Differential Equation Model Research Articles

Optimal Control of Collective Electrotaxis in Epithelial Monolayers

Epithelial monolayers are some of the best-studied models for collective cell migration due to their abundance in multicellular systems and their tractability. Experimentally, the collective migration of epithelial monolayers can be robustly steered e.g. using electric fields, via a process termed electrotaxis. Theoretically, however, the question of how to design an electric field to achieve a desired spatiotemporal movement pattern is underexplored. In this work, we construct and calibrate an ordinary differential equation model to predict the average velocity of the centre of mass of a cellular monolayer in response to stimulation with an electric field. We use this model, in conjunction with optimal control theory, to derive physically realistic optimal electric field designs to achieve a variety of aims, including maximising the total distance travelled by the monolayer, maximising the monolayer velocity, and keeping the monolayer velocity constant during stimulation. Together, this work is the first to present a unified framework for optimal control of collective monolayer electrotaxis and provides a blueprint to optimally steer collective migration using other external cues.

Hybrid carbon nanotubes flow and heat transfer over a vertical thin needle with suction effect: A numerical and optimisation analysis

The present study aims to provide a numerical and optimisation analysis of the hybrid carbon nanotubes flow over a vertical thin needle under the suction effect. The thin needle is suspended in water-based hybrid nanofluids containing a combination of single- and multi-walled carbon nanotubes. The heat is transported using the mixed convection flow. A mathematical model of partial differential equations (PDEs) is developed subject to boundary conditions. The PDEs system is converted to non-dimensional ordinal differential equations (ODEs) by using the similarity solution method. Then, the first order of the ODEs system is solved in a MATLAB bvp4c function. We innovatively carry out an optimisation process using response surface methodology (RSM) to enhance the efficiency of the numerical experiment data and fill a gap in the optimised analysis. To observe the variation solutions for the reduced skin friction and heat transfer coefficients, several parameters, such as the mixed convection and suction parameters, are altered into several values. The solutions are essential for accurately forecasting the occurrence of boundary layer separation, particularly in the case of dual solutions. Our analysis reveals that the model generates multiple solutions for the opposing flow, and the boundary layer separates slowly due to the suction effect. To complete the research, RSM reveals that the highest value of the nanoparticle volume fraction and suction parameters, as well as the smallest value of the needle size, generate the maximum transmitting heat through the slender needle. Furthermore, both the numerical and RSM results show that hybrid carbon nanotubes perform better than mono‑carbon nanotubes for better practical reference.

Deciphering oxygen distribution and hypoxia profiles in the tumor microenvironment: a data-driven mechanistic modeling approach

Objective. The distribution of hypoxia within tissues plays a critical role in tumor diagnosis and prognosis. Recognizing the significance of tumor oxygenation and hypoxia gradients, we introduce mathematical frameworks grounded in mechanistic modeling approaches for their quantitative assessment within a tumor microenvironment. By utilizing known blood vasculature, we aim to predict hypoxia levels across different tumor types. Approach. Our approach offers a computational method to measure and predict hypoxia using known blood vasculature. By formulating a reaction-diffusion model for oxygen distribution, we derive the corresponding hypoxia profile. Main results. The framework successfully replicates observed inter- and intra-tumor heterogeneity in experimentally obtained hypoxia profiles across various tumor types (breast, ovarian, pancreatic). Additionally, we propose a data-driven method to deduce partial differential equation models with spatially dependent parameters, which allows us to comprehend the variability of hypoxia profiles within tissues. The versatility of our framework lies in capturing diverse and dynamic behaviors of tumor oxygenation, as well as categorizing states of vascularization based on the dynamics of oxygen molecules, as identified by the model parameters. Significance. The proposed data-informed mechanistic method quantitatively assesses hypoxia in the tumor microenvironment by integrating diverse histopathological data and making predictions across different types of data. The framework provides valuable insights from both modeling and biological perspectives, advancing our comprehension of spatio-temporal dynamics of tumor oxygenation.

