Send more data: a systematic review of mathematical models of antimicrobial resistance

The discovery of antimicrobials in medicine in the 1920s was regarded as a miracle. Since then, millions of lives have been saved as a result of this treatment. However, history has shown that the introduction of any kind of antimicrobial compound into human or veterinary medicine is swiftly followed by emerging resistance to that compound [1]. Antimicrobial resistance (AMR) is threatening our ability to treat common infectious diseases, resulting in prolonged illness, disability and death [2]. Multidrug and even pan-resistant organisms are now a worldwide problem. Despite the difficulty in estimating the actual costs of AMR, the true economic burden is substantial [3]. The estimated economic consequences of AMR in Europe in 2007 were at least €1.5 billion, while they were estimated to be $55 billion in the US in 2000 (cited from Gandra et al., 2014 [3]). It is therefore of utmost importance to limit the emergence and spread of AMR.

AMR is spreading globally - not just in the human population, but also in animal populations and the environment. Furthermore, there is consistent evidence that an exchange of bacteria resistant to antimicrobials and AMR determinants exists between these different compartments [4]. AMR determinants have been shown to survive in environments such as sludge and wastewater treatment systems [5, 6], thus allowing for the transmission of infectious bacteria and accelerating the problem of AMR.

Mathematical models have become important decision support tools in medicine and public health [7]. They have helped in improving our understanding of the development, emergence and spread of AMR [7, 8]. In addition, they can identify gaps in our knowledge, and direct research towards missing information for important parameters and processes in the modelled system. However, in 2006, Opatowski et al. [7] published a review on mathematical models on AMR and concluded that there was still a need for major improvements of AMR models such as regarding implementing important features of pathogen including resistance mechanisms and inter-species cooperation. Continual evaluation of published mathematical models is therefore necessary for us to recognise the progress in AMR modelling. Gaps in our knowledge can be identified, and this can be used to set the agenda and form suitable hypotheses for future research in the fight against AMR.

Grimm et al. [9] updated the TRACE paradigm that was established in 2010 with the aim of developing guidelines to produce useful models. The TRACE paradigm includes eight elements that, when followed, ensure that models are clearly communicated when published. These elements are: 1) Problem formulation (clear formulation of the objective and a description of the context of the model); 2) Model description (written description of model elements to allow readers to understand and replicate the model); 3) Data evaluation (an assessment of the quality of data used to parameterise the model); 4) Conceptual model evaluation (a list and explanation of the most important conceptual design decisions); 5) Implementation verification (internal validation of the model, testing for programming errors and assessing model performance); 6) Model output verification (external validation, testing whether the model output matches the observations); 7) Model analysis (mainly sensitivity analysis); 8) Model output corroboration (a comparison of model output with data that were not used to create the model). For a full description of the TRACE elements, see Grimm et al. [9].

Since the comprehensive systematic review of mathematical models between 1993 and 2006 was conducted by Temime et al. [10], a number of additional reviews have been published [7, 8, 11]. However, these reviews either focused on models linking antibiotic use to AMR [11] or modelling AMR in populations (humans and bacteria) and hospitals [7], and did not include exclusively within-host models [8]. These systematic reviews did not examine models of AMR in relation to animal populations and the environment. However, a comprehensive review of mathematical models of AMR should consider models of all relevant populations and ecosystems in order to target the AMR problem from a One-Health perspective. In this way, researchers from different fields could benefit from experiences and advances in the other fields.

The objective of this review was to assess the usefulness of mathematical and simulation models of AMR development and/or spread in individuals and/or populations of humans, animals and bacteria, as well as in the environment. We also aimed to identify gaps in the knowledge needed to provide useful models of AMR. The assessment was achieved using a systematic review. The published models were then summarised and compared using an adapted version of previously developed frameworks [7, 8]. Furthermore, the strengths and weaknesses of the models were discussed using the TRACE paradigm [9] .

This is a systematic review following the PRISMA guidelines [12] without the use of an existing review protocol.

Recently, Heesterbeek et al. [25] reviewed the importance of mathematical modelling of infectious disease dynamics in terms of improving public health. The authors concluded that, mathematical models can provide inside that can be used in public health policies through the use of new data.

AMR is a major threat to public health, and the fight against it could benefit from the use of mathematical modelling. It could play an important role in providing an insight into the dynamics of AMR, quantifying the effect of factors that influence it and providing tools for its control and prevention. Furthermore, modelling can present an opportunity to elucidate potential gaps in our knowledge.

