Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness

Global phase-space approach to rate-induced tipping: A brief review

In nonautonomous dynamical systems, rate-induced tipping (R-tipping) is a critical transition triggered by the rate of change of a time-varying parameter, rather than its absolute value. In recent years, there is a growing interest in R-tipping due to its relevance to significant problems of current interest, such as potential, catastrophic collapse of various ecosystems induced by climate change. This brief review provides an overview of the basic concept, theory, and real-world implications of R-tipping from a global phase-space point of view. The key quantity underlying the global approach is the probability of R-tipping defined with respect to initial conditions in the phase space. A recently discovered scaling law governing this probability and the rate of parameter change is introduced, which has so far been restricted to a class of high-dimensional, complex, and empirical ecological networks: pollinator-plant mutualistic networks. Issues such as prediction of tipping and protection of ecosystems from R-tipping are discussed.

Threshold uncertainty, early warning signals and the prevention of dangerous climate change

The goal of the Paris Agreement is to keep global temperature rise well below 2°C. In this agreement-and its antecedents negotiated in Copenhagen and Cancun-the fear of crossing a dangerous climate threshold is supposed to serve as the catalyst for cooperation among countries. However, there are deep uncertainties about the location of the threshold for dangerous climate change, and recent evidence indicates this threshold uncertainty is a major impediment to collective action. Early warning signals of approaching climate thresholds are a potential remedy to this threshold uncertainty problem, and initial experimental evidence suggests such early detection systems may improve the prospects of cooperation. Here, we provide a direct experimental assessment of this early warning signal hypothesis. Using a catastrophe avoidance game, we show that large initial-and subsequently unreduced-threshold uncertainty undermines cooperation, consistent with earlier studies. An early warning signal that reduced uncertainty to within 10% (but not 30%) of the threshold value catalysed cooperation and reduced the probability of catastrophe occurring, albeit not reliably so. Our findings suggest early warning signals can trigger action to avoid a dangerous threshold, but additional mechanisms may be required to foster the cooperation needed to ensure the threshold is not breached.

Transitions in intensive care: Investigating critical slowing down post extubation

Complex biological systems undergo sudden transitions in their state, which are often preceded by a critical slowing down of dynamics. This results in longer recovery times as systems approach transitions, quantified as an increase in measures such as the autocorrelation and variance. In this study, we analysed paediatric patients in intensive care for whom mechanical ventilation was discontinued through removal of the endotracheal tube (extubation). Some patients failed extubation, and required a re-intubation within 48 hours. We investigated whether critical slowing down could be observed post failed extubations, prior to re-intubation. We tested for significant increases (p <.05) between extubation and re-intubation, in the variance and autocorrelation, over the time series data of heart rate, respiratory rate and mean blood pressure. The autocorrelation of the heart rate showed a significantly higher proportion of increases in the group that failed extubation, compared who those who did not. It also showed a significantly higher magnitude of increase for the failed extubation group in a t-test. Moreover, incorporating these magnitudes significantly improved the fit of a logistic regression model when compared to a model that solely used the mean and standard deviation of the vital signs. While immediate clinical utility is limited, the work marks an important first step towards using dynamical systems theory to understand the dynamics of signals measured at the bedside during intensive care.

Understanding pesticide-induced tipping in plant-pollinator networks across geographical scales: Prioritizing richness and modularity over nestedness

Mutually beneficial interactions between plants and pollinators are crucial for biodiversity, ecosystem stability, and crop production. A threat to a mutualistic network is the occurrence of a tipping point at which the species abundances collapse to a near zero level. In modern agriculture, there is widespread use of pesticides. What are the effects of extensive pesticide use on mutualistic networks? We develop a plant-pollinator-pesticide model and study its dynamics using 123 mutualistic networks across the globe. We demonstrate that pesticide exposure can lead to a tipping point. Furthermore, while the network characteristics such as richness and modularity exhibit a strong association with pesticide-induced tipping, nestedness shows a weak association. A surprising finding is that the mutualistic networks in the African continent are less pesticide tolerant than those in Europe. We articulate and test a pragmatic intervention strategy through targeted management of pesticide levels within specific plant species to delay or avert the tipping point. Our study provides quantitative insights into the phenomenon of pesticide-induced tipping for safeguarding mutualistic networks that are fundamental to agriculture and ecosystems.

Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green's Functions, and Nonlinear Fingerprints

Detection and attribution (DA) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method's assumptions, advantages, and potential weaknesses. We use our theory to perform DA for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others.

Early Warning Signals of the Tipping Point in Strongly Interacting Rydberg Atoms

The identification of tipping points is essential for the prediction of collapses or other sudden changes in complex systems. Applications include studies of ecology, thermodynamics, climatology, and epidemiology. However, detecting early signs of proximity to a tipping is made challenging by complexity and nonlinearity. Strongly interacting Rydberg atom gases offer model systems that offer both complexity and nonlinearity, including phase transition and critical slowing down. Here, via an external probe we observe prior warning of the proximity of a phase transition of Rydberg thermal gases. This warning signal is manifested as a deviation from linear growth of the variance with increasing probe intensity. We also observed the dynamics of the critical slowing down behavior versus different timescales and atomic densities, thus providing insights into the study of a Rydberg atom system's critical behavior. Our experiment suggests that the full critical slowing down dynamics of strongly interacting Rydberg atoms can be probed systematically, thus providing a benchmark with which to identify critical phenomena in quantum many-body systems.

Resilience of the Atlantic meridional overturning circulation

We address the issue of resilience of the Atlantic Meridional Overturning Circulation (AMOC) given the many indications that this dynamical system is in a multi-stable regime. A novel approach to resilience based on rare event techniques is presented, which leads to a measure capturing "resistance to change" and "ability to return" aspects in a probabilistic way. The application of this measure to a conceptual model demonstrates its suitability for assessing AMOC resilience but also shows its potential use in many other non-autonomous dynamical systems. This framework is then extended to compute the probability that the AMOC undergoes a transition conditioned on an external forcing. Such conditional probability can be estimated by exploiting the information available when computing the resilience of this system. This allows us to provide a probabilistic view on safe operating spaces by defining a conditional safe operating space as a subset of the parameter space of the (possibly transient) imposed forcing.