Quantifying the effectiveness of brucellosis control strategies in northern China using a mechanistic and data-driven model

In recent years, with the frequent movement of livestock and related products on the Chinese mainland that have resulted in a rise in brucellosis cases in certain regions, the Chinese government has implemented “vaccination + culling” strategy to control the spread of brucellosis, but the epidemiological situation and the intensity of prevention and control measures vary from province to province, and much of the situation has not yet been reversed. Thus, it is crucial to understand and assess the combined effects of vaccination and culling on brucellosis transmission behind the data from multiple provinces and cities. In this paper, we combine an ordinary differential equation model and a deep neural network model to develop a hybrid model that not only characterizes the propagation mechanism of brucellosis, but also describes control strategies as time-varying parameters to track the dynamic evolution of brucellosis transmission under control measures with varying intensity. In addition, the real-time reproduction number is calculated and taken as an evaluation index. We then train and test the hybrid model using actual data from four provincial administrative regions in Northern China from 2010 to 2020. It is found that in Inner Mongolia, the sheep vaccination rate (θ(t)) and the culling rate of infected sheep (c(t)) have a weak effect on reducing the brucellosis transmission. In Gansu, the incomplete culling of infected sheep was the main reason for the increase in human brucellosis cases after the introduction of whole-flock vaccination in 2016. Culling and vaccination contributed to the decrease in human cases in both Xinjiang and Jilin. Prevention and control measures for human brucellosis are inadequate in Xinjiang. However, prevention and control measures for human brucellosis are inadequate, while indirect transmission of Brucella from the environment to humans and sheep poses a greater risk than direct contact with infected sheep in Jilin. The above analysis can provide theoretical foundation for decision-making on public health policies to control the pandemic.

Stable predictive control of continuous stirred-tank reactors using deep learning

In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.

Digital twin mathematical models suggest individualized hemorrhagic shock resuscitation strategies

BackgroundOptimizing resuscitation to reduce inflammation and organ dysfunction following human trauma-associated hemorrhagic shock is a major clinical hurdle. This is limited by the short duration of pre-clinical studies and the sparsity of early data in the clinical setting.MethodsWe sought to bridge this gap by linking preclinical data in a porcine model with clinical data from patients from the Prospective, Observational, Multicenter, Major Trauma Transfusion (PROMMTT) study via a three-compartment ordinary differential equation model of inflammation and coagulation.ResultsThe mathematical model accurately predicts physiologic, inflammatory, and laboratory measures in both the porcine model and patients, as well as the outcome and time of death in the PROMMTT cohort. Model simulation suggests that resuscitation with plasma and red blood cells outperformed resuscitation with crystalloid or plasma alone, and that earlier plasma resuscitation reduced injury severity and increased survival time.ConclusionsThis workflow may serve as a translational bridge from pre-clinical to clinical studies in trauma-associated hemorrhagic shock and other complex disease settings.

Bridging the gap between agent based models and continuous opinion dynamics

There is a rich literature on microscopic models for opinion dynamics; most of them fall into one of two categories – agent-based models or differential equation models – with a general understanding that the two are connected in certain scaling limits. In this paper we show rigorously this is indeed the case. In particular we show that both ordinary and stochastic differential equations can be obtained as a limit of agent-based models by simultaneously rescaling time and the extent to which an agent updates their opinion after an interaction. This approach provides a pathway to analyse much more diverse modelling paradigms, for example: the motivation behind several possible multiplicative noise terms in stochastic differential equation models; the connection between selection noise and the mollification of the discontinuous bounded confidence interaction function; and how the method for selecting interacting pairs can determine the normalisation in the corresponding differential equation. Our computational experiments confirm our findings, showing excellent agreement of solutions to the two classes of models in a variety of settings.

