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Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students

Roles Conceptualization, Investigation, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliations Department of Biological Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada, Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada

Affiliation Department of Biology, Memorial University of Newfoundland, St John’s, Newfoundland, Canada

• Korryn Bodner, 
• Chris Brimacombe, 
• Emily S. Chenery, 
• Ariel Greiner, 
• Anne M. McLeod, 
• Stephanie R. Penk, 
• Juan S. Vargas Soto

Published: January 14, 2021

• https://doi.org/10.1371/journal.pcbi.1008539
• Reader Comments

Citation: Bodner K, Brimacombe C, Chenery ES, Greiner A, McLeod AM, Penk SR, et al. (2021) Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students. PLoS Comput Biol 17(1): e1008539. https://doi.org/10.1371/journal.pcbi.1008539

Editor: Scott Markel, Dassault Systemes BIOVIA, UNITED STATES

Copyright: © 2021 Bodner et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Biologists spend their time studying the natural world, seeking to understand its various patterns and the processes that give rise to them. One way of furthering our understanding of natural phenomena is through laboratory or field experiments, examining the effects of changing one, or several, variables on a measured response. Alternatively, one may conduct an observational study, collecting field data and comparing a measured response along natural gradients. A third and complementary way of understanding natural phenomena is through mathematical models. In the life sciences, more scientists are incorporating these quantitative methods into their research. Given the vast utility of mathematical models, ranging from providing qualitative predictions to helping disentangle multiple causation (see Hurford [ 1 ] for a more complete list), their increased adoption is unsurprising. However, getting started with mathematical models may be quite daunting for those with traditional biological training, as in addition to understanding new terminology (e.g., “Jacobian matrix,” “Markov chain”), one may also have to adopt a different way of thinking and master a new set of skills.

Here, we present 10 simple rules for tackling your first mathematical models. While many of these rules are applicable to basic scientific research, our discussion relates explicitly to the process of model-building within ecological and epidemiological contexts using dynamical models. However, many of the suggestions outlined below generalize beyond these disciplines and are applicable to nondynamic models such as statistical models and machine-learning algorithms. As graduate students ourselves, we have created rules we wish we had internalized before beginning our model-building journey—a guide by graduate students, for graduate students—and we hope they prove insightful for anyone seeking to begin their own adventures in mathematical modelling.

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Boxes represent susceptible, infected, and recovered compartments, and directed arrows represent the flow of individuals between these compartments with the rate of flow being controlled by the contact rate, c , the probability of infection, γ , and the recovery rate, θ .

https://doi.org/10.1371/journal.pcbi.1008539.g001

Rule 1: Know your question

“All models are wrong, some are useful” is a common aphorism, generally attributed to statistician George Box, but determining which models are useful is dependent upon the question being asked. The practice of clearly defining a research question is often drilled into aspiring researchers in the context of selecting an appropriate research design, interpreting statistical results, or when outlining a research paper. Similarly, the practice of defining a clear research question is important for mathematical models as their results are only as interesting as the questions that motivate them [ 5 ]. The question defines the model’s main purpose and, in all cases, should extend past the goal of merely building a model for a system (the question can even answer whether a model is even necessary). Ultimately, the model should provide an answer to the research question that has been proposed.

When the research question is used to inform the purpose of the model, it also informs the model’s structure. Given that models can be modified in countless ways, providing a purpose to the model can highlight why certain aspects of reality were included in the structure while others were ignored [ 6 ]. For example, when deciding whether we should adopt a more realistic model (i.e., add more complexity), we can ask whether we are trying to inform general theory or whether we are trying to model a response in a specific system. For example, perhaps we are trying to predict how fast an epidemic will grow based on different age-dependent mixing patterns. In this case, we may wish to adapt our basic SIR model to have age-structured compartments if we suspect this factor is important for the disease dynamics. However, if we are exploring a different question, such as how stochasticity influences general SIR dynamics, the age-structured approach would likely be unnecessary. We suggest that one of the first steps in any modelling journey is to choose the processes most relevant to your question (i.e., your hypothesis) and the direct and indirect causal relationships among them: Are the relationships linear, nonlinear, additive, or multiplicative? This challenge can be aided with a good literature review. Depending on your model purpose, you may also need to spend extra time getting to know your system and/or the data before progressing forward. Indeed, the more background knowledge acquired when forming your research question, the more informed your decision-making when selecting the structure, parameters, and data for your model.

Rule 2: Define multiple appropriate models

Natural phenomena are complicated to study and often impossible to model in their entirety. We are often unsure about the variables or processes required to fully answer our research question(s). For example, we may not know how the possibility of reinfection influences the dynamics of a disease system. In cases such as these, our advice is to produce and sketch out a set of candidate models that consider alternative terms/variables which may be relevant for the phenomena under investigation. As in Fig 2 , we construct 2 models, one that includes the ability for recovered individuals to become infected again, and one that does not. When creating multiple models, our general objective may be to explore how different processes, inputs, or drivers affect an outcome of interest or it may be to find a model or models that best explain a given set of data for an outcome of interest. In our example, if the objective is to determine whether reinfection plays an important role in explaining the patterns of a disease, we can test our SIR candidate models using incidence data to determine which model receives the most empirical support. Here we consider our candidate models to be alternative hypotheses, where the candidate model with the least support is discarded. While our perspective of models as hypotheses is a view shared by researchers such as Hilborn and Mangel [ 7 ], and Penk and colleagues [ 8 ], please note that others such as Oreskes and colleagues [ 9 ] believe that models are not subject to proof and hence disagree with this notion. We encourage modellers who are interested in this debate to read the provided citations.

(A) A susceptible/infected/recovered model where individuals remain immune (gold) and (B) a susceptible/infected/recovered model where individuals can become susceptible again (blue). Arrows indicate the direction of movement between compartments, c is the contact rate, γ is the infection rate given contact, and θ is the recovery rate. The text below each conceptual model are the hypotheses ( H1 and H2 ) that represent the differences between these 2 SIR models.

https://doi.org/10.1371/journal.pcbi.1008539.g002

Finally, we recognize that time and resource constraints may limit the ability to build multiple models simultaneously; however, even writing down alternative models on paper can be helpful as you can always revisit them if your primary model does not perform as expected. Of course, some candidate models may not be feasible or relevant for your system, but by engaging in the activity of creating multiple models, you will likely have a broader perspective of the potential factors and processes that fundamentally shape your system.

Rule 3: Determine the skills you will need (and how to get them)

Equipping yourself with the necessary analytical tools that form the basis of all quantitative techniques is essential. As Darwin said, those that have knowledge of mathematics seem to be endowed with an extra sense [ 10 ], and having a background in calculus, linear algebra, and statistics can go a long way. Thus, make it a habit to set time for yourself to learn these mathematical skills, and do not treat all your methods like a black box. For instance, if you plan to use ODEs, consider brushing up on your calculus, e.g., using Stewart [ 11 ]. If you are working with a system of ODEs, also read up on linear algebra, e.g., using Poole [ 12 ]. Some universities also offer specialized math biology courses that combine topics from different math courses to teach the essentials of mathematical modelling. Taking these courses can help save time, and if they are not available, their syllabi can help focus your studying. Also note that while narrowing down a useful skillset in the early stages of model-building will likely spare you from some future headaches, as you progress in your project, it is inevitable that new skills will be required. Therefore, we advise you to check in at different stages of your modelling journey to assess the skills that would be most relevant for your next steps and how best to acquire them. Hopefully, these decisions can also be made with the help of your supervisor and/or a modelling mentor. Building these extra skills can at first seem daunting but think of it as an investment that will pay dividends in improving your future modelling work.

When first attempting to tackle a specific problem, find relevant research that accomplishes the same tasks and determine if you understand the processes and techniques that are used in that study. If you do, then you can implement similar techniques and methods, and perhaps introduce new methods. If not, then determine which tools you need to add to your toolbox. For instance, if the problem involves a system of ODEs (e.g., SIR models, see above), can you use existing symbolic software (e.g., Maple, Matlab, Mathematica) to determine the systems dynamics via a general solution, or is the complexity too great that you will need to create simulations to infer the dynamics? Figuring out questions like these is key to understanding what skills you will need to work with the model you develop. While there is a time and a place for involving collaborators to help facilitate methods that are beyond your current reach, we strongly advocate that you approach any potential collaborator only after you have gained some knowledge of the methods first. Understanding the methodology, or at least its foundation, is not only crucial for making a fruitful collaboration, but also important for your development as a scientist.

Rule 4: Do not reinvent the wheel

While we encourage a thorough understanding of the methods researchers employ, we simultaneously discourage unnecessary effort redoing work that has already been done. Preventing duplication can be ensured by a thorough review of the literature (but note that reproducing original model results can advance your knowledge of how a model functions and lead to new insights in the system). Often, we are working from established theory that provides an existing framework that can be applied to different systems. Adapting these frameworks can help advance your own research while also saving precious time. When digging through articles, bear in mind that most modelling frameworks are not system-specific. Do not be discouraged if you cannot immediately find a model in your field, as the perfect model for your question may have been applied in a different system or be published only as a conceptual model. These models are still useful! Also, do not be shy about reaching out to authors of models that you think may be applicable to your system. Finally, remember that you can be critical of what you find, as some models can be deceptively simple or involve assumptions that you are not comfortable making. You should not reinvent the wheel, but you can always strive to build a better one.

Rule 5: Study and apply good coding practices

The modelling process will inevitably require some degree of programming, and this can quickly become a challenge for some biologists. However, learning to program in languages commonly adopted by the scientific community (e.g., R, Python) can increase the transparency, accessibility, and reproducibility of your models. Even if you only wish to adopt preprogrammed models, you will likely still need to create code of your own that reads in data, applies functions from a collection of packages to analyze the data, and creates some visual output. Programming can be highly rewarding—you are creating something after all—but it can also be one of the most frustrating parts of your research. What follows are 3 suggestions to avoid some of the frustration.

Organization is key, both in your workflow and your written code. Take advantage of existing software and tools that facilitate keeping things organized. For example, computational notebooks like Jupyter notebooks or R-Markdown documents allow you to combine text, commands, and outputs in an easily readable and shareable format. Version control software like Git makes it simple to both keep track of changes as well as to safely explore different model variants via branches without worrying that the original model has been altered. Additionally, integrating with hosting services such as Github allows you to keep your changes safely stored in the cloud. For more details on learning to program, creating reproducible research, programming with Jupyter notebooks, and using Git and Github, see the 10 simple rules by Carey and Papin [ 13 ], Sandve and colleagues [ 14 ], Rule and colleagues [ 15 ], and Perez-Riverol and colleagues [ 16 ], respectively.

Comment your code and comment it well (see Fig 3 ). These comments can be the pseudocode you have written on paper prior to coding. Assume that when you revisit your code weeks, months, or years later, you will have forgotten most of what you did and why you did it. Good commenting can also help others read and use your code, making it a critical part of increasing scientific transparency. It is always good practice to write your comments before you write the code, explaining what the code should do. When coding a function, include a description of its inputs and outputs. We also encourage you to publish your commented model code in repositories such that they are easily accessible to others—not only to get useful feedback for yourself but to provide the modelling foundation for others to build on.

Two functionally identical codes in R [ 17 ] can look very different without comments (left) and with descriptive comments (right). Writing detailed comments will help you and others understand, adapt, and use your code.

https://doi.org/10.1371/journal.pcbi.1008539.g003

When writing long code, test portions of it separately. If you are writing code that will require a lot of processing power or memory to run, use a simple example first, both to estimate how long the project will take, and to avoid waiting 12 hours to see if it works. Additionally, when writing code, try to avoid too many packages and “tricks” as it can make your code more difficult to understand. Do not be afraid of writing 2 separate functions if it will make your code more intuitive. As with writing, your skill as a writer is not dependent on your ability to use big words, but instead about making sure your reader understands what you are trying to communicate.

Rule 6: Sweat the “right” small stuff

By “sweat the ‘right’ small stuff,” we mean considering the details and assumptions that can potentially make or break a mathematical model. A good start would be to ensure your model follows the rules of mass and energy conservation. In a closed system, mass and energy cannot be created nor destroyed, and thus, the left side of the mathematical equation must equal the right under all circumstances. For example, in Eq 2 , if the number of susceptible individuals decreases due to infection, we must include a negative term in this equation (− cγIS ) to indicate that loss and its conjugate (+ cγIS ) to the infected individuals equation, Eq 3 , to represent that gain. Similarly, units of all terms must also be balanced on both sides of the equation. For example, if we wish to add or subtract 2 values, we must ensure their units are equivalent (e.g., cannot add day −1 and year −1 ). Simple oversights in units can lead to major setbacks and create bizarre dynamics, so it is worth taking the time to ensure the units match up.