A non-autonomous framework for climate change and extreme weather events increase in a stochastic energy balance model

We develop a three-timescale framework for modeling climate change and introduce a space-heterogeneous one-dimensional energy balance model. This model, addressing temperature fluctuations from rising carbon dioxide levels and the super-greenhouse effect in tropical regions, fits within the setting of stochastic reaction-diffusion equations. Our results show how both mean and variance of temperature increase, without the system going through a bifurcation point. This study aims to advance the conceptual understanding of the extreme weather events frequency increase due to climate change.

Early warning signals for bifurcations embedded in high dimensions

Recent work has highlighted the utility of methods for early warning signal detection in dynamic systems approaching critical tipping thresholds. Often these tipping points resemble local bifurcations, whose low dimensional dynamics can play out on a manifold embedded in a much higher dimensional state space. In many cases of practical relevance, the form of this embedding is poorly understood or entirely unknown. This paper explores how measurement of the critical phenomena that generically precede such bifurcations can be used to make inferences about some properties of their embeddings, and, conversely, how prior knowledge about the mechanism of bifurcation can robustify predictions of an oncoming tipping event. These modes of analysis are first demonstrated on a simple fluid flow system undergoing a Hopf bifurcation. The same approach is then applied to data associated with the West African monsoon shift, with results corroborated by existing models of the same system. This example highlights the effectiveness of the methodology even when applied to complex climate data, and demonstrates how a well-resolved spatial structure associated with the onset of atmospheric instability can be inferred purely from time series measurements.

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

On the persistence of near-surface temperature dynamics in a warming world

We consider issues related to the effect of climate change on the persistence of (trend-corrected) temperatures using global gridded data for both land and oceans. We first discuss how the presence of trends and additive noise affects inference about persistence. Ignoring a trend induces an upward bias, while not accounting for noise induces a downward bias. We show that the increase in persistence in the commonly used Warm Spell Duration Index is simply an artifact of increasing temperatures. To purge the impact of both trends and noise, we adopt a simple state-space model. Of separate interest, we document a much larger noise component for land than for oceans. The estimates of the persistence are much larger for oceans than for land. Inspection of the estimates across various subsamples and the application of tests for structural changes suggest the same pattern of persistence for both land and oceans across time, with few minor exceptions. Hence, our results show that surface temperature persistence has remained constant during the observed period.

Rate-induced tipping in complex high-dimensional ecological networks

In an ecosystem, environmental changes as a result of natural and human processes can cause some key parameters of the system to change with time. Depending on how fast such a parameter changes, a tipping point can occur. Existing works on rate-induced tipping, or R-tipping, offered a theoretical way to study this phenomenon but from a local dynamical point of view, revealing, e.g., the existence of a critical rate for some specific initial condition above which a tipping point will occur. As ecosystems are subject to constant disturbances and can drift away from their equilibrium point, it is necessary to study R-tipping from a global perspective in terms of the initial conditions in the entire relevant phase space region. In particular, we introduce the notion of the probability of R-tipping defined for initial conditions taken from the whole relevant phase space. Using a number of real-world, complex mutualistic networks as a paradigm, we find a scaling law between this probability and the rate of parameter change and provide a geometric theory to explain the law. The real-world implication is that even a slow parameter change can lead to a system collapse with catastrophic consequences. In fact, to mitigate the environmental changes by merely slowing down the parameter drift may not always be effective: Only when the rate of parameter change is reduced to practically zero would the tipping be avoided. Our global dynamics approach offers a more complete and physically meaningful way to understand the important phenomenon of R-tipping.

Early warning signals have limited applicability to empirical lake data

Research aimed at identifying indicators of persistent abrupt shifts in ecological communities, a.k.a regime shifts, has led to the development of a suite of early warning signals (EWSs). As these often perform inaccurately when applied to real-world observational data, it remains unclear whether critical transitions are the dominant mechanism of regime shifts and, if so, which EWS methods can predict them. Here, using multi-trophic planktonic data on multiple lakes from around the world, we classify both lake dynamics and the reliability of classic and second generation EWSs methods to predict whole-ecosystem change. We find few instances of critical transitions, with different trophic levels often expressing different forms of abrupt change. The ability to predict this change is highly processing dependant, with most indicators not performing better than chance, multivariate EWSs being weakly superior to univariate, and a recent machine learning model performing poorly. Our results suggest that predictive ecology should start to move away from the concept of critical transitions, developing methods suitable for predicting resilience loss not limited to the strict bounds of bifurcation theory.

Network percolation provides early warnings of abrupt changes in coupled oscillatory systems: An explanatory analysis

Functional networks are powerful tools to study statistical interdependency structures in spatially extended or multivariable systems. They have been used to get insights into the dynamics of complex systems in various areas of science. In particular, percolation properties of correlation networks have been employed to identify early warning signals of critical transitions. In this work, we further investigate the corresponding potential of percolation measures for the anticipation of different types of sudden shifts in the state of coupled irregularly oscillating systems. As a paradigmatic model system, we study the dynamics of a ring of diffusively coupled noisy FitzHugh-Nagumo oscillators and show that, when the oscillators are nearly completely synchronized, the percolation-based precursors successfully provide very early warnings of the rapid switches between the two states of the system. We clarify the mechanisms behind the percolation transition by separating global trends given by the mean-field behavior from the synchronization of individual stochastic fluctuations. We then apply the same methodology to real-world data of sea surface temperature anomalies during different phases of the El Niño-Southern Oscillation. This leads to a better understanding of the factors that make percolation precursors effective as early warning indicators of incipient El Niño and La Niña events.

Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect

We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted εa. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2. (Note that the physical range of values for εa is (0,1].) Next, we explain the long time dynamics for εa∈(0,2), and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove, in particular, that the surface temperature increases monotonically with respect to εa. This is the key mathematical manifestation of the greenhouse effect.

