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Royal Society Open Science RSS feed -- recent Mathematics articles2054-5703Royal Society Open Science<![CDATA[Surrogate modelling for the prediction of spatial fields based on simultaneous dimensionality reduction of high-dimensional input/output spaces]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/4/171933?rss=1
Time-consuming numerical simulators for solving groundwater flow and dissolution models of physico-chemical processes in deep aquifers normally require some of the model inputs to be defined in high-dimensional spaces in order to return realistic results. Sometimes, the outputs of interest are spatial fields leading to high-dimensional output spaces. Although Gaussian process emulation has been satisfactorily used for computing faithful and inexpensive approximations of complex simulators, these have been mostly applied to problems defined in low-dimensional input spaces. In this paper, we propose a method for simultaneously reducing the dimensionality of very high-dimensional input and output spaces in Gaussian process emulators for stochastic partial differential equation models while retaining the qualitative features of the original models. This allows us to build a surrogate model for the prediction of spatial fields in such time-consuming simulators. We apply the methodology to a model of convection and dissolution processes occurring during carbon capture and storage.
]]>2018-04-25T00:10:32-07:00info:doi/10.1098/rsos.171933hwp:master-id:royopensci;rsos.1719332018-04-25Mathematics54171933171933<![CDATA[Noise-induced transitions and shifts in a climate-vegetation feedback model]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/4/171531?rss=1
Motivated by the extremely important role of the Earth’s vegetation dynamics in climate changes, we study the stochastic variability of a simple climate–vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate–vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate–vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth’s vegetation cover is revealed. Our analysis demonstrates that the climate–vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.
]]>2018-04-11T00:05:23-07:00info:doi/10.1098/rsos.171531hwp:master-id:royopensci;rsos.1715312018-04-11Mathematics54171531171531<![CDATA[Bayesian model evidence as a practical alternative to deviance information criterion]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/3/171519?rss=1
While model evidence is considered by Bayesian statisticians as a gold standard for model selection (the ratio in model evidence between two models giving the Bayes factor), its calculation is often viewed as too computationally demanding for many applications. By contrast, the widely used deviance information criterion (DIC), a different measure that balances model accuracy against complexity, is commonly considered a much faster alternative. However, recent advances in computational tools for efficient multi-temperature Markov chain Monte Carlo algorithms, such as steppingstone sampling (SS) and thermodynamic integration schemes, enable efficient calculation of the Bayesian model evidence. This paper compares both the capability (i.e. ability to select the true model) and speed (i.e. CPU time to achieve a given accuracy) of DIC with model evidence calculated using SS. Three important model classes are considered: linear regression models, mixed models and compartmental models widely used in epidemiology. While DIC was found to correctly identify the true model when applied to linear regression models, it led to incorrect model choice in the other two cases. On the other hand, model evidence led to correct model choice in all cases considered. Importantly, and perhaps surprisingly, DIC and model evidence were found to run at similar computational speeds, a result reinforced by analytically derived expressions.
]]>2018-03-21T00:06:16-07:00info:doi/10.1098/rsos.171519hwp:master-id:royopensci;rsos.1715192018-03-21Mathematics53171519171519<![CDATA[How successful are mutants in multiplayer games with fluctuating environments? Sojourn times, fixation and optimal switching]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/3/172176?rss=1
Using a stochastic model, we investigate the probability of fixation, and the average time taken to achieve fixation, of a mutant in a population of wild-types. We do this in a context where the environment in which the competition takes place is subject to stochastic change. Our model takes into account interactions which can involve multiple participants. That is, the participants take part in multiplayer games. We find that under certain circumstances, there are environmental switching dynamics which minimize the time that it takes for the mutants to fixate. To analyse the dynamics more closely, we develop a method by which to calculate the sojourn times for general birth–death processes in fluctuating environments.
