingdom, QI focuses on continually enhancing care standards by emphasizing safety, timeliness, effectiveness, efficiency, equity, and patient-centeredness—collectively known as the STEEEP framework. 5

STEEEP serves as a comprehensive guide for QI in health care. It aims to ensure care that avoids harm (Safety), provides timely services without unnecessary delays (Timeliness), delivers evidence-based care to improve outcomes (Effectiveness), optimizes resources and minimizes waste (Efficiency), guarantees consistent care regardless of socioeconomic or demographic factors (Equity), and prioritizes care that respects individual patient needs and preferences (Patient-Centeredness).

To enhance the quality of care in Emergency Departments (EDs), the NHS and the United Kingdom government conceived a comprehensive strategy to improve patient experience, reduce waiting times, and restore high-quality care. Specifically, the proposal emphasizes workforce support and flexibility, with a target of a 76% of 4-hour waiting time by March 2024. 6 Extended waiting times are an issue as current guidelines mandate that 95% of patients must be admitted, transferred, or discharged within 4 hours of their arrival. 7 However, recent statistics unveiled a concerning decline in ED efficiency: Less than half of patients achieved the 4-hour benchmark in December 2022, and are increasingly likely to leave the ED without being examined. 8

The major challenges facing EDs are access blocking and overcrowding, which are correlated with increased mortality rates, longer lengths of stay, and higher costs for admitted patients. 9 10 Several industrial models (e.g., modelling and system analysis tools) are currently being implemented to improve the quality of care provided by health care systems and have reported many benefits. A study by Angelo et al. 11 has highlighted the efficacy of mathematical modelling in optimizing hospital resource allocation, reducing patient wait times, and improving resource utilization.

The health care sector employs mathematical modelling, including mechanistic, statistical, and simulation-based approaches, to enhance decision-making and operational processes. 12 Mathematical modelling employs equations and formal structures to depict real-world systems or processes. This extensive category encompasses mechanistic models, which illustrate causal mechanisms through theoretical constructs, and statistical models, which depend on empirical data to estimate parameters and reveal patterns. 13 Simulation modelling is sometimes considered a subset of mathematical modelling; however, it differs in that it represents system behaviors through computational algorithms rather than closed-form equations. Simulation modelling recreates complex systems by replicating processes and scenarios over time, allowing dynamic testing of interventions in environments where real-world experimentation may be impractical or unethical. 14 Simulation is a valuable technique utilized across disciplines, yet its purpose varies significantly between the natural and social sciences. In natural sciences and engineering, simulations are grounded in established theories, enabling precise modelling and prediction. 15 Conversely, in social sciences such as health care and ED management, simulation serves as a tool for theory development rather than validation, addressing social systems' complexities and adaptive nature where universal laws are absent. In ED management, simulations help explore hypothetical interventions, like changes in staffing or triage protocols, without disrupting actual patient care.

Simulation enables decision-makers to model system behavior, evaluates the impacts of operational and policy changes, and explores alternative solutions. Optimization techniques are frequently integrated within simulation models to identify the most effective resource configurations or policy scenarios under constraints. Optimization refers to determining the best-case outcome, typically a minimum or maximum value across simulated scenarios. However, the usefulness of optimization depends entirely on the quality and representativeness of the underlying simulation model. A poorly specified model will yield optimal solutions that may not translate meaningfully to the real world, highlighting the need for critical evaluation of both the modelling and optimization components. The health care literature frequently combines optimization and simulation methods to effectively address capacity challenges, including hospital bed availability, ED throughput, and resource allocation. For example, simulations can determine the required bed capacity 16 or evaluate staffing needs in medical assessment units. 17 18 In crisis scenarios, Farahi and Salimifard 19 used a hybrid discrete-event simulation (DES) and optimization model to assess ED resilience. Their model examined capacity constraints across various conditions and identified a critical threshold interarrival time of 13.8 minutes, below which the ED faced a heightened risk of operational failure. This case study illustrates how simulation and optimization can be employed to anticipate capacity limits and explore preemptive strategies in crisis scenarios.