Modellers should also consider the fundamental boundary conditions of each parameter to determine if there are some values that are illogical. Logical constraints and boundaries can be developed for each parameter using prior knowledge and assumptions (e.g., Huntley [ 18 ]). For example, when considering an SIR model, there are 2 parameters that comprise the transmission rate—the contact rate, c , and the probability of infection given contact, γ . Using our intuition, we can establish some basic rules: (1) the contact rate cannot be negative; (2) the number of susceptible, infected, and recovered individuals cannot be below 0; and (3) the probability of infection given contact must fall between 0 and 1. Keeping these in mind as you test your model’s dynamics can alert you to problems in your model’s structure. Finally, simulating your model is an excellent method to obtain more reasonable bounds for inputs and parameters and ensure behavior is as expected. See Otto and Day [ 5 ] for more information on the “basic ingredients” of model-building.

Rule 7: Simulate, simulate, simulate

Even though there is a lot to be learned from analyzing simple models and their general solutions, modelling a complex world sometimes requires complex equations. Unfortunately, the cost of this complexity is often the loss of general solutions [ 19 ]. Instead, many biologists must calculate a numerical solution, an approximate solution, and simulate the dynamics of these models [ 20 ]. Simulations allow us to explore model behavior, given different structures, initial conditions, and parameters ( Fig 4 ). Importantly, they allow us to understand the dynamics of complex systems that may otherwise not be ethical, feasible, or economically viable to explore in natural systems [ 21 ].

Gold lines represent the SIR structure ( Fig 2A ) where lifelong immunity of individuals is inferred after infection, and blue lines represent an SIRS structure ( Fig 2B ) where immunity is lost over time. The solid lines represent model dynamics assuming a recovery rate ( θ ) of 0.05, while dotted lines represent dynamics assuming a recovery rate of 0.1. All model runs assume a transmission rate, cγ , of 0.2 and an immunity loss rate, ψ , of 0.01. By using simulations, we can explore how different processes and rates change the system’s dynamics and furthermore determine at what point in time these differences are detectable. SIR, Susceptible-Infected-Recovered; SIRS, Susceptible-Infected-Recovered-Susceptible.

https://doi.org/10.1371/journal.pcbi.1008539.g004

One common method of exploring the dynamics of complex systems is through sensitivity analysis (SA). We can use this simulation-based technique to ascertain how changes in parameters and initial conditions will influence the behavior of a system. For example, if simulated model outputs remain relatively similar despite large changes in a parameter value, we can expect the natural system represented by that model to be robust to similar perturbations. If instead, simulations are very sensitive to parameter values, we can expect the natural system to be sensitive to its variation. Here in Fig 4 , we can see that both SIR models are very sensitive to the recovery rate parameter ( θ ) suggesting that the natural system would also be sensitive to individuals’ recovery rates. We can therefore use SA to help inform which parameters are most important and to determine which are distinguishable (i.e., identifiable). Additionally, if observed system data are available, we can use SA to help us establish what are the reasonable boundaries for our initial conditions and parameters. When adopting SA, we can either vary parameters or initial conditions one at a time (local sensitivity) or preferably, vary multiple of them in tandem (global sensitivity). We recognize this topic may be overwhelming to those new to modelling so we recommend reading Marino and colleagues [ 22 ] and Saltelli and colleagues [ 23 ] for details on implementing different SA methods.

Simulations are also a useful tool for testing how accurately different model fitting approaches (e.g., Maximum Likelihood Estimation versus Bayesian Estimation) can recover parameters. Given that we know the parameter values for simulated model outputs (i.e., simulated data), we can properly evaluate the fitting procedures of methods when used on that simulated data. If your fitting approach cannot even recover simulated data with known parameters, it is highly unlikely your procedure will be successful given real, noisy data. If a procedure performs well under these conditions, try refitting your model to simulated data that more closely resembles your own dataset (i.e., imperfect data). If you know that there was limited sampling and/or imprecise tools used to collect your data, consider adding noise, reducing sample sizes, and adding temporal and spatial gaps to see if the fitting procedure continues to return reasonably correct estimates. Remember, even if your fitting procedures continue to perform well given these additional complexities, issues may still arise when fitting to empirical data. Models are approximations and consequently their simulations are imperfect representations of your measured outcome of interest. However, by evaluating procedures on perfectly known imperfect data, we are one step closer to having a fitting procedure that works for us even when it seems like our data are against us.

Rule 8: Expect model fitting to be a lengthy, arduous but creative task

Model fitting requires an understanding of both the assumptions and limitations of your model, as well as the specifics of the data to be used in the fitting. The latter can be challenging, particularly if you did not collect the data yourself, as there may be additional uncertainties regarding the sampling procedure, or the variables being measured. For example, the incidence data commonly adopted to fit SIR models often contain biases related to underreporting, selective reporting, and reporting delays [ 24 ]. Taking the time to understand the nuances of the data is critical to prevent mismatches between the model and the data. In a bad case, a mismatch may lead to a poor-fitting model. In the worst case, a model may appear well-fit, but will lead to incorrect inferences and predictions.

Model fitting, like all aspects of modelling, is easier with the appropriate set of skills (see Rule 2). In particular, being proficient at constructing and analyzing mathematical models does not mean you are prepared to fit them. Fitting models typically requires additional in-depth statistical knowledge related to the characteristics of probability distributions, deriving statistical moments, and selecting appropriate optimization procedures. Luckily, a substantial portion of this knowledge can be gleaned from textbooks and methods-based research articles. These resources can range from covering basic model fitting, such as determining an appropriate distribution for your data and constructing a likelihood for that distribution (e.g., Hilborn and Mangel [ 7 ]), to more advanced topics, such as accounting for uncertainties in parameters, inputs, and structures during model fitting (e.g., Dietze [ 25 ]). We find these sources among others (e.g., Hobbs and Hooten [ 26 ] for Bayesian methods; e.g., Adams and colleagues [ 27 ] for fitting noisy and sparse datasets; e.g., Sirén and colleagues [ 28 ] for fitting individual-based models; and Williams and Kendall [ 29 ] for multiobject optimization—to name a few) are not only useful when starting to fit your first models, but are also useful when switching from one technique or model to another.

After you have learned about your data and brushed up on your statistical knowledge, you may still run into issues when model fitting. If you are like us, you will have incomplete data, small sample sizes, and strange data idiosyncrasies that do not seem to be replicated anywhere else. At this point, we suggest you be explorative in the resources you use and accept that you may have to combine multiple techniques and/or data sources before it is feasible to achieve an adequate model fit (see Rosenbaum and colleagues [ 30 ] for parameter estimation with multiple datasets). Evaluating the strength of different techniques can be aided by using simulated data to test these techniques, while SA can be used to identify insensitive parameters which can often be ignored in the fitting process (see Rule 7).

Model accuracy is an important metric but “good” models are also precise (i.e., reliable). During model fitting, to make models more reliable, the uncertainties in their inputs, drivers, parameters, and structures, arising due to natural variability (i.e., aleatory uncertainty) or imperfect knowledge (i.e., epistemic uncertainty), should be identified, accounted for, and reduced where feasible [ 31 ]. Accounting for uncertainty may entail measurements of uncertainties being propagated through a model (a simple example being a confidence interval), while reducing uncertainty may require building new models or acquiring additional data that minimize the prioritized uncertainties (see Dietze [ 25 ] and Tsigkinopoulou and colleagues [ 32 ] for a more thorough review on the topic). Just remember that although the steps outlined in this rule may take a while to complete, when you do achieve a well-fitted reliable model, it is truly something to be celebrated.

Rule 9: Give yourself time (and then add more)

Experienced modellers know that it often takes considerable time to build a model and that even more time may be required when fitting to real data. However, the pervasive caricature of modelling as being “a few lines of code here and there” or “a couple of equations” can lead graduate students to hold unrealistic expectations of how long finishing a model may take (or when to consider a model “finished”). Given the multiple considerations that go into selecting and implementing models (see previous rules), it should be unsurprising that the modelling process may take weeks, months, or even years. Remembering that a published model is the final product of long and hard work may help reduce some of your time-based anxieties. In reality, the finished product is just the tip of the iceberg and often unseen is the set of failed or alternative models providing its foundation. Note that taking time early on to establish what is “good enough” given your objective, and to instill good modelling practices, such as developing multiple models, simulating your models, and creating well-documented code, can save you considerable time and stress.

Rule 10: Care about the process, not just the endpoint

As a graduate student, hours of labor coupled with relative inexperience may lead to an unwillingness to change to a new model later down the line. But being married to one model can restrict its efficacy, or worse, lead to incorrect conclusions. Early planning may mitigate some modelling problems, but many issues will only become apparent as time goes on. For example, perhaps model parameters cannot be estimated as you previously thought, or assumptions made during model formulation have since proven false. Modelling is a dynamic process, and some steps will need to be revisited many times as you correct, refine, and improve your model. It is also important to bear in mind that the process of model-building is worth the effort. The process of translating biological dynamics into mathematical equations typically forces us to question our assumptions, while a misspecified model often leads to novel insights. While we may wish we had the option to skip to a final finished product, in the words of Drake, “sometimes it’s the journey that teaches you a lot about your destination”.

There is no such thing as a failed model. With every new error message or wonky output, we learn something useful about modelling (mostly begrudgingly) and, if we are lucky, perhaps also about the study system. It is easy to cave in to the ever-present pressure to perform, but as graduate students, we are still learning. Luckily, you are likely surrounded by other graduate students, often facing similar challenges who can be an invaluable resource for learning and support. Finally, remember that it does not matter if this was your first or your 100th mathematical model, challenges will always present themselves. However, with practice and determination, you will become more skilled at overcoming them, allowing you to grow and take on even greater challenges.

Acknowledgments

We thank Marie-Josée Fortin, Martin Krkošek, Péter K. Molnár, Shawn Leroux, Carina Rauen Firkowski, Cole Brookson, Gracie F.Z. Wild, Cedric B. Hunter, and Philip E. Bourne for their helpful input on the manuscript.

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Mathematical modeling for theory-oriented research in educational technology

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• Volume 70 , pages 149–167, ( 2022 )

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• Elena Novak   ORCID: orcid.org/0000-0003-0873-2081 1  

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Mathematical modeling describes how events, concepts, and systems of interest behave in the world using mathematical concepts. This research approach can be applied to theory construction and testing by using empirical data to evaluate whether the specific theory can explain the empirical data or whether the theory fits the data available. Although extensively used in the physical sciences and engineering, as well as some social and behavioral sciences to examine theoretical claims and form predictions of future events and behaviors, theory-oriented mathematical modeling is less common in educational technology research. This article explores the potential of using theory-oriented mathematical modeling for theory construction and testing in the field of educational technology. It presents examples of how this approach was used in social, behavioral, and educational disciplines, and provides rationale for why educational technology research can benefit from a theory-oriented model-testing approach.

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Mathematical modelling for health systems research: a systematic review of system dynamics and agent-based models

• Rachel Cassidy   ORCID: orcid.org/0000-0002-4824-0260 1 ,
• Neha S. Singh 1 ,
• Pierre-Raphaël Schiratti 2 , 3 ,
• Agnes Semwanga 4 ,
• Peter Binyaruka 5 ,
• Nkenda Sachingongu 6 ,
• Chitalu Miriam Chama-Chiliba 7 ,
• Zaid Chalabi 8 ,
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BMC Health Services Research volume  19 , Article number:  845 ( 2019 ) Cite this article

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Mathematical modelling has been a vital research tool for exploring complex systems, most recently to aid understanding of health system functioning and optimisation. System dynamics models (SDM) and agent-based models (ABM) are two popular complementary methods, used to simulate macro- and micro-level health system behaviour. This systematic review aims to collate, compare and summarise the application of both methods in this field and to identify common healthcare settings and problems that have been modelled using SDM and ABM.

We searched MEDLINE, EMBASE, Cochrane Library, MathSciNet, ACM Digital Library, HMIC, Econlit and Global Health databases to identify literature for this review. We described papers meeting the inclusion criteria using descriptive statistics and narrative synthesis, and made comparisons between the identified SDM and ABM literature.

We identified 28 papers using SDM methods and 11 papers using ABM methods, one of which used hybrid SDM-ABM to simulate health system behaviour. The majority of SDM, ABM and hybrid modelling papers simulated health systems based in high income countries. Emergency and acute care, and elderly care and long-term care services were the most frequently simulated health system settings, modelling the impact of health policies and interventions such as those targeting stretched and under resourced healthcare services, patient length of stay in healthcare facilities and undesirable patient outcomes.

Conclusions

Future work should now turn to modelling health systems in low- and middle-income countries to aid our understanding of health system functioning in these settings and allow stakeholders and researchers to assess the impact of policies or interventions before implementation. Hybrid modelling of health systems is still relatively novel but with increasing software developments and a growing demand to account for both complex system feedback and heterogeneous behaviour exhibited by those who access or deliver healthcare, we expect a boost in their use to model health systems.

