Ordinary Differential Equation Model Research Articles

African swine fever in wild boar: investigating model assumptions and structure.

African swine fever (ASF) is a highly virulent viral disease that affects domestic pigs and wild boar. Current ASF transmission in Europe is in part driven by wild boar populations, which act as a disease reservoir. Wild boar are abundant throughout Europe and are highly social animals with complex social organization. Despite the known importance of wild boar in ASF spread and persistence, knowledge gaps remain surrounding wild boar transmission. We developed a wild boar modelling framework to investigate the influence of contact-density functions and wild boar social structure on disease dynamics. The framework included an ordinary differential equation model, a homogeneous stochastic model and various network-based stochastic models that explicitly included wild boar social grouping. We found that power-law functions (transmission density0.5) and frequency-based contact-density functions were best able to reproduce recent Baltic outbreaks; however, power-law function models predicted considerable carcass transmission, while frequency-based models had negligible carcass transmission. Furthermore, increased model heterogeneity caused a decrease in the relative importance of carcass-based transmission. The transmission pathways predicted by each model type affected the efficacy of idealized interventions, which highlights the importance of evaluating model type and structure when modelling systems with significant uncertainties.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Network model of skeletal muscle cell signalling predicts differential responses to endurance and resistance exercise training.

Exercise-induced muscle adaptations vary based on exercise modality and intensity. We constructed a signalling network model from 87 published studies of human or rodent skeletal muscle cell responses to endurance or resistance exercise in vivo or simulated exercise in vitro. The network comprises 259 signalling interactions between 120 nodes, representing eight membrane receptors and eight canonical signalling pathways regulating 14 transcriptional regulators, 28 target genes and 12 exercise-induced phenotypes. Using this network, we formulated a logic-based ordinary differential equation model predicting time-dependent molecular and phenotypic alterations following acute endurance and resistance exercises. Compared with nine independent studies, the model accurately predicted 18/21 (85%) acute responses to resistance exercise and 12/16 (75%) acute responses to endurance exercise. Detailed sensitivity analysis of differential phenotypic responses to resistance and endurance training showed that, in the model, exercise regulates cell growth and protein synthesis primarily by signalling via mechanistic target of rapamycin, which is activated by Akt and inhibited in endurance exercise by AMP-activated protein kinase. Endurance exercise preferentially activates inflammation via reactive oxygen species and nuclear factor κB signalling. Furthermore, the expected preferential activation of mitochondrial biogenesis by endurance exercise was counterbalanced in the model by protein kinase C in response to resistance training. This model provides a new tool for investigating cross-talk between skeletal muscle signalling pathways activated by endurance and resistance exercise, and the mechanisms of interactions such as the interference effects of endurance training on resistance exercise outcomes.

Adaptive composite learning control of a flexible two‐link manipulator with unknown spatiotemporally varying disturbance

AbstractThis article presents a novel adaptive composite learning (ACL) control strategy combining reinforcement learning and a disturbance observer (DOB) to address vibration issues in a flexible two‐link manipulator (FTLM) system affected by unknown spatiotemporally varying disturbances. Based on the assumed mode method, the FTLM system is initially transformed into an ordinary differential equation model, while effectively capturing the elastic deformation and vibration characteristics of the flexible link. A composite learning controller, based on the actor‐critic algorithm and DOB, is then developed to achieve trajectory tracking and vibration suppression in the FTLM system. The DOB in the controller compensates for unknown disturbances resulting in reduced system error. It is noting that the proposed optimal control strategy is continuously gathering system experience and evaluating the current policy's effectiveness. The stability and robustness of the closed‐loop system incorporating the composite controller are analyzed using Lyapunov's direct method, and the semi‐global uniform ultimate boundedness of the tracking and vibration errors are also demonstrated. To validate the effectiveness and superiority of the proposed ACL controller, comparative simulations and experiments are conducted on the Quanser experimental platform.

Characterization of partial wetting by CMAS droplets using multiphase many-body dissipative particle dynamics and data-driven discovery based on PINNs

The molten sand that is a mixture of calcia, magnesia, alumina and silicate, known as CMAS, is characterized by its high viscosity, density and surface tension. The unique properties of CMAS make it a challenging material to deal with in high-temperature applications, requiring innovative solutions and materials to prevent its buildup and damage to critical equipment. Here, we use multiphase many-body dissipative particle dynamics simulations to study the wetting dynamics of highly viscous molten CMAS droplets. The simulations are performed in three dimensions, with varying initial droplet sizes and equilibrium contact angles. We propose a parametric ordinary differential equation (ODE) that captures the spreading radius behaviour of the CMAS droplets. The ODE parameters are then identified based on the physics-informed neural network (PINN) framework. Subsequently, the closed-form dependency of parameter values found by the PINN on the initial radii and contact angles are given using symbolic regression. Finally, we employ Bayesian PINNs (B-PINNs) to assess and quantify the uncertainty associated with the discovered parameters. In brief, this study provides insight into spreading dynamics of CMAS droplets by fusing simple parametric ODE modelling and state-of-the-art machine-learning techniques.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Building erudition in the wound healing process: an inflammation model analysis

