Ordinary Differential Equation Model Research Articles

The Mathematical Model Comparison Between Regular Immunogenic Tumor and Chronic Myelogenous Leukemia (CML)

T cell plays important role in lymphocytic leukemia. And the understanding of T cells would help people to modify the growth of leukemia and administrate the dosage to patients. This paper discussed the use of two different models of ordinary differential equations to analyze leukemia. Although the models may have some differences, both of them focused on analyzing the model to calculate the equilibrium state of the tumor in its absence and presence. Regular Immunogenic tumors use a more basic mathematical model, while the chronic myelogenous leukemia (CML) model adds more complex deformations to it. This study compares the steady states of the models proposed by Kuznetsov et al. and Moore and Li, and incorporates the parameter values used in Moore and Li to compare the disparity between them. The research concluded that there were no significant differences in health trials but there were substantial disparities in disease trials between two models

Catching a wave: On the suitability of traveling-wave solutions in epidemiological modeling.

Ordinary differential equation models such as the classical SIR model are widely used in epidemiology to study and predict infectious disease dynamics. However, these models typically assume that populations are homogeneously mixed, ignoring possible variations in disease prevalence due to spatial heterogeneity. To address this issue, reaction-diffusion models have been proposed as an alternative approach to modeling spatially continuous populations in which individuals move in a diffusive manner. In this study, we explore the conditions under which such spatial structure must be explicitly considered to accurately predict disease spread, and when the assumption of homogeneous mixing remains adequate