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Introduction

Health systems are complex adaptive systems [ 1 ]. As such, they are characterised by extraordinary complexity in relationships among highly heterogeneous groups of stakeholders and the processes they create [ 2 ]. Systems phenomena of massive interdependencies, self-organising and emergent behaviour, non-linearity, time lags, feedback loops, path dependence and tipping points make health system behaviour difficult and sometimes impossible to predict or manage [ 3 ]. Conventional reductionist approaches using epidemiological and implementation research methods are inadequate for tackling the problems health systems pose [ 4 ]. It is increasingly recognised that health systems and policy research need a special set of approaches, methods and tools that derive from systems thinking perspectives [ 5 ]. Health systems encompass a many tiered system providing services to local, district and national populations, from community health centres to tertiary hospitals. Attempting to evaluate the performance of such a multi-faceted organisation presents a daunting task. Mathematical modelling, capable of simulating the behaviour of complex systems, is therefore a vital research tool to aid our understanding of health system functioning and optimisation.

System dynamics model (SDM)

System dynamics models (SDM) and agent-based models (ABM) are the two most popular mathematical modelling methods for evaluating complex systems; while SDM are used to study macro-level system behaviour such as the movement of resources or quantities in a system over time, ABM capture micro-level system behaviour, such as human decision-making and heterogeneous interactions between humans.

While use of SDM began in business management [ 6 , 7 ] it now has wide spread application from engineering to economics, from environmental science to waste and recycling research [ 8 , 9 , 10 , 11 , 12 , 13 ]. A SDM simulates the movement of entities in a system, using differential equations to model over time changes to system state variables. A stock and flow diagram can be used to provide a visual representation of a SDM, describing the relationships between system variables using stocks, rates and influencing factors. The diagram can be interpreted as mimicking the flow of water in and out of a bath tub [ 7 ]; the rates control how much ‘water’ (some quantifiable entity, resource) can leave or enter a ‘bath tub’ (a stock, system variable) which changes over time depending on what constraints or conditions (e.g. environmental or operational) are placed on the system. Often before the formulation of a stock and flow diagram, a causal loop diagram is constructed which can be thought of as a ‘mental model’ of the system [ 14 ], representing key dynamic hypotheses.

Agent-based model (ABM)

Unlike SDM, ABM is a ground-up representation of a system, simulating the changing states of individual ‘agents’ in a system rather than the broad entities or aggregate behaviour modelled in SDM. Aggregate system behaviour can however be inferred from ABM. Use of ABM to model system behaviour has been trans-disciplinary, with application in economics to ecology, from social sciences to engineering [ 15 , 16 , 17 , 18 , 19 ]. There can be multiple types of agent modelled, each assigned their own characteristics and pattern of behaviour [ 20 , 21 ]. Agents can learn from their own experiences, make decisions and perform actions based on set rules (e.g. heuristics), informed by their interactions with other agents, their own assigned attributes or based on their interaction with the modelled environment [ 22 ]. The interactions between agents can result in three levels of communication between agents; one-to-one communication between agents, one-to-many communication between agents and one-to-location communication where an agent can influence other agents contained in a particular location [ 22 ].

Why use SDM and ABM to model health systems?

ABM and SDM, with their ability to simulate micro- and macro-level behaviour, are complementary instruments for examining the mechanisms in complex systems and are being recognised as crucial tools for exploratory analysis. Their use in mapping health systems, for example, has steadily risen over the last three decades. ABM is well-suited to explore systems with dynamic patient or health worker activity, a limitation of other differential equation or event-based simulation tools [ 23 , 24 , 25 ]. Unlike discrete-event simulation (DES) for example, which simulates a queue of events and agent attributes over time [ 26 ], the agents modelled in ABM are decision makers rather than passive individuals. Closer to the true system modelled, ABM can also incorporate ongoing learning from events whereby patients can be influenced by their interactions with other patients or health workers and by their own personal experience with the health system [ 21 ]. SDM has also been identified as a useful tool for simulating feedback and activity across the care continuum [ 27 , 28 , 29 , 30 ] and is highly adept at capturing changes to the system over time [ 31 ]. This is not possible with certain ‘snapshot in time’ modelling approaches such as DES [ 32 ]. SDM is best implemented where the aim of the simulation is to examine aggregate flows, trends and sub-system behaviour as opposed to intricate individual flows of activity which are more suited to ABM or DES [ 33 ].

There are also models that can accommodate two or more types of simulation, known as hybrid models. Hybrid models produce results closer to true system behaviour by drawing on the strengths of one or more modelling methods while reducing the limitations associated with using a single simulation type [ 27 ]. The activity captured in such models emulates the individual variability of patients and health professionals while retaining the complex, aggregate behaviour exhibited in health systems.

Health scientists and policy makers alike have recognised the potential of using SDM and ABM to model all aspects of health systems in support of decision making from emergency department (ED) optimisation [ 34 ] to policies that support prevention or health promotion [ 35 ]. Before implementing or evaluating costly health policy interventions or health service re-structuring in the real world, modelling provides a relatively risk-free and low budget method of examining the likely impact of potential health system policy changes. They allow the simulation of ‘what if’ scenarios to optimise an intervention [ 36 ]. They can help identify sensitive parameters in the system that can impede the success of initiatives and point to possible spill-over effects of these initiatives to other departments, health workers or patients. Perhaps most important of all, these modelling methods allow researchers to produce simulations, results and a graphical-user interface in relation to alternative policy options that are communicable to stakeholders in the health system [ 37 ], those responsible for implementing system-wide initiatives and changes.

Study aim and objectives

Given the increasing amount of literature in this field, the main aim of the study was to examine and describe the use of SDM and ABM to model health systems. The specific objectives were as follows: (1) Determine the geographical, and healthcare settings in which these methods have been used (2) Identify the purpose of the research, particularly the health policies or interventions tested (3) Evaluate the limitations of these methods and study validation, and (4) Compare the use of SDM and ABM in health system research.

Although microsimulation, DES and Markov models have been widely used in disease health modelling and health economic evaluation, our aim in this study was to review the literature on mathematical methods which are used to model complex dynamic systems, SDM and ABM. These models represent two tenants of modelling: macroscopic (top-level) and microscopic (individual-level) approaches. Although microsimulation and DES are individual-based models like ABM, individuals in ABM are “active agents” i.e. decision-makers rather than “passive agents” which are the norm in microsimulation and DES models. Unlike Markov models which are essentially one-dimensional, unidirectional and linear, SDM are multi-dimensional, nonlinear with feedback mechanisms. We have therefore focussed our review on SDM and ABM because they are better suited to characterise the complexity of health systems. This study reviews the literature on the use of SDM and ABM in modelling health systems, and identifies and compares the key characteristics of both modelling approaches in unwrapping the complexity of health systems. In identifying and summarising this literature, this review will shed light on the types of health system research questions that these methods can be used to explore, and what they add to more traditional methods of health system research. By providing an over overview of how these models can be used within health system research, this paper is also expected to encourage wider use and uptake of these methods by health system researchers and policy makers.

The review was conducted in compliance with the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) statement [ 38 ].

Search strategy and information sources

The literature on ABM and SDM of health systems has not been confined to a single research discipline, making it necessary to widen the systematic review to capture peer-reviewed articles found in mathematical, computing, medicine and health databases. Accordingly, we searched MEDLINE, EMBASE, Cochrane Library, MathSciNet, ACM Digital Library, HMIC, Econlit and Global Health databases for literature. The search of health system literature was narrowed to identify articles that were concerned with modelling facility-based healthcare, services and related healthcare financing agreements which had been excluded or were not the focus of previous reviews [ 34 , 35 , 39 , 40 , 41 ]. The search criteria used for MEDLINE was as follows, with full search terms for each database and search terms used to locate SDM and ABM literature found in Additional file  1 :

(health system* OR health care OR healthcare OR health service* OR health polic* OR health facil* OR primary care OR secondary care OR tertiary care OR hospital*).ab,ti. AND (agent-based OR agent based).ab,ti. AND (model*).ab,ti.

In addition, the reference list of papers retained in the final stage of the screening process, and systematic reviews identified in the search, were reviewed for relevant literature.

Data extraction and synthesis

The screening process for the review is given in Fig.  1 (adapted from [ 38 ]). All search results were uploaded to Mendeley reference software where duplicate entries were removed. The remaining records were screened using their titles and abstracts, removing entries based on eligibility criteria given in Table  1 . Post-abstract review, the full text of remaining articles was screened. Papers retained in final stage of screening were scrutinised, with data imported to Excel based on the following categories; publication date, geographical and healthcare setting modelled, purpose of research in addition to any policies or interventions tested, rationale for modelling method and software platform, validation and limitations of model. The results were synthesised using descriptive statistics and analysis of paper content that were used to answer the objectives.

a Flow-chart for systematic review of SDMs and b ABMs of health systems (Database research discipline is identified by colour; mathematical and computing (red), medicine (blue) and health (green) databases). Adapted from PRISMA [ 38 ]

The studies were first described by three characteristics: publication date, geographical setting, and what aspect of the health system was modelled and why. These characteristics were chosen for the following reasons. Publication date (Fig.  2 ) allows us to examine the quantity of SDM and ABM studies over time. Geographical settings (Fig. 2 , top) allows us to see which health systems have been studied, as health systems in LMIC are very different from those in developed countries. Studies are classified as modelling health systems in high, upper middle, lower middle and low income countries as classified by The World Bank based on economy, July 2018 [ 42 ]. Finally, we examined which aspects of the health system have been modelled and the types of research/policy questions that the models were designed to address, to shed light on the range of potential applications of these models, and also potential gaps in their application to date.

Number of articles in the final review by year of publication and economic classification

The analysis of paper content was split into three sections; SDM use in health system research (including hybrid SDM-DES), ABM use in health system research (including hybrid ABM-DES) and hybrid SDM-ABM use in health system research. The quality of selected studies will not be presented as our aim was to compare and summarise the application of SDM and ABM in modelling health systems rather than a quality appraisal of studies.

Study selection

The search initially yielded 535 citations for ABM and 996 citations for SDM of facility-based healthcare and services (see Fig. 1 ). Post-full text screening 11 ABM and 28 SDM papers were retained for analysis, six of which utilised hybrid modelling methods. Three of the hybrid modelling papers integrated SDM with DES [ 43 , 44 , 45 ], two integrated ABM with DES [ 24 , 46 ] and one integrated SDM with ABM [ 47 ]. A summary table of selected papers is given in Table  2 .

Descriptive statistics

Publication date.

The first SDM paper to model health systems was published in 1998 [ 56 ] whilst the first publication [ 66 ] utilising ABM came almost a decade later (Fig. 2 ). We found an increasing trend in publications for both modelling approaches, with 90.9% (10/11) and 71.4% (20/28) of all ABM and SDM articles, respectively, having been published in the last decade. The first hybrid modelling article was published in 2010 [ 43 ], using SDM and DES to model the impact of an intervention to aid access to social care services for elderly patients in Hampshire, England.

Geographical setting

The proportion of papers that modelled health systems in high, upper middle, lower middle and low income countries is presented in Fig. 2 . Eighteen (18/28) papers that employed SDM simulated health systems in high income countries including England [ 33 , 36 , 43 , 45 , 50 , 54 , 56 , 57 ] and Canada [ 28 , 51 , 62 ]. Four SDM papers simulated upper middle income country health systems, including Turkey [ 52 , 59 ] and China [ 64 ], with a nominal number of papers (5/28) focussing on lower middle or low income countries (West Bank and Gaza [ 48 , 55 ], Indonesia [ 37 ], Afghanistan [ 30 ] and Uganda [ 60 ]). Almost all ABM papers (9/11) modelled a high income country health system, including the US [ 20 , 23 , 25 ] and Austria [ 65 ]. Two (2/11) ABM papers described an upper-middle income based health system (Brazil [ 22 , 67 ]). All six articles that implemented a hybrid SDM or ABM simulated health systems based in high income countries, including Germany [ 44 ] and Poland [ 47 ].

Healthcare setting and purpose of research

The healthcare settings modelled in the SDM, ABM and hybrid simulation papers are presented in Fig.  3 . Healthcare settings modelled using SDM included systems that were concerned with delivering emergency or acute care (11/28) [ 28 , 31 , 36 , 45 , 47 , 50 , 56 , 57 , 58 , 61 , 62 ], elderly or long-term care services (LTC)(12/28) [ 28 , 31 , 36 , 43 , 44 , 45 , 49 , 50 , 51 , 54 , 61 , 62 ] and hospital waste management (4/28) [ 37 , 48 , 52 , 55 ]. Twenty of the SDM papers selected in this review assessed the impact of health policy or interventions on the modelled system. Common policy targets included finding robust methods to relieve stretched healthcare services, ward occupancy and patient length of stay [ 28 , 31 , 36 , 43 , 49 , 50 , 54 , 58 , 62 ], reducing the time to patient admission [ 33 , 53 , 61 ], targeting undesirable patient health outcomes [ 47 , 58 , 60 , 63 ], optimising performance-based incentive health system policies [ 30 , 59 ] and reducing the total cost of care [ 33 , 54 , 61 ]. The remaining eight papers explored factors leading to undesirable emergency care system behaviour [ 56 , 57 ], simulating hospital waste management systems and predicting future waste generation [ 37 , 48 , 55 ], estimating future demand for cardiac care [ 44 ], exploring the impact of patient admission on health professionals stress level in an integrated care system [ 45 ], and variation in physician decision-making [ 32 ].