The main four stages of wound healing are hemostasis, inflammation, proliferation, and remodelling. In the inflammation stage, the body rids of pathogens and debris, and prepares for tissue regrowth. An ordinary differential equation model of the inflammation is constructed that expands on previous models and incorporates dynamics reported in the updated literature. The interaction between debris and apoptotic neutrophils, along with classical and alternative macrophages are included. In all, the model incorporates pathogens, debris, apoptotic neutrophils, neutrophil, M1 macrophages, and M2 macrophages. Parameter values are analysed based on 1. experimental data and 2. reported ranges and expected peak times of white blood cells. Next, global sensitivity analyses which is able to test change in parameter values in tandem is conducted to interpret the overall influence each parameter has on white blood cell, debris, and pathogen output.

A Parameter Identification Approach towards Hemodynamics based on Capnography

Research question Pulse detection decisions during cardiopulmonary resuscitation (CPR) are commonly augmented by end-tidal CO2-concentration (etCO2) levels and trend as etCO2 is a surrogate measure of systemic blood flow during (CPR)1. However, ventilatory and circulatory parameters influence the interpretability of etCO2 heavily. Recently developed methods of continuous cardiac output measurement based on capnography are not applicable during CPR due to the required controlled ventilation settings2. In this study, a parameter identification for a model of CO2 transport during resuscitation is used to estimate cardiac output based on capnography and tidal flow. Methodology After development of a compartmental ordinary differential equation (ODE) model to represent CO2 transport during CPR, parameters such as airway dead space and the level of systemic perfusion are estimated by a multishooting parameter identification approach3. To evaluate the method, tests are conducted on experimental data from a porcine model (approved by an ethics committee vote of the Austrian Federal Ministry of Education, Science and Research (GZ: 2021-0.895.386)). The estimated level of systemic perfusion is compared visually to invasively measured mean arterial pressure due to a lack of continuous cardiac output measurements as a ground truth. Results For exemplary data from a female pig (weight 50.4 kg), airway resistance was estimated to 20.8 hPa s /L, compliance: 25.2 mL/hPa, and dead space: 109 mL. Changes in cardiac output can be identified qualitatively as shown in Fig. 1. Conclusion A validated, simple ODE model for CO2 extraction during CPR could contribute quantifying the impact of ventilatory parameters on etCO2 and elucidate CO2 production, storage and transport during cardiac arrest. Assumptions on CO2 production during cardiac arrest and redistribution in the body, and the lacking cardiac output measurements complicate the analysis. Conflicts of Interest All authors have no conflict of interest to declare.

Economic optimization of Wolbachia-infected Aedes aegypti release to prevent dengue.

Dengue virus, primarily transmitted by the Aedes aegypti mosquito, is a major public health concern affecting ≈3.83 billion people worldwide. Recent releases of Wolbachia-transinfected Ae. aegypti in several cities worldwide have shown that it can reduce dengue transmission. However, these releases are costly, and, to date, no framework has been proposed for determining economically optimal release strategies that account for both costs associated with disease risk and releases. We present a flexible stochastic dynamic programming framework for determining optimal release schedules for Wolbachia-transinfected mosquitoes that balances the cost of dengue infection with the costs of rearing and releasing transinfected mosquitoes. Using an ordinary differential equation model of Wolbachia and dengue in a hypothetical city loosely describing areas at risk of new dengue epidemics, we determined that an all-or-nothing release strategy that quickly brings Wolbachia to fixation is often the optimal solution. Based on this, we examined the optimal facility size, finding that it was inelastic with respect to the mosquito population size, with a 100% increase in population size resulting in a 50-67% increase in optimal facility size. Furthermore, we found that these results are robust to mosquito life-history parameters and are mostly determined by the mosquito population size and the fitness costs associated with Wolbachia. These results reinforce that Wolbachia-transinfected mosquitoes can reduce the cost of dengue epidemics. Furthermore, they emphasize the importance of determining the size of the target population and fitness costs associated with Wolbachia before releases occur. © 2024 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Computational modelling of dynamic cAMP responses to GPCR agonists for exploration of GLP-1R ligand effects in pancreatic β-cells and neurons