However, it is crucial to recognize the limitations inherent in simulation modelling. As George Box famously noted, “All models are wrong, but some are useful.” 20 This perspective is particularly relevant in health care, where models simplify reality to offer actionable insights rather than absolute truths. In ED simulations, the focus is not on perfectly replicating the system but on developing robust frameworks to inform decision-making and policy planning.

This study reviews and critically analyzes the current literature concerning the use and objectives of mathematical and simulation models in improving the quality of care in the ED and explores their application in this context. Specifically, it investigates how these modelling approaches have been applied to address the dimensions of quality of care and their proposed potential improvements, as defined by the STEEEP principles. Additionally, it provides a comprehensive analysis of mathematical and simulation models (applicable to ED management) to assist ED managers in activity planning and policymaking. Finally, this study identifies existing gaps in the literature and provides recommendations for future studies designed to optimize the quality of care in the ED.

Methods

We employed the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) method. 21

Eligibility Criteria

The rigorous inclusion and exclusion criteria for the study were established using the Population, Intervention, Comparator, and Outcomes (PICO) framework.

Inclusion Criteria

• Population: Adult individuals using ED services.

• Intervention: All modelling and simulation approaches to improve health care provision in ED settings.

• Comparison: Any or no comparator.

• Outcome: The quality-of-care dimensions are identified as Safe, Timely, Effective, Efficient, Equitable, and Patient-centered care (STEEEP).

Information Sources and Search Strategy

Searches for this systematic literature review were conducted using a search strategy developed with expert input from the field. These terms were included in the search strategy: ED OR emergency department AND improve patient outcomes; OR patient outcomes; OR optimizing quality of care; OR demand forecasting AND modelling; OR simulation; OR operational research. Table 1 details the search strategies for each database. In March 2024, these searches were conducted on the three most extensive health education databases: Scopus, Web of Science, and PubMed. The reference lists and citations of each study were examined to identify additional relevant articles.

Table 1. Planning—search strategy.

1. Web of Science

ED or Emergency department or Emergency Room (All Fields) and improve patient outcomes or patient outcomes or optimising quality of care or optimizing quality of care or Demand Forecasting (All Fields) and modelling or modeling or simulation or operational research or operations research (All Fields) and Article or Review Article (Document Types) and Emergency Medicine (Web of Science Categories) last 5 years = 1,230 papers

2. PubMed

((ED [Title/Abstract] OR Emergency department [Title/Abstract] OR Emergency Room [Title/Abstract]) AND (improve patient outcomes [Title/Abstract] OR patient outcomes [Title/Abstract] OR optimising quality of care [Title/Abstract] OR optimizing quality of care [Title/Abstract] OR Demand Forecasting [Title/Abstract])) AND (modelling [Title/Abstract] OR modelling [Title/Abstract] OR simulation [Title/Abstract] OR operational research [Title/Abstract]) = 89 papers

3. Scopus

(ALL [ed OR emergency department OR emergency room] AND ALL [improve patient outcomes OR patient outcomes OR optimising quality of care OR optimizing quality of care OR demand forecasting] AND ALL [modelling OR modeling OR simulation OR operational research OR operations research]) Limit-to English Journal = 450 papers

Selection Process

To provide a comprehensive review, all relevant papers were collected, and their details were transposed into an Excel spreadsheet to exclude duplicates. The first author (T.A.A.) applied the inclusion/exclusion criteria to screen the titles and abstracts for potentially relevant papers and the accuracy of the data extraction was confirmed by the co-authors (J.B. and E.A.). Studies that employed simulation techniques and modelling to improve the quality of care in the ED were considered. Finally, the full texts of the selected papers were evaluated for further analysis, and the data were extracted and recorded in a spreadsheet.

