Guide Mathematical Modelling: Concepts and Case Studies

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We identified 20 relevant articles see Figure 1 for a flow chart of the identification of eligible articles 25 , 27 — Most gave recommendations for good modelling practice and were compiled by a task force in a consensus process or based on a systematic or narrative review of the literature. A questionnaire or checklist was not included. A subsequent series of seven articles 25 , 38 — 42 , 44 by the joint International Society for Pharmacoeconomics and Outcomes Research ISPOR and Society for Medical Decision Making SMDM task force elaborated upon these recommendations, providing detailed advice on conceptualizing the model, state transition models, discrete event simulations, dynamic transmission models, parameter estimation and uncertainty, and transparency and validation.

The 79 recommendations are summarized in the first article of the series We identified four articles 32 , 34 , 37 , 43 that present comprehensive frameworks of good modelling practice, with detailed justifications of the items covered and attributes of good practice. They include signalling or helper questions to facilitate the critical appraisal of published modelling studies: the number of questions ranges from 38 in Caro et al.

Fees and funding

The four frameworks cover similar territory, including items related to the problem concept, model structure, data sources and synthesis of the evidence, model uncertainty, consistency, transparency and validation Table 2. Two of the frameworks include sponsorship and conflicts of interest 32 , In a qualitative study Chilcot et al.

Respondents agreed that developing an understanding of the clinical situation or disease process being investigated is paramount in ensuring model credibility, highlighting the importance of clinical input during the model development process Published mathematical models addressing the same issue may reach contrasting conclusions. In this situation, careful comparison of the models may lead to a deeper understanding of the factors that drive outputs and conclusions. Ideally, the different modelling groups come together to explore the importance of differences in the type and structure of their models, and of the data used to parameterize them 19 , 45 , For example, several groups of modellers have investigated the impact of expanding access to antiretroviral therapy ART on new HIV infections.

The HIV Modelling Consortium compared the predictions of several mathematical models simulating the same ART intervention programs to determine the extent to which models agree on the epidemiological impact of expanded ART The consortium concluded that although models vary substantially in structure, complexity, and parameter choices, all suggested that ART, at high levels of access and with high adherence, has the potential to substantially reduce new HIV infections in the population There was broad agreement regarding the short-term epidemiologic impact of ART scale-up, but more variation in longer-term projections and in the efficiency with which treatment can reduce new infections.

The impact of ART on HIV incidence long-term is expected to be lower if models: i allow for heterogeneity in sexual risk behaviour; ii are age-structured; iii estimate a low proportion of HIV transmission from individuals not on ART with advanced disease at low CD4 counts ; iv are compared to what would be expected in the presence of HIV counselling and testing compared to no counselling and testing ; v assume relatively high infectiousness on ART; and vi consider drug resistance 19 , 47 , GRADE was conceived with the intention of creating a uniform system to assess a body of evidence to support guideline development in response to a confusing array of different systems in use at that time It has since been adopted by over 90 organisations, including WHO.

GRADE addresses clinical management questions, including the impact of therapies and diagnostic strategies, diagnostic accuracy questions i. The GRADE approach encompasses two main considerations: the degree of certainty in the evidence used to support a decision and the strength of the recommendation. The degree of certainty, i. The initial assessment is based on the study design: RCTs start as high certainty and observational studies as low certainty.

Based on the assessments of the five dimensions, RCTs may be down-rated and observational studies up- or down-rated. Judgment is required when assessing the certainty of the evidence, taking into account the number of studies of higher and lower quality and the relative importance of the different dimensions in a given context.

We believe that evidence from mathematical modelling studies could be assessed within the GRADE framework and included in the guideline development process. Specifically, guideline groups might include mathematical modelling studies as an additional study category, in addition to the categories of RCTs and observational studies currently defined in GRADE. The dimensions of indirectness, inconsistency, imprecision and publication bias are applicable to mathematical modelling studies, but criteria may need to be adapted. The concept of bias relates to results or inferences from empirical studies, including RCTs and observational studies 51 , 52 and is too narrow in the context of assessing mathematical modelling studies The assessment of the credibility of a model is informed by a comprehensive quality framework and should cover the conceptualization of the problem, model structure, input data and their risk of bias, different dimensions of uncertainty, as well as transparency and validation Table 2.

The framework should be tailored to each set of modelling studies by adding or omitting questions and developing review-specific guidance on how to assess each criterion. The certainty of the body of evidence from modelling studies can then be classified as high, moderate, low, or very low.

1st Edition

In the evidence-to-decision framework a distinction should be made between observed outcomes from empirical studies and modelled outcomes from modelling studies see the Meeting Report 4 for an example. Based on the discussions and presentations at the workshop in Geneva, the survey and rapid systematic review, we believe a number of conclusions can be formulated. The use of modelling studies should routinely be considered in the process of developing WHO guidelines.

Findings of mathematical modelling studies can provide important evidence that may be highly relevant.

3rd Edition

Evidence from modelling studies should be considered specifically in the absence of empirical data directly addressing the question of interest, when modelling based on appropriate indirect evidence may be indicated. Examples for such situations include the evaluation of long-term effectiveness, and the impact of one or several interventions comparative effectiveness , for example in the context of public health programmes where RCTs are rarely available. Modelling may be more acceptable and more influential in situations where immediate action is called for, but little direct empirical evidence is available, and may arguably be more acceptable in public health than in clinical decision making.

In these situations for example, the HIV, Ebola, or Zika epidemics funding is also likely to become available to support dedicated modelling studies. The use of evidence from mathematical models should be carefully considered and there should be a systematic and transparent approach to identifying existing models that may be relevant, and to commissioning new models.

