On Ordinary Differential Equations Research Articles

GraFITi: Graphs for Forecasting Irregularly Sampled Time Series

Forecasting irregularly sampled time series with missing values is a crucial task for numerous real-world applications such as healthcare, astronomy, and climate sciences. State-of-the-art approaches to this problem rely on Ordinary Differential Equations (ODEs) which are known to be slow and often require additional features to handle missing values. To address this issue, we propose a novel model using Graphs for Forecasting Irregularly Sampled Time Series with missing values which we call GraFITi. GraFITi first converts the time series to a Sparsity Structure Graph which is a sparse bipartite graph, and then reformulates the forecasting problem as the edge weight prediction task in the graph. It uses the power of Graph Neural Networks to learn the graph and predict the target edge weights. GraFITi has been tested on 3 real-world and 1 synthetic irregularly sampled time series dataset with missing values and compared with various state-of-the-art models. The experimental results demonstrate that GraFITi improves the forecasting accuracy by up to 17% and reduces the run time up to 5 times compared to the state-of-the-art forecasting models.

Enhancing Robustness of Medical Image Segmentation Model with Neural Memory Ordinary Differential Equation.

Deep neural networks (DNNs) have emerged as a prominent model in medical image segmentation, achieving remarkable advancements in clinical practice. Despite the promising results reported in the literature, the effectiveness of DNNs necessitates substantial quantities of high-quality annotated training data. During experiments, we observe a significant decline in the performance of DNNs on the test set when there exists disruption in the labels of the training dataset, revealing inherent limitations in the robustness of DNNs. In this paper, we find that the neural memory ordinary differential equation (nmODE), a recently proposed model based on ordinary differential equations (ODEs), not only addresses the robustness limitation but also enhances performance when trained by the clean training dataset. However, it is acknowledged that the ODE-based model tends to be less computationally efficient compared to the conventional discrete models due to the multiple function evaluations required by the ODE solver. Recognizing the efficiency limitation of the ODE-based model, we propose a novel approach called the nmODE-based knowledge distillation (nmODE-KD). The proposed method aims to transfer knowledge from the continuous nmODE to a discrete layer, simultaneously enhancing the model's robustness and efficiency. The core concept of nmODE-KD revolves around enforcing the discrete layer to mimic the continuous nmODE by minimizing the KL divergence between them. Experimental results on 18 organs-at-risk segmentation tasks demonstrate that nmODE-KD exhibits improved robustness compared to ODE-based models while also mitigating the efficiency limitation.

A Recursive tree-structured neural network with goal forgetting and information aggregation for solving math word problems

Most existing state-of-the-art neural network models for math word problems use the Goal-driven Tree-Structured decoder (GTS) to generate expression trees. However, we found that GTS does not provide good predictions for longer expressions, mainly because it does not capture the relationships among the goal vectors of each node in the expression tree and ignores the position order of the nodes before and after the operator. In this paper, we propose a novel Recursive tree-structured neural network with Goal Forgetting and information aggregation (RGFNet) to address these limits. The goal forgetting and information aggregation module is based on ordinary differential equations (ODEs) and we use it to build a sub-goal information feedback neural network (SGIFNet). Unlike GTS, which uses two-layer gated-feedforward networks to generate goal vectors, we introduce a novel sub-goal generation module. The sub-goal generation module could capture the relationship among the related nodes (e.g. parent nodes, sibling nodes) using attention mechanism. Experimental results on two large public datasets i.e. Math23K and Ape-clean show that our tree-structured model outperforms the state-of-the-art models and obtains answer accuracy over 86%. Furthermore, the performance on long-expression problems is promising.11The source code of this paper is available at: https://github.com/SCNU203/RGFNet.

Modeling antibody dynamics following herpes zoster indicates that higher varicella-zoster virus viremia generates more VZV-specific antibodies.

Studying antibody dynamics following re-exposure to infection and/or vaccination is crucial for a better understanding of fundamental immunological processes, vaccine development, and health policy research. We adopted a nonlinear mixed modeling approach based on ordinary differential equations (ODE) to characterize varicella-zoster virus specific antibody dynamics during and after clinical herpes zoster. Our ODEs models convert underlying immunological processes into mathematical formulations, allowing for testable data analysis. In order to cope with inter- and intra-individual variability, mixed models include population-averaged parameters (fixed effects) and individual-specific parameters (random effects). We explored the use of various ODE-based nonlinear mixed models to describe longitudinally collected markers of immunological response in 61 herpes zoster patients. Starting from a general formulation of such models, we study different plausible processes underlying observed antibody titer concentrations over time, including various individual-specific parameters. Among the converged models, the best fitting and most parsimonious model implies that once Varicella-zoster virus (VZV) reactivation is clinically apparent (i.e., Herpes-zoster (HZ) can be diagnosed), short-living and long-living antibody secreting cells (SASC and LASC, respectively) will not expand anymore. Additionally, we investigated the relationship between age and viral load on SASC using a covariate model to gain a deeper understanding of the population's characteristics. The results of this study provide crucial and unique insights that can aid in improving our understanding of VZV antibody dynamics and in making more accurate projections regarding the potential impact of vaccines.