The glucagon-like peptide-1 receptor (GLP-1R) is a class B G protein-coupled receptor (GPCR) which plays important physiological roles in insulin release and promoting fullness. GLP-1R agonists initiate cellular responses by cyclic AMP (cAMP) pathway signal transduction. Understanding of the potential of GLP-1R agonists in the treatment of type 2 diabetes may be advanced by considering the cAMP dynamics for agonists at GLP-1R in both pancreatic β-cells (important in insulin release) and neurons (important in appetite regulation). Receptor desensitisation in the cAMP pathway is known to be an important regulatory mechanism, with different ligands differentially promoting G protein activation and desensitisation. Here, we use mathematical modelling to quantify and understand experimentally obtained cAMP timecourses for two GLP-1R agonists, exendin-F1 (ExF1) and exendin-D3 (ExD3), which give markedly different signals in β-cells and neurons. We formulate an ordinary differential equation (ODE) model for the dynamics of cAMP signalling in response to G protein-coupled receptor (GPCR) ligands, encompassing ligand binding, receptor activation, G protein activation, desensitisation and second messenger generation. We validate our model initially by fitting to timecourse data for HEK293 cells, then proceed to parameterise the model for β-cells and neurons. Through numerical simulation and sensitivity studies, our analysis adds support to the hypothesis that ExF1 offers more potential glucose regulation benefit than ExD3 over long timescales via signalling in pancreatic β-cells, but that there is little difference between the two ligands in the potential appetite suppression effects offered via long-time signalling in neurons on the same timescales.

Physics-informed data-driven model of dehydration reaction stage in the sintering process of ternary cathode materials

The sintering process is the main process that affects the performance of ternary cathode materials. Due to the closed, continuous, and long-term characteristics of the sintering process carried out in a roller kiln, the reaction state including reaction degree, particle size, and etc., of the raw materials cannot be directly obtained through sampling detection, resulting in ineffective temperature control. This article mainly focuses on modeling the dehydration reaction stage of the sintering process to obtain the reaction state of the raw materials. Firstly, a two-sphere model is established based on Newton’s second law and the particle volume conservation equation to describe the relationship between the particle radius and sintering temperature of lithium hydroxide and ternary precursor; Secondly, the chemical kinetic parameters of the dehydration reaction in the two-sphere model are obtained by combining the single heating rate scanning method with the multiple heating rate scanning method; Then, in order to improve the accuracy of the two-sphere model and reduce the impact of external environmental temperature changes, a physics-informed neural network model is constructed by combining the two-sphere model with the neural ordinary differential equation model. Finally, the effectiveness of the modeling method is verified through numerical simulation and empirical data.

Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study

In this work, we rigorously investigate the robustness of graph neural fractional-order differential equation (FDE) models. This framework extends beyond traditional graph neural (integer-order) ordinary differential equation (ODE) models by implementing the time-fractional Caputo derivative. Utilizing fractional calculus allows our model to consider long-term memory during the feature updating process, diverging from the memoryless Markovian updates seen in traditional graph neural ODE models. The superiority of graph neural FDE models over graph neural ODE models has been established in environments free from attacks or perturbations. While traditional graph neural ODE models have been verified to possess a degree of stability and resilience in the presence of adversarial attacks in existing literature, the robustness of graph neural FDE models, especially under adversarial conditions, remains largely unexplored. This paper undertakes a detailed assessment of the robustness of graph neural FDE models. We establish a theoretical foundation outlining the robustness characteristics of graph neural FDE models, highlighting that they maintain more stringent output perturbation bounds in the face of input and graph topology disturbances, compared to their integer-order counterparts. Our empirical evaluations further confirm the enhanced robustness of graph neural FDE models, highlighting their potential in adversarially robust applications.

Operator-Learning-Inspired Modeling of Neural Ordinary Differential Equations

Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for various downstream tasks, e.g., image classification, time series classification, image generation, etc. Its key part is how to model the time-derivative of the hidden state, denoted dh(t)/dt. People have habitually used conventional neural network architectures, e.g., fully-connected layers followed by non-linear activations. In this paper, however, we present a neural operator-based method to define the time-derivative term. Neural operators were initially proposed to model the differential operator of partial differential equations (PDEs). Since the time-derivative of NODEs can be understood as a special type of the differential operator, our proposed method, called branched Fourier neural operator (BFNO), makes sense. In our experiments with general downstream tasks, our method significantly outperforms existing methods.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Mathematical Modeling of Intra-Communal Violence and Risk-Level Analysis. Case Study: Obiaruku Community in Delta State, Nigeria