The health system sector locations modelled in the SDM, ABM and hybrid modelling literature. Long-term care (LTC); Accountable care organisation (ACO); Maternal, newborn and child health (MNCH)

ABM papers modelled systems focussed on delivering emergency or acute care (4/11) [ 21 , 22 , 47 , 67 ] and accountable care organisations (ACO) or health insurance reimbursement schemes (3/11) [ 23 , 25 , 65 ]. Nine of the ABM papers assessed the impact of health policy or interventions on the modelled system. Common policy targets included decreasing the time agents spent performing tasks, waiting for a service or residing in parts of the system [ 20 , 22 , 24 , 67 ], reducing undesirable patient outcomes [ 23 , 25 , 47 , 67 ], reducing the number of patients who left a health facility without being seen by a physician [ 22 , 67 ] and optimising resource utility (beds and healthcare staff) [ 46 , 66 , 67 ]. The remaining two papers described simulation tools capable of comparing health insurance reimbursement schemes [ 65 ] and assessing risk, allocation of resources and identifying weaknesses in emergency care services [ 21 ].

Papers that utilised hybrid simulation, combining the strengths of two modelling approaches to capture detailed individual variability, agent-decision making and patient flow, modelled systems focussed on delivering elderly care or LTC services [ 43 , 44 , 45 ] and emergency or acute care [ 45 , 47 ]. Four of the hybrid simulation papers assessed the impact of policy or intervention on the modelled system. Policy targets included improving access to social support and care services [ 43 ], reducing undesirable patient outcomes [ 47 ], decreasing patient waiting time to be seen by a physician [ 24 ] and improving patient flow through the system by optimising resource allocation [ 46 ]. The remaining two papers used hybrid simulation to estimate the future demand for health care from patients with cardiac disease [ 44 ] and model patient flow through an integrated care system to estimate impact of patient admission on health care professionals wellbeing [ 45 ].

SDM use in health systems research (including hybrid SDM-DES)

Rationale for using model.

Gaining a holistic system perspective to facilitate the investigation of delays and bottlenecks in health facility processes, exploring counter-intuitive behaviour and monitoring inter-connected processes between sub-systems was cited frequently as reasons for using SDM to model health systems [ 28 , 36 , 37 , 48 , 56 ]. SDM was also described as a useful tool for predicting future health system behaviour and demand for care services, essential for health resource and capacity planning [ 48 , 60 ]. Configuration of the model was not limited by data availability [ 28 , 52 , 64 ] and could integrate data from various sources when required [ 51 ].

SDM was described as a tool for health policy exploration and optimising system interventions [ 33 , 36 , 51 , 54 , 58 , 64 ], useful for establishing clinical and financial ramifications on multiple groups (such as patients and health care providers) [ 63 ], identifying policy resistance or unintended system consequences [ 59 , 61 ] and quantifying the impact of change to the health system before real world implementation [ 62 ]. The modelling platform also provided health professionals, stakeholders and decision makers with an accessible visual learning environment that enabled engagement with experts necessary for model conception and validation [ 48 , 50 , 55 , 57 ]. The model interface could be utilised by decision makers to develop and test alternative policies in a ‘real-world’ framework that strengthened their understanding of system-wide policy impact [ 31 , 49 , 58 , 61 ].

SDM-DES hybrid models enabled retention of deterministic and stochastic system variability and preservation of unique and valuable features of both methods [ 44 ], capable of describing the flow of entities through a system and rapid insight without the need for large data collection [ 43 ], while simulating individual variability and detailed interactions that influence system behaviour [ 43 ]. SDM-DES offered dual model functionality [ 44 ] vital for simulating human-centric activity [ 45 ], reducing the practical limitations that come with using either SDM or DES to model health systems such as attempting to use SDM to model elements which have non-aggregated values (e.g. patient arrival time) [ 45 ] which is better suited for DES.

Healthcare setting

Sixteen papers that utilised SDM modelled systems that were concerned with the delivery of emergency or acute care, or elderly care or LTC services.

Ten of the reviewed papers primarily modelled sectors of the health system that delivered emergency or acute care Footnote 1 , Footnote 2 . Brailsford et al. [ 50 ], Lane et al. [ 56 ], Lane et al. [ 57 ] and Lattimer et al. [ 36 ] simulated the delivery of emergency care in English cities, specifically in Nottingham and London. Brailsford et al. [ 50 ] and Lattimer et al. [ 36 ] created models that replicated the entire emergency care system for the city of Nottingham, from primary care (i.e. General Practice surgeries) to secondary care (i.e. hospital admissions wards), to aid understanding of how emergency care was delivered and how the system would need to adapt to increasing demand. Lane et al. [ 56 ] and Lane et al. [ 57 ] modelled the behaviour of an ED in an inner-London teaching hospital, exploring the knock on effects of ED performance to hospital ward occupancy and elective admissions. Esensoy et al. [ 28 ] and Wong et al. [ 62 ] both modelled emergency care in Canada, Esensoy et al. [ 28 ] focussing on six sectors of the Ontario health system that cared for stroke patients while Wong et al. [ 62 ] simulated the impact of delayed transfer of General Internal Medicine patients on ED occupancy. Rashwan et al. [ 31 ], Walker et al. [ 61 ] and Mahmoudian-Dehkordi et al. [ 58 ] modelled patient flow through a generic emergency care facility with six possible discharge locations in Ireland, a sub-acute extended care hospital with patient flow from feeder facilities in Australia and an intensive care unit, ED and general wards in a generic facility.

Five of the SDM papers primarily simulated the behaviour of LTC facilities or care services for elderly patients Footnote 3 . Ansah et al. [ 49 ] modelled the demand and supply of general LTC services in Singapore with specific focus on the need for LTC and acute health care professionals. Desai et al. [ 54 ] developed a SDM that investigated future demand of care services for older people in Hampshire, England which simulated patient flow through adult social care services offering 13 different care packages. In modelling complex care service demand, Cepoiu-Martin et al. [ 51 ] explored patient flow within the Alberta continuing care system in Canada which offered supportive living and LTC services for patients with dementia. Brailsford et al. [ 43 ] used a hybrid SDM-DES model to investigate how local authorities could improve access to services and support for older people, in particular the long term impact of a new contact centre for patients. The SDM replicated the whole system for long term care, simulating the future demography and demand for care services and the nested DES model simulated the operational issues and staffing of the call centre in anticipation of growing demand for services. Zulkepli et al. [ 45 ] also used SDM-DES to model the behaviour of an integrated care system in the UK, modelling patient flow (DES) and intangible variables (SDM) related to health professionals such as motivation and stress levels.

Policy impact evaluation/testing

Twenty papers that utilised SDM tested the impact of policy or interventions on key health system performance or service indicators. The intended target of these policies ranged from relieving strained and under resourced healthcare services, decreasing healthcare costs to reducing patient mortality rates.

Ansah et al. [ 49 ], Brailsford et al. [ 50 ] and Desai et al. [ 54 ] aimed to reduce occupancy in acute or emergency care departments through policies that targeted elderly utilisation of these services. While demand for LTC services is expected to exponentially increase in Singapore, focus has been placed on expanding the acute care sector. Ansah et al. [ 49 ] simulated various LTC service expansion policies (static ‘current’ policy, slow adjustment, quick adjustment, proactive adjustment) and identified that proactive expansion of LTC services stemmed the number of acute care visits by elderly patients over time and required only a modest increase in the number of health professionals when compared with other policies. In Brailsford et al. [ 50 ] simulation of the entire emergency care system for Nottingham, England, policy testing indicated that while the emergency care system is operating near full capacity, yearly total occupancy of hospital beds could be reduced by re-directing emergency admissions from patients over 60 years of age (who make up around half of all admissions) to more appropriate services, such as those offered by community care facilities. To explore challenges that accompany providing care for an ageing population subject to budget restraints, Desai et al. [ 54 ] simulated the delivery and demand for social care services in Hampshire over a projected 5 year period. In offering care packages to only critical need clients and encouraging extra care services at home rather than offering residential care, the number of patients accessing acute care services reduced over the observed period.

Desai et al. [ 54 ], in addition to Taylor et al. [ 33 ] and Walker et al. [ 61 ], also examined policies that could reduce the total cost of care. Increasing the proportion of hired unqualified care workers (over qualified care workers who are employed at a higher cost rate) resulted in savings which could be fed back into care funding, although Desai et al. [ 54 ] remarked on the legal and practical limitations to this policy. Taylor et al. [ 33 ] examined the impact of shifting cardiac catheterization services from tertiary to secondary level hospitals for low risk investigations and explored how improvements could be made to services. Significant and stable improvements in service, including reduced waiting list and overall cost of service, were achieved with the implementation of strict (appropriate referral) guidelines for admitting patients. Walker et al. [ 61 ] modelled patient flow from feeder hospitals to a single sub-acute extended care facility in Victoria, Australia, to assess the impact of local rules used by the medical registrar for admission. The local admission policy which prioritised admissions from patients under the care of private doctors pushed the total cost of care over the facility budget by 6% whereas employing no prioritisation rule reduced the total cost of care to 3% under budget.

Semwanga et al. [ 60 ], Mahmoudian-Dehkordi et al. [ 58 ] and Worni et al. [ 63 ] evaluated the impact of health policy on undesirable patient outcomes (mortality and post-treatment complication rates). Semwanga et al. [ 60 ] tested the effectiveness of policies designed to promote maternal and neonatal care in Uganda, established from the literature. Policies that enabled service uptake, such as community health education, free delivery kits and motorcycle coupons were significant in reducing neonatal death over the simulated period. Mahmoudian-Dehkordi et al. [ 58 ] explored the intended and unintended consequences of intensive care unit resource and bed management policies on system performance indicators, including patient mortality. During a simulated crisis scenario, prioritising intensive care unit patient admission to general wards over emergency admissions was found to be the most effective policy in reducing total hospital mortality. Worni et al. [ 63 ] estimated the impact of a policy to reduce venous thromboembolism rates post-total knee arthroplasty surgery and identified unintentional consequences of the strategy. The policy prevented the reimbursement of patient care fees in the event that a patient was not taking the recommended prophylaxis medication and consequently develops venous thromboembolism. Simulation results indicated a positive 3-fold decrease in venous thromboembolism rates but an unintended 6-fold increase in the number of patients who develop bleeding complications as a result of compulsory prophylaxis treatment.

Validation (including sensitivity analysis)

Statistically-based models are usually used in quantitative data rich environments where model parameters are estimated through maximum likelihood or least-squares estimation methods. Bayesian methods can also be used to compare alternative statistical model structures. SDMs and ABMs on the other hand are not fitted to data observations in the traditional statistical sense. The data are used to inform model development. Both quantitative data and qualitative data (e.g. from interviews) can be used to inform the structure of the model and the parameters of the model. Furthermore, model structure and parameter values can also be elicited from expert opinion. This means that the nature of validation of ABMs and SDMs requires more scrutiny than that of other types of models.

With increasing complexity of such models, and to strengthen confidence in their use particularly for decision support, models are often subjected to sensitivity analysis and validation tests. Twenty-two papers that utilised SDM undertook model validation, the majority having performed behavioural validity tests (see Additional file  2 for details of validation methods for each model). Key model output such as bed occupancy [ 36 , 50 ], department length of stay [ 62 ] and number of department discharges [ 31 ] were compared with real system performance data from hospitals [ 32 , 33 , 36 , 48 , 50 , 54 , 58 , 59 , 61 , 62 ], local councils [ 54 ], nationally reported figs [ 31 , 64 ]. as well being reviewed by experts [ 57 , 60 ] as realistic. Others performed more structure orientated validity tests. Model conception [ 28 , 60 ], development [ 30 , 36 , 50 , 53 , 54 , 57 , 62 ] and formulation [ 54 , 56 , 59 ] were validated by a variety of experts including health professionals [ 47 , 53 , 54 , 57 , 59 , 62 ], community groups [ 56 ] and leaders [ 60 ], steering committees [ 36 ], hospital and care representatives [ 50 , 56 , 59 ], patient groups [ 60 ] and healthcare policy makers [ 60 ]. Further tests for structural validity included checking model behaviour when subjected to extreme conditions or extreme values of parameters [ 30 , 31 , 52 , 57 , 59 , 60 , 64 ], model dimensional consistency [ 31 , 52 , 57 , 59 , 60 ], model boundary adequacy [ 31 ] and mass balance [ 54 ] and integration error checks [ 31 , 52 ]. Sensitivity analysis was performed to assess how sensitive model output was to changes in key parameters [ 49 , 51 , 57 , 60 , 64 ], to test the impact of parameters that had been based on expert opinion on model output [ 28 ] and varying key system parameters to test the robustness and effectiveness of policies [ 28 , 30 , 52 , 53 , 58 ] (on the assumption of imperfect policy implementation [ 28 ]).