Systematic analysis and optimization of early warning signals for critical transitions using distribution data

Abrupt shifts between alternative regimes occur in complex systems, from cell regulation to brain functions to ecosystems. Several model-free early warning signals (EWS) have been proposed to detect impending transitions, but failure or poor performance in some systems have called for better investigation of their generic applicability. Notably, there are still ongoing debates whether such signals can be successfully extracted from data in particular from biological experiments. In this work, we systematically investigate properties and performance of dynamical EWS in different deteriorating conditions, and we propose an optimized combination to trigger warnings as early as possible, eventually verified on experimental data from microbiological populations. Our results explain discrepancies observed in the literature between warning signs extracted from simulated models and from real data, provide guidance for EWS selection based on desired systems and suggest an optimized composite indicator to alert for impending critical transitions using distribution data.

Improving the Process of Early-Warning Detection and Identifying the Most Affected Markets: Evidence from Subprime Mortgage Crisis and COVID-19 Outbreak-Application to American Stock Markets

Stock-market-crash predictability is of particular interest in the field of financial time-series analysis. Famous examples of major stock-market crashes are the real-estate bubble in 2008 and COVID-19 in 2020. Several studies have studied the prediction process without taking into consideration which markets might be falling into a crisis. To this end, a combination analysis is utilized in this manuscript. Firstly, the auto-regressive estimation (ARE) algorithm is successfully applied to electroencephalography (EEG) brain data for detecting diseases. The ARE algorithm is employed based on state-space modelling, which applies the expectation-maximization algorithm and Kalman filter. This manuscript introduces its application, for the first time, to stock-market data. For this purpose, a time-evolving interaction surface is constructed to observe the change in the surface topology. This enables tracking of the stock market's behavior over time and differentiates between different states. This provides a deep understanding of the underlying system behavior before, during, and after a crisis. Different patterns of the stock-market movements are recognized, providing novel information regarding detecting an early-warning sign. Secondly, a Granger-causality time-domain technique, called directed partial correlation, is employed to infer the underlying interconnectivity structure among markets. This information is crucial for investors and market players, enabling them to differentiate between those markets which will fall in a catastrophic loss, and those which will not. Consequently, they can make successful decisions towards selecting less risky portfolios, which guarantees lower losses. The results showed the effectiveness of the use of this methodology in the framework of the process of early-warning detection.

Dynamical stability indicator based on autoregressive moving-average models: Critical transitions and the Atlantic meridional overturning circulation

A statistical indicator for dynamic stability, known as the Υ indicator, is used to gauge the stability and, hence, detect approaching tipping points of simulation data from a reduced five-box model of the North Atlantic Meridional Overturning Circulation (AMOC) exposed to a time-dependent hosing function. The hosing function simulates the influx of fresh water due to the melting of the Greenland ice sheet and increased precipitation in the North Atlantic. The Υ indicator is designed to detect changes in the memory properties of the dynamics and is based on fitting auto-regressive moving-average models in a sliding window approach to time series data. An increase in memory properties is interpreted as a sign of dynamical instability. The performance of the indicator is tested on time series subject to different types of tipping, namely, bifurcation-induced, noise-induced, and rate-induced tipping. The numerical analysis shows that the indicator indeed responds to the different types of induced instabilities. Finally, the indicator is applied to two AMOC time series from a full complexity Earth systems model (CESM2). Compared with the doubling _CO scenario, the quadrupling _CO scenario results in stronger dynamical instability of the AMOC during its weakening phase.

Detecting Metabolic Thresholds from Nonlinear Analysis of Heart Rate Time Series: A Review

Heart rate time series are widely used to characterize physiological states and athletic performance. Among the main indicators of metabolic and physiological states, the detection of metabolic thresholds is an important tool in establishing training protocols in both sport and clinical fields. This paper reviews the most common methods, applied to heart rate (HR) time series, aiming to detect metabolic thresholds. These methodologies have been largely used to assess energy metabolism and to identify the appropriate intensity of physical exercise which can reduce body weight and improve physical fitness. Specifically, we focused on the main nonlinear signal evaluation methods using HR to identify metabolic thresholds with the purpose of identifying a method which can represent a useful tool for the real-time settings of wearable devices in sport activities. While the advantages and disadvantages of each method, and the possible applications, are presented, this review confirms that the nonlinear analysis of HR time series represents a solid, robust and noninvasive approach to assess metabolic thresholds.

The fundamental benefits of multiplexity in ecological networks

A tipping point presents perhaps the single most significant threat to an ecological system as it can lead to abrupt species extinction on a massive scale. Climate changes leading to the species decay parameter drifts can drive various ecological systems towards a tipping point. We investigate the tipping-point dynamics in multi-layer ecological networks supported by mutualism. We unveil a natural mechanism by which the occurrence of tipping points can be delayed by multiplexity that broadly describes the diversity of the species abundances, the complexity of the interspecific relationships, and the topology of linkages in ecological networks. For a double-layer system of pollinators and plants, coupling between the network layers occurs when there is dispersal of pollinator species. Multiplexity emerges as the dispersing species establish their presence in the destination layer and have a simultaneous presence in both. We demonstrate that the new mutualistic links induced by the dispersing species with the residence species have fundamental benefits to the well-being of the ecosystem in delaying the tipping point and facilitating species recovery. Articulating and implementing control mechanisms to induce multiplexity can thus help sustain certain types of ecosystems that are in danger of extinction as the result of environmental changes.

Practical guide to using Kendall's τ in the context of forecasting critical transitions

Recent studies demonstrate that trends in indicators extracted from measured time series can indicate an approach of an impending transition. Kendall's τ coefficient is often used to study the trend of statistics related to the critical slowing down phenomenon and other methods to forecast critical transitions. Because statistics are estimated from time series, the values of Kendall's τ are affected by parameters such as window size, sample rate and length of the time series, resulting in challenges and uncertainties in interpreting results. In this study, we examine the effects of different parameters on the distribution of the trend obtained from Kendall's τ, and provide insights into how to choose these parameters. We also suggest the use of the non-parametric Mann-Kendall test to evaluate the significance of a Kendall's τ value. The non-parametric test is computationally much faster compared with the traditional parametric auto-regressive, moving-average model test.

