Spatio-temporal Model Research Articles

A novel spatio-temporal prediction model of epidemic spread integrating cellular automata with agent-based modeling

Since 2019, major infectious disease outbreaks have placed tremendous pressure on global public health systems, triggering extensive research on the predictive modeling of infectious diseases. Cellular Automaton (CA) is primarily used in the spatial prediction of infectious diseases to establish a model to for simulating the interaction between different regions and the infection risk to simulate the transmission process of the disease and predict its development trend. However, CA models are governed by initial fixed rules and local interactions, and often fail to capture the complex dynamics of epidemic transmission, which are influenced by factors such as public behavior and government intervention. In view of these limitations, we propose a factorial simulation model for the spatial spread of epidemics, the CA-ABM, which divides agents into three categories–public, government, and hospital agents–to comprehensively express the macro factors that affect the development of epidemics. Agent-Based Modeling (ABM) influences the transition rules of the CA through agent choices, constraints and supporting behaviors. Focusing on the COVID-19 pandemic in mainland China from February 6 to March 20, 2020, we simulate its spread. The results showed an average improvement of 8.4 % in prediction accuracy, with few errors, RMSE under 200, and R2 values over 0.9 in most provinces, demonstrating strong macro-scale stability. This approach helps regions to understand influencing factors and enables targeted infection risk assessment and prevention. In addition, scenario analysis based on CA-ABM model changes epidemic decision-making from “prediction-response” to “scenario-response” and provides theoretical reference for future epidemic management.

Prenatal exposure to ambient fine particulate matter and child lung function in the CANDLE cohort

Background Ambient fine particulate matter (PM2.5) exposure adversely impacts child airway health; however, research on prenatal PM2.5 exposure, and child lung function is limited. We investigated these associations in the ECHO-PATHWAYS Consortium, focusing on the role of exposure timing during different phases of fetal lung development. Methods We included 675 children in the CANDLE cohort born between 2007 and 2011 in Memphis, TN, USA. Prenatal exposure to ambient PM2.5 was estimated using a spatiotemporal model based on maternal residential history and averaged over established prenatal periods of lung development. Forced expiratory volume in the first second (FEV1) and forced vital capacity (FVC) were measured by spirometry at age 8–9 years. We used linear regression and Bayesian Distributed Lag Interaction Models (BDLIM) to estimate associations between exposure and lung function z-scores, adjusting for maternal/child characteristics, prenatal/postnatal tobacco exposure, and birth year/season, and evaluating effect modification by child sex and allergic sensitization. Results The average ambient concentration of PM2.5 during pregnancy was 11.1 µg/m3 (standard deviation:1.0 µg/m3). In the adjusted linear regression and BDLIM models, adverse, but not statistically significant, associations were observed between exposure during the pseudoglandular (5–16 weeks of gestation) and saccular (24–36 weeks) phases of lung development and FEV1 and FVC. The strongest association was between a 2 μg/m3 higher concentration of PM2.5 during the saccular phase and FEV1 z-score (−0.176, 95% Confidence Interval [CI]: −0.361, 0.010). The FEV1/FVC ratio was not associated with PM2.5 in any exposure window. No effect modification by child sex or allergic sensitization was observed. Conclusions We did not find strong evidence of associations between prenatal ambient PM2.5 exposure and child lung function in a large, well-characterized study sample. However, there was a suggested adverse association between FEV1 and exposure during late pregnancy. The saccular phase of lung development might be an important window for exposure to PM2.5.

Bayesian hierarchical spatiotemporal modelling of tuberculosis – Human immunodeficiency virus co-infection in Ethiopia