The reviewed papers varied in their choice of model structure and complexity – from simple deterministic models to advanced mechanistic models (agent-based, individual and nested models). However, they generally provided little justification for the model type and structure that was chosen. In addition, the majority of studies focused on modelling only one unit (Table 2), a single strain of a pathogen (Table 5), assumed homogeneous mixing (Table 4), and ignored uncertainty and stochasticity in the development and/or spread of AMR (Table 3). AMR is a multifactorial problem with several elements – including external factors and interactions within and between populations (microbiota, animal and human populations) – able to affect its development and spread [26]. This creates nonlinearity, heterogeneity, and stochasticity that should be considered when mathematical models of AMR are developed. Opatowski et al. [7] wrote that models should take into account the specific pathogen characteristics such as the resistance mechanism of the pathogen and cooperation among species. They concluded that this would provide major improvements of models.. However, in the 6 years since their review was published, only one paper has described a truly nested model [27] (Table 2), modelling multiple bacterial strains within individuals (pigs) that interact as a population with a heterogeneous structure. Unfortunately, this model was not validated and does not allow conversion of the pathogens. Furthermore, one article [28] published a framework to cope with multiple nested layers from the genetic composition of cells, to the environment of cells, the host of the cells, and the environment of the host. This type of models is clearly something to be striving for in the future as the AMR problem is highly complex, and the interaction on many levels require a deep understanding. It would also be very helpful if the community could commit to using this type of standard models, so that the huge work of parameterise these models could begin. That in the future we could stand on the shoulders of each other instead of trying to building new models for every single problem.

Mechanistic modelling using stochastic processes can describe complex heterogeneous structures and processes, multiple pathogens/genes simultaneously, and model biological interactions that may affect AMR such as the immune system, the dosing effect of antibiotics, the microbiome and variabilities involved in the system. In addition, these models can provide insights into the temporal dynamics of AMR, both in the individual and the population. Arepeva et al. [11] also point to the advantage of this class of models over simpler types of models such as deterministic differential equations. Nine models used analytical solutions to solve the modelled system (Table 2), providing extensive mathematical solutions with a limited interpretation of the applicability of the outcomes to real life. In fact, only two papers [29, 30] attempted to validate the models using data. Analytical solutions can be useful when trying to avoid time-consuming and computer-intensive simulations. Nevertheless, from a practical point of view, the high complexity of AMR and limited translation of analytical solutions to real life can call the usefulness of this approach to solve and/or limit the AMR problem into question.

Ideally, models of AMR should be validated by data. However, many of the published models represent hypothetical situations in hospitals or communities with no supporting data [31‐35]. Such models are only useful in the event that a similar hospital or environment can be located. If this is the case, experiments or observational studies can be carried out to validate the models. In addition, there seems to be a lack of knowledge of how to implement different typical parameters and how to relate these to reality. For instance, what is the carrying capacity of a human patient for different types of AMR bacteria or genes, and how do levels of AMR relate to transmission rates under different circumstances in a hospital or community? This highlights the necessity for further fundamental and conceptual research to provide information and data to develop useful simulation models of AMR processes.

Validation is an essential factor when developing a mathematical model. Validation can be both internal (conducted to ensure that the model is doing what it should) and external (conducted to assess whether the model outcomes resemble real life). Models were externally validated in only 13 of the studies; ten studies used data and three were dependent on literature. The absence of validation in many of the published models (Table 5) could be due to a lack of usable data. There is a large gap in our knowledge when it comes to the dynamics of AMR inside a host, especially in terms of genotypic AMR. Interestingly, none of the studies indicated that internal validation had been conducted. Several methods can be used to internally validate the models, such as the rationalism method, tracing method and face validity [36]. Internal validation is important to ensure that the code is free from errors, satisfying the fifth criteria of the TRACE method [9]. It is possible that internal validation has been conducted, even if it is not mentioned in the paper. Nevertheless, we believe that it is important to describe the methods and steps used for internal validation in order to ensure confidence in the predictions. A lack of model validation may increase the risk of erroneous outcomes and conclusions, which in turn may reduce any confidence the scientific community and decision makers have in the predictions. Strict internal validation of the models must therefore be conducted and reported. Furthermore, additional research should be conducted to provide data to externally validate the models, resulting in models that can provide trustworthy recommendations. There exist papers on mathematical models where the TRACE criteria are fulfilled. A good example hereof is written by Foddai et al. [37].