Evaluating the performance of temporal and spatial early warning statistics of algal blooms

Regime shifts have large consequences for ecosystems and the services they provide. However, understanding the potential for, causes of, proximity to, and thresholds for regime shifts in nearly all settings is difficult. Generic statistical indicators of resilience have been proposed and studied in a wide range of ecosystems as a method to detect when regime shifts are becoming more likely without direct knowledge of underlying system dynamics or thresholds. These early warning statistics (EWS) have been studied separately but there have been few examples that directly compare temporal and spatial EWS in ecosystem-scale empirical data. To test these methods, we collected high-frequency time series and high-resolution spatial data during a whole-lake fertilization experiment while also monitoring an adjacent reference lake. We calculated two common EWS, standard deviation and autocorrelation, in both time series and spatial data to evaluate their performance prior to the resulting algal bloom. We also applied the quickest detection method to generate binary alarms of resilience change from temporal EWS. One temporal EWS, rolling window standard deviation, provided advanced warning in most variables prior to the bloom, showing trends and between-lake patterns consistent with theory. In contrast, temporal autocorrelation and both measures of spatial EWS (spatial SD, Moran's  I) provided little or no warning. By compiling time series data from this and past experiments with and without nutrient additions, we were able to evaluate temporal EWS performance for both constant and changing resilience conditions. True positive alarm rates were 2.5-8.3 times higher for rolling window standard deviation when a lake was being pushed towards a bloom than the rate of false positives when it was not. For rolling window autocorrelation, alarm rates were much lower and no variable had a higher true positive than false positive alarm rate. Our findings suggest temporal EWS provide advanced warning of algal blooms and that this approach could help managers prepare for and/or minimize negative bloom impacts.

Practical guide to using Kendall's τ in the context of forecasting critical transitions

Recent studies demonstrate that trends in indicators extracted from measured time series can indicate an approach of an impending transition. Kendall's τ coefficient is often used to study the trend of statistics related to the critical slowing down phenomenon and other methods to forecast critical transitions. Because statistics are estimated from time series, the values of Kendall's τ are affected by parameters such as window size, sample rate and length of the time series, resulting in challenges and uncertainties in interpreting results. In this study, we examine the effects of different parameters on the distribution of the trend obtained from Kendall's τ, and provide insights into how to choose these parameters. We also suggest the use of the non-parametric Mann-Kendall test to evaluate the significance of a Kendall's τ value. The non-parametric test is computationally much faster compared with the traditional parametric auto-regressive, moving-average model test.

Towards a Data-Driven Estimation of Resilience in Networked Dynamical Systems: Designing a Versatile Testbed

Estimating resilience of adaptive, networked dynamical systems remains a challenge. Resilience refers to a system's capacity "to absorb exogenous and/or endogenous disturbances and to reorganize while undergoing change so as to still retain essentially the same functioning, structure, and feedbacks." The majority of approaches to estimate resilience requires exact knowledge of the underlying equations of motion; the few data-driven approaches so far either lack appropriate strategies to verify their suitability or remain subject of considerable debate. We develop a testbed that allows one to modify resilience of a multistable networked dynamical system in a controlled manner. The testbed also enables generation of multivariate time series of system observables to evaluate the suitability of data-driven estimators of resilience. We report first findings for such an estimator.

Performance of early warning signals for disease re-emergence: A case study on COVID-19 data

Developing measures for rapid and early detection of disease re-emergence is important to perform science-based risk assessment of epidemic threats. In the past few years, several early warning signals (EWS) from complex systems theory have been introduced to detect impending critical transitions and extend the set of indicators. However, it is still debated whether they are generically applicable or potentially sensitive to some dynamical characteristics such as system noise and rates of approach to critical parameter values. Moreover, testing on empirical data has, so far, been limited. Hence, verifying EWS performance remains a challenge. In this study, we tackle this question by analyzing the performance of common EWS, such as increasing variance and autocorrelation, in detecting the emergence of COVID-19 outbreaks in various countries. Our work illustrates that these EWS might be successful in detecting disease emergence when some basic assumptions are satisfied: a slow forcing through the transitions and not-fat-tailed noise. In uncertain cases, we observe that noise properties or commensurable time scales may obscure the expected early warning signals. Overall, our results suggest that EWS can be useful for active monitoring of epidemic dynamics, but that their performance is sensitive to certain features of the underlying dynamics. Our findings thus pave a connection between theoretical and empirical studies, constituting a further step towards the application of EWS indicators for informing public health policies.

Overlapping timescales obscure early warning signals of the second COVID-19 wave

Early warning indicators based on critical slowing down have been suggested as a model-independent and low-cost tool to anticipate the (re)emergence of infectious diseases. We studied whether such indicators could reliably have anticipated the second COVID-19 wave in European countries. Contrary to theoretical predictions, we found that characteristic early warning indicators generally decreased rather than increased prior to the second wave. A model explains this unexpected finding as a result of transient dynamics and the multiple timescales of relaxation during a non-stationary epidemic. Particularly, if an epidemic that seems initially contained after a first wave does not fully settle to its new quasi-equilibrium prior to changing circumstances or conditions that force a second wave, then indicators will show a decreasing rather than an increasing trend as a result of the persistent transient trajectory of the first wave. Our simulations show that this lack of timescale separation was to be expected during the second European epidemic wave of COVID-19. Overall, our results emphasize that the theory of critical slowing down applies only when the external forcing of the system across a critical point is slow relative to the internal system dynamics.

Machine learning methods trained on simple models can predict critical transitions in complex natural systems

Forecasting sudden changes in complex systems is a critical but challenging task, with previously developed methods varying widely in their reliability. Here we develop a novel detection method, using simple theoretical models to train a deep neural network to detect critical transitions-the Early Warning Signal Network (EWSNet). We then demonstrate that this network, trained on simulated data, can reliably predict observed real-world transitions in systems ranging from rapid climatic change to the collapse of ecological populations. Importantly, our model appears to capture latent properties in time series missed by previous warning signals approaches, allowing us to not only detect if a transition is approaching, but critically whether the collapse will be catastrophic or non-catastrophic. These novel properties mean EWSNet has the potential to serve as an indicator of transitions across a broad spectrum of complex systems, without requiring information on the structure of the system being monitored. Our work highlights the practicality of deep learning for addressing further questions pertaining to ecosystem collapse and has much broader management implications.

Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model

We study the mitigation of climate tipping point transitions using an energy balance model. The evolution of the global mean surface temperature is coupled with the CO 2 concentration through the green-house effect. We model the CO 2 concentration with a stochastic delay differential equation (SDDE), accounting for various carbon emission and capture scenarios. The resulting coupled system of SDDEs exhibits a tipping point phenomena: if CO 2 concentration exceeds a critical threshold (around 478   ppm ), the temperature experiences an abrupt increase of about six degrees Celsius. We show that the CO 2 concentration exhibits a transient growth which may cause a climate tipping point, even if the concentration decays asymptotically. We derive a rigorous upper bound for the CO 2 evolution which quantifies its transient and asymptotic growths, and provides sufficient conditions for evading the climate tipping point. Combining this upper bound with Monte Carlo simulations of the stochastic climate model, we investigate the emission reduction and carbon capture scenarios that would avert the tipping point.

Complexity Economics in a Time of Crisis: Heterogeneous Agents, Interconnections, and Contagion

In this article, we consider a variety of different mechanisms through which crises such as COVID-19 can propagate from the micro-economic behaviour of individual agents through to an economy’s aggregate dynamics and subsequently spill over into the global economy. Our central theme is one of changes in the behaviour of heterogeneous agents, agents who differ in terms of some measure of size, wealth, connectivity, or behaviour, in different parts of an economy. These are illustrated through a variety of case studies, from individuals and households with budgetary constraints, to financial markets, to companies composed of thousands of small projects, to companies that implement single multi-billion dollar projects. In each case, we emphasise the role of data or theoretical models and place them in the context of measuring their inter-connectivity and emergent dynamics. Some of these are simple models that need to be ‘dressed’ in socio-economic data to be used for policy-making, and we give an example of how to do this with housing markets, while others are more similar to archaeological evidence; they provide hints about the bigger picture but have yet to be unified with other results. The result is only an outline of what is possible but it shows that we are drawing closer to an integrated set of concepts, principles, and models. In the final section, we emphasise the potential as well as the limitations and what the future of these methods hold for economics.

Early warning signals of infectious disease transitions: a review

Early warning signals (EWSs) are a group of statistical time-series signals which could be used to anticipate a critical transition before it is reached. EWSs are model-independent methods that have grown in popularity to support evidence of disease emergence and disease elimination. Theoretical work has demonstrated their capability of detecting disease transitions in simple epidemic models, where elimination is reached through vaccination, to more complex vector transmission, age-structured and metapopulation models. However, the exact time evolution of EWSs depends on the transition; here we review the literature to provide guidance on what trends to expect and when. Recent advances include methods which detect when an EWS becomes significant; the earlier an upcoming disease transition is detected, the more valuable an EWS will be in practice. We suggest that future work should firstly validate detection methods with synthetic and historical datasets, before addressing their performance with real-time data which is accruing. A major challenge to overcome for the use of EWSs with disease transitions is to maintain the accuracy of EWSs in data-poor settings. We demonstrate how EWSs behave on reported cases for pertussis in the USA, to highlight some limitations when detecting disease transitions with real-world data.

Effects of stochasticity on the length and behaviour of ecological transients

There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system's transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.

Sources of Greenhouse Gas Emissions in Agriculture, with Particular Emphasis on Emissions from Energy Used

The relationship between agriculture and climate change is two-sided. Agriculture is the branch of the economy most affected by the ongoing processes. It is also a large emitter of greenhouse gases and there are more and more voices about the need to reduce emissions. The purpose of the study was, based on FADN (Farm Accountancy Data Network) data, to determine the structure of greenhouse gas emissions in farms and to identify types of farms where it is possible to reduce GHG (greenhouse gas) emissions through better energy use. The emission volume was determined on the basis of the IPCC (Intergovernmental Panel on Climate Change) methodology modified for the FADN data. The emissions related to the production of energy were found to be of minor importance compared to other emission sources. Only in the horticultural crop type is the emission from the Energy section the dominant stream of GHG emission. The greatest emissions come from livestock production. Therefore, the emphasis on reducing emissions should not be placed on the Energy sector because, except for the type of horticultural farm, there is not much potential for reduction. The introduction of taxes for GHG emissions at the level of 27.31 EUR/t would reduce farm income from 21% for the type of field crops to 40% for the type of herbivorous animals. The exception is low-emission permanent crops, where the decrease in income would be only 3.85%.

Early warning signals of failures in building management systems

In the context of sensor data generated by Building Management Systems (BMS), early warning signals are still an unexplored topic. The early detection of anomalies can help preventing malfunctions of key parts of a heating, cooling and air conditioning (HVAC) system that may lead to a range of BMS problems, from important energy waste to fatal errors in the worst case. We analyse early warning signals in BMS sensor data for early failure detection. In this paper, the studied failure is a malfunction of one specific Air Handling Unit (AHU) control system that causes temperature spikes of up to 30 degrees Celsius due to overreaction of the heating and cooling valves in response to an anomalous temperature change caused by the pre-heat coil in winter period in a specific area of a manufacturing facility. For such purpose, variance, lag-1 autocorrelation function (ACF1), power spectrum (PS) and variational autoencoder (VAE) techniques are applied to both univariate and multivariate scenarios. The univariate scenario considers the application of these techniques to the control variable only (the one that displays the failure), whereas the multivariate analysis considers the variables affecting the control variable for the same purpose. Results show that anomalies can be detected up to 32 hours prior to failure, which gives sufficient time to BMS engineers to prevent a failure and therefore, an proactive approach to BMS failures is adopted instead of a reactive one.