Data Extraction Process

The relevant papers were identified, and their details were recorded in a spreadsheet. Data extraction encompassed the author's name, publication date, country, study aims, outcomes (the problems addressed), and modelling techniques. The authors resolved any discrepancies via discussion.

Synthesis Methods

This review conducted a narrative synthesis of findings from the selected studies, with a focus on the classification of optimization outcomes achieved through mathematical and simulation models. These were categorized into six dimensions: safety, effectiveness, patient-centeredness, timeliness, efficiency, and equity.

Results

Search Results

In total, 1,849 records were retrieved from Web of Science (1,230 records), PubMed (89), Scopus (450), and reference lists (see Fig. 1 ). After removing duplicates, the total number of records was 1,834. Based on titles and abstracts, and the inclusion criteria, 151 articles were selected. A second screening of the potentially relevant full texts led to the exclusion of 39 articles (ineligible for review), leaving 112 studies to be evaluated.

Fig. 1.

PRISMA flow diagram of the literature search and publication selection. ED, emergency department; PRISMA, Preferred Reporting Items for Systematic Reviews and Meta-Analyses; QI, quality improvement.

Study Selection

The studies were categorized to highlight the common themes and connections. This approach aimed to understand how simulation techniques and modelling impact the quality of care in ED and comprised 6 key dimensions, each containing 13 subdimensions (as detailed in the “Results of Syntheses” section).

A considerable emphasis was placed on timeliness and efficiency: 59 studies investigated strategies to reduce waiting times or lengths of stay, while 35 examined patient flow dynamics. ED crowding and bottlenecks were addressed in 28 studies, and demand forecasting for EDs was explored in 14 studies. Resource utilization was another primary focus, with 43 studies assessing staff allocation, bed management, room occupancy, and equipment distribution.

Additional themes included work pressure and staff burnout (14 studies), self-discharge rates (4 studies), staffing patterns and scheduling (7 studies), patient safety (3 studies), cost overruns (4 studies), and patient satisfaction and experience (8 studies).

The review highlighted a range of mathematical and simulation models applied across these studies, including DES (43 studies), Agent-Based Modelling (ABM; 14 studies), System Dynamics (SD; 9 studies), queueing models (15 studies), time series analyses (14 studies), Markov models (6 studies), and optimization techniques such as genetic algorithms (7 studies). QI tools—including Lean, Six Sigma, and Value Stream Mapping—were employed in eight studies ( Fig. 2 ).

Fig. 2.

Distribution of mathematical and simulation models in studies.

An overview of the relationship between quality dimensions and simulation models is presented in Fig. 3 , where the circle sizes correspond to the number of studies focussing on each dimension: timeliness (86), efficiency (52), effectiveness (24), safety (3), and patient-centeredness (4). While the figure provides a comprehensive overview, it reveals a notable research bias towards timeliness and efficiency, often at the expense of safety and patient-centeredness.

Fig. 3.

Intersection of mathematical and simulation models with quality-of-care dimensions.

This imbalance raises concerns about current research priorities and underscores the need for future studies to explore underrepresented areas. A thorough examination of methodological assumptions and potential biases is essential to enhance the application of simulation and modelling in EDs, ensuring a more balanced and comprehensive approach to improving health care quality.

Study Characteristics

This review includes a systematic analysis of 112 studies originating from countries such as the United States (29), the United Kingdom (17), Canada (12), the Kingdom of Saudi Arabia (1), and other countries across North America, Europe, Asia, and Australia (53). Notably, the majority of studies (77 out of 112) undertook examinations across multiple dimensions of quality of care and 28 studies demonstrated significant flexibility by incorporating multiple models or techniques. This diversity in both temporal and geographical aspects, and the extensive utilization of multiple models, underscores the richness and complexity of the research landscape within the ED context.

Results of Syntheses

Based on the definitions in the introduction, this review categorized specific performance measures according to their relevant dimensions to ensure that each measure aligned directly with its respective dimension. This section highlights key findings and identifies areas meriting further exploration to provide a sound perspective regarding the measures frequently employed to assess specific outcomes in ED care. As defined by the IOM, 1 these outcomes encompass several dimensions of quality of care, known as the STEEEP dimensions.