Simple model to incorporate statistical noise based on a modified hodgkin-huxley approach for external electrical field driven neural responses

Noise activity is known to affect neural networks, enhance the system response to weak external signals, and lead to stochastic resonance phenomenon that can effectively amplify signals in nonlinear systems. In most treatments, channel noise has been modeled based on multi-state Markov descriptions or the use stochastic differential equation models. Here we probe a computationally simple approach based on a minor modification of the traditional Hodgkin-Huxley approach to embed noise in neural response. Results obtained from numerous simulations with different excitation frequencies and noise amplitudes for the action potential firing show very good agreement with output obtained from well-established models. Furthermore, results from the Mann–Whitney U Test reveal a statistically insignificant difference. The distribution of the time interval between successive potential spikes obtained from this simple approach compared very well with the results of complicated Fox and Lu type methods at much reduced computational cost. This present method could also possibly be applied to the analysis of spatial variations and/or differences in characteristics of random incident electromagnetic signals.

Stochastic Latent Transformer: Efficient Modeling of Stochastically Forced Zonal Jets

AbstractWe present a novel probabilistic deep learning approach, the “stochastic latent transformer” (SLT), designed for the efficient reduced‐order modeling of stochastic partial differential equations. Stochastically driven flow models are pertinent to a diverse range of natural phenomena, including jets on giant planets, ocean circulation, and the variability of midlatitude weather. However, much of the recent progress in deep learning has predominantly focused on deterministic systems. The SLT comprises a stochastically‐forced transformer paired with a translation‐equivariant autoencoder, trained toward the Continuous Ranked Probability Score. We showcase its effectiveness by applying it to a well‐researched zonal jet system, where the interaction between stochastically forced eddies and the zonal mean flow results in a rich low‐frequency variability. The SLT accurately reproduces system dynamics across various integration periods, validated through quantitative diagnostics that include spectral properties and the rate of transitions between distinct states. The SLT achieves a five‐order‐of‐magnitude speedup in emulating the zonally‐averaged flow compared to direct numerical simulations. This acceleration facilitates the cost‐effective generation of large ensembles, enabling the exploration of statistical questions concerning the probabilities of spontaneous transition events.

Fuzzy Boundary Sampled-Data Control for Nonlinear Parabolic DPSs.

For a nonlinear parabolic distributed parameter system (DPS), a fuzzy boundary sampled-data (SD) control method is introduced in this article, where distributed SD measurement and boundary SD measurement are respected. Initially, this nonlinear parabolic DPS is represented precisely by a Takagi-Sugeno (T-S) fuzzy parabolic partial differential equation (PDE) model. Subsequently, under distributed SD measurement and boundary SD measurement, a fuzzy boundary SD control design is obtained via linear matrix inequalities (LMIs) on the basis of the T-S fuzzy parabolic PDE model to guarantee exponential stability for closed-loop parabolic DPS by using inequality techniques and a acrlong LF. Furthermore, respecting the property of membership functions, we present some LMI-based fuzzy boundary SD control design conditions. Finally, the effectiveness of the designed fuzzy boundary SD controller is demonstrated via two simulation examples.

Estimating the region of attraction of wind integrated power systems based on improved expanding interior algorithm

AbstractThis paper proposes an improved expanding interior algorithm (EIA) to estimate the region of attraction (ROA) of power systems with wind power generation based on sum of squares (SOS) programming. An ordinary differential equation (ODE) model is derived for the doubly‐fed induction generator‐based wind turbine (DFIGWT), which is named as an enhanced synchronous‐generator‐mimicking (ESGM) model. The ESGM model bridges the gap between the requirement of an ODE model in ROA estimation and the conventional differential‐algebraic equation (DAE) model of the DFIGWT system. The ESGM model is able to accurately reflect the low frequency dynamics of the DFIGWT. Moreover, an improved EIA is designed to estimate the ROA based on SOS programming, which has higher efficiency than the existing ROA estimation algorithms based on SOS programming. It is able to adaptively search for the Lyapunov function and obtain an optimal estimation of the ROA in an iterative process. The accuracy and efficiency of this algorithm are verified in three test systems composed of DFIGWTs and synchronous generators (SGs). The morphological changes in the ROA of the test systems caused by the penetration of DFIGWT are examined.