No warning for slow transitions

A rise in fragility as a system approaches a tipping point may be sometimes estimated using dynamical indicators of resilience (DIORs) that measure the characteristic slowing down of recovery rates before a tipping point. A change in DIORs could be interpreted as an early warning signal for an upcoming critical transition. However, in order to be able to estimate the DIORs, observational records need to be long enough to capture the response rate of the system. As we show here, the required length of the time series depends on the response rates of the system. For instance, the current rate of anthropogenic climate forcing is fast relative to the response rate of some parts of the climate system. Therefore, we may expect difficulties estimating the resilience from modern time series. So far, there have been no systematic studies of the effects of the response rates of the dynamical systems and the rates of forcing on the detectability trends in the DIORs prior to critical transitions. Here, we quantify the performance of the resilience indicators variance and temporal autocorrelation, in systems with different response rates and for different rates of forcing. Our results show that the rapid rise of anthropogenic forcing to the Earth may make it difficult to detect changes in the resilience of ecosystems and climate elements from time series. These findings suggest that in order to determine with models whether the use of the DIORs is appropriate, we need to use realistic models that incorporate the key processes with the appropriate time constants.

Statistical physics approaches to the complex Earth system

Global warming, extreme climate events, earthquakes and their accompanying socioeconomic disasters pose significant risks to humanity. Yet due to the nonlinear feedbacks, multiple interactions and complex structures of the Earth system, the understanding and, in particular, the prediction of such disruptive events represent formidable challenges to both scientific and policy communities. During the past years, the emergence and evolution of Earth system science has attracted much attention and produced new concepts and frameworks. Especially, novel statistical physics and complex networks-based techniques have been developed and implemented to substantially advance our knowledge of the Earth system, including climate extreme events, earthquakes and geological relief features, leading to substantially improved predictive performances. We present here a comprehensive review on the recent scientific progress in the development and application of how combined statistical physics and complex systems science approaches such as critical phenomena, network theory, percolation, tipping points analysis, and entropy can be applied to complex Earth systems. Notably, these integrating tools and approaches provide new insights and perspectives for understanding the dynamics of the Earth systems. The overall aim of this review is to offer readers the knowledge on how statistical physics concepts and theories can be useful in the field of Earth system science.

Detecting Early-Warning Signals in Time Series of Visits to Points of Interest to Examine Population Response to COVID-19 Pandemic

The objective of this paper is to examine population response to COVID-19 and associated policy interventions through detecting early-warning signals in time series of visits to points of interest (POIs). Complex systems, such as cities, would demonstrate early-warning signals (e.g., increased autocorrelation and standard deviation) when they approach phase transitions responding to external perturbation, such as crises, policy changes, and human behavior changes. In urban systems, population visits to POIs, such as restaurants, museums, and hospitals, represent a state of cities as complex systems. These states may undergo phase transitions due to population response to pandemic risks and intervention policies (e.g., social distancing and shelter-in-place orders). In this study, we conducted early-warning signal detection on population visits to POIs to examine population response to pandemic risks, and we evaluated time lags between detected early-warning dates and dates of first cases and policy interventions. We examined two early-warning signals, the increase of autocorrelation at-lag-1 and standard deviation, in time series of population visits to POIs in 17 metropolitan cities in the United States of America. We examined visits to grouped POIs according to two categories of essential services and non-essential services. The results show that: (1) early-warning signals for population response to COVID-19 were detected between February 14 and March 11, 2020 in 17 cities; (2) detected population response had started prior to shelter-in-place orders in 17 cities; (3) early-warning signals detected from the essential POIs visits appeared earlier than those from non-essential POIs; and 4) longer time lags between detected population response and shelter-in-place orders led to a less decrease in POI visits. The results show the importance of detecting early-warning signals during crises in cities as complex systems. Early-warning signals could provide important insights regarding the timing and extent of population response to crises to inform policymakers.

Temperature variance portends and indicates the extent of abrupt climate shifts

Here, we show a discernable increase in temperature variance before a glacial termination by both the Ansari-Bradley test and the moving variance methods plus introduce the idea that there is a correlation between the peak variance and peak temperature increase. The behavior of temperature variance shows potential as a useful tool in analyzing time series data of Earth systems to assess the risk and extent of an upcoming abrupt climate transition.

Response-shift effects in neuromyelitis optica spectrum disorder: a secondary analysis of clinical trial data

Background

Researchers have long posited that response-shift effects may obfuscate treatment effects. The present work investigated possible response-shift effects in a recent clinical trial testing a new treatment for Neuromyelitis Optica Spectrum Disorder (NMOSD). This pivotal trial provided impressive support for the drug Eculizumab in preventing relapse, but less strong or null results as the indicators became more subjective or evaluative. This pattern of results suggests that response-shift effects are present.

Methods

This secondary analysis utilized data from a randomized, double-blind trial evaluating the impact of Eculizumab in preventing relapses in 143 people with NMOSD. Treatment arm and then relapse status were hypothesized ‘catalysts’ of response shift in two series of analyses. We devised a “de-constructed” version of Oort structural-equation modeling using random-effects modeling for use in small samples. This method begins by testing an omnibus response-shift hypothesis and then, pending a positive result, implements a series of random-effects models to elucidate specific response-shift effects.

Results

In the omnibus test, the ‘standard quality-of-life (QOL) model’ captured substantially less well the experience of placebo as compared to Eculizumab group. Recalibration and reconceptualization response-shift effects were detected. Detected relapse-related response shifts included recalibration, reprioritization, and reconceptualization.

Conclusions

Trial patients experienced response shifts related to treatment- and relapse-related experiences. Published trial results likely under-estimated Eculizumab vs. Placebo differences due to recalibration and reconceptualization, and relapse effects due to recalibration, reprioritization, and reconceptualization. This novel random-effects- model application builds on response-shift theory and provides a small-sample method for better estimating treatment effects in clinical trials.

Electronic supplementary material

The online version of this article (10.1007/s11136-020-02707-y) contains supplementary material, which is available to authorized users.

Tipping point and noise-induced transients in ecological networks

A challenging and outstanding problem in interdisciplinary research is to understand the interplay between transients and stochasticity in high-dimensional dynamical systems. Focusing on the tipping-point dynamics in complex mutualistic networks in ecology constructed from empirical data, we investigate the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are studied: environmental (Gaussian white) noise and state-dependent demographic noise. The dynamical mechanism responsible for both phenomena is a transition from one stable steady state to another driven by stochastic forcing, mediated by an unstable steady state. Exploiting a generic and effective two-dimensional reduced model for real-world mutualistic networks, we find that the average transient lifetime scales algebraically with the noise amplitude, for both environmental and demographic noise. We develop a physical understanding of the scaling laws through an analysis of the mean first passage time from one steady state to another. The phenomena of noise-induced collapse and recovery and the associated scaling laws have implications for managing high-dimensional ecological systems.

