Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

Novel methods improve prediction of species' distributions from occurrence data

Prediction of species' distributions is central to diverse applications in ecology, evolution and conservation science. There is increasing electronic access to vast sets of occurrence records in museums and herbaria, yet little effective guidance on how best to use this information in the context of numerous approaches for modelling distributions. To meet this need, we compared 16 modelling methods over 226 species from 6 regions of the world, creating the most comprehensive set of model comparisons to date. We used presence-only data to fit models, and independent presence-absence data to evaluate the predictions. Along with well-established modelling methods such as generalised additive models and GARP and BIOCLIM, we explored methods that either have been developed recently or have rarely been applied to modelling species' distributions. These include machine-learning methods and community models, both of which have features that may make them particularly well suited to noisy or sparse information, as is typical of species' occurrence data. Presence-only data were effective for modelling species' distributions for many species and regions. The novel methods consistently outperformed more established methods. The results of our analysis are promising for the use of data from museums and herbaria, especially as methods suited to the noise inherent in such data improve.     

Non-stationarity and local approaches to modelling the distributions of wildlife

Despite a growing interest in species distribution modelling, relatively little attention has been paid to spatial autocorrelation and non-stationarity. Both spatial autocorrelation (the tendency for adjacent locations to be more similar than distant ones) and non-stationarity (the variation in modelled relationships over space) are likely to be common properties of ecological systems. This paper focuses on non-stationarity and uses two local techniques, geographically weighted regression (GWR) and varying coefficient modelling (VCM), to assess its impact on model predictions. We extend two published studies, one on the presence–absence of calandra larks in Spain and the other on bird species richness in Britain, to compare GWR and VCM with the more usual global generalized linear modelling (GLM) and generalized additive modelling (GAM). For the calandra lark data, GWR and VCM produced better-fitting models than GLM or GAM. VCM in particular gave significantly reduced spatial autocorrelation in the model residuals. GWR showed that individual predictors became stationary at different spatial scales, indicating that distributions are influenced by ecological processes operating over multiple scales. VCM was able to predict occurrence accurately on independent data from the same geographical area as the training data but not beyond, whereas the GAM produced good results on all areas. Individual predictions from the local methods often differed substantially from the global models. For the species richness data, VCM and GWR produced far better predictions than ordinary regression. Our analyses suggest that modellers interpolating data to produce maps for practical actions (e.g. conservation) should consider local methods, whereas they should not be used for extrapolation to new areas. We argue that local methods are complementary to global methods, revealing details of habitat associations and data properties which global methods average out and miss.     

Robust, quantitative tools for modelling ex-vivo expansion of haematopoietic stem cells and progenitors

Despite substantial research activity on bioreactor design and experiments, there are very few reports of modelling tools that can be used to generate predictive models describing how bioreactor parameters affect performance. New developments in mathematics, such as sparse Bayesian feature selection methods and nonlinear model-free modelling regression methods, offer considerable promise for modelling diverse types of data. The utility of these mathematical tools in stem cell biology are demonstrated by analysis of a large set of bioreactor data derived from the literature. In spite of the diversity of the data sources, and the inherent difficulty in representing bioreactor variables, these modelling methods were able to develop robust, quantitative, predictive models. These models relate bioreactor operational parameters to the degree of expansion of haematopoietic stem cells or their progenitors, and also identify the bioreactor variables that are most likely to affect performance across many experiments. These methods show substantial promise in assisting the design and optimisation of stem cell bioreactors.     

An analysis of interfilament shear in flagella

A method for systematic analysis and modelling of interacting species is developed. An iterative algorithm for solving the inverse problem in ecological modelling (the problem of using empirical population data to evaluate the parameters of a given model) is first presented. It is then shown that certain results from the theory of statistical hypothesis testing provide a method by which the many possible interspecies interactions in an ecosystem can be examined and the dominant interactions can be identified. This analysis leads to the development of mathematical models which are optimum for a given set of empirical data. An example analysis of a four-species webbed ecosystem is discussed.     