Research on Power Angle Characteristics of One-Machine Infinite-Bus Power Systems of Mixed Gauss and Poisson Stochastic Excitation

Background: A high percentage of renewable energy and a high percentage of power electronic devices are connected to the power system, which leads to the diversification and complexity of stochastic excitation, and the traditional single-excitation stochastic model is no longer applicable. Objective: The study aimed to solve the problem that the high proportion of renewable energy and the high proportion of power electronic equipment are connected to the power system, which leads to the diversification and complexity of stochastic excitation and makes the traditional stochastic model of single excitation no longer applicable. Methods: Firstly, stochastic differential equations for power systems have been modelled with mixed Gaussian white noise and Poisson white noise excitation. Secondly, the Milstein-Euler predictor-corrector method has been developed to solve the stochastic differential equation model of the power system. Finally, the influence of Gauss white noise and Poisson white noise on the power system stability under different excitation intensities has been analyzed. The rationality and correctness of the model have been verified by the simulation of a one-machine infinitebus (OMIB) system. Results: The stochastic differential equation model of a power system with Gauss white noise and Poisson white noise excitation has been established and its angle stability has been analyzed. Increasing the Gaussian white noise and Poisson white noise excitation intensity can lead to an increase in the fluctuation of the power angle curve, as well as an increase in the standard deviation and expected value of the power angle mean curve, which may decrease the stability of the power system. Conclusion: This study provides a reference for stochastic power systems modeling and efficient simulation, and has important application value for power system stability assessment and safety evaluation as well as related patent applications.

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Intracellular "In Silico Microscopes"-Comprehensive 3D Spatio-Temporal Virus Replication Model Simulations.

Despite their small and simple structure compared with their hosts, virus particles can cause severe harm and even mortality in highly evolved species such as humans. A comprehensive quantitative biophysical understanding of intracellular virus replication mechanisms could aid in preparing for future virus pandemics. By elucidating the relationship between the form and function of intracellular structures from the host cell and viral components, it is possible to identify possible targets for direct antiviral agents and potent vaccines. Biophysical investigations into the spatio-temporal dynamics of intracellular virus replication have thus far been limited. This study introduces a framework to enable simulations of these dynamics using partial differential equation (PDE) models, which are evaluated using advanced numerical mathematical methods on leading supercomputers. In particular, this study presents a model of the replication cycle of a specific RNA virus, the hepatitis C virus. The diffusion-reaction model mimics the interplay of the major components of the viral replication cycle, including non structural viral proteins, viral genomic RNA, and a generic host factor. Technically, surface partial differential equations (sufPDEs) are coupled on the 3D embedded 2D endoplasmic reticulum manifold with partial differential equations (PDEs) in the 3D membranous web and cytosol volume. The membranous web serves as a viral replication factory and is formed on the endoplasmic reticulum after infection and in the presence of nonstructural proteins. The coupled sufPDE/PDE model was evaluated using realistic cell geometries based on experimental data. The simulations incorporate the effects of non structural viral proteins, which are restricted to the endoplasmic reticulum surface, with effects appearing in the volume, such as host factor supply from the cytosol and membranous web dynamics. Because the spatial diffusion properties of genomic viral RNA are not yet fully understood, the model allows for viral RNA movement on the endoplasmic reticulum as well as within the cytosol. Visualizing the simulated intracellular viral replication dynamics provides insights similar to those obtained by microscopy, complementing data from in vitro/in vivo viral replication experiments. The output data demonstrate quantitative consistence with the experimental findings, prompting further advanced experimental studies to validate the model and refine our quantitative biophysical understanding.

Mathematical and computationalmodeling of migrations, population spread, and Location-dependent carrying capacities

In single-variable calculus, differential equation models for population dynamics are of high importance and by developing these ordinary differential equation (ODE) models, a practical and intelligible approach can be created to partial differential equations (PDEs). This paper researches focus on PDE models for migrations, population spread and location-dependent carrying capacities. However, logistic equation from which the PDE models develop include the diffusion equation, traveling wave equation, and building blocks with location-dependent parameters. This methodology is suitable for multivariable calculus lecture, evaluation task, and problem activity. The text is accompanied by interactive examples, and the article is formatted to be used as a CDF document in which some of the input can be hidden.