Intrinsic and Extrinsic Thermodynamics for Stochastic Population Processes with Multi-Level Large-Deviation Structure

A set of core features is set forth as the essence of a thermodynamic description, which derive from large-deviation properties in systems with hierarchies of timescales, but which are not dependent upon conservation laws or microscopic reversibility in the substrate hosting the process. The most fundamental elements are the concept of a macrostate in relation to the large-deviation entropy, and the decomposition of contributions to irreversibility among interacting subsystems, which is the origin of the dependence on a concept of heat in both classical and stochastic thermodynamics. A natural decomposition that is known to exist, into a relative entropy and a housekeeping entropy rate, is taken here to define respectively the intensive thermodynamics of a system and an extensive thermodynamic vector embedding the system in its context. Both intensive and extensive components are functions of Hartley information of the momentary system stationary state, which is information about the joint effect of system processes on its contribution to irreversibility. Results are derived for stochastic chemical reaction networks, including a Legendre duality for the housekeeping entropy rate to thermodynamically characterize fully-irreversible processes on an equal footing with those at the opposite limit of detailed-balance. The work is meant to encourage development of inherent thermodynamic descriptions for rule-based systems and the living state, which are not conceived as reductive explanations to heat flows.

Testing Critical Slowing Down as a Bifurcation Indicator in a Low-Dissipation Dynamical System

We study a two-dimensional low-dissipation nonautonomous dynamical system, with a control parameter that is swept linearly in time across a transcritical bifurcation. We investigate the relaxation time of a perturbation applied to a variable of the system and we show that critical slowing down may occur at a parameter value well above the bifurcation point. We test experimentally the occurrence of critical slowing down by applying a perturbation to the accessible control parameter and we find that this perturbation leaves the system behavior unaltered, thus providing no useful information on the occurrence of critical slowing down. The theoretical analysis reveals the reasons why these tests fail in predicting an incoming bifurcation.

Transient indicators of tipping points in infectious diseases

The majority of known early warning indicators of critical transitions rely on asymptotic resilience and critical slowing down. In continuous systems, critical slowing down is mathematically described by a decrease in magnitude of the dominant eigenvalue of the Jacobian matrix on the approach to a critical transition. Here, we show that measures of transient dynamics, specifically, reactivity and the maximum of the amplification envelope, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases. Furthermore, we introduce indicators designed to detect trends in these measures and find that they reliably classify time series of case notifications simulated from stochastic models according to levels of vaccine uptake. Greater attention should be focused on the potential for systems to exhibit transient amplification of perturbations as a critical threshold is approached, and should be considered when searching for generic leading indicators of tipping points. Awareness of this phenomenon will enrich understanding of the dynamics of complex systems on the verge of a critical transition.

Multiple metastable network states in urban traffic

While abrupt regime shifts between different metastable states have occurred in natural systems from many areas including ecology, biology, and climate, evidence for this phenomenon in transportation systems has been rarely observed so far. This limitation might be rooted in the fact that we lack methods to identify and analyze possible multiple states that could emerge at scales of the entire traffic network. Here, using percolation approaches, we observe such a metastable regime in traffic systems. In particular, we find multiple metastable network states, corresponding to varying levels of traffic performance, which recur over different days. Based on high-resolution global positioning system (GPS) datasets of urban traffic in the megacities of Beijing and Shanghai (each with over 50,000 road segments), we find evidence supporting the existence of tipping points separating three regimes: a global functional regime and a metastable hysteresis-like regime, followed by a global collapsed regime. We can determine the intrinsic critical points where the metastable hysteresis-like regime begins and ends and show that these critical points are very similar across different days. Our findings provide a better understanding of traffic resilience patterns and could be useful for designing early warning signals for traffic resilience management and, potentially, other complex systems.

Modeling competitive evolution of multiple languages

Increasing evidence demonstrates that in many places language coexistence has become ubiquitous and essential for supporting language and cultural diversity and associated with its financial and economic benefits. The competitive evolution among multiple languages determines the evolution outcome, either coexistence, or decline, or extinction. Here, we extend the Abrams-Strogatz model of language competition to multiple languages and then validate it by analyzing the behavioral transitions of language usage over the recent several decades in Singapore and Hong Kong. In each case, we estimate from data the model parameters that measure each language utility for its speakers and the strength of two biases, the majority preference for their language, and the minority aversion to it. The values of these two biases decide which language is the fastest growing in the competition and what would be the stable state of the system. We also study the system convergence time to stable states and discover the existence of tipping points with multiple attractors. Moreover, the critical slowdown of convergence to the stable fractions of language users appears near and peaks at the tipping points, signaling when the system approaches them. Our analysis furthers our understanding of evolution of various languages and the role of tipping points in behavioral transitions. These insights may help to protect languages from extinction and retain the language and cultural diversity.

A behavioral approach to instability pathways in financial markets

We introduce an indicator that aims to detect the emergence of market instabilities by quantifying the intensity of self-organizing processes arising from stock returns' co-movements. In financial markets, phenomena like imitation, herding and positive feedbacks characterize the emergence of endogenous instabilities, which can modify the qualitative and quantitative behavior of the underlying system. The impossibility to formalize ex-ante the dynamic laws that rule the evolution of financial systems motivates the use of a parsimonious synthetic indicator to detect the disruption of an existing equilibrium configuration. Here we show that the emergence of an interconnected sub-graph of stock returns co-movements from a broader market index is a signal of an out-of-equilibrium transition of the underlying system. To test the validity of our approach, we propose a model-free application that builds on the identification of up and down market phases.