Selection of line-transect methods for estimating the density of group-living animals: lessons from the primates

We review the four major contemporary methods for estimating density of group-living animals from line-transect sampling: perpendicular modelling of group centers, perpendicular modelling of center of measurable individuals, strip transects and animal-observer distance. The efficacy of each method is evaluated to produce a simple selection guide. We review the literature and use field data from the Udzungwa Mountains, Tanzania. The review is relevant to all group-living animals; however, examples are drawn from the primates. Perpendicular methods have better mathematical justification than non-perpendicular methods. For perpendicular methods using detection function models, it is preferable to measure group location using center of measurable individuals, as group centers are hard to estimate. The assumptions of detection function models are often broken in poor visibility habitats or with unhabituated animals. Alternatively strip transects may be used where there are reliable data on group spread and/or visibility. Strip transects are also the most practical, along with the animal-observer method; however, the latter lacks mathematical justification. We conclude that there are arguments for continued use of all four methods. In certain situations the use of raw encounter rates may also be considered. The appropriate method is determined by minimizing bias and considering time, resources and field conditions.     

An Algorithm for Calculation of n-Dimensional Nonlinear Dependencies – Behaviour Tests on Rats

The evaluation of the data obtained during the behaviour tests always leads to the problem of multiple correlation, very often with non-linear dependencies on the target. All mathematical and statistical procedures that have been used so far are based on the assumption of an equation for the desired correlation for which parameters and related statistical equivalents are determined eventually. The MODAK system applied here (MODAK = algorithms of modelling for the calculation of multi-dimensional non-linear mathematical models) breaks down a complex correlation into individual dependencies in a mathematical and statistical way and selects suitable equations for each of them independently and determines the corresponding parameters. The numerical example evaluates data of behaviour tests on rats. First results obtained on the correlations of various behaviour tests indicate both the possibility of selecting suitable tests independent of each other and a better interpretation of the observed patterns of behaviour taking into account the interrelations between the tests. In addition, MODAK is a method which can be applied as a matter of course in a general way to all cases which call for the reduction and analysis of data occurring in process and system analysis and in the evaluation of test results requiring statistical modelling. So far, MODAK applications range from engineering sciences to medicine.     

A mathematical and statistical framework for modelling dispersal

Mechanistic and phenomenological dispersal modelling of organisms has long been an area of intensive research. Recently, there has been an increased interest in intermediate models between the two. Intermediate models include major mechanisms that affect dispersal, in addition to the dispersal curve of a phenomenological model. Here we review and describe the mathematical and statistical framework for phenomenological dispersal modelling. In the mathematical development we describe modelling of dispersal in two dimensions from a point source, and in one dimension from a line or area source. In the statistical development we describe applicable observation distributions, and the procedures of model fitting, comparison, checking, and prediction. The procedures are also demonstrated using data from dispersal experiments. The data are hierarchically structured, and hence, we fit hierarchical models. The Bayesian modelling approach is applied, which allows us to show the uncertainty in the parameter estimates and in predictions. Finally, we show how to account for the effect of wind speed on the estimates of the dispersal parameters. This serves as an example of how to strengthen the coupling in the modelling between the phenomenon observed in an experiment and the underlying process – something that should be striven for in the statistical modelling of dispersal.     

Models of classical conditioning

Classical conditioning is a basic form of associative learning in the animal kingdom. Many paradigmatic features of classical conditioning appear to be conserved throughout species and phyla and are independent of stimulus nature. This paper presents an analysis of trial-based and real-time models of classical conditioning which are mathematical abstractions of the underlying processing principles. Various models are reviewed and in a formal analysis, their capability of simulating and explaining classical conditioning is investigated. Since every existing model fails to simulate some particular conditioning phenomena and since some modelling approaches are not appropriate for detailed mathematical analysis, new model components will be introduced that overcome most of the weaknesses observed in the other models.     