Existing frameworks and checklists may be adapted to a set of modelling studies by adding or omitting questions. In some situations, the approach will need to be developed de novo. Additional expertise will typically be required in the systematic review groups or guideline development groups to appropriately assess the credibility of modelling studies and interpret their results.

The credibility of the models should not be evaluated only by modellers, and not only by modellers involved in the development of these models. The inclusion of evidence from modelling studies into the GRADE process is possible and desirable, with relatively few adaptations. GRADE is simply rating the certainty of evidence to support a decision and any type of evidence can in principle be included.

The certainty of the evidence for modelling studies should be assessed and presented separately in summaries of the evidence GRADE evidence profiles , and classified as high, moderate, low, or very low certainty. The GRADE dimensions of certainty imprecision, indirectness, inconsistency and publication bias and the criteria defined for their assessment are also relevant to modelling studies.

When summarizing the evidence, a distinction should be made between observed and modelled outcomes. It should then be possible to increase or decrease the certainty of modelling studies based on a set of criteria. We look forward to discussing these recommendations with experts and stakeholders and to developing exact procedures and criteria for the assessment of modelling studies and their inclusion in the GRADE process.

Mathematical modelling : concepts and case studies - Deakin University Library

All authors of this article also participated. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Table S1. Search strategy in MEDLINE from inception to January without language restrictions, combining terms for mathematical models with terms for quality assessment and health care decision-making. Figure S1. Questionnaire of the online survey on the use of mathematical modelling in guidelines for public health decision making.

Thanks for the detailed replies; the revision addresses all of my comments. I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. The authors propose to incorporate findings from mathematical modeling studies into the development of WHO guidelines and other processes related to evaluating and developing public health policies. The authors argue that model credibility is more appropriate than risk of bias for evaluating strength of evidence generated by modeling studies.

The paper is based on discussions and findings from a meeting of modeling experts in Geneva in ; the authors were also participants in the meeting. The paper lays out a structured argument for incorporating modeling studies into the evidence base, particularly for formulating WHO recommendations related to treatment of HIV. The authors start by providing a review of how models are used in various fields, with suggestions about how they can inform guideline development.

They address the question of what constitutes a modeling study. A comprehensive accounting of published literature on assessment of models is provided. Finally, they give recommendations for how models can be evaluated using the GRADE approach, with specific conclusions about important issues such as when modeling studies should be used as part of the evidence base and how their credibility should be assessed.

Given the sweeping variety of models used in published studies about HIV treatments, policies and interventions, the authors are to be applauded for putting forward a framework for having this conversation.

Mathematical model

It will promote broader understanding of how models work and how they can be most optimally used for informing treatment guidelines. At the crux of their argument is the claim that evidence generated from models should be judged in terms of model credibility rather than on risk of bias. This argument raises several important issues. First, what constitutes a mathematical model? If a model is to be evaluated on its credibility, we need a definition to work from. Second, what kinds of output from models should be considered evidence, and how should the quality of that evidence be judged and ultimately weighed against or combined with evidence generated from randomized trials and observational cohort data?

This definition covers a vast assortment of mathematical models, ranging from validated descriptions of natural phenomena where the mathematical relationships are known and directly observable to representations of progression from HIV infection to death comprising known and unknown mathematical components, many of which cannot be directly observed. The mathematical representation of radioactive decay is known and can be written down explicitly. The model enables accurate and replicable predictions of future observations. The mathematical model for absorption of a specific drug is typically not known, but empirical studies have shown that it is possible to approximate the systematic variation using nonlinear equations.

These models incorporate known information about physiology and properties of a specific drug, but are necessarily oversimplified representations of drug absorption because there are unobservable characteristics of individuals that affect absorption. The models can be used to make reliable predictions on average, but require unexplained variation to be reflected in terms of prediction intervals. Now consider a model of the population dynamics of HIV infection and disease progression.


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This process also follows a mathematical model, but the model itself is highly complex. Unlike radioactive decay or rate of drug absorption, the mathematical representations of several components of the underlying processes are essentially unknown. Moreover, much of the data needed to inform the models are either unobserved e. All of these are mathematical models, but definitions must distinguish between them. Otherwise there is an implied equivalence that lends more credibility than is deserved to models that are heavily reliant on unverified assumptions about the mathematical structure underlying the dynamic system being modeled.

A more systematic classification of model types would therefore be helpful. While the authors' definition of mathematical model is overly broad, the definition of statistical model, used to contrast with mathematical models, is too narrow. A statistical model is used to characterize sources of variation in observed data. It is based on a probabilistic representation of the data generating mechanism, which is itself a mathematical model.

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  • Large Print. Title Author Advanced Search. The course consists of 5 compulsory modules, 3 optional modules, plus an individual project. Each module corresponds to approximately 30 hours of lectures.

    Mathematical Modelling: Concepts and Case Studies (Mathematical Modelling: Theory and Applications)

    The majority of the compulsory modules are held in the first term from late September to mid December. The optional modules are taught mostly in the second term from early January to mid March, but it is also possible take some options that are taught in the first term. Some modules may be assessed by coursework only, some by examination only, and some by a combination of both coursework and examination. All students then embark on an individual project with submission early in September. A postgraduate diploma for the taught component is available as an option for those who do not wish to take the individual summer project.

    While a strong background in mathematics will be important, applications from students whose qualifications are in physics or other areas will also be welcomed and considered on individual merit. The utilisation of computers for simulation and visualisation will form a part of the course. Thus a familiarity with computers is desirable but is not essential. The Masters level course in Mathematical Modelling has three main aims:- To provide an understanding of the processes undertaken to arrive at a suitable mathematical model.