Understanding the epidemiological patterns of tuberculosis-human immunodeficiency virus (TB-HIV) co-infection over space and time is crucial because it assists in identifying areas with high risks that need special control strategies. This article aimed to determine districts in Ethiopia that are most vulnerable to TB-HIV co-infection by examining the spatiotemporal patterns of the co-infection across four years, from 2015 to 2018. The study’s data came from Ethiopia’s Federal Ministry of Health. The data was analysed by applying the Bayesian hierarchical spatiotemporal modelling. We considered four models with different space–time interaction structures via the Integrated Nested Laplace Approximation (INLA) in the R-INLA package. In addition, we have applied the Deviance Information Criterion to select the most suitable model. The mean raw annual TB-HIV relative risk (RR) continuously decreased from 2015 to 2018, and the raw RRs of co-infection varied over districts and years. The spatiotemporal model, which allows for space–time interaction with independent spatial random effect and dependent temporal random effect, was the preferred model for describing the variations in TB-HIV co-infection across different districts over time. The prior variance for the spatial structured random effect had a smaller precision mode than the spatial unstructured random effect. This difference reveals that the former accounted for more spatial autocorrelation than the latter, indicating an information-borrowing effect amongst districts. Furthermore, the findings exhibit that the relative risk of TB-HIV co-infection had significant spatiotemporal variation and clustering. Through this research, further information was obtained regarding the temporal evolution of the geographical spread of TB and HIV co-infection at the district level in the country. It also made it possible to determine districts that should receive priority for control actions because of their high risk of co-infection.

Spatiotemporal Topic Modeling Reveals Storm‐Driven Advection and Stirring Control Plankton Community Variability in an Open Ocean Eddy

AbstractPhytoplankton communities in the open ocean are high‐dimensional, sparse, and spatiotemporally heterogeneous. The advent of automated imaging systems has enabled high‐resolution observation of these communities, but the amounts of data and their statistical properties make analysis with traditional approaches challenging. Spatiotemporal topic models offer an unsupervised and interpretable approach to dimensionality reduction of sparse, high‐dimensional categorical data. Here we use topic modeling to analyze neural‐network‐classified phytoplankton imagery taken in and around a retentive eddy during the 2021 North Atlantic EXport Processes in the Ocean from Remote Sensing (EXPORTS) field campaign. We investigate the role physical‐biological interactions play in altering plankton community composition within the eddy. Analysis of a water mass mixing framework suggests that storm‐driven surface advection and stirring were major drivers of the progression of the eddy plankton community away from a diatom bloom over the course of the cruise.

Behavioural Change Piecewise Constant Spatial Epidemic Models

Human behaviour significantly affects the dynamics of infectious disease transmission as people adjust their behavior in response to outbreak intensity, thereby impacting disease spread and control efforts. In recent years, there have been efforts to incorporate behavioural change into spatio-temporal individual-level models within a Bayesian MCMC framework. In this past work, parametric spatial risk functions were employed, depending on strong underlying assumptions regarding disease transmission mechanisms within the population. However, selecting appropriate parametric functions can be challenging in real-world scenarios, and incorrect assumptions may lead to erroneous conclusions. As an alternative, non-parametric approaches offer greater flexibility. The goal of this study is to investigate the utilization of semi-parametric spatial models for infectious disease transmission, integrating an “alarm function” to account for behavioural change based on infection prevalence over time within a Bayesian MCMC framework. In this paper, we discuss findings from both simulated and real-life epidemics, focusing on constant piecewise distance functions with fixed change points. We also demonstrate the selection of the change points using the Deviance Information Criteria (DIC).

The Polar Vortex Hypothesis: Evolving, Spectrally Distinct Polar Regions Explain Short- and Long-term Light-curve Evolution and Color–Inclination Trends in Brown Dwarfs and Giant Exoplanets

Recent studies revealed viewing-angle-dependent color and spectral trends in brown dwarfs, as well as long-term photometric variability (∼100 hr). The origins of these trends are yet unexplained. Here, we propose that these seemingly unrelated sets of observations stem from the same phenomenon: the polar regions of brown dwarfs and directly imaged exoplanets are spectrally different from lower-latitude regions, and they evolve over longer timescales, possibly driven by polar vortices. We explore this hypothesis via a spatiotemporal atmosphere model capable of simulating time series and disk-integrated spectra of ultracool atmospheres. We study three scenarios with different spectral and temporal components: a null hypothesis without polar vortex, and two scenarios with polar vortices. We find that the scenarios with polar vortex can explain the observed infrared color–inclination trend and the variability amplitude–inclination trend. The presence of spectrally distinct, time-evolving polar regions in brown dwarfs and giant exoplanet atmospheres raises the possibility that one-dimensional static atmospheric models may be insufficient for reproducing ultracool atmospheres in detail.