Limitations of research

Most of the model limitations reported were concerned with missing parameters, feedback or inability to simulate all possible future health system innovations. Mielczarek et al. [ 44 ], Cepoiu-Martin et al. [ 51 ], Ansah et al. [ 49 ] and Rashwan et al. [ 31 ] did not take into account how future improvements in technology or service delivery may have impacted results, such as the possibility of new treatment improving patient health outcomes [ 51 ] and how this could impact the future utilisation of acute care services [ 49 ]. Walker et al. [ 61 ] and Alonge et al. [ 30 ] described how the models may not simulate all possible actions or interactions that occurred in the real system, such as all proactive actions taken by hospital managers to achieve budget targets [ 61 ] or all unintended consequences of a policy on the system [ 30 ]. De Andrade et al. [ 53 ] and Rashwan et al. [ 31 ] discussed the reality of model boundaries, that SDMs cannot encapsulate all health sub-sector behaviour and spill-over effects. Although these have been listed here as limitations, not accounting for possible future improvements in healthcare service or not simulating all possible actions in the modelled system did not prevent authors from fulfilling study objectives. When developing a SDM, it is not possible to account for all possible spill-over effects to other healthcare departments and this should not be attempted; model boundaries are set to only include variables and feedback that are pertinent to exploring the defined problem.

Simplification of model parameters was another common limitation. Wong et al. [ 62 ] stated that this would result in some model behaviour not holding in the real system, such as using weekly hospital admission and discharge averages in place of hourly rates due to the hospital recording aggregated data. This aggregation of model parameters may not have reflected real system complexity; Eleyan et al. [ 55 ] did not differentiate between service level and type of hospital when modelling health care waste production (described as future work) and Worni et al. [ 63 ] refrained from stratifying post-surgery complications by severity, potentially combining lethal and less harmful complications within the same stock (although this did not detract from the study conclusion that the rate of complications would increase as a result of the tested policy).

Data availability, lack of costing analysis and short time horizons were also considered credible limitations. Models that had been calibrated with real data were at risk of using datasets that contained measurement errors or incomplete datasets lacking information required to inform model structure or feedback [ 32 ]. Routine facility data required for model conception and formulation was unavailable which restricted the replication of facility behaviour in the model [ 36 ] and restricted validation of model behaviour [ 59 ], although it should be noted that this is only one method among many for SDM validation and the author was able to use other sources of data for this purpose. Lack of costing or cost effectiveness analysis when testing policies [ 60 ], particularly policies that required significant investment or capacity expansion [ 58 ], limited discussion on their feasibility in the real system. Models that simulated events over short time scales did not evaluate long term patient outcomes [ 33 ] or the long term effects of facility policies on certain groups of patient [ 57 ].

ABM use in health system research (including hybrid ABM-DES)

The model’s ability to closely replicate human behaviour that exists in the real system was frequently cited [ 20 , 21 , 22 , 25 , 66 ], providing a deeper understanding of multiple agent decision-making [ 23 , 67 ], agent networks [ 25 ] and interactions [ 21 , 22 ]. The modelling method was described as providing a flexible framework capable of conveying intricate system structures [ 20 ], where simulations captured agent capacity for learning and adaptive behaviour [ 20 , 25 ] and could incorporate stochastic processes that mimicked agent transition between states [ 25 ]. ABM took advantage of key individual level agent data [ 25 ] and integrated information from various sources including demographic, epidemiological and health service data [ 65 ]. The visualisation of systems and interface available with ABM software packages facilitated stakeholder understanding of how tested policies could impact financial and patient health outcomes [ 23 ], particularly those experts in the health industry with minimal modelling experience [ 67 ].

Integrating DES and ABM within a single model ensured an intelligent and flexible approach for simulating complex systems, such as the outpatient clinic described in Kittipittayakorn et al. [ 24 ]. The hybrid model captured both orthopaedic patient flow and agent decision-making that enabled identification of health care bottlenecks and optimum resource allocation [ 24 ].

Seven papers that utilised ABM modelled systems that were either concerned with delivering emergency or acute care 2 , ACOs or health insurance reimbursement schemes.

Liu et al. [ 21 ] and Yousefi et al. [ 22 ] modelled behaviour in EDs in Spanish and Brazilian tertiary hospitals. Liu et al. [ 21 ] simulated the behaviour of eleven key agents in the ED including patients, admission staff, doctors, triage nurses and auxiliary staff. Patients were admitted to the ED and triaged before tests were requested and a diagnosis issued. Over time, agent states changed based on their interaction with other agents such as when a doctor decided upon a course of action for a patient (sending the patient home, to another ward, or continue with diagnosis and treatment). For further details of agent type and model rules for each paper, see Additional file  3 .

Yousefi et al. [ 22 ] modelled the activities of patients, doctors, nurses and receptionists in a ED. Agents could communicate with each other, to a group of other agents or could send a message to an area of the ED where other agents reside. They made decisions based on these interactions and the information available to them at the time. The main focus of the simulation was on patients who left the ED without being seen by a physician; patients decided whether to leave the ED based on a ‘tolerance’ time extracted from the literature, which changed based on their interaction with other agents. In an additional paper, Yousefi et al. [ 67 ] simulated decision-making by patients, doctors, nurses and lab technicians within a generic ED informed from the literature. Group decision-making was employed, whereby facility staff could interact with each other and reach a common solution for improving the efficacy of the department such as re-allocating staff where needed. Yousefi et al. [ 67 ], Yousefi et al. [ 22 ] and Liu et al. [ 21 ] each used a finite state machine (a computational model which describes an entity that can be in one of a finite number of states) to model interactions between agents and their states.

Liu et al. [ 25 ] and Alibrahim et al. [ 23 ] modelled the behaviour of patients, health providers and payers using series of conditional probabilities, where health providers had participated in an ACO in the United States. Liu et al. [ 25 ] presented a model where health providers within an ACO network worked together to reduce congestive heart failure patient healthcare costs and were consequently rewarded a portion of the savings from the payer agent (hypothetically, the Centers for Medicare and Medicaid Services). Patients were Medicare beneficiaries over the age of 65 who developed diabetes, hypertension and/or congestive heart failure and sought care within the network of health providers formed of three hospitals and 15 primary care physician clinics. Alibrahim et al. [ 23 ] adapted Liu et al. [ 25 ] ACO network model to allow patients to bypass their nearest medical provider in favour of an alternative provider. The decision for a patient to bypass their nearest health centre was influenced by patient characteristics, provider characteristics and the geographical distance between health providers. Providers were also given a choice on whether to participate in an ACO network, where they would then need to implement a comprehensive congestive heart failure disease management programme.

Einzinger et al. [ 65 ] created a tool that could be used to compare different health insurance reimbursement schemes in the Austrian health sector. The ABM utilised anonymous routine data from practically all persons with health insurance in Austria, pertaining to medical services accessed in the outpatient sector. In the simulation, patients developed a chronic medical issue (such as coronary heart disease) that required medical care and led to the patient conducting a search of medical providers through the health market. The patient then accessed care at their chosen provider where the reimbursement system, notified of the event via a generic interface, reimbursed the medical provider for patients care.

Nine papers tested the impact of policy on key health system performance or service indicators. The intended target of these policies ranged from decreasing patient length of stay, to reducing the number of patients who leave without being seen by a physician to reducing patient mortality and hospitalisation rates.

Huynh et al. [ 20 ], Yousefi et al. [ 22 ], Yousefi et al. [ 67 ] and Kittipittayakorn et al. [ 24 ] tested policies to reduce the time agents spent performing tasks, waiting for a service or residing in parts of the system. Huynh et al. [ 20 ] modelled the medication administration workflow for registered nurses at an anonymous medical centre in the United States and simulated changes to the workflow to improve medication administration safety. Two policies were tested; establishing a rigid order for tasks to be performed and for registered nurses to perform tasks in the most frequently observed order (observed in a real medical centre) to see if this improved the average amount of time spent on tasks. Yousefi et al. [ 67 ] modelled the effects of group decision-making in ED compared with the standard approach for resource allocation (where a single supervisor allocates resources) to assess which policy resulted in improved ED performance. Turning ‘on’ group decision-making and starting the simulation with a higher number of triage staff and receptionists resulted in the largest reduction of average patient length of stay and number of patients who left without being seen. This last performance indicator was the subject of an additional paper [ 22 ], with focus on patient-to-patient interactions and how this impacted their decision to leave the ED before being seen by a physician. Four policies adapted from case studies were simulated to reduce the number of patients leaving the ED without being seen and average patient length of stay. The policy of fast-tracking patients who were not acutely unwell during triage performed well as opposed to baseline, where acutely ill patients were always given priority. Kittipittayakorn et al. [ 24 ] used ABM-DES to identify optimal scheduling for appointments in an orthopaedic outpatient clinic, with average patient waiting time falling by 32% under the tested policy.

Liu et al. [ 25 ], Alibrahim et al. [ 23 ] and Yousefi et al. [ 67 ] tested the impact of health policy on undesirable patient outcomes (patient mortality and hospitalisation rates). Liu et al. [ 25 ] modelled health care providers who operated within an ACO network and outside of the network and compared patient outcomes. Providers who operated within the ACO network worked together to reduce congestive heart failure patient healthcare costs and were then rewarded with a portion of the savings. As part of their membership, providers implemented evidence-based interventions for patients, including comprehensive discharge planning with post-discharge follow-up; this intervention was identified in the literature as key to reducing congestive heart failure patient hospitalisation and mortality, leading to a reduction in patient care fees without compromising the quality of care. The ACO network performed well, with a 10% reduction observed in hospitalisation compared with the standard care network. In another study [ 23 ] six scenarios were simulated with combinations of patient bypass capability (turned “on” or “off”) and provider participation in the ACO network (no ACO present, optional participation in ACO or compulsory participation in ACO). Provider participation in the ACO, in agreement with Liu et al. [ 25 ], led to reduced mortality and congestive heart failure patient hospitalisation, with patient bypass capability marginally increasing provider ACO participation. Yousefi et al. [ 67 ] also modelled the impact of group decision-making in ED on the number of patient deaths and number of wrong discharges i.e. patients sent to the wrong sector for care after triage and are then discharged before receiving correct treatment.

Nine of the 11 papers that utilised ABM undertook model validation, consisting almost exclusively of behavioural validity tests. Model output, such as patient length of stay and mortality rates, was reviewed by health professionals [ 46 , 66 ] and compared with data extracted from pilot studies [ 20 ], health facilities (historical) [ 22 , 24 , 46 , 65 , 66 ], national health surveys [ 65 ] and relevant literature [ 23 , 25 ]. Papers presented the results of tests to determine the equivalence of variance [ 20 ] and difference in mean [ 20 , 24 ] between model output and real data. Structural validity tests included extreme condition testing [ 23 , 46 ] and engaging health care experts to ensure the accuracy of model framework [ 22 , 47 ]. Sensitivity analysis was performed to determine how variations or uncertainty in key parameters (particularly where they had not been derived from historical or care data [ 65 ]) affected model outcomes [ 23 , 25 ].

The majority of model limitations reported were concerned the use or availability of real system or case data. Huynh et al. [ 20 ], Yousefi et al. [ 67 ] and Liu et al. [ 25 ] formulated their models using data that was obtainable, such as limited sample data extracted from a pilot study [ 20 ], national average trends [ 25 ] and data from previous studies [ 67 ]. Yousefi et al. [ 22 ] case study dataset did not contain key system feedback, such as the tolerance time of patients waiting to be seen by a physician in the ED, although authors were able to extract this data from a comparable study identified in the literature.

Missing model feedback or parameters, strict model boundaries and simplification of system elements were also considered limitations. Huynh et al. [ 20 ], Hutzschenreuter et al. [ 66 ] and Einzinger et al. [ 65 ] did not model all the realistic complexities of their system, such as all possible interruptions to tasks that occur in patient care units [ 20 ], patient satisfaction of admission processes [ 66 ] (which will be addressed in future work), how treatment influences the course of disease or that morbid patients are at higher risk of developing co-morbidity than healthier patients, which would affect the service needs and consumption needs of the patient [ 65 ]. To improve the accuracy of the model, Huynh et al. stated that further research is taking place to obtain real, clinical data (as opposed to clinical simulation lab results) to assess the impact of interruptions on workflow. Liu et al.’s [ 21 ] model boundary did not include other hospital units that may have been affected by ED behaviour and they identify this as future work, for example to include hospital wards that are affected by ED behaviour. Alibrahim et al. [ 23 ] and Einzinger et al. [ 65 ] made simplifications to the health providers and networks that were modelled, such as assuming equal geographical distances and identical care services between health providers in observed networks [ 23 ], limiting the number of factors that influenced a patients decision to bypass their nearest health provider [ 65 ] and not simulating changes to health provider behaviour based on service utilisation or reimbursement scheme in place [ 23 ]. Alibrahim et al. [ 23 ] noted that although the model was constrained by such assumptions, the focus of future work would be to improve the capability of the model to accurately study the impact of patient choice on economic, health and health provider outcomes.