Integration of a System Dynamics Model and 3D Tree Rendering—VISmaF Part II: Model Development, Results and Potential Agronomic Applications

Biological–mathematical models of trees can be exploited for a wide range of agronomic applications including crop management, visualization of ecosystem changes over time, in-field phenotyping, crop load effects, testing of plant functions, biomechanics, and many others. Some models propose a 3D output of tree that, in addition to having functionality to visualize the result, offers an additional tool for the evaluation of some parameters of the model itself (interception and amount of light, temperature, obstacles, physical competition between multiple trees). The present study introduces a biological–mathematical model of tree growth with a 3D output of its structure in a realtime 3D rendering environment (Unity©). Thanks to the virtual environment created in Unity©, it was possible to obtain variable environmental parameters (amount of light, temperature) used as inputs to the mathematical simulation of growth. The model is based on ordinary differential equations (ODEs) that compute the growth of each single internode in length (primary growth) and width (secondary growth) and the accumulation of growth inhibitors regulating the seasonal cyclicity of the tree. Virtual experiments were conducted varying environmental conditions (amount of light and temperature), and the species-specific characteristics of the simulated tree (number of buds, branching angle). The results have been analyzed showing also how the model can be adapted for the creation of different tree species and discussing the potential agronomic applications of model.

Construction and management of smart campus: Anti-disturbance control of flexible manipulator based on PDE modeling.

With the rapid development of smart campus, this paper studies the attitude tracking control of flexible manipulator (FM) in colleges and universities under elastic vibration and external disturbances. First, different from the traditional modeling based on ordinary differential equations (ODEs), the partial differential equations (PDEs) dynamic model of a manipulator system is established based on the Hamilton principle (HP). Second, the boundary control condition of the end system of the manipulator is introduced to adjust the vibration of the manipulator. Furthermore, a Proportional-Derivative (PD) boundary control (PDBC) strategy is proposed by the Lyapunov function to suppress the vibration of the manipulator. Finally, a numerical comparison simulation based on MATLAB/SIMULINK further verifies the robustness and anti-disturbance performance of the control method proposed in this paper.

A mathematical modelling portrait of Wnt signalling in early vertebrate embryogenesis

There are two phases of Wnt signalling in early vertebrate embryogenesis: very early, maternal Wnt signalling promotes dorsal development, and slightly later, zygotic Wnt signalling promotes ventral and lateral mesoderm induction. However, recent molecular biology analysis has revealed more complexity among the direct Wnt target genes, with at least five classes. Here in order to test the logic and the dynamics of a new Gene Regulatory Network model suggested by these discoveries we use mathematical modelling based on ordinary differential equations (ODEs). Our mathematical modelling of this Gene Regulatory Network reveals that a simplified model, with one “super-gene” for each class is sufficient to a large extent to describe the regulatory behaviour previously observed experimentally.

CAM Models: Lessons and Implications for CAM Evolution.

The evolution of Crassulacean acid metabolism (CAM) by plants has been one of the most successful strategies in response to aridity. On the onset of climate change, expanding the use of water efficient crops and engineering higher water use efficiency into C3 and C4 crops constitute a plausible solution for the problems of agriculture in hotter and drier environments. A firm understanding of CAM is thus crucial for the development of agricultural responses to climate change. Computational models on CAM can contribute significantly to this understanding. Two types of models have been used so far. Early CAM models based on ordinary differential equations (ODE) reproduced the typical diel CAM features with a minimal set of components and investigated endogenous day/night rhythmicity. This line of research brought to light the preponderant role of vacuolar malate accumulation in diel rhythms. A second wave of CAM models used flux balance analysis (FBA) to better understand the role of CO2 uptake in flux distribution. They showed that flux distributions resembling CAM metabolism emerge upon constraining CO2 uptake by the system. We discuss the evolutionary implications of this and also how CAM components from unrelated pathways could have integrated along evolution.