The STAP-Net: A new health perception and prediction framework for bearing-rotor systems under special working conditions

The health perception of bearing-rotor systems and their remaining useful life prediction has been a critical and challenging theme in the field of Prognostic and Health Management (PHM). Deep learning has become a prominent area of PHM research. However, current models have difficulty in adequately extracting the deep degradation characteristics of bearings and effectively capturing time-series information during the failure process. Also, most remaining useful life (RUL) prediction methods focus on point estimation, limiting their ability to quantify prediction uncertainty. To address these shortcomings, this study proposes a novel health perception and prediction framework, the Spatiotemporal Self-Attention Mechanism Probabilistic model (STAP-Net). The framework embodies the principles of lightweight design, focusing, and probabilistic approaches, and is tailored for bearing rotor systems operating under unique conditions. The key innovation of STAP-Net is the integration of a modified gate recurrent unit, known as the Weight Diminish Recurrent Unit (WDRU). It greatly reduces the training parameters of the proposed STAP-Net framework and improves the convergence speed of the framework while ensuring the prediction accuracy. Through analyzing the bearing-rotor system degradation data, the efficacy of STAP-Net is validated under special operating conditions such as misalignment and abrasive wear. The superior performance of the proposed framework is evaluated and confirmed based on 3 key metrics: high-precision point prediction, suitable prediction intervals, and reliable probabilistic prediction results.

Economical-epidemiological analysis of the coffee trees rust pandemic.

Coffee leaf rust is a prevalent botanical disease that causes a worldwide reduction in coffee supply and its quality, leading to immense economic losses. While several pandemic intervention policies (PIPs) for tackling this rust pandemic are commercially available, they seem to provide only partial epidemiological relief for farmers. In this work, we develop a high-resolution spatiotemporal economical-epidemiological model, extending the Susceptible-Infected-Removed model, that captures the rust pandemic's spread in coffee tree farms and its associated economic impact. Through extensive simulations for the case of Colombia, a country that consists mostly of small-size coffee farms and is the second-largest coffee producer in the world, our results show that it is economically impractical to sustain any profit without directly tackling the rust pandemic. Furthermore, even in the hypothetical case where farmers perfectly know their farm's epidemiological state and the weather in advance, any rust pandemic-related efforts can only amount to a limited profit of roughly 4% on investment. In the more realistic case, any rust pandemic-related efforts are expected to result in economic losses, indicating that major disturbances in the coffee market are anticipated.

Multi-relation spatiotemporal graph residual network model with multi-level feature attention: A novel approach for landslide displacement prediction

Accurate prediction of landslide displacement is crucial for effective early warning of landslide disasters. While most existing prediction methods focus on time-series forecasting for individual monitoring points, there is limited research on the spatiotemporal characteristics of landslide deformation. This paper proposes a novel Multi-Relation Spatiotemporal Graph Residual Network with Multi-Level Feature Attention (MFA-MRSTGRN) that effectively improves the prediction performance of landslide displacement through spatiotemporal fusion. This model integrates internal seepage factors as data feature enhancements with external triggering factors, allowing for accurate capture of the complex spatiotemporal characteristics of landslide displacement and the construction of a multi-source heterogeneous dataset. The MFA-MRSTGRN model incorporates dynamic graph theory and four key modules: multi-level feature attention, temporal-residual decomposition, spatial multi-relational graph convolution, and spatiotemporal fusion prediction. This comprehensive approach enables the efficient analyses of multi-source heterogeneous datasets, facilitating adaptive exploration of the evolving multi-relational, multi-dimensional spatiotemporal complexities in landslides. When applying this model to predict the displacement of the Liangshuijing landslide, we demonstrate that the MFA-MRSTGRN model surpasses traditional models, such as random forest (RF), long short-term memory (LSTM), and spatial temporal graph convolutional networks (ST-GCN) models in terms of various evaluation metrics including mean absolute error (MAE = 1.27 mm), root mean square error (RMSE = 1.49 mm), mean absolute percentage error (MAPE = 0.026), and R-squared (R2 = 0.88). Furthermore, feature ablation experiments indicate that incorporating internal seepage factors improves the predictive performance of landslide displacement models. This research provides an advanced and reliable method for landslide displacement prediction.