SDM-ABM use in health system research

A single paper used hybrid SDM-ABM to model health system behaviour. Djanatliev et al. [ 47 ] developed a tool that could be used to assess the impact of new health technology on performance indicators such as patient health and projected cost of care. A modelling method that could reproduce detailed, high granularity system elements in addition to abstract, aggregate health system variables was sought and a hybrid SDM-ABM was selected. The tool nested an agent-based human decision-making module (regarding healthcare choices) within a system dynamics environment, simulating macro-level behaviour such as health care financing and population dynamics. A case study was presented to show the potential impact of Mobile Stroke Units (MSU) on patient morbidity in Berlin, where stroke diagnosis and therapy could be initiated quickly as opposed to standard care. The model structure was deemed credible after evaluation by experts, including doctors and health economists.

Comparison of SDM and ABM papers

The similarities and differences among the SDM and ABM body of literature are described in this section and shown in Table  3 . A high proportion of papers across both modelling methods simulated systems that were concerned with emergency or acute care. A high number of SDM papers (11/28) simulated patient flow and pathways through emergency care [ 28 , 31 , 36 , 45 , 47 , 50 , 56 , 57 , 58 , 61 , 62 ] with a subset evaluating the impact of policies that relieved pressure on at capacity ED’s [ 28 , 36 , 50 , 58 , 62 ]. ABM papers simulated micro-level behaviour associated with emergency care, such as health professional and patient behaviour in EDs and what impact agent interactions have on actions taken over time [ 21 , 22 , 47 , 67 ]. ACOs and health insurance reimbursement schemes, a common modelled healthcare setting among the ABM papers [ 23 , 25 , 65 ] was the focus of a single SDM paper [ 63 ] while health care waste management, a popular healthcare setting for SDM application [ 37 , 48 , 52 , 55 ] was entirely absent among the selected ABM literature. SDM and ABM were both used to test the impact of policy on undesirable patient outcomes, including patient mortality [ 23 , 25 , 58 , 60 , 67 ] and hospitalisation rates [ 23 , 25 ]. Interventions for reducing patient waiting time for services [ 24 , 33 , 53 , 61 , 67 ] and patient length of stay [ 22 , 31 , 67 ] were also tested using these methods, while policy exploration to reduce the total cost of care was more frequent among SDM studies [ 33 , 54 , 61 ].

SDM and ABM software platforms provide accessible, user-friendly visualisations of systems that enable engagement with health experts necessary for model validation [ 48 , 50 , 55 , 57 ] and facilitate stakeholder understanding of how alternative policies can impact health system performance under a range conditions [ 31 , 49 , 58 , 61 ]. The ability to integrate information and data from various sources was also cited as rationale for using SDM and ABM [ 51 ]. Reasons for using SDM to model health systems, as opposed to other methods, included gaining a whole-system perspective crucial for investigating undesirable or counter-intuitive system behaviour across sub-systems [ 28 , 36 , 37 , 48 , 56 ] and identifying unintended consequences or policy resistance with tested health policies [ 59 , 61 ]. The ability to replicate human behaviour [ 20 , 21 , 22 , 25 , 66 ] and capacity for learning and adaptive behaviour [ 20 , 25 ] was frequently cited as rationale for using ABM to simulate health systems.

Validation of SDMs and ABMs consisted mostly of behavioural validity tests where model output was reviewed by experts and compared to real system performance data or to relevant literature. Structural validity tests were uncommon among ABM papers while expert consultation on model development [ 30 , 36 , 50 , 53 , 54 , 57 , 62 , 63 ], extreme condition [ 30 , 31 , 52 , 57 , 59 , 60 , 64 ] and dimensional consistency tests [ 31 , 52 , 57 , 59 , 60 ] were frequently reported in the SDM literature. The inability to simulate all actions or interactions that occur in the real system [ 20 , 30 , 61 , 65 , 66 ] and simplification of model parameters [ 23 , 55 , 62 , 63 , 65 ] were described as limitations in both SDM and ABM papers. Data availability for model conception and formulation [ 20 , 22 , 25 , 32 , 36 , 67 ] and the impact of model boundaries (restricting exploration of interconnected sub-system behaviour [ 21 , 31 , 53 ]) were also cited limitations common to both sets of literature. Lack of costing analysis [ 58 , 60 ], short time horizons [ 33 , 57 ] and an inability to model future improvements in technology or service delivery [ 31 , 44 , 49 , 51 ] were additionally cited among the SDM papers.

Statement of principal findings

Our review has confirmed that there is a growing body of research demonstrating the use of SDM and ABM to model health care systems to inform policy in a range of settings. While the application of SDM has been more widespread (with 28 papers identified) there are also a growing number of ABM being used (11), just over half of which used hybrid simulation. A single paper used hybrid SDM-ABM to model health system behaviour. To our knowledge this is the first review to identify and compare the application of both SDM and ABM to model health systems. The first ABM article identified in this review was published almost a decade after the first SDM paper; this reflects to a certain extent the increasing availability of SDM and ABM dedicated software tools with the developments in ABM software lagging behind their SDM modelling counterparts.

Emergency and acute care, and elderly care and LTC services were the most frequently simulated health system setting. Both sets of services are facing exponential increases in demand with constraints on resources, presenting complex issues ideal for evaluation through simulation. Models were used to explore the impact and potential spill over effects of alternative policy options, prior to implementation, on patient outcomes, service use and efficiency under various structural and financial constraints.

Strengths and weaknesses of the study

To ensure key papers were identified, eight databases across four research areas were screened for relevant literature. Unlike other reviews in the field [ 39 , 40 ], there was no restriction placed on publication date. The framework for this review was built to provide a general overview of the SDM and ABM of healthcare literature, capturing papers excluded in other published reviews as a result of strict inclusion criteria. These include reviews that have focussed specifically on compiling examples of modelled health policy application in the literature [ 35 ] or have searched for papers with a particular health system setting, such as those that solely simulate the behaviour of emergency departments [ 34 ]. One particularly comprehensive review of the literature had excluded papers that simulated hospital systems, which we have explicitly included as part of our search framework [ 39 ].

The papers presented in this review, with selection restricted by search criteria, provide a broad picture of the current health system modelling landscape. The focus of this review was to identify models of facility-based healthcare, purposely excluding literature where the primary focus is on modelling disease progression, disease transmission or physiological disorders which can be found in other reviews such as Chang et al. [ 39 ] and Long et al. [ 41 ]. The data sources or details of how data was used to conceptualise and formulate models are not presented in this paper; this could on its own be the focus of another study and we hope to publish these results as future work. This information would be useful for researchers who want to gain an understanding of the type and format of data used to model health systems and best practice for developing and validating such models.

Literature that was not reported in English was excluded from the review which may have resulted in a small proportion of relevant papers being missed. Papers that described DES models, the other popular modelling method for simulating health system processes, were not included in this review (unless DES methods are presented as part of a hybrid model integrated with SDM or ABM) but have been compiled elsewhere [ 68 , 69 , 70 ]. Finally, the quality of the papers was not assessed.

Implications for future research

A nominal number of SDM papers (9/28), an even lower proportion of ABM papers (2/11) and none of the hybrid methods papers simulated health systems based in low- or middle-income countries (LMICs). The lower number of counterpart models in LMICs can be attributed to a lack of capacity in modelling methods and perhaps the perceived scarcity of suitable data; however, the rich quantitative and qualitative primary data collected in these countries for other types of evaluation could be used to develop such models. Building capacity for using these modelling methods in LMICs should be a priority and generating knowledge of how and which secondary data to use in these settings for this purpose. In this review, we observed that it is feasible to use SDM to model low-income country health systems, including those in Uganda [ 60 ] and Afghanistan [ 30 ]. The need to increase the use of these methods within LMICs is paramount; even in cases where there is an absence of sufficient data, models can be formulated for LMICs and used to inform on key data requirements through sensitivity analysis, considering the resource and healthcare delivery constraints experienced by facilities in these settings. This research is vital for our understanding of health system functioning in LMICs, and given the greater resource constraints, to allow stakeholders and researchers to assess the likely impact of policies or interventions before their costly implementation, and to shed light on optimised programme design.

Health system professionals can learn greatly from using modelling tools, such as ABM, SDM and hybrid models, developed originally in non-health disciplines to understand complex dynamic systems. Understanding the complexity of health systems therefore require collaboration between health scientists and scientists from other disciplines such as engineering, mathematics and computer science. Discussion and application of hybrid models is not a new phenomenon in other fields but their utilisation in exploring health systems is still novel; the earliest article documenting their use in this review was published in 2010 [ 43 ]. Five of the six hybrid modelling papers [ 43 , 44 , 45 , 46 , 47 ] were published as conference proceedings (the exception Kittipittayakorn et al. [ 24 ]), demonstrating the need to include conference articles in systematic reviews of the literature in order to capture new and evolving applications of modelling for health systems research.

The configuration and extent to which two distinct types of models are combined has been described in the literature [ 71 , 72 , 73 , 74 , 75 ]. The hybrid modelling papers selected in this review follow what is described as ‘hierarchical’ or ‘process environment’ model structures, the former where two distinct models pass information to each other and the latter where one model simulates system processes within the environment of another model [ 72 ]. Truly ‘integrated’ models, considered the ‘holy grail’ [ 43 ] of hybrid simulation, where elements of the system are simulated by both methods of modelling with no clear distinction, were not identified in this review and in the wider literature remain an elusive target. In a recent review of hybrid modelling in operational research only four papers were identified to have implemented truly integrated hybrid simulation and all used bespoke software, unrestricted by the current hybrid modelling environments [ 76 ].

Of the six hybrid modelling papers, only Djanatliev et al. [ 47 ] presented a model capable of both ABM and SDM simulation. The crucial macro- and micro- level activity captured in such models represent feedback in the wider, complex system while retaining the variable behaviour exhibited by those who access or deliver healthcare. With increasing software innovation and growing demand for multi-method modelling in not only in healthcare research but in the wider research community, we need to increase their application to modelling health systems and progress towards the ‘holy grail’ of hybrid modelling.

We identified 28 papers using SDM methods and 11 papers using ABM methods to model health system behaviour, six of which implemented hybrid model structures with only a single paper using SDM-ABM. Emergency and acute care, and elderly care and LTC services were the most frequently simulated health system settings, modelling the impact of health policies and interventions targeting at-capacity healthcare services, patient length of stay in healthcare facilities and undesirable patient outcomes. A high proportion of articles modelled health systems in high income countries; future work should now turn to modelling healthcare settings in LMIC to support policy makers and health system researchers alike. The utilisation of hybrid models in healthcare is still relatively new but with an increasing demand to develop models that can simulate the macro- and micro-level activity exhibited by health systems, we will see an increase in their use in the future.

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

One of the elderly or LTC services papers also modelled emergency or acute care but it was not the primary focus and is therefore not discussed here.

The single SDM-ABM paper that modelled the delivery of emergency or acute care is discussed in section ‘SDM-ABM use in health system research’.

Six of the emergency or acute care review papers and one of the cardiology care papers also modelled elderly or LTC services but it was not the primary focus and are therefore not discussed here.

Abbreviations

Accountable care organisation

Agent-based model

Discrete-event simulation

Emergency Department

Long-term care

Low- and middle-income countries

System dynamics model

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The work described in this paper was funded by the Health Systems Research Initiative (HSRI). MRC Grant Reference Number: MR/R013454/1.

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Additional file 1..

Search criteria used for each database.

Additional file 2.

Descriptive table of validation methods used in SDM and ABM literature.

Additional file 3.

Descriptive table of ABM model rules.

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Cassidy, R., Singh, N.S., Schiratti, PR. et al. Mathematical modelling for health systems research: a systematic review of system dynamics and agent-based models. BMC Health Serv Res 19 , 845 (2019). https://doi.org/10.1186/s12913-019-4627-7

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Abstract: The remarkable progress of Multi-modal Large Language Models (MLLMs) has garnered unparalleled attention, due to their superior performance in visual contexts. However, their capabilities in visual math problem-solving remain insufficiently evaluated and understood. We investigate current benchmarks to incorporate excessive visual content within textual questions, which potentially assist MLLMs in deducing answers without truly interpreting the input diagrams. To this end, we introduce MathVerse, an all-around visual math benchmark designed for an equitable and in-depth evaluation of MLLMs. We meticulously collect 2,612 high-quality, multi-subject math problems with diagrams from publicly available sources. Each problem is then transformed by human annotators into six distinct versions, each offering varying degrees of information content in multi-modality, contributing to 15K test samples in total. This approach allows MathVerse to comprehensively assess whether and how much MLLMs can truly understand the visual diagrams for mathematical reasoning. In addition, we propose a Chain-of-Thought (CoT) evaluation strategy for a fine-grained assessment of the output answers. Rather than naively judging True or False, we employ GPT-4(V) to adaptively extract crucial reasoning steps, and then score each step with detailed error analysis, which can reveal the intermediate CoT reasoning quality by MLLMs. We hope the MathVerse benchmark may provide unique insights to guide the future development of MLLMs. Project page: this https URL

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Mathematical Models of Infectious Diseases

Humanity has the ability to control the environment within which it resides. In day-to-day life, humans interact with different beings cohabiting their world. These interactions can sometimes be harmful or destructive to living beings and humans, and as a result, humanity has developed techniques, weapons and sophisticated technological instruments to help reduce the threat. Despite technological advances, we are continuously exposed to new challenges, and constantly face biological threats within our environment. Viruses are one such threat. Invisible to the human eye, they live in the air, soil, and water and on material surfaces and are responsible for a number of diseases that kill millions of people. Most recently, the rise of a new strain of coronavirus SARS-COV-2 developed into a pandemic that claimed over 200,000 lives between its first documented case in December 2019 in Wuhan, China, and May 1, 2020.