In the studies reviewed, the total number of instances of measures identified was 222. Of these, 111 were associated with timeliness, 75 with efficiency, 3 with safety, 4 with patient-centeredness, and 29 with effectiveness. These measures were specifically selected to assess the success of various intervention methods (including simulation and modelling) in ED settings, emphasizing their significance in understanding ED quality of care and how to promote improvement.

This section is organized into three subsections to provide a structured, comprehensive analysis.

Performance Measures Categorized by STEEEP Dimensions

Timeliness

To measure the timeliness of care, the literature primarily focused on measures such as waiting time (time before care initiation), length of stay (total time in the ED), and patient flow (movement through ED stages). Broader concepts, such as work pressure (stress and demands placed on the ED staff) and the overall ED performance (based on the holistic functionality and efficiency of the department). Patient flow is quantified by tracking the movement and progression of patients throughout the various stages of the ED; work pressure and burnout are gauged via staff surveys and workload intensity assessments, while ED performance encompasses metrics such as patient satisfaction, treatment success rates, and staff feedback. Distinct mathematical and simulation models have been employed to enhance ED performance concerning these measures. These include simulation combined with optimization, 22 DES, 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ABM, 39 40 time series modelling, 41 genetic algorithms, 42 multiobjective optimization, 43 and lean process mapping. 44 Health care managers adopting these methodologies can generate data-driven decisions and identify areas for improvement.

It should be acknowledged that although these techniques present solutions, their results depend on the underlying modelling assumptions and how well the models represent real-world dynamics.

Efficiency

ED efficiency has been scrutinized via various measures that quantify elements such as crowding, bottlenecks, which measure points in the ED process where patient flow is obstructed, resource utilization, which assesses the efficiency with which resources are used (including staff, beds, rooms and other equipment), and cost overruns, which measure instances where costs exceed the budget. Studies have explored a diverse range of mathematical and simulation models to enhance efficiency. For example, one study utilized a combination of simulation and optimization, suggesting its potential utility in determining optimal staffing levels and maximizing patient throughput, especially when considering budget constraints. 22 An additional study highlighted the potential of such techniques in identifying optimal resource arrangements (staff, equipment, and space) to achieve desired ED outcomes. 45 An SD model can provide insight into the impact that increasing volumes of emergency patient arrivals have on ED waiting times. 46 The Markov models are utilized to predict patient flow and crowding in EDs. However, their reliance on simplified probabilistic transitions may obscure the complexity of patient care pathways and may require additional modelling approaches that better capture system heterogeneity and care dynamics. 47 Time series analysis techniques, although promising in predicting patient volumes and offering insights into demand dynamics, must be approached with caution, primarily because they rely on the assumption of stable underlying SD, which may not apply in rapidly evolving ED contexts. 48

Safety

The literature revealed a variety of measures to evaluate the safety of care (with a specific focus on patient safety measures, focusing on incidents that could harm patients, such as medication errors or procedural complications). One study 49 conducted at Greenville Memorial Hospital, SC, identified beneficial outcomes by incorporating two distinct models (queueing theory and ABM) to assess the patient handoff processes (the transfer of patient care between health care professionals). The duration patients spend in the ED and the number of clinical working hours—termed “full-time equivalents”—were also examined. The study observed a 41.5% decrease in handoffs and a 14% reduction in patient waiting time in the ED; however, there was a minor rise in the number of clinical working hours. Additionally, the study introduced the concept of “overlapping staffing policies” to create a 1-hour overlap between the incoming and outgoing clinical shifts in the ED. This approach could permit the ED to effectively process a 10-to-15% increase in patient volume.