Changes in the European Union households' consumption structure and the sustainable development

The subject of this study is the relationship between the households' consumption structure changes and sustainable development. This research was conducted for the European Union (EU) countries in the years 2008-2020. Additionally, in the investigation, the considered area is divided into two separate regimes: countries that joined the EU before 2000 (referred to as Old EU Member States) and countries that joined the EU later (called New EU Member States). It is widely known that the territorial units in the first regime are relatively wealthier than those in the second group. As a result of differences in wealth between the defined regimes, consumer behaviours can vary. Sustainable consumption (SC) is one of the major aspects of sustainable development. SC indicates consumer choices that are both intentional and effective, and these choices demonstrate a positive impact on environmental and/or social conditions. In this research, the influence of households' consumption changes on the environmental pillar of sustainable development is evaluated. Caring for the natural environment is one of the main challenges today. In this research, another approach to measure environmental development is proposed. So far, different types of footprints have been used. This analysis considers most of the environmental indicators defined in the Sustainable Development Goals (SDG's). The environmental development of EU countries is evaluated based on the synthetic measure of development proposed by Hellwig. Moreover, the spatio-temporal sensitivity models are estimated and verified. The sensitivity parameter of these models shows the impact of consumption structure changes on environmental development, wherein the consumption structure is categorized into four groups of goods: services, non-durable, semi-durable, and durable. The spatial autoregressive (SAR) models are also estimated and verified. The investigation results point out the differences in the influence of consumption structure changes on sustainable development depending on the considered group of countries.

Combining residual convolutional LSTM with attention mechanisms for spatiotemporal forest cover prediction

Understanding spatiotemporal variations in forest cover is crucial for effective forest resource management. However, existing models often lack accuracy in simultaneously capturing temporal continuity and spatial correlation. To address this challenge, we developed ResConvLSTM-Att, a novel hybrid model integrating residual neural networks, Convolutional Long Short-Term Memory (ConvLSTM) networks, and attention mechanisms. We evaluated ResConvLSTM-Att against four deep learning models: LSTM, combined convolutional neural network and LSTM (CNN-LSTM), ConvLSTM, and ResConvLSTM for spatiotemporal prediction of forest cover in Tasmania, Australia. ResConvLSTM-Att achieved outstanding prediction performance, with an average root mean square error (RMSE) of 6.9% coverage and an impressive average coefficient of determination of 0.965. Compared with LSTM, CNN-LSTM, ConvLSTM, and ResConvLSTM, ResConvLSTM-Att achieved RMSE reductions of 31.2%, 43.0%, 10.1%, and 6.5%, respectively. Additionally, we quantified the impacts of explanatory variables on forest cover dynamics. Our work demonstrated the effectiveness of ResConvLSTM-Att in spatiotemporal data modelling and prediction.

Comparison of K-Means and FCM Algorithms to Optimize Spatiotemporal Pore Pressure Prediction of Earth Dams

To identify the behavior and health monitoring of dams, it is necessary to correctly interpret the results of instrumentation in different phases of construction, impounding, and operation. Therefore, the approach based on spatiotemporal prediction is presented to improve the interpretation of pore pressure behavior of Eyvashan Earth Dam. In this research, using the results of other existing healthy piezometers, a spatiotemporal distribution model is proposed using panel data, which can be effective for predicting and reconstructing missing data. The optimal spatiotemporal clustering of pore pressure changes monitoring with K-Means and Fuzzy C-means (FCM) algorithms will enable the monitoring of points of the dam where instrumentations are not designed and installed or defective instrumentations. In predicting the pore pressure of dams, the input data is classified based on the pore pressure monitoring data, but with the use of clustering algorithms, the classification after the cluster analysis steps will lead to the proper resolution of the pore pressure clustering. According to the validation results of each of the clustering algorithms, the FCM clustering algorithm has more suitable results than the K-Means algorithm in determining the pore pressure clusters. In general, FCM clustering and K-Means algorithms are suitable and efficient tools in the field of more accurate monitoring of earth dams, and by using the proposed method, the detection of unusual areas of pore pressure and the related safety diagnosis is facilitated.