To combat these invisible enemies, we rely on the study of their behaviors in laboratories, analysis, and prediction. To perform the analysis and prediction, observed facts are converted into models using mathematical tools, including, differentiation, integration and statistical approaches. These models are analyzed and solved analytically or numerically for prediction using some obtained parameters and initial conditions. This present special issue is devoted to a collection of latest results from theoretical to application on research based on infectious diseases.

Edited by: Abdon Atangana, Muhammad Altaf Khan, Jose Francisco Gomez Aguila, Dumitru Baleanu, Emile Franc Doungmo Goufo, Abdullahi Yusuf

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• Published: 11 September 2023

Analysis of the research progress on the deposition and drift of spray droplets by plant protection UAVs

• Qin Weicai 1 , 2 &
• Chen Panyang 3  

Scientific Reports volume  13 , Article number:  14935 ( 2023 ) Cite this article

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• Plant sciences

Plant protection unmanned aerial vehicles (UAVs), which are highly adapted to terrain and capable of efficient low-altitude spraying, will be extensively used in agricultural production. In this paper, single or several independent factors influencing the deposition characteristics of droplets sprayed by plant protection UAVs, as well as the experimental methods and related mathematical analysis models used to study droplet deposition and drift, are systematically investigated. A research method based on farmland environmental factors is proposed to simulate the deposition and drift characteristics of spray droplets. Moreover, the impacts of multiple factors on the droplet deposition characteristics are further studied by using an indoor simulation test system for the spraying flow field of plant protection UAVs to simulate the plant protection UAVs spraying flow field, temperature, humidity and natural wind. By integrating the operation parameters, environmental conditions, crop canopy characteristics and rotor airflow, the main effects and interactive effects of the factors influencing the deposition of spray droplets can be explored. A mathematical model that can reflect the internal relations of multiple factors and evaluate and analyze the droplet deposition characteristics is established. A scientific and effective method for determining the optimal spray droplet deposition is also proposed. In addition, this research method can provide a necessary scientific basis for the formulation of operating standards for plant protection UAVs, inspection and evaluation of operating tools at the same scale, and the improvement and upgrading of spraying systems.

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Introduction

In agriculture, aerial spray is widely used to spray fertilizers, herbicides, fungicides and other materials used for crop protection 1 . Compared with large fixed-wing agricultural aircraft, small unmanned aerial vehicles (UAVs) are particularly advantageous because they are highly maneuverable and do not need any airport for taking off or landing 2 . In recent years, aerial machinery for plant protection, especially aerial spray by small plant protection UAVs, has developed rapidly 3 . Small plant protection UAVs have greater application prospects in agricultural production because of their better terrain adaptability and low-altitude spraying capability (Figs. 1 and 2 ) 4 , 5 , 6 , 7 . However, as an emerging technology, UAV spraying technology in agricultural pest control are not common due to the lack of operational standards and uncertainty about the best spraying parameters, which leads to a series of problems, such as the poor uniformity of droplet deposition distribution and low levels of fog deposition.

Single-rotor UAV spraying.

Multirotor UAV spraying.

Some studies have shown that if the aerial spraying parameters are not set scientifically, it will lead to not only repeated spraying and missed spraying, degrading the effect of pest control but also pesticide drift 8 . The use of new pesticide additives and the innovative research and development of precise spraying equipment of plant protection UAVs along with its safe and efficient use in the prevention and control of diseases, pests and weeds are indispensable means to increase the pesticide deposition amount and reduce drift. Studying the deposition characteristics of spray droplets is not only of scientific significance for the development of new pesticide formulations and precise spraying equipment of plant protection UAVs but also of practical guiding significance for the safe and efficient use of pesticides in farmland. Due to many factors, such as the natural environment, pesticide characteristics, crop canopy characteristics, and plant protection UAV operating parameters, it is a complicated process to study the uniformity and penetration of spray droplets. To improve the spraying effect and reduce drift, scientific and technological staff all over the world have carried out a large number of exploratory studies on the deposition and drift characteristics of spray droplets through field or wind tunnel experiments or mathematical model analysis 9 , 10 , 11 , 12 , 13 . The main factors and secondary factors influencing the characteristics of droplet deposition and drift are organized from the many influencing factors (nozzle, droplet, aircraft type, weather factors, etc.), and the functional relationship between the amount of different droplet deposition and drift and their influencing factors are determined. However, there are not sufficient deposition models for plant protection rotor UAVs, and the existing models consider only a few influencing factors, which need to be further modified.

With the development of UAV technology, there are an increasing number of studies on the droplet deposition rules, operation parameter optimization and evaluation methods of pesticides applied by plant protection UAVs in rice fields and maize fields 14 , 15 , 16 , 17 ; however, these studies have defects in that the meteorological factors in the farmland environment are unstable and uncontrollable, the UAV track easily deviates, resulting in the poor uniformity of droplet deposition distribution (the coefficient of variation may be above 40% 16 , while it is usually below 10% for spraying by ground equipment), the test result cannot be well repeated, and different types of UAVs cannot be easily evaluated at the same scale. Thus, it is difficult to evaluate the droplet deposition characteristics of different types of UAVs scientifically. Some research has established mathematical models to study the impact of plant protection UAV operating parameters (operating height, operating speed, and spraying flow rate) on droplet deposition and drift characteristics 18 , 19 , 20 and determined the main effects influencing droplet deposition. However, due to the lack of conformity between the assumptions of these models and farmland practice, they neglected the influence of the characteristics of the crop canopy and the interaction of multiple factors such as the environment, crops, and operating parameters of application equipment on the droplet deposition characteristics (uniformity of distribution and penetration), making the results obtained through analysis with existing mathematical models highly deviate from practice.

In this paper, the current status and problems of research on the deposition and drift of spray droplets from plant protection drones are introduced, and the importance of research in this area to improve the effectiveness of pesticide application and reduce drift hazards is emphasized. The need for more in depth, comprehensive and systematic research on the deposition and drift of spray droplets from plant protection drones is highlighted, and the problems and challenges of the current research are pointed out, providing important guidance and references for future research.

Research on the influencing factors of spray droplet deposition characteristics

Studying droplet deposition characteristics (uniformity and penetration) is always a major subject in pesticide application technology research 21 . The deposition characteristics of spray droplets are influenced by application techniques and equipment, crops, the environment, etc. Detailed influencing factors include the wind speed, wind direction, leaf area index, target crop canopy structure, leaf inclination, leaf surface characteristics, and characteristics of the spray droplet population (release height, release rate, application liquid volume, spray droplet particle size spectrum) 22 , 23 , 24 .

Several studies have investigated the influence of various factors on droplet deposition characteristics in plant protection UAV spraying. Diepenbrock noted that plant leaf characteristics, such as size, inclination angle, drooping degree, and spatial arrangement, impact the composition quantity and distribution quality within the crop canopy structure, subsequently affecting droplet penetration and deposition 25 . Song et al. found that altering the initial velocity of droplets increases deposition amounts on horizontal and vertical targets. Factors like flying altitude and speed of different aircraft types have been extensively studied for their influence on droplet deposition and drift 26 . Qiu et al. used an orthogonal experimental method to study the deposition distribution rules of droplets sprayed by unmanned helicopters at different flying heights and speeds under field conditions. They established a relationship model that clarifies the interactions between deposition concentration, uniformity, flying speed, and flying height, providing valuable insights for optimizing spray operation parameters 18 . Chen et al. investigated the pattern of aerial spray droplet deposition in the rice canopy using a small unmanned helicopter. They explored the effects of different spraying parameters on droplet distribution, specifically analyzing the deposition of growth regulator spraying 27 . Wang et al. proposed a method for testing the spatial mass balance of UAV-applied droplets and conducted field experiments on three types of UAVs to accurately determine the spatial distribution of the droplets and the downdraft field. They also conducted an experimental study on the droplet deposition pattern of hovering UAV variable spraying and highlighted the significant impact of downward swirling airflow on droplet deposition distribution 14 . Qin et al. focused on the influence of spraying parameters, such as operation height and velocity, of the UAV on droplet deposition on the rice canopy and protection efficacy against plant hoppers, using water-sensitive paper to collect droplets and statistically analyzing their coverage rates. The findings indicated that UAV spraying exhibited a low-volume and highly concentrated spray pattern 19 .

In summary, there are many factors influencing the deposition characteristics (uniformity and penetration) of spray droplets. However, in most of the current research on spraying by plant protection UAVs, only the influence of factors such as the flying height and flying speed on droplet deposition in the field environment is taken into consideration. Considering the influence of the interaction between environmental factors, crop canopy characteristics (growth stage, leaf area index, leaf inclination angle) and plant protection UAV spraying parameters on droplet deposition characteristics, there is neither in-depth understanding nor relevant reports, especially under controllable environmental conditions (Fig.  3 ). To promote high-efficiency spraying technology for plant protection UAVs, targeted basic research should be carried out on the analysis of the influencing factors of plant protection UAV spraying and the optimal deposition of droplets.

Description of the deposition and drift with rotor UAV spraying.

Research on the experimental means and testing methods of droplet deposition and drift

At present, the deposition and drift of droplets are mainly researched by field tests and wind tunnel tests 28 , 29 , 30 , 31 , 32 . Field test research on pesticide deposition and drift is similar to the actual situation, but it is quite difficult to acquire correct data due to the constant changes in meteorological factors such as the wind speed, wind direction, temperature and humidity. In addition, Emilia et al. noted that the terrain and plant morphology also influence the wind flow and droplet deposition, leading to considerable deviation among repeated test results 33 . Therefore, it is difficult to accurately determine the total amount and distribution of pesticides drifting in the air 34 . The wind tunnel laboratory can provide a controllable environment to simulate the external spraying conditions, and the wind speed and direction can be easily controlled. Therefore, it is an important means to study the drift characteristics of spraying components and avoid many defects in field test research 10 , 35 . The typical wind tunnels that are widely used in agricultural aviation spraying technology are shown in Table 1 36 , 37 .

Internationally well-known professional research institutions for pesticide application, such as the Julius Kuehn Institute-Federal Research Centre for Cultivated Plants (JKI, formerly BBA) and USDA-Agricultural Research Service, Application Technology Research Unit (USDA-ARS-ATRU), have a circular closed low-speed standard wind tunnel (Fig.  4 ). This wind tunnels are widely used to assess the distribution, degradation and drift of pesticide sprays, simulating real crop and environmental conditions. The advantages are that they provide accurate measurements of pesticide distribution and drift and are able to reproduce wind field conditions in realistic environments. However, circular low-speed wind tunnels have limitations when it comes to parameters such as spray particle size, density and flow rate for different pesticides. The Silsoe Research Institute, UK (SRI) has a standard linear low-speed wind tunnel. This wind tunnel can be used to test the performance of agricultural mechanised sprayers and the design of sprayers. The advantage is that they can simulate actual operating conditions and can accurately test the performance and flow rate of agricultural mechanised sprayers. However, linear low speed wind tunnels are typically more expensive than circular wind tunnels and can only simulate a single environmental condition. The Center for Pesticide Application and Safety (CPAS) of the University of Queensland in Australia has an open-path wind tunnel (Fig.  5 ). This type of wind tunnel can be used to test aspects such as drift and particle distribution of agricultural sprayers. The advantages are ease of operation, low cost and the ability to reproduce wind fields under different environmental conditions. However, open path wind tunnels do not simulate realistic crop environments and have unstable wind speeds. In 2014, the Nanjing Institute of Agricultural Mechanization, Ministry of Agriculture and Rural Affairs, built the NJS-1 plant protection direct flow closed wind tunnel (Fig.  6 ). This type of wind tunnel is mainly used to evaluate different sprayers in terms of performance and droplet distribution. The advantages are the ability to simulate a realistic farm environment with high accuracy and the ability to test different types and brands of sprayers. However, straight-through enclosed wind tunnels are only suitable for small equipment and small-scale trials and are costly. In 2018, the National Center for International Collaboration Research on Precision Agricultural Aviation Pesticide Spraying Technology of South China Agricultural University built a high- and low-speed composite wind tunnel for agricultural aviation research (Fig.  7 ). This wind tunnel is suitable for agricultural aerial research and can simulate the effects of spraying at different heights and wind speeds. The advantage is that it can accurately test the effects of pesticide spraying at different heights and speeds, and can improve the efficiency and accuracy of agricultural aerial spraying. However, high and low speed composite wind tunnels are relatively costly and require a high level of technology and equipment requirements. As the basic conditions for technical research, these wind tunnels have made great contributions to the study of pesticide deposition and drift rules, product testing, and product optimization 38 , 39 , 40 , 41 , 42 . However, for the study of spray droplet deposition and drift under the disturbance of the wind field of plant protection UAVs, the single-direction wind tunnel simulation test is still insufficient to simulate the combined effect of the downward swirl flow under the rotor and the natural wind. In addition, the existing agricultural wind tunnels are limited in size, so plant protection UAVs cannot be placed. In the military, a scaled model method is used to put UAVs into wind tunnels for research 43 , 44 , but it is not suitable for research on pesticide spraying with plant protection UAVs, and the airflow will rebound from the tunnel wall.