The vast majority of the papers focused on modelling AMR in relation to humans, either directly by modelling human populations (in hospitals or communities) or in bacteria related directly to human health. Only four models relating to animals were conducted (Table 1). Animals might constitute a reservoir of AMR that can be spread to humans through their products (e.g. meat [38‐41]), the environment (faeces used as fertilisers [41‐43]), or direct contact [41, 44, 45], so more attention should be paid to improving our understanding of AMR dynamics within livestock production systems and the environment.

All studies included in this review report that an increase in antimicrobial use increases AMR in general. Some papers report that certain strategies show relatively smaller increases in AMR, which could be due to reducing contact rates or cycling different kinds of antimicrobial products [13, 15, 20, 27, 31‐34, 46‐56]. One paper reported a decrease in AMR when using an antimicrobial against which bacteria have no resistance [35]. However, as the authors report, such a property is transient and will diminish in time proportional to the extent to which that drug is used. Some papers construct several pathways to achieving AMR (i.e. hospital- versus community-acquired AMR) and deduce the parameter values at which the R0 (basic reproduction number, denoting how infectious the disease is) is above 1 [20, 31, 48, 52, 55, 35, 57]. However, no papers actually fit epidemiological data to determine parameters or validate their model. There are many studies looking at the epidemiological spread of specific resistant pathogens (e.g. MRSA), but these studies were excluded from this review, as we are interested in the spread of resistance rather than specific pathogens. In stating that there are no data of epidemiological spread, we mean spread of resistance between bacteria in an in vivo situation. There is an abundance of papers describing spread in in vitro experiments [19, 58‐61], yet we believe that such parameters can at best be a starting point for estimating parameters in vivo, as the natural environment is much more complex and competitive than a petri dish.

To improve our understanding of AMR, we might need to specifically understand the mechanisms that generate resistance. Some papers in our review modelled specific mechanisms (i.e. efflux pumps, senescence, indoles, or influence of the normal flora) [29, 53, 59, 61]. However, only one of these papers was actually validated by data [29]. Modelling specific mechanisms might be a way to better understand the behaviours and interactions of bacteria using these methods, and it may also give us a better understanding of how AMR interacts when multidrug resistance is considered.

The environmental impact of AMR was modelled in four papers: three of the papers modelled rivers and described the accumulation and survival of AMR [17, 62, 63]. One of the papers also included the effects of metals on the development of AMR [17]. One paper described AMR bacteria growth in slurry [64], showing that AMR bacteria can thrive in this medium. The aggregation and possible growth of AMR bacteria in the environment might be of great concern if bacteria are exposed to a mixture of AM from several sources e.g. in rivers or slurry, they may acquire multiple resistances. If there is a chance that these can then transfer back into the animal or human population, these types of models may be very useful.

The studies originated from 16 different countries and were published in 30 different journals. This indicates that a relatively large number of journals are interested in modelling AMR. Furthermore, it highlights that mathematical modelling is a relevant subject for a broad section of the scientific community. When screening and excluding papers, we might have excluded papers based on a misinterpretation of the title or abstract. However, to minimise such mistakes, we strived to include papers in cases where there was any doubt.

The majority of the studies modelled phenotypic AMR, while few models represented genotypic AMR (Table 5). Modelling genotypic AMR can be more complicated as many genes can be linked to a specific antibiotic, and the relationship between gene abundance and antibiotic use seems to be more complex than first anticipated [65]. Modelling genotypic AMR requires the relevant genes for the modelled AMR to be represented, as well as circumstances allowing for the genetic AMR to be expressed as phenotypic AMR, leading to a spread of the resistant pathogen within the population. Published models of genotypic AMR do not link this AMR type to the development of phenotypic AMR and the subsequent spread of the resistant pathogen between individuals [18]. This is perhaps due to a lack of information on the necessary circumstances for the phenotypic expression of genetic resistance determinants, thus emphasising the need for more research to better understand this process. Understanding the process is essential in the prevention of AMR development and spread.

In our opinion, the following elements should be considered when developing future models of AMR:

The current study provides a comprehensive review of published models of AMR spread and development since 2006. Although the study focuses on providing insights into the technical elements of and differences between the models, it also provides an insight into the elements that should be included when AMR is modelled.

Many mathematical models of AMR development and spread exist. However, there is still a lack of knowledge regarding the underlying mechanisms at work, thus limiting the true usefulness of the developed models. Furthermore, few models complied with the TRACE criteria. Future AMR models should elucidate the dynamics and variability of AMR occurrence and spread in order to investigate ways of effectively influencing these dynamics to prevent and control AMR. In addition, it is of utmost importance to focus research on providing data to parameterise and validate AMR models in order to extract useful conclusions from them. There is a need for more rigorous model development and testing and more abundant experimental and observational data to support model validation.