The role of mathematical models of host-pathogen interactions for livestock health and production - a review

Compared with the application of mathematical models to study human diseases, models that describe animal responses to pathogen challenges are relatively rare. The aim of this review is to explain and show the role of mathematical host-pathogen interaction models in providing underpinning knowledge for improving animal health and sustaining livestock production. Existing host-pathogen interaction models can be assigned to one of three categories: (i) models of the infection and immune system dynamics, (ii) models that describe the impact of pathogen challenge on health, survival and production and (iii) models that consider the co-evolution of host and pathogen. State-of-the-art approaches are presented and discussed for models belonging to the first two categories only, as they concentrate on the host-pathogen dynamics within individuals. Models of the third category fall more into the class of epidemiological models, which deserve a review by themselves. An extensive review of published models reveals a rich spectrum of methodologies and approaches adopted in different modelling studies, and a strong discrepancy between models concerning diseases in animals and models aimed at tackling diseases in humans (most of which belong to the first category), with the latter being generally more sophisticated. The importance of accounting for the impact of infection not only on health but also on production poses a considerable challenge to the study of host-pathogen interactions in livestock. This has led to relatively simplistic representations of host-pathogen interaction in existing models for livestock diseases. Although these have proven appropriate for investigating hypotheses concerning the relationships between health and production traits, they do not provide predictions of an animal's response to pathogen challenge of sufficient accuracy that would be required for the design of appropriate disease control strategies. A synthesis between the modelling methodologies adopted in categories 1 and 2 would therefore be desirable. The progress achieved in mathematical modelling to study immunological processes relevant to human diseases, together with the current advances in the generation and analysis of biological data related to animal diseases, offers a great opportunity to develop a new generation of host-pathogen interaction models that take on a fundamental role in the study and control of disease in livestock.     

Hybrid semi-parametric mathematical systems: bridging the gap between systems biology and process engineering

Systems biology is an integrative science that aims at the global characterization of biological systems. Huge amounts of data regarding gene expression, proteins activity and metabolite concentrations are collected by designing systematic genetic or environmental perturbations. Then the challenge is to integrate such data in a global model in order to provide a global picture of the cell. The analysis of these data is largely dominated by nonparametric modelling tools. In contrast, classical bioprocess engineering has been primarily founded on first principles models, but it has systematically overlooked the details of the embedded biological system. The full complexity of biological systems is currently assumed by systems biology and this knowledge can now be taken by engineers to decide how to optimally design and operate their processes. This paper discusses possible methodologies for the integration of systems biology and bioprocess engineering with emphasis on applications involving animal cell cultures. At the mathematical systems level, the discussion is focused on hybrid semi-parametric systems as a way to bridge systems biology and bioprocess engineering.     

Using Hawkes Processes to model imported and local malaria cases in near-elimination settings

Developing new methods for modelling infectious diseases outbreaks is important for monitoring transmission and developing policy. In this paper we propose using semi-mechanistic Hawkes Processes for modelling malaria transmission in near-elimination settings. Hawkes Processes are well founded mathematical methods that enable us to combine the benefits of both statistical and mechanistic models to recreate and forecast disease transmission beyond just malaria outbreak scenarios. These methods have been successfully used in numerous applications such as social media and earthquake modelling, but are not yet widespread in epidemiology. By using domain-specific knowledge, we can both recreate transmission curves for malaria in China and Eswatini and disentangle the proportion of cases which are imported from those that are community based.     

Nonlinear regression methods in design of experiments and mathematical modelling. Applications to the analysis of the steady-state kinetics of glutathione reductase.

A branching reaction pathway involving a ping pong and a sequential loop has been proposed for glutathione reductase (Biochem. Biophys. Res. Commun. 53 (1973) 1151). In the present investigation nonlinear regression methods have been applied in the fitting of rate equations to experimental data to test the validity of the model proposed and to discriminate between alternative mathematical models (cf. FEBS Lett. 26 (1972) 252). In the best rate law, some of the parameters were numerically redundant. Therefore, a feature-wise analysis of the rate equation was carried out by varying one substrate concentration at a time. The overall strategy used was a cyclic procedure involving: experimentation - analysis of data - modelling - design of experiments - new experimentation etc. Consideration was given to the experimental error structure and to the importance of weighting in the regression analysis. In the design of experiments for discrimination between rival models, a previously defined discrimination function was used. The results of the analysis support the branching reaction scheme proposed for glutathione reductase.     