Mathematical Modeling of Three - Dimensional Genetic Regulatory Networks Using Logistic and Gompertz Functions

Mathematical modeling is a method of cognition of the surrounding world in which the description of the object is carried out in the language of mathematics, and the study of the model is performed using certain mathematical methods. Mathematical models based on ordinary differential equations (ODE) are used in the study of networks of different kinds, including the study of genetic regulatory networks (GRN). The use of ODE makes it possible to predict the evolution of GRN in time. Nonlinearity in these models is included in the form of a sigmoidal function. There are many of them, and in the literature, there are models that use different sigmoidal functions. The article discusses the models that use the logistic function and Gompertz function. The comparison of the results, related to three-dimensional networks, has been made. The text is accompanied by examples and illustrations.

Petri nets and ODEs as complementary methods for comprehensive analysis on an example of the ATM\u2013p53\u2013NF-kappaB signaling pathways

Intracellular processes are cascades of biochemical reactions, triggered in response to various types of stimuli. Mathematical models describing their dynamics have become increasingly popular in recent years, as tools supporting experimental work in analysis of pathways and regulatory networks. Not only do they provide insights into general properties of these systems, but also help in specific tasks, such as search for drug molecular targets or treatment protocols. Different tools and methods are used to model complex biological systems. In this work, we focus on ordinary differential equations (ODEs) and Petri nets. We consider specific methods of analysis of such models, i.e., sensitivity analysis (SA) and significance analysis. So far, they have been applied separately, with different goals. In this paper, we show that they can complement each other, combining the sensitivity of ODE models and the significance analysis of Petri nets. The former is used to find parameters, whose change results in the greatest quantitative and qualitative changes in the model response, while the latter is a structural analysis and allows indicating the most important subprocesses in terms of information flow in Petri net. Ultimately, both methods facilitate finding the essential processes in a given signaling pathway or regulatory network and may be used to support medical therapy development. In the paper, the use of dual modeling is illustrated with an example of ATM/p53/NF-kappaB pathway. Each method was applied to analyze this system, resulting in finding different subsets of important processes that might be prospective targets for changing this system behavior. While some of the processes were indicated in each of the approaches, others were found by one method only and would be missed if only that method was applied. This leads to the conclusion about the complementarity of the methods under investigation. The dual modeling approach of comprehensive structural and parametric analysis yields results that would not be possible if these two modeling approaches were applied separately. The combined approach, proposed in this paper, facilitates finding not only key processes, with which significant parameters are associated, but also significant modules, corresponding to subsystems of regulatory networks. The results provide broader insight into therapy targets in diseases in which the natural control of intracellular processes is disturbed, leading to the development of more effective therapies in medicine.

Simple epidemic models with segmentation can be better than complex ones.

Given a sequence of epidemic events, can a single epidemic model capture its dynamics during the entire period? How should we divide the sequence into segments to better capture the dynamics? Throughout human history, infectious diseases (e.g., the Black Death and COVID-19) have been serious threats. Consequently, understanding and forecasting the evolving patterns of epidemic events are critical for prevention and decision making. To this end, epidemic models based on ordinary differential equations (ODEs), which effectively describe dynamic systems in many fields, have been employed. However, a single epidemic model is not enough to capture long-term dynamics of epidemic events especially when the dynamics heavily depend on external factors (e.g., lockdown and the capability to perform tests). In this work, we demonstrate that properly dividing the event sequence regarding COVID-19 (specifically, the numbers of active cases, recoveries, and deaths) into multiple segments and fitting a simple epidemic model to each segment leads to a better fit with fewer parameters than fitting a complex model to the entire sequence. Moreover, we propose a methodology for balancing the number of segments and the complexity of epidemic models, based on the Minimum Description Length principle. Our methodology is (a) Automatic: not requiring any user-defined parameters, (b) Model-agnostic: applicable to any ODE-based epidemic models, and (c) Effective: effectively describing and forecasting the spread of COVID-19 in 70 countries.

A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics.

In the context of 2019 coronavirus disease (COVID-19), considerable attention has been paid to mathematical models for predicting country- or region-specific future pandemic developments. In this work, we developed an SVICDR model that includes a susceptible, an all-or-nothing vaccinated, an infected, an intensive care, a deceased, and a recovered compartment. It is based on the susceptible-infectious-recovered (SIR) model of Kermack and McKendrick, which is based on ordinary differential equations (ODEs). The main objective is to show the impact of parameter boundary modifications on the predicted incidence rate, taking into account recent data on Germany in the pandemic, an exponential increasing vaccination rate in the considered time window and trigonometric contact and quarantine rate functions. For the numerical solution of the ODE systems a model-specific non-standard finite difference (NSFD) scheme is designed, that preserves the positivity of solutions and yields the correct asymptotic behaviour.