Circle closed low-speed wind tunnel.

Open wind tunnel.

NJS-1DC closed wind tunnel.

High and close speed composite wind.

Another important test technique for drift research is the sampling and analysis of droplet drift. Test studies on the drift of aerial mist in developed countries such as the United States and Germany are carried out with advanced test instruments, including automatic air samplers, gas or liquid chromatography, fluorescence analyzers, and electronic scanners. to collect and analyze the droplet deposition amount, the number of droplets, the coverage density of droplets, and the content of substances and study the correlation between additive concentration, spraying height and drift 4 , 45 , 46 . However, these traditional methods involve a long collecting and processing cycle, samples have to be processed in the lab, and it is difficult to express the dynamics of droplets in air. Particle image velocimetry (PIV) and LIDAR scanning test methods can solve the above problems, and each has its own advantages. PIV can obtain the three-dimensional spatial velocity vector of droplets and droplet size with a high sampling accuracy but limited spatial measurement scale 47 , 48 , 49 ; the LIDAR scanning method, realized by layered scanning, can quickly and accurately obtain the large-scale spatial droplet point cloud data and inversely form the three-dimensional distribution and temporal-spatial change of the droplets, but cannot reflect the spatial velocity vector change of the droplets 50 . The advantages, disadvantages and applications of droplet deposition and drift measurement methods are shown in Table 2 51 .

Overall, the sampling and analysis of droplet drift, along with techniques such as PIV and LIDAR scanning, play a crucial role in studying and understanding the behavior of droplets during aerial spraying. These methods provide valuable insights into droplet deposition, drift patterns, and the effects of various factors, enabling researchers to optimize spray practices, minimize drift, and enhance the efficiency and effectiveness of plant protection UAV applications.

Research on the mathematical analysis model of spray droplet deposition characteristics

In the development of spraying equipment and the determination of the optimal deposition conditions for spray, a large amount of data and information are needed to explain the influence of different factors on the spraying performance and the relationship between variables. At present, spraying drift modeling can be divided into models based on mechanics and models based on statistics 52 , 53 , 54 .

One of the models based on mechanics analyzes the movement of a single droplet in the airflow field by the Lagrangian trajectory tracking analysis method. Teske et al. established the AGDISP model by the analytical Lagrangian method to describe aerial spraying under the condition of ignoring the influence of aircraft wake and atmospheric turbulence 46 . This model takes not only the aircraft type, environmental conditions, and droplet properties but also the influencing factors of the nozzle model into consideration. The user can input the parameters of the nozzle, droplet spectrum, aircraft type and weather factors. from an internal database and predict the drift potential. It can effectively and accurately predict a range of 20 km but is generally used for fixed-wing aircraft. Duga et al. and Gregorio et al. also studied the deposition distribution of aerial spray in orchards with the Lagrangian discrete phase model, and the result of the numerical model showed that the prediction error of total deposition on the fruit tree canopy is above 30% 48 , 51 . Dorr et al. developed a spray deposition model for whole plants based on L-studio, which takes into account the plant leaf wettability, impact angle, droplet break-up and rebound behavior, and the number of sub-droplets produced 55 . In 2020, Zabkiewicz et al. used an updated version of the software based on this model, developing a new user interface and refining the droplet fragmentation model 56 .

Another model based on mechanics is realized with the CFD (Computational Fluid Dynamics) method 57 , 58 , but there are still large errors between the simulated value and the real value of some models due to various factors. Holterman et al. carried out a series of cross-wind single nozzle field experiments in consideration of the traveling speed, entrained airflow, geometric parameters of the farmland, sprayer system setting parameters and environmental factors when studying the droplet deposition drift model of ground boom sprayers to calibrate the mathematical model. The results showed that when the height from the crop canopy is less than or equal to 0.7 m, the error between the test and the model simulation is within 10%, but the error between droplet deposition and drift prediction gradually increases as the height of the spray boom increases 59 , 60 , 61 .

Chinese scientific and technological staff have conducted experimental research and numerical analysis on the numerical simulation and mathematical modeling of spraying droplet deposition and drift prediction of ground plant protection equipment and have drawn some conclusions that physical quantities such as the operating speed, droplet size and crosswind impact the droplet deposition and drift process (Figs. 8 and 9 ) 62 , 63 . Zhu et al. developed the DRIFTSIM based on CFD and Lagrangian methods with a CFD simulation database for ground drift prediction and a user interface to access drift-related data 64 . Hong et al. constructed an integrated computational hydrodynamic model to predict the deposition and transport of pesticide sprays under the canopy in apple orchards during different growth periods 65 .

Rotor wind field test platform based on a wind tunnel.

Layout scene of droplet drift.

The above research proves that computer simulation technologies are widely applicable to the prediction research of droplet deposition under various complicated wind-supply airflow conditions 66 . The existing AGDISP model is relatively developed and only suitable for research on fixed-wing aircraft, which is very different from research on plant protection UAVs. The current plant protection UAV spraying prediction model still has problems such as large relative errors between the experimental value and simulation value of the deposition and drift at each measurement point. Therefore, the prediction accuracy of the numerical model for the spray droplet deposition of plant protection UAVs is still low and needs to be improved, and there is a lack of in-depth basic research on analyzing the rotor flow field and establishing mathematical analysis models for droplet deposition 67 .

The rotor wind field test platform and droplet drift

The use of UAVs for crop spraying has become increasingly popular due to its efficiency and effectiveness. However, accurately analyzing the spraying process is challenging due to the complex flow field of the droplets in the air and the multitude of factors that can affect their deposition characteristics. Current testing systems rely on simple methods such as static targets or trays, which do not accurately represent the dynamic and complex nature of the real environment. To better study the UAV spraying flow field, a corresponding indoor simulation test system is needed. The indoor simulation system proposed in this study combines a natural wind simulation system and a rotor simulation system that can simulate several factors present in the natural environment that affect droplet deposition characteristics. The natural wind simulation system can effectively replicate wind speed variations, which is a key factor influencing droplet dispersion and deposition. By adjusting the settings of the wind simulation system, it is possible to replicate a range of wind speeds encountered in the field, allowing researchers to study the effects of different wind speeds on droplet behaviour and deposition. By adjusting the settings of the rotor simulation system, it is possible to demonstrate the magnitude of the downwash airflow at different speeds of the UAV rotor. However, it is important to note that while wind speed variations can be simulated, other factors, such as wind direction and turbulence, may have limitations in being accurately replicated in an indoor simulation system. These factors may require further development of simulation techniques to achieve more accurate replication. Nevertheless, the inclusion of natural wind simulation systems and rotor simulation systems in indoor simulation setups provides a valuable tool for studying the effects of wind speed.

The fluorescence tracer method involves adding a fluorescent dye or tracer to the liquid spray mixture used in the UAV spraying process. When these droplets containing the fluorescent tracer are released into the air, they can be illuminated with a specific wavelength of light, typically ultraviolet (UV) light. The fluorescent dye absorbs this UV light and re-emits it at a longer wavelength, usually in the visible range.

The high-speed camera is synchronized with the UV light source and captures the emitted fluorescent signals from the droplets. By analyzing the recorded video footage, researchers can precisely track the movement and behavior of the fluorescent droplets in the air. The high-speed camera captures images at a rapid frame rate, allowing for the visualization and analysis of the droplet flow field in detail.

The proposed indoor simulation test system for the spraying flow field of plant protection UAVs is a comprehensive and innovative method that combines the fluorescence tracer method and high-speed camera method to accurately track the dynamic changes in the local droplet flow field in the air. This system also includes a natural wind simulation system, which allows for the more realistic simulation of the actual environment, and thus more accurately reproduces the complex factors that affect droplet deposition characteristics. This method represents a significant improvement over existing testing systems, as it provides a more accurate and comprehensive analysis of the deposition process of droplets affected by multiple factors, enabling researchers to more effectively study the flow field and optimize the spraying process for plant protection UAVs. Overall, this proposed system has the potential to revolutionize the study of UAV spraying flow fields and could lead to significant advancements in the field of plant protection. Therefore, the method proposed in this paper is superior to the methods currently in use (Fig.  10 ).

Diagram of the rotor wind field test platform and droplet drift.

In conclusion, existing studies on plant protection UAV spraying have primarily focused on isolated factors, such as flying height, flying speed, and nozzle flow, without considering the interaction effects among other influential factors. This limitation calls for the need to conduct experimental research that combines spray droplet deposition characteristics with crop canopy characteristics in a controllable environment, encompassing environmental conditions and operating parameters. The proposed research aims to address this gap by developing an indoor simulation system that incorporates a natural wind simulation system. This innovative system enables the study of droplet deposition characteristics influenced by multiple factors in a realistic environment. By statistically analyzing the factors affecting droplet deposition and establishing a multivariable relationship model, optimal droplet deposition suitable for field operation decision-making of plant protection UAVs can be quantified and evaluated. This research presents an effective technical pathway for understanding the deposition patterns of droplets sprayed by plant protection UAVs and supports the formulation of relevant pesticide application standards for plant protection UAVs.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

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This research was funded by the National Natural Science Foundation of China (Grant No. 31971804); Independent Innovation Project of Agricultural Science and Technology in Jiangsu Province (CX(21)3091); Suzhou Agricultural Independent Innovation Project (SNG2022061); and Suzhou Agricultural Vocational and Technical College Landmark Achievement Cultivation Project (CG[2022]02).

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Q.W. conceived and designed the study. Q.W. and C.P. performed most experiments. Q.W. analyzed the data and wrote the first draft of the manuscript. C.P. revised the manuscript. Q.W. supervised the project and reviewed the manuscript.

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Weicai, Q., Panyang, C. Analysis of the research progress on the deposition and drift of spray droplets by plant protection UAVs. Sci Rep 13 , 14935 (2023). https://doi.org/10.1038/s41598-023-40556-0

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Two articles in this Special Issue focus on the modeling of infection transmission dynamics. The article by Isa Abdullahi Baba et al. presents a fractional-order cholera model that is an extension of the Susceptible–Infected–Recovered epidemic model [ 8 ]. The model incorporates the saturated incidence rate to accurately represent the transmission dynamics of the disease. The article by Hosam Alhakami et al. uses a deterministic mathematical model of vector-borne viral plant disease dynamics to train a feed-forward neural network using Levenberg–Marquardt backpropagation algorithm [ 9 ]. The neural network is then used to study the implication of fluctuations on natural plant mortality and vector mortality rates.

Mathematical models of biological systems usually describe many interacting components and involve many parameters. Furthermore, it is common that only limited experimental data are available to calibrate the models. Therefore, reliable mathematical models of biological systems can only be developed with rigorous parameter estimation and model validation techniques. Samaneh Gholami and Silvana Ilie propose a parameter estimation method for stochastic discrete models of biochemical networks [ 10 ]. The method utilizes finite-difference approximations of the parameter sensitivities and the singular value decomposition of the sensitivity matrix. Several models of biochemical systems are used to demonstrate the advantages of the proposed method.

The article by Christopher Parker, Erik Nelson, and Tongli Zhang presents a computational framework named VeVaPy, which is designed to verify and validate mathematical models comprising many interacting components and parameters [ 11 ]. VeVaPy is a publicly available Python library that can help determine which model from the literature is the best for fitting new experimental data. The authors use several hypothalamic–pituitary–adrenal (HPA) axis models from the literature to demonstrate the way in which VeVaPy can help to verify and validate these models against new data: VeVaPy runs the differential evolution parameter optimization algorithm on each model against several novel datasets and ranks the models based on their average cost function value. In their demonstration, two out of five HPA models performed the best in elucidating the novel datasets. Overall, the model validation process is able to operate with significantly less effort when using VeVaPy.

Funding Statement

This research received no external funding.

Conflicts of Interest

The author declares no conflict of interest.

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