Dynamic causal models of neural system dynamics: current state and future extensions

Complex processes resulting from interaction of multiple elements can rarely be understood by analytical scientific approaches alone; additional, mathematical models of system dynamics are required. This insight, which disciplines like physics have embraced for a long time already, is gradually gaining importance in the study of cognitive processes by functional neuroimaging. In this field, causal mechanisms in neural systems are described in terms of effective connectivity. Recently, dynamic causal modelling (DCM) was introduced as a generic method to estimate effective connectivity from neuroimaging data in a Bayesian fashion. One of the key advantages of DCM over previous methods is that it distinguishes between neural state equations and modality-specific forward models that translate neural activity into a measured signal. Another strength is its natural relation to Bayesian model selection (BMS) procedures. In this article, we review the conceptual and mathematical basis of DCM and its implementation for functional magnetic resonance imaging data and event-related potentials. After introducing the application of BMS in the context of DCM, we conclude with an outlook to future extensions of DCM. These extensions are guided by the long-term goal of using dynamic system models for pharmacological and clinical applications, particularly with regard to synaptic plasticity.     

Does probabilistic modelling of linkage disequilibrium evolution improve the accuracy of QTL location in animal pedigree?

Background

Since 2001, the use of more and more dense maps has made researchers aware that combining linkage and linkage disequilibrium enhances the feasibility of fine-mapping genes of interest. So, various method types have been derived to include concepts of population genetics in the analyses. One major drawback of many of these methods is their computational cost, which is very significant when many markers are considered. Recent advances in technology, such as SNP genotyping, have made it possible to deal with huge amount of data. Thus the challenge that remains is to find accurate and efficient methods that are not too time consuming. The study reported here specifically focuses on the half-sib family animal design. Our objective was to determine whether modelling of linkage disequilibrium evolution improved the mapping accuracy of a quantitative trait locus of agricultural interest in these populations. We compared two methods of fine-mapping. The first one was an association analysis. In this method, we did not model linkage disequilibrium evolution. Therefore, the modelling of the evolution of linkage disequilibrium was a deterministic process; it was complete at time 0 and remained complete during the following generations. In the second method, the modelling of the evolution of population allele frequencies was derived from a Wright-Fisher model. We simulated a wide range of scenarios adapted to animal populations and compared these two methods for each scenario.

Results

Our results indicated that the improvement produced by probabilistic modelling of linkage disequilibrium evolution was not significant. Both methods led to similar results concerning the location accuracy of quantitative trait loci which appeared to be mainly improved by using four flanking markers instead of two.

Conclusions

Therefore, in animal half-sib designs, modelling linkage disequilibrium evolution using a Wright-Fisher model does not significantly improve the accuracy of the QTL location when compared to a simpler method assuming complete and constant linkage between the QTL and the marker alleles. Finally, given the high marker density available nowadays, the simpler method should be preferred as it gives accurate results in a reasonable computing time.     

Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends

Complex models of biochemical reaction systems have become increasingly common in the systems biology literature. The complexity of such models can present a number of obstacles for their practical use, often making problems difficult to intuit or computationally intractable. Methods of model reduction can be employed to alleviate the issue of complexity by seeking to eliminate those portions of a reaction network that have little or no effect upon the outcomes of interest, hence yielding simplified systems that retain an accurate predictive capacity. This review paper seeks to provide a brief overview of a range of such methods and their application in the context of biochemical reaction network models. To achieve this, we provide a brief mathematical account of the main methods including timescale exploitation approaches, reduction via sensitivity analysis, optimisation methods, lumping, and singular value decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed.     

Bayesian inference of biochemical kinetic parameters using the linear noise approximation

Background  

Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.     

A new design of stochastic partnership models for epidemics of sexually transmitted diseases with stages

A problem of importance in modelling epidemics of sexually transmitted diseases is the development of mathematical structures accommodating sexual and other contacts among members of a population. Because these models may be complex, it is often necessary to use computer intensive methods in their analysis, which raises questions on the design of computer models. In this paper a new approach to designing models sexual contacts is presented within the context of a stochastic model accommodating the formation and dissolution of partnerships in heterosexual populations. Emphasis will be placed on the development of algorithms with a view towards developing software to implement computer intensive methods. Unlike previous formulations, rather than using rejection methods in Monte Carlo simulations to impose necessary constraints on random functions describing partnership formation, in the new formulation all constraints are satisfied with probability one.