]]>2018-03-21T00:06:16-07:00info:doi/10.1098/rsos.172176hwp:master-id:royopensci;rsos.1721762018-03-21Mathematics53172176172176<![CDATA[Explicit solutions to correlation matrix completion problems, with an application to risk management and insurance]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/3/172348?rss=1
We derive explicit solutions to the problem of completing a partially specified correlation matrix. Our results apply to several block structures for the unspecified entries that arise in insurance and risk management, where an insurance company with many lines of business is required to satisfy certain capital requirements but may have incomplete knowledge of the underlying correlation matrix. Among the many possible completions, we focus on the one with maximal determinant. This has attractive properties and we argue that it is suitable for use in the insurance application. Our explicit formulae enable easy solution of practical problems and are useful for testing more general algorithms for the maximal determinant correlation matrix completion problem.
]]>2018-03-14T00:05:32-07:00info:doi/10.1098/rsos.172348hwp:master-id:royopensci;rsos.1723482018-03-14Mathematics53172348172348<![CDATA[Implications of asymptomatic carriers for infectious disease transmission and control]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/2/172341?rss=1
For infectious pathogens such as Staphylococcus aureus and Streptococcus pneumoniae, some hosts may carry the pathogen and transmit it to others, yet display no symptoms themselves. These asymptomatic carriers contribute to the spread of disease but go largely undetected and can therefore undermine efforts to control transmission. Understanding the natural history of carriage and its relationship to disease is important for the design of effective interventions to control transmission. Mathematical models of infectious diseases are frequently used to inform decisions about control and should therefore accurately capture the role played by asymptomatic carriers. In practice, incorporating asymptomatic carriers into models is challenging due to the sparsity of direct evidence. This absence of data leads to uncertainty in estimates of model parameters and, more fundamentally, in the selection of an appropriate model structure. To assess the implications of this uncertainty, we systematically reviewed published models of carriage and propose a new model of disease transmission with asymptomatic carriage. Analysis of our model shows how different assumptions about the role of asymptomatic carriers can lead to different conclusions about the transmission and control of disease. Critically, selecting an inappropriate model structure, even when parameters are correctly estimated, may lead to over- or under-estimates of intervention effectiveness. Our results provide a more complete understanding of the role of asymptomatic carriers in transmission and highlight the importance of accurately incorporating carriers into models used to make decisions about disease control.
]]>2018-02-14T00:05:35-08:00info:doi/10.1098/rsos.172341hwp:master-id:royopensci;rsos.1723412018-02-14Mathematics52172341172341<![CDATA[Quantification of plasma cell dynamics using mathematical modelling]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/170759?rss=1
Plasma cells (PCs) are the main antibody-producing cells in humans. They are long-lived so that specific antibodies against either pathogens or vaccines are produced for decades. PC longevity is attributed to specific areas within the bone marrow micro-environment, the so-called ‘niche’, providing the cells with required growth and survival factors. With antigen encounters, e.g. infection or vaccination, new PCs are generated and home to the bone marrow where they compete with resident PCs for the niche. We propose a parametrized mathematical model describing healthy PC dynamics in the bone marrow. The model accounts for competition for the niche between newly produced PCs owing to vaccination and resident PCs. Mathematical analysis and numerical simulations of the model allow explanation of the recovery of PC homoeostasis after a vaccine-induced perturbation, and the fraction of vaccine-specific PCs inside the niche. The model enables quantification of the niche-related dynamics of PCs, i.e. the duration of PC transition into the niche and the impact of different rates for PC transitions into and out of the niche on the observed cell dynamics. Ultimately, it provides a potential basis for further investigations in health and disease.
]]>2018-01-24T00:05:38-08:00info:doi/10.1098/rsos.170759hwp:master-id:royopensci;rsos.1707592018-01-24Mathematics51170759170759<![CDATA[Trans-heteroclinic bifurcation: a novel type of catastrophic shift]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/171304?rss=1
Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication–mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication–mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.