Conservation water environment of a lagoon system by controlling manure distribution

The Ebrie lagoon system comprises the lagoons of Ebrie, Potou, and Adjin. In the villages that are around the edges of this lagoon system, there are several farms, such as pigsty, poultry and enclosures. Thus, the excessive use of manure on farmland creates a serious environmental problem. The excessive manure discharge that crops need as fertilizer leads to excess manure in these fields. As Ivory Coast is a country where it rains a lot, rainwater drains these excessive quantities from the surface into the Ebrie, Potou, and Adjin lagoons, where they pollute the water, feed the algae, and harm the fish. Our first aim is to develop an optimization method based on ordinary differential equations (ODE) to determine the right field at the right time for environmental damage to be limited. We collect current data in a region of topologically connected lagoons and their adjacent farmlands. To numerically solve this mathematical problem, the software used is Matlab. Using real data, this solution improves the quality of the water by drastically reducing in a continuous-time, the quantity of manure and its principal components such as nitrogen, phosphorus, and potassium.

A general ODE-based model to describe the physiological age structure of ectotherms: Description and application to Drosophila suzukii

This paper introduces a novel general model based on Ordinary Differential Equations (ODEs) which is able to describe the population dynamics of a large class of insect pests. The proposed model is a physiologically-inspired generalization of a number of ad hoc models presented through the years in the literature. Its main feature is that it allows the systematic generation of a population model for a species by simply defining its key features, namely the sex ratio and the development, fertility, and mortality rates. The first part of the paper provides a detailed description of the model and shows that most ODE-based models existing in literature can be obtained as a special case of the proposed model. The second part of the paper shows an application of the model to the spotted wing drosophila Drosophila suzukii, which is a highly relevant pest in agriculture. The biological features of this species, i.e., the sex ratio and the various rates, were retrieved from the existing scientific literature. The obtained model was validated using data from a three-year survey conducted in two experimental fields. Results show that the model describes faithfully the experimental populations, although the simulations were performed completely in open-loop and without any adaptation of the parameters extracted from the existing literature.

Scaling effect in COVID-19 spreading: The role of heterogeneity in a hybrid ODE-network model with restrictions on the inter-cities flow

The new Covid-19 pandemic has left traces of suffering and devastation to individuals of almost all countries worldwide and severe impact on the global economy. Understanding the clinical characteristics, interactions with the environment, and the variables that favor or hinder its dissemination help the public authorities in the fight and prevention, leading for a rapid response in society. Using models to estimate contamination scenarios in real time plays an important role. Population compartments models based on ordinary differential equations (ODE) for a given region assume two homogeneous premises, the contact mechanisms and diffusion rates, disregarding heterogeneous factors as different contact rates for each municipality and the flow of contaminated people among them. This work considers a hybrid model for covid-19, based on local SIR models and the population flow network among municipalities, responsible for a complex lag dynamic in their contagion curves. Based on actual infection data, local contact rates (β) are evaluated. The epidemic evolution at each municipality depends on the local SIR parameters and on the inter-municipality transport flow. When heterogeneity of β values and flow network are included, forecasts differ from those of the homogeneous ODE model. This effect is more relevant when more municipalities are considered, hinting that the latter overestimates new cases. In addition, mitigation scenarios are assessed to evaluate the effect of earlier interventions reducing the inter-municipality flux. Restricting the flow between municipalities in the initial stage of the epidemic is fundamental for flattening the contamination curve, highlighting advantages of a contamination lag between the capital curve and those of other municipalities in the territories.

Software Supporting a Workflow of Quantitative Dynamic Flux Maps Estimation in Central Metabolism from SIRM Experimental Data.

Stable isotope-resolved metabolomics (SIRM), based on the analysis of biological samples from living cells incubated with artificial isotope enriched substrates, enables mapping the rates of biochemical reactions (metabolic fluxes). We developed software supporting a workflow of analysis of SIRM data obtained with mass spectrometry (MS). The evaluation of fluxes starting from raw MS recordings requires at least three steps of computer support: first, extraction of mass spectra of metabolites of interest, then correction of the spectra for natural isotope abundance, and finally, evaluation of fluxes by simulation of the corrected spectra using a corresponding mathematical model. A kinetic model based on ordinary differential equations (ODEs) for isotopomers of metabolites of the corresponding biochemical network supports the final part of the analysis, which provides a dynamic flux map.