Patient-Centeredness

To measure patient-centeredness, the literature frequently employs measures such as self-discharge (patients leaving before treatment completion) and voluntary departure, which are crucial indicators of patient satisfaction and trust in the health care system. They reflect the patient's decision to leave the hospital before receiving the necessary treatment (often due to prolonged wait times or perceived inadequacies in care). The literature discusses the efficacy of two different models in ED operations. A study conducted 50 in an urban ED located in northern Manhattan, which serves approximately 25,000 patients annually, employed queueing theory to reallocate providers. Despite a 6.3% escalation in patient arrivals (over 6 months in 2004), the adjustments resulted in a 22.9% reduction in the percentage of patients who departed before receiving care. The reallocation of providers based on queueing model resulted in a 21.7% (161 patients) decrease in the number of patients leaving without care, despite a 5.5% (548 patients) increase in admissions (during the second half of the study). This proactive approach to adjusting staffing schedules (based on modelling results) significantly improved patient care and reduced wait times. Meanwhile, the behavioral Markov chain (BMC) model, as examined by Ricciardi et al. 51 in “A. Cardarelli” hospital in Naples, describes the emergency care process from a patient's perspective, noting that most risk of self-discharge occurs during initial wait times, while the BMC model can be adapted to numerous hospital settings by adjusting the transition probabilities.

A study by Cochran and Broyles 52 emphasizes the requirement for a comprehensive assessment method for managing ED capacity and validate the implementation of queuing theory in capturing patient safety concerns via the concept of reneging (which describes the behavior of ED patients who decide to “leave without treatment”) and serves as a measure of patient safety concerns and ED overcrowding. However, the authors emphasize caution when generalizing their findings as variations in patient behavior and the dynamic operational environments of different EDs can influence outcomes, underscoring the need for context-specific interpretations.

Effectiveness

To assess the effectiveness of care within the ED, the literature employs measures such as patient satisfaction (the contentment of patients with the care they received); patient experience (capturing the entirety of patient encounters throughout their visit); efficient staffing patterns (ensuring the availability of doctors and nurses during peak times); allocation of staff (strategically positioning staff where they are most needed); adjusting nurses' schedules (modifying work timings based on patient influx); and predicting future demand in the ED (forecasting potential patient arrivals). In this research, effectiveness is defined as the ED's capability to achieve desired outcomes. For example, elevated patient satisfaction levels and positive patient experiences are indicative of high-quality care, while well-organized staffing patterns guarantee that sufficient resources are available to cater to patient demand.

Numerous studies have employed mathematical and simulation models to explore the effectiveness of interventions in the ED. Techniques such as DES and ABM have frequently been employed. 26 36 53 54 55 56 However, this review indicates that, although the effectiveness dimension was considered in some studies, it was often addressed implicitly and not systematically integrated as a primary modelling objective. This limited methodological focus reduces the capacity of these studies to demonstrate how simulation models interact with effectiveness-related outcomes. Future research should give greater attention to explicitly incorporating the effectiveness dimension within the design of simulation models to ensure a more balanced and transparent analysis of all quality-of-care dimensions.

Equity

None of the studies evaluated equitability.

Synthesis of Modelling Approaches across STEEEP Dimensions

Table 2 illustrates the alignment between the STEEEP quality dimensions in EDs and various simulation and mathematical models. DES was predominantly applied to address timeliness and efficiency, particularly in studies aimed at reducing waiting times and improving patient flow, underscoring its strength in operational optimization. Although used less frequently, ABM was more closely associated with patient-centeredness, as it effectively captures complex interactions between patients and staff, and insights into the overall dynamics of patient–staff interactions within system-level simulations. SD primarily contributed to effectiveness, providing a macro-level perspective on long-term policy impacts and resource allocation strategies.

Table 2. Mathematical and simulation models employed across the included studies.