A Novel Spatio-Temporal Vision Transformer Model for Improving Wetland Mapping Using Multi-Seasonal Sentinel Data

A Novel Spatio-Temporal Vision Transformer Model for Improving Wetland Mapping Using Multi-Seasonal Sentinel Data

Multivariate spatiotemporal models with low rank coefficient matrix

Multivariate spatiotemporal models with low rank coefficient matrix

Non-stationary spatio-temporal modeling using the stochastic advection-diffusion equation

We construct flexible spatio-temporal models through stochastic partial differential equations (SPDEs) where both diffusion and advection can be spatially varying. Computations are done through a Gaussian Markov random field approximation of the solution of the SPDE, which is constructed through a finite volume method. The new flexible non-separable model is compared to a flexible separable model both for reconstruction and forecasting, and evaluated in terms of root mean square errors and continuous rank probability scores. A simulation study demonstrates that the non-separable model performs better when the data is simulated from a non-separable model with diffusion and advection. Further, we estimate surrogate models for emulating the output of a ocean model in Trondheimsfjorden, Norway, and simulate observations of autonomous underwater vehicles. The results show that the flexible non-separable model outperforms the flexible separable model for real-time prediction of unobserved locations.

Spatiotemporal Ecologic Analysis of COVID-19 Vaccination Coverage and Outcomes, Oklahoma, USA, February 2020-December 2021.

Data on COVID-19 cases, deaths, hospitalizations, and vaccinations in Oklahoma, USA, have not been systematically described. The relationship between vaccination and COVID-19-related outcomes over time has not been investigated. We graphically described data collected during February 2020-December 2021 and conducted spatiotemporal modeling of monthly increases in COVID-19 cumulative death and hospitalization rates, adjusting for cumulative case rate, to explore the relationship. A 1 percentage point increase (absolute change) in the cumulative vaccination rate was associated with a 6.3% (95% CI 1.4%-10.9%) relative decrease in death outcome during April-June 2021, and a 1.9% (95% CI 1.1%-2.6%) relative decrease in death outcome and 1.1% (95% CI 0.5%-1.7%) relative decrease in hospitalization outcome during July-December 2021; the effect on hospitalizations was driven largely by data from urban counties. Our findings from Oklahoma suggest that increasing cumulative vaccination rates might reduce the increase in cumulative death and hospitalization rates from COVID-19.

Air pollution exposure in vitrified oocyte donors and male recipient partners in relation to fertilization and embryo quality.

Studies on air pollution and outcomes of in vitro fertilization (IVF) have focused on couples undergoing autologous IVF, in which it is challenging to disentangle maternal and paternal exposures during gametogenesis. We sought to evaluate the independent associations between air pollution exposure during oogenesis and spermatogenesis on fertilization and embryo quality in non-identified donor oocyte IVF cycles. Our study included 500 oocyte donors and 915 male recipient partners who contributed 1,095 oocyte thaw cycles (2008-2019). Daily ambient air pollutant exposure was estimated using spatio-temporal models based on residential address and averaged over folliculogenesis (i.e., three months prior to initiation of controlled ovarian stimulation), controlled ovarian stimulation, and spermatogenesis (i.e., 72days prior to oocyte thaw). We used multivariable generalized estimating equations to estimate the adjusted odds ratios (aOR) and 95% confidence intervals (CI) for an interquartile range increase in pollutant exposure in relation to the proportion of oocytes surviving thaw, oocytes fertilized, and usable embryos. Oocyte donors with higher exposure to organic carbon (OC) (aOR=0.86 95%CI 0.79,0.94) and particulate matter<10µm (aOR=0.69 95%CI 0.54,0.90) during folliculogenesis had a lower proportion of oocytes surviving thaw. During ovarian stimulation, higher particulate matter<2.5µm (aOR=0.78 95%CI 0.66, 0.91), nitrate (aOR=0.83 95% CI 0.69,0.99), and OC (aOR=0.86 95% CI 0.80,0.93) exposure was associated with a lower proportion of surviving oocytes while nitrogen dioxide (aOR=1.11 95%CI 1.00,1.23) and ozone (aOR=1.19 95%CI 1.04,1.37) exposure was associated with a higher proportion of fertilized oocytes and usable embryos. Elemental carbon (aOR=0.93 95%CI 0.87,1.00) and OC (aOR=0.95 95%CI 0.90,1.00) exposure during spermatogenesis was associated with a slightly lower proportion of usable embryos. On the day of oocyte thaw, higher ambient OC at the IVF clinic was associated with lower oocyte survival and higher ozone was associated with lower fertilization. Both maternal and paternal air pollution exposures during gametogenesis have independent, largely detrimental, effects on early embryological outcomes.