This paper aims to capture the dynamics of intra-communal violence in a deterministic model of ordinary differential equations, accordingly, the Authors found some interesting results. Lack of quality education, insecurity, bad roads, drugs and alcoholism, unequal representation in government and religious decay have been identified as key factors supporting intra-communal violence over the years. In this research work we built all these factors into a deterministic model describing intra-communal violence and performed some basic mathematical analysis such as positivity of solutions, existence of invariant region, violence-free equilibrium, violence-persistent equilibrium, basic reproduction number, sensitivity analysis, stability analysis and bifurcation analysis. It was revealed that the violence-free equilibrium is globally asymptotically stable. The model exhibits a forward bifurcation. The sensitivity analysis revealed that injustice and insecurity are highly sensitive parameters of the basic reproduction number. We also designed a questionnaire to ascertain the violence risk level of Obiaruku community in Delta State, Nigeria and the analysis revealed that the community is at the medium high risk level and thus violence may occur in most cases in the community. The results of the stability analysis and the sensitivity analysis showed that under certain conditions, a community can be brought to the maximum low risk level and the maximum high peace level.

Properties and practicability of convergence-guaranteed optimization methods derived from weak discrete gradients

The ordinary differential equation (ODE) models of optimization methods allow for concise proofs of convergence rates through discussions based on Lyapunov functions. The weak discrete gradient (wDG) framework discretizes ODEs while preserving the properties of convergence, serving as a foundation for deriving optimization methods. Although various optimization methods have been derived through wDG, their properties and practical applicability remain underexplored. Hence, this study elucidates these aspects through numerical experiments. Particularly, although wDG yields several implicit methods, we highlight the potential utility of these methods in scenarios where the objective function incorporates a regularization term.

Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

Boundary Layer Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Carbon Nanotubes: Numerical Analysis and Response Surface Methodology under the Influence of Magnetohydrodynamics

The present study aims to offer new numerical solutions and optimisation strategies for the fluid flow and heat transfer behaviour at a stagnation point through a nonlinear sheet that is expanding or contracting in water-based hybrid nanofluids. Most hybrid nanofluids typically use metallic nanoparticles. However, we deliver a new approach by combining single- and multi-walled carbon nanotubes (SWCNTs-MWCNTs). The flow is presumptively steady, laminar, and surrounded by a constant temperature of the ambient and body walls. By using similarity variables, a model of partial differential equations (PDEs) with the magnetohydrodynamics (MHD) effect on the momentum equation is converted into a model of non-dimensional ordinary differential equations (ODEs). Then, the dimensionless first-order ODEs are solved numerically using the MATLAB R2022b bvp4C program. In order to explore the range of computational solutions and physical quantities, several dimensionless variables are manipulated, including the magnetic parameter, the stretching/shrinking parameter, and the volume fraction parameters of hybrid and mono carbon nanotubes. To enhance the originality and effectiveness of this study for practical applications, we optimise the heat transfer coefficient via the response surface methodology (RSM). We apply a face-centred central composite design (CCF) and perform the CCF using Minitab. All of our findings are presented and illustrated in tabular and graphic form. We have made notable contributions in the disciplines of mathematical analysis and fluid dynamics. From our observations, we find that multiple solutions appear when the magnetic parameter is less than 1. We also detect double solutions in the shrinking region. Furthermore, the increase in the magnetic parameter and SWCNTs-MWCNTs volume fraction parameter increases both the skin friction coefficient and the local Nusselt number. To compare the performance of hybrid nanofluids and mono nanofluids, we note that hybrid nanofluids work better than single nanofluids both in skin friction and heat transfer coefficients.

A hybrid neural ordinary differential equation model of the cardiovascular system.

In the human cardiovascular system (CVS), the interaction between the left and right ventricles of the heart is influenced by the septum and the pericardium. Computational models of the CVS can capture this interaction, but this often involves approximating solutions to complex nonlinear equations numerically. As a result, numerous models have been proposed, where these nonlinear equations are either simplified, or ventricular interaction is ignored. In this work, we propose an alternative approach to modelling ventricular interaction, using a hybrid neural ordinary differential equation (ODE) structure. First, a lumped parameter ODE model of the CVS (including a Newton-Raphson procedure as the numerical solver) is simulated to generate synthetic time-series data. Next, a hybrid neural ODE based on the same model is constructed, where ventricular interaction is instead set to be governed by a neural network. We use a short range of the synthetic data (with various amounts of added measurement noise) to train the hybrid neural ODE model. Symbolic regression is used to convert the neural network into analytic expressions, resulting in a partially learned mechanistic model. This approach was able to recover parsimonious functions with good predictive capabilities and was robust to measurement noise.

Understanding the impact of numerical solvers on inference for differential equation models.

Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers that seem sufficiently accurate for the forward problem, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the inverse problem, i.e. for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which might become jagged, causing inference algorithms to get stuck in local 'phantom' optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyse an ODE changepoint model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfallrun-off model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that, when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.