Quality dimensions STEEEP	ED area studied (problem)		Model									Total
			Discrete-event simulation	Agent-based modelling	System dynamics	Optimization approach	Queueing theory	Time series	Markov	Stochastic control approach	Quality approaches
Timely	Waiting time/length of stay		30	10	3	2	8	3	4	2	10	72
	Patient flow		14	3	5	4	3	3	3	0	4	39
	Work pressure and burnout/emergency department's performance		6	5	3	1	2	2	0	0	0	19
Efficient	Crowding/bottlenecks		9	1	0	2	0	6	2	2	8	30
	Resource utilization	Staff	6	4	1	1	2	0	2	2		18
		Beds/Rooms	10	2	3	1	3	2	1	1	2	25
		Equipment	1	1	1	1	1	0	0	1	0	6
	Cost overruns		3	1	0	1	1	0	1	0	2	9
Safety	Patient safety		0	1	0	0	3	0	0	0	0	4
Patient-centeredness	Self-discharge/Voluntary departure		0	0	0	0	1	0	1	0	0	2
Effective	Staffing pattern/allocation of staff/adjust nurses' schedules		1	1	0	1	0	0	1	1	0	5
	Demand in the ED/forecasts of ED arrivals/predict the number of required ED		3	0	0	0	0	6	0	0	2	11
	Patient satisfaction/patient experience		3	0	0	2	0	0	2	0	3	10
Total			86	29	16	16	24	22	17	9	31	250

Abbreviations: ED, emergency department; STEEEP, Safe, Timely, Effective, Efficient, Equitable, and Patient-centered care.

Models such as Queueing Theory and Time Series Analysis were often employed to enhance timeliness by predicting patient arrivals and optimizing scheduling. At the same time, Markov models and stochastic control approaches focused on efficiency and cost management. Equity remains largely underexplored across all modelling approaches, with limited efforts to address disparities in care delivery, highlighting a significant gap in the current research landscape. This uneven distribution of focus across the STEEEP dimensions reveals a prevailing bias towards operational metrics, often at the expense of broader patient-centeredness and equity considerations.

An Overview of the Range of Simulation and Modelling Approaches Applied in Emergency Departments

This review explores the diverse range of simulation and modelling approaches applied in EDs, examining their various applications and contributions to improving ED operations. Among these approaches, some of the most prominent and widely used include DES, ABM, and SD. DES models systems as sequences of discrete events over time, making it particularly effective for analyzing patient flow and resource utilization, and identifying bottlenecks within EDs. 57 ABM focuses on simulating the actions and interactions of individual agents—such as patients and staff—to capture complex and decentralized behaviors. However, its high computational demands can limit its use in structured ED workflows. 58 SD examines system-wide behaviors through feedback loops and stock-and-flow structures, which is valuable for assessing policy impacts and long-term trends, albeit with less detail at the patient level. 59

These models provide dynamic tools for evaluating complex ED operations. They complement traditional mathematical and statistical methods by simulating scenarios to guide optimization strategies. Together, they enable decision-makers to explore interventions, assess potential outcomes, and enhance system performance without disrupting patient care.

Discussion

This systematic literature review aims to critically evaluate the degree to which mathematical and simulation models provide an authentic and insightful portrayal of care quality in the ED. While aiming for comprehensiveness, the “Results” section is focused on exploring the model's potential for improving the quality of care, potentially dismissing other influential factors such as the settings or designs of the studies involved, which could affect the study outcomes. This section will identify significant knowledge gaps and provide a baseline for future research; however, there was limited evidence supporting the accuracy of the data collection process, the refinement of input data, or the creation of accurate simulation models based on a detailed analysis of ED settings. While simulation models can function without direct dependence on empirical data, the lack of accurate and contextual input data can limit their practical relevance.