Alcohol outlet density and adolescent drinking behaviors in Thailand, 2007-2017: A spatiotemporal mixed model analysis.

This study aimed to explore the relationship between alcohol outlet density and the proportion of alcohol consumption among Thai adolescents. We utilized the alcohol consumption data from the 2007, 2011, and 2017 Tobacco and Alcohol Consumption Survey in Thailand. We analyzed the relationships between alcohol sales license figures and consumption behavior using a spatiotemporal mixed model. Our data had two levels. The upper (provincial) level featured alcohol sales license density (spatial effect), the years of survey (temporal effect), and the social deprivation index. The lower (individual) level included the demographic data of the adolescents. A total of 9,566 Thai adolescents participated in this study, based on surveys conducted in 2007 (n = 4,731), 2011 (n = 3,466), and 2017 (n = 1,369). The density of alcohol sales licenses increased the odds for the one-year current drinker category (odds ratio [OR] = 1.08, 95% confidence interval [CI], 1.04-1.45), especially in male adolescents (OR = 1.09, 95% CI, 1.04-1.14). Furthermore, it increased the odds for the heavy episodic drinker category for female adolescents (OR = 1.23, 95% CI, 1.05-1.44). Increased alcohol sales licenses are associated with higher alcohol consumption among Thai adolescents. This highlights the need for government organizations to develop and apply strategies to reduce the number of licenses for the sale of alcohol.

A Novel Centralized Allocation Data Envelopment Analysis Model for Carbon Emission Allocation Under a Heterogeneous Abatement Cost: Application Within the Chinese Industrial Sector

This paper presents a mathematical approach to analyzing carbon abatement costs and the allocation of carbon emission allowances in China’s industrial sectors. We utilize input–output data from 30 Chinese provinces between 2009 and 2018 to estimate carbon abatement costs by applying the slack-based measure (SBM) efficiency model and its dual form. The SBM model captures inefficiencies and offers a rigorous framework for measuring abatement costs. Using these costs, we develop a centralized allocation data envelopment analysis (DEA) model, which maximizes sectoral benefits through optimal reallocation. This DEA model is formalized as a linear programming problem, with the aim of determining the efficient allocations of carbon allowances while maintaining the system’s economic productivity. Furthermore, we construct intertemporal, interregional, and spatiotemporal allocation DEA models to examine the dynamics of carbon emission allowance allocation over time, space, and combined spatiotemporal dimensions. These models offer insights into the efficiency of carbon markets under varying conditions. Our proposed new mathematical formulations reveal optimal allocation strategies that can balance emission reductions with industrial productivity. This study also provides novel mathematical frameworks for analyzing the carbon allowance distribution and contributions to both the theory and application of mathematical optimization in environmental policy design. Our findings reveal that China’s industrial carbon abatement costs exhibit significant interprovincial and regional differences. Developed provinces with higher levels of industrial development have higher carbon abatement costs, while provinces with less-developed industrial sectors have lower costs. Under the interregional allocation scenario of carbon emission allowances that consider abatement costs, developed provinces have smaller industrial carbon emission reductions, whereas less-developed provinces have larger reductions. In the intertemporal allocation scenario, provinces with larger industrial economies face greater emission reduction tasks. Under the combined interregional and intertemporal allocation scenario, industrial sectors in coastal developed provinces have lower carbon emission reductions, while those in inland less-developed provinces have higher reductions, mirroring the spatial allocation results of carbon emission allowances.

A parsimonious setup for streamflow forecasting using CNN-LSTM

ABSTRACT Significant strides have been made in advancing streamflow predictions, notably with the introduction of cutting-edge machine-learning models. Predominantly, long short-term memories (LSTMs) and convolution neural networks (CNNs) have been widely employed in this domain. While LSTMs are applicable in both rainfall-runoff and time-series settings, CNN–LSTMs have primarily been utilized in rainfall-runoff scenarios. In this study, we extend the application of CNN–LSTMs to time-series settings, leveraging lagged streamflow data in conjunction with precipitation and temperature data to predict streamflow. Our results show a substantial improvement in predictive performance in 21 out of 32 HUC8 basins in Nebraska, showcasing noteworthy increases in the Kling–Gupta efficiency values. These results highlight the effectiveness of CNN–LSTMs in time-series settings, particularly for spatiotemporal hydrological modeling, for more accurate and robust streamflow predictions.