'12 tips for teaching environmental sustainability to health professionals'

Background: As recognition of the health impacts of climate change and other environmental challenges increases, so too does the need for health care professionals to practice healthcare sustainably. Environmental sustainability in healthcare extends beyond our traditional understanding of environmental health, which is often limited to environmental hazards and disease. Health services, professional organizations, and training institutions are increasingly forming climate and sustainability position statements and policies accordingly. To prepare future health professionals for global environmental change, environmental sustainability must be meaningfully integrated into health curricula.Aim: To provide educators with 12 tips for integrating environmental sustainability into health professional education.Methods: The authors reviewed the literature relating to climate change, environmental sustainability and health, and health professional education. By combining findings from this search with reflections on their own experience in clinical and public health teaching across nursing and midwifery, paramedicine, medicine, and public health, the authors developed recommendations for integrating environmental sustainability into health professional education.Results: These 12 tips can be used to teach students and qualified health professionals in nursing, allied health, and medicine to practice healthcare in an environmentally sustainable manner.Conclusions: Empowering health professionals to practice environmentally sustainable healthcare has economic, social, health, and environmental benefits. Teaching environmental sustainability to health professionals enhances existing learning by updating curricula with the latest evidence of how environmental determinants of health are rapidly changing and enables both educators and students to make an important contribution to safeguarding human health, the environment, and healthcare for future generations.

Noise-enabled species recovery in the aftermath of a tipping point

The beneficial role of noise in promoting species coexistence and preventing extinction has been recognized in theoretical ecology, but previous studies were mostly concerned with low-dimensional systems. We investigate the interplay between noise and nonlinear dynamics in real-world complex mutualistic networks with a focus on species recovery in the aftermath of a tipping point. Particularly, as a critical parameter such as the mutualistic interaction strength passes through a tipping point, the system collapses and approaches an extinction state through a dramatic reduction in the species populations to near-zero values. We demonstrate the striking effect of noise: when the direction of parameter change is reversed through the tipping point, noise enables species recovery which otherwise would not be possible. We uncover an algebraic scaling law between the noise amplitude and the parameter distance from the tipping point to the recovery point and provide a physical understanding through analyzing the nonlinear dynamics based on an effective, reduced-dimension model. Noise, in the form of small population fluctuations, can thus play a positive role in protecting high-dimensional, complex ecological networks.

Detecting Critical Transitions in the Human Innate Immune System Post-cardiac Surgery

Coronary artery bypass grafting with cardiopulmonary bypass activates the human innate immune system (HIIS) and invokes a vigorous inflammatory response that is systemic. This massive inflammatory reaction can contribute to the development of postoperative complications that could topple the state of the system from health to disease, or even to some extent, death. The body, after all, is in a state where majority of its immune cell populations have been depleted, and sometimes needs days or even longer to recuperate. To obtain a deeper understanding on how HIIS responds to complications after cardiac surgery, we perturb the immune system model that we have developed in an earlier work in-silico by adding another source of inflammation triggering moieties (ITMs) hours after surgery in various regimes. A critical transition occurs upon the addition of a critical concentration of ITMs when the insult is sustained for approximately 3 h – a total concentration that corresponds to the fatal concentration of ITMs documented in literature. By perturbing HIIS in-silico with additional sources of ITMs to mimic persistent and recurring episodes of post-surgery complications, we are able to specify under which conditions critical transitions occur in HIIS, as well as pinpoint important blood parameters that exhibit critical transitions in our model. More importantly, by applying early warning signals on the clinical trial data used to calibrate and validate HIIS model, we are able to detect blood parameters that exhibit critical transitions in patients who died post-surgery, where pro-inflammatory cytokines are deemed potential markers for critical transitions.

Anticipating critical transitions in epithelial-hybrid-mesenchymal cell-fate determination

In the vicinity of a tipping point, critical transitions occur when small changes in an input condition cause sudden, large, and often irreversible changes in the state of a system. Many natural systems ranging from ecosystems to molecular biosystems are known to exhibit critical transitions in their response to stochastic perturbations. In diseases, an early prediction of upcoming critical transitions from a healthy to a disease state by using early-warning signals is of prime interest due to potential application in forecasting disease onset. Here, we analyze cell-fate transitions between different phenotypes (epithelial, hybrid-epithelial/mesenchymal [E/M], and mesenchymal states) that are implicated in cancer metastasis and chemoresistance. These transitions are mediated by a mutually inhibitory feedback loop-microRNA-200/ZEB-driven by the levels of transcription factor SNAIL. We find that the proximity to tipping points enabling these transitions among different phenotypes can be captured by critical slowing down-based early-warning signals, calculated from the trajectory of ZEB messenger RNA level. Further, the basin stability analysis reveals the unexpectedly large basin of attraction for a hybrid-E/M phenotype. Finally, we identified mechanisms that can potentially elude the transition to a hybrid-E/M phenotype. Overall, our results unravel the early-warning signals that can be used to anticipate upcoming epithelial-hybrid-mesenchymal transitions. With the emerging evidence about the hybrid-E/M phenotype being a key driver of metastasis, drug resistance, and tumor relapse, our results suggest ways to potentially evade these transitions, reducing the fitness of cancer cells and restricting tumor aggressiveness.

The problem of detrending when analysing potential indicators of disease elimination

As we strive towards the elimination of many burdensome diseases, the question of when intervention efforts may cease is increasingly important. It can be very difficult to know when prevalences are low enough that the disease will die out without further intervention, particularly for diseases that lack accurate tests. The consequences of stopping an intervention prematurely can put back elimination efforts by decades. Critical slowing down theory predicts that as a dynamical system moves through a critical transition, deviations from the steady state return increasingly slowly. We study two potential indicators of disease elimination predicted by this theory, and investigate their response using a simple stochastic model. We compare our dynamical predictions to simulations of the fluctuation variance and coefficient of variation as the system moves through the transition to elimination. These comparisons demonstrate that the primary challenge facing the analysis of early warning signs in timeseries data is that of accurately 'detrending' the signal, in order to preserve the statistical properties of the fluctuations. We show here that detrending using the mean of even just four realisations of the process can give a significant improvement when compared to using a moving window average. Taking this idea further, we consider a 'metapopulation' model of an endemic disease, in which infection spreads in various separated areas with some movement between the subpopulations. We successfully predict the behaviour of both variance and the coefficient of variation in a metapopulation by using information from the other subpopulations to detrend the system.