]]>2018-01-24T00:05:39-08:00info:doi/10.1098/rsos.171304hwp:master-id:royopensci;rsos.1713042018-01-24Mathematics51171304171304<![CDATA[A prediction model of compressor with variable-geometry diffuser based on elliptic equation and partial least squares]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/171468?rss=1
To achieve a much more extensive intake air flow range of the diesel engine, a variable-geometry compressor (VGC) is introduced into a turbocharged diesel engine. However, due to the variable diffuser vane angle (DVA), the prediction for the performance of the VGC becomes more difficult than for a normal compressor. In the present study, a prediction model comprising an elliptical equation and a PLS (partial least-squares) model was proposed to predict the performance of the VGC. The speed lines of the pressure ratio map and the efficiency map were fitted with the elliptical equation, and the coefficients of the elliptical equation were introduced into the PLS model to build the polynomial relationship between the coefficients and the relative speed, the DVA. Further, the maximal order of the polynomial was investigated in detail to reduce the number of sub-coefficients and achieve acceptable fit accuracy simultaneously. The prediction model was validated with sample data and in order to present the superiority of compressor performance prediction, the prediction results of this model were compared with those of the look-up table and back-propagation neural networks (BPNNs). The validation and comparison results show that the prediction accuracy of the new developed model is acceptable, and this model is much more suitable than the look-up table and the BPNN methods under the same condition in VGC performance prediction. Moreover, the new developed prediction model provides a novel and effective prediction solution for the VGC and can be used to improve the accuracy of the thermodynamic model for turbocharged diesel engines in the future.
]]>2018-01-24T00:05:39-08:00info:doi/10.1098/rsos.171468hwp:master-id:royopensci;rsos.1714682018-01-24Mathematics51171468171468<![CDATA[Ebola could be eradicated through voluntary vaccination]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/171591?rss=1
Ebola virus disease (EVD) is a severe infection with an extremely high fatality rate spread through direct contact with body fluids. A promising Ebola vaccine (rVSV-ZEBOV) may soon become universally available. We constructed a game-theoretic model of Ebola incorporating individual decisions to vaccinate. We found that if a population adopts selfishly optimal vaccination strategies, then the population vaccination coverage falls negligibly short of the herd immunity level. We concluded that eradication of Ebola is feasible if voluntary vaccination programmes are coupled with focused public education efforts. We conducted uncertainty and sensitivity analysis to demonstrate that our findings do not depend on the choice of the epidemiological model parameters.
]]>2018-01-24T00:05:39-08:00info:doi/10.1098/rsos.171591hwp:master-id:royopensci;rsos.1715912018-01-24Mathematics51171591171591<![CDATA[Beyond 'significance: principles and practice of the Analysis of Credibility]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/171047?rss=1
The inferential inadequacies of statistical significance testing are now widely recognized. There is, however, no consensus on how to move research into a ‘post p < 0.05’ era. We present a potential route forward via the Analysis of Credibility, a novel methodology that allows researchers to go beyond the simplistic dichotomy of significance testing and extract more insight from new findings. Using standard summary statistics, AnCred assesses the credibility of significant and non-significant findings on the basis of their evidential weight, and in the context of existing knowledge. The outcome is expressed in quantitative terms of direct relevance to the substantive research question, providing greater protection against misinterpretation. Worked examples are given to illustrate how AnCred extracts additional insight from the outcome of typical research study designs. Its ability to cast light on the use of p-values, the interpretation of non-significant findings and the so-called ‘replication crisis’ is also discussed.
]]>2018-01-17T00:05:51-08:00info:doi/10.1098/rsos.171047hwp:master-id:royopensci;rsos.1710472018-01-17Mathematics51171047171047<![CDATA[Modelling science trustworthiness under publish or perish pressure]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/1/171511?rss=1
Scientific publication is immensely important to the scientific endeavour. There is, however, concern that rewarding scientists chiefly on publication creates a perverse incentive, allowing careless and fraudulent conduct to thrive, compounded by the predisposition of top-tier journals towards novel, positive findings rather than investigations confirming null hypothesis. This potentially compounds a reproducibility crisis in several fields, and risks undermining science and public trust in scientific findings. To date, there has been comparatively little modelling on factors that influence science trustworthiness, despite the importance of quantifying the problem. We present a simple phenomenological model with cohorts of diligent, careless and unethical scientists, with funding allocated by published outputs. This analysis suggests that trustworthiness of published science in a given field is influenced by false positive rate, and pressures for positive results. We find decreasing available funding has negative consequences for resulting trustworthiness, and examine strategies to combat propagation of irreproducible science.
]]>2018-01-10T00:05:22-08:00info:doi/10.1098/rsos.171511hwp:master-id:royopensci;rsos.1715112018-01-10Mathematics51171511171511