Further advancements in modelling and simulation research in ED are required to address these limitations and improve the efficacy and reliability of the existing models. Notably, several studies 58 60 61 highlighted the inadequacies of current data collection methods. While these studies have promoted the use of advanced techniques, they also acknowledge practical challenges such as time and resource constraints. Earlier research by Konrad et al., 62 has also emphasized the need to clean and structure input data to ensure simulations accurately reflect real ED situations. Challenges in cleaning and structuring data often stem from inconsistent data formats, missing or incomplete records and variability in data sources, leading to inaccuracies in simulation outputs. Supporting these findings, recent research by Castanheira-Pinto et al. 63 further underscores the crucial role of meticulous data management. This includes maintaining data integrity over time, ensuring real-time updates, and implementing strict data validation protocols to avoid errors during simulation runs. Castanheira-Pinto et al. 63 showed how structured input can lead to better designs for emergency rooms. 64 These external studies reinforce the ongoing relevance of the issues identified in our included studies and highlight the necessity of addressing these challenges to advance the field. However, it is important to acknowledge that while empirical data can improve the calibration and validation of simulation models, it is not essential for constructing the models themselves. 64 Data quality alone cannot offset the inherent simplifications and assumptions embedded in simulation approaches.

Cross-sectional studies in EDs are most useful for understanding patient flows and waiting times 65 66 : gathering data at specific intervals provides a clear picture of current and developing conditions. Future research concerning medical record systems, extended wait times, and patient flow issues must consider a broad range of factors, such as the availability of professional staff and ED resources. Such approaches will enhance understanding of the related challenges and lead to increasingly effective mathematical and simulation models.

This review identified several other gaps concerning critical factors and patient behaviors within an ED context. Studies 36 67 68 identified these elements (including organizational and human factors such as workload, shift patterns, policies, procedures, resource management and language barriers) and emphasized the pivotal role of such issues in a robust and holistic health care system. While this review does not evaluate the empirical validity of the models, it is important to highlight that certain studies 22 46 69 seem to have neglected the complexities of the ED context in their modelling assumptions, which could restrict the contextual relevance of their findings.

This implies a need to consider such issues and the potential development of alternative methodologies and tools to bolster the precision and depth of health care system representations and enhance decision-making capabilities. Future research seeking to address these limitations and enhance the understanding and effectiveness of future models should incorporate direct observations conducted in EDs. 70 While data analytics can provide a snapshot of the ED setting, enriching interpretation and context are paramount and can be achieved via observational research and analysis of patient journeys to ensure the development of simulation models that are pertinent to clinical practice. Research methods may entail shadowing health care professionals, chronicling patient trajectories, documenting communication patterns among staff, and decoding patient behaviors. Direct immersion in an ED context provides researchers with detailed insights into the extant pathways, challenges, and models, and provides a profound understanding of the dynamism of the underlying processes. 49 Such approaches can identify areas for enhancement and establish effective mathematical and simulation solutions.

The included studies highlighted several key points regarding modelling and simulation research within an ED context. As with all modelling approaches, simulation models simplify complex real-world systems, limiting their ability to predict real-world outcomes fully. Numerous studies have recognized the need for further research to validate and refine these models to enhance their accuracy, applicability, and effectiveness. 34 41 67 69 71 72 73 74 75 Other studies 74 75 recommend validating models in different health care settings, accurate data collection, and incorporating real-world data into the validation process. This recommendation recognizes that while simulation models do not require empirical data, integrating real-world data is crucial for enhancing contextual validity and informing practical QI initiatives. Additionally, the models possess limitations, including their simplified nature and an inability to consider critical factors such as limited capacity and the boarding of admitted patients in the ED. 26 Thus, these should be targets for future modelling work to attempt to capture. 52 59 76 77 78 79

Adopting these suggestions will contribute to advancing the field of ED operations modelling and simulation. By combining these strategies, the ED can effectively allocate resources, reduce waiting times, improve patient flow and decision-making, and enhance the overall quality of care.