Research on Quantifying Sea Lamprey Ecological Impacts Based on Differential Equation Modeling and AHP-TOPSIS Framework

Research on Quantifying Sea Lamprey Ecological Impacts Based on Differential Equation Modeling and AHP-TOPSIS Framework

Structural and Organizational Strategies of Locomotor Modules during Landing in Patients with Chronic Ankle Instability.

Human locomotion involves the coordinated activation of a finite set of modules, known as muscle synergy, which represent the motor control strategy of the central nervous system. However, most prior studies have focused on isolated muscle activation, overlooking the modular organization of motor behavior. Therefore, to enhance comprehension of muscle coordination dynamics during multi-joint movements in chronic ankle instability (CAI), exploring muscle synergies during landing in CAI patients is imperative. A total of 22 patients with unilateral CAI and 22 healthy participants were recruited for this research. We employed a recursive model for second-order differential equations to process electromyographic (EMG) data after filtering preprocessing, generating the muscle activation matrix, which was subsequently inputted into the non-negative matrix factorization model for extraction of the muscle synergy. Muscle synergies were classified utilizing the K-means clustering algorithm and Pearson correlation coefficients. Statistical parameter mapping (SPM) was employed for temporal modular parameter analyses. Four muscle synergies were identified in both the CAI and healthy groups. In Synergy 1, only the gluteus maximus showed significantly higher relative weight in CAI compared to healthy controls (p = 0.0035). Synergy 2 showed significantly higher relative weights for the vastus lateralis in the healthy group compared to CAI (p = 0.018), while in Synergy 4, CAI demonstrated significantly higher relative weights of the vastus lateralis compared to healthy controls (p = 0.030). Furthermore, in Synergy 2, the CAI group exhibited higher weights of the tibialis anterior compared to the healthy group (p = 0.042). The study suggested that patients with CAI exhibit a comparable modular organizational framework to the healthy group. Investigation of amplitude adjustments within the synergy spatial module shed light on the adaptive strategies employed by the tibialis anterior and gluteus maximus muscles to optimize control strategies during landing in patients with CAI. Variances in the muscle-specific weights of the vastus lateralis across movement modules reveal novel biomechanical adaptations in CAI, offering valuable insights for refining rehabilitation protocols.

Inverse Scheme to Locally Determine Nonlinear Magnetic Material Properties: Numerical Case Study

We are interested in the determination of the local nonlinear magnetic material behaviour in electrical steel sheets due to cutting and punching effects. For this purpose, the inverse problem has to be solved, where the objective function, which penalises the difference between the measured and the simulated magnetic flux density, has to be minimised under a constraint defined according to the corresponding partial differential equation model. We use the adjoint method to efficiently obtain the gradients of the objective function with respect to the material parameters. The optimisation algorithm is low-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS), the forward and adjoint formulations are solved using the finite element (FE) method and the ill-posedness is handled via Tikhonov regularisation, in combination with the discrepancy principle. Realistic numerical case studies show promising results.

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems.

In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

Research on Locating and Searching for Missing Submersibles Based on Differential Equations and SAR-A* Algorithm

In recent years, the application of submersibles has become increasingly widespread. The current focus of research is on real-time and accurate positioning of submersibles to cope with potential emergencies. This paper aims to establish a trajectory model for faulty submersibles. Firstly, a differential equation model for submersible dynamics is established, numerical solutions are obtained and fitted to determine the exponential relationship between velocity and time, resulting in a theoretical formula for submersible displacement. Subsequently, based on the trajectory, a probability model for the presence of targets is established using a two-dimensional Gaussian distribution. The SAR-A* algorithm is employed for regional search, combined with a greedy algorithm to obtain initial deployment points for search and rescue, thus deriving the search and rescue trajectory. This study provides theoretical support for locating and rescuing missing submersibles and is expected to play an important role in future submersible applications.