Pleione: A tool for statistical and multi-objective calibration of Rule-based models

Mathematical models based on Ordinary Differential Equations (ODEs) are frequently used to describe and simulate biological systems. Nevertheless, such models are often difficult to understand. Unlike ODE models, Rule-Based Models (RBMs) utilise formal language to describe reactions as a cumulative number of statements that are easier to understand and correct. They are also gaining popularity because of their conciseness and simulation flexibility. However, RBMs generally lack tools to perform further analysis that requires simulation. This situation arises because exact and approximate simulations are computationally intensive. Translating RBMs into ODEs is commonly used to reduce simulation time, but this technique may be prohibitive due to combinatorial explosion. Here, we present the software called Pleione to calibrate RBMs. Parameter calibration is essential given the incomplete experimental determination of reaction rates and the goal of using models to reproduce experimental data. The software distributes stochastic simulations and calculations and incorporates equivalence tests to determine the fitness of RBMs compared with data. The primary features of Pleione were thoroughly tested on a model of gene regulation in Escherichia coli. Pleione yielded satisfactory results regarding calculation time and error reduction for multiple simulators, models, parameter search strategies, and computing infrastructures.

Parameter estimation and mathematical modeling for the quantitative description of therapy failure due to drug resistance in gastrointestinal stromal tumor metastasis to the liver.

In this work we develop a general mathematical model and devise a practical identifiability approach for gastrointestinal stromal tumor (GIST) metastasis to the liver, with the aim of quantitatively describing therapy failure due to drug resistance. To this end, we have modeled metastatic growth and therapy failure produced by resistance to two standard treatments based on tyrosine kinase inhibitors (Imatinib and Sunitinib) that have been observed clinically in patients with GIST metastasis to the liver. The parameter identification problem is difficult to solve, since there are no general results on this issue for models based on ordinary differential equations (ODE) like the ones studied here. We propose a general modeling framework based on ODE for GIST metastatic growth and therapy failure due to drug resistance and analyzed five different model variants, using medical image observations (CT scans) from patients that exhibit drug resistance. The associated parameter estimation problem was solved using the Nelder-Mead simplex algorithm, by adding a regularization term to the objective function to address model instability, and assessing the agreement of either an absolute or proportional error in the objective function. We compared the goodness of fit to data for the proposed model variants, as well as evaluated both error forms in order to improve parameter estimation results. From the model variants analyzed, we identified the one that provides the best fit to all the available patient data sets, as well as the best assumption in computing the objective function (absolute or proportional error). This is the first work that reports mathematical models capable of capturing and quantitatively describing therapy failure due to drug resistance based on clinical images in a patient-specific manner.

Modeling Oil Spills in the Sea

A model of the spreading of an oil spill based on ordinary differential equations (ODE) of the balance of forces acting on an axially symmetric slick is proposed. The proposed model differs from similar models due to the special choice of coefficients of enforcement members in the ODE. These coefficients are chosen according to the following condition: the Fay relation is obtained from the main equation of the model by equating force pairs at each spreading stage; at each stage, the prevailing (according to the Fay approach) force pair is selected for it. An advantage of the proposed model is the more consistent and natural passage from the finite Fay relations containing constants determined by the experimental data to the more general ODE describing the spreading. Additionally, the model considers the main weathering processes, using the method of pseudocomponents and the general posing of the problem in the form of the Cauchy problem for a set of differential equations and additional relationships. The simulation of the transformation of an oil spill for various parameters and a comparison with observations are implemented.

Accounting for Space—Quantification of Cell-To-Cell Transmission Kinetics Using Virus Dynamics Models.

Mathematical models based on ordinary differential equations (ODE) that describe the population dynamics of viruses and infected cells have been an essential tool to characterize and quantify viral infection dynamics. Although an important aspect of viral infection is the dynamics of viral spread, which includes transmission by cell-free virions and direct cell-to-cell transmission, models used so far ignored cell-to-cell transmission completely, or accounted for this process by simple mass-action kinetics between infected and uninfected cells. In this study, we show that the simple mass-action approach falls short when describing viral spread in a spatially-defined environment. Using simulated data, we present a model extension that allows correct quantification of cell-to-cell transmission dynamics within a monolayer of cells. By considering the decreasing proportion of cells that can contribute to cell-to-cell spread with progressing infection, our extension accounts for the transmission dynamics on a single cell level while still remaining applicable to standard population-based experimental measurements. While the ability to infer the proportion of cells infected by either of the transmission modes depends on the viral diffusion rate, the improved estimates obtained using our novel approach emphasize the need to correctly account for spatial aspects when analyzing viral spread.