This review followed PRISMA guidelines and offered a detailed narrative analysis of intervention types and outcomes. Although the search across three major databases ensured broad coverage, relevant studies from other sources may have been missed. The restriction to English-language articles also introduced potential language bias. A significant limitation of ED modelling is the data cleaning and structuring complexity. When empirical data are used primarily for model initialization, calibration, or validation, inconsistent data entries, missing values, and unstandardized metrics can undermine simulation accuracy, resulting in biased or incomplete outcomes. Furthermore, the heterogeneity of study designs and modelling approaches hindered formal quality assessments, limiting the comparability and generalizability of the findings. Future research should focus on improving data management practices, expanding literature searches to include non-English sources and adopting standardized modelling frameworks to strengthen the reliability and applicability of simulation studies in ED contexts.

Conclusion

This systematic review critically evaluated the application of mathematical and simulation models in ED settings, revealing substantial strengths alongside critical limitations, particularly the absence of equity evaluations. DES proved the modelling approach more suited for trying to model enhancing efficiency and timeliness, especially in high-volume EDs where optimizing patient flow is essential. ABM offered valuable insights into patient-centeredness by simulating complex patient–staff interactions but faced scalability challenges in more significant EDs. SD was the most appropriate approach for assessing long-term policy impacts and how to improve overall effectiveness, though its limited granularity reduced its utility for real-time decision-making.

A significant challenge identified across studies was the lack of attention to equity, with many models failing to consider how optimization strategies might inadvertently disadvantage vulnerable patient groups. Proper data cleaning and organization remain crucial to ensure that simulation models accurately reflect real-world ED contexts, as inconsistencies or gaps in data can undermine model validity. Nevertheless, these data-driven improvements should be viewed as complementary to, rather than a substitute for, the models' conceptual soundness and contextual alignment. Cross-sectional studies have also been encouraged to better analyze patient flow and waiting times, providing valuable insights into operational bottlenecks.

Future research should prioritize understanding the real-world settings being modelled, including direct observation or engagement with clinical environments, since data alone does not always capture contextual dynamics. This may involve using observational insights to complement empirical and systematically compare various simulation tools and modelling algorithms to enhance the precision and applicability of simulation models. Moreover, incorporating equity-focused parameters such as socioeconomic status and access disparities into simulation frameworks is essential to ensure that efficiency improvements do not come at the expense of fairness. Finally, considering ED characteristics like size, type, and location, greater emphasis on context-specific modelling is warranted, as these factors significantly influence model outcomes. Addressing these gaps will better align simulation efforts with STEEEP goals, ultimately improving emergency care delivery quality, efficiency, and equity.

Clinical Relevance Statement

This systematic review highlights the potential of mathematical and simulation models to enhance the quality of care in EDs, providing actionable insights for improving operational efficiency and patient outcomes.

Multiple-Choice Questions

1. What is the primary focus of the systematic literature review?

   1. Implementation of new simulation tools

   2. Enhancing emergency department quality of care

   3. Reducing health care costs

   4. Improving patient satisfaction only

   Correct Answer : The correct answer is option b. Enhancing emergency department quality of care.

2. Which performance measure was most frequently examined in the reviewed studies?

   1. Safety

   2. Effectiveness

   3. Timeliness

   4. Equity

   Correct Answer : The correct answer is options a, b, and c: Timeliness, Safety, Effectiveness

3. What methodology was primarily used to synthesize the findings of the review?

   1. Observational analysis

   2. Randomized control trials

   3. Narrative approach

   4. Quantitative meta-analysis

   Correct Answer : The correct answer is option c. Narrative approach.

Funding Statement

Funding None.

Summary Points.

• This systematic review rigorously evaluated the application of mathematical and simulation models within ED settings, revealing substantial strengths alongside critical limitations, particularly the absence of equity evaluations.

• Correctly cleaning and organizing data are crucial to ensuring that simulation models accurately reflect a practical ED context.

• Cross-sectional studies were encouraged to help analyze and understand patient flow and wait times.

• Additionally, the integration of observational methodologies and a comparison of various simulation tools and modelling algorithms are recommended to enhance the precision and applicability of these models.

Addressing these gaps will optimize operational efficiency, ensure equitable patient-centered care and enhance public health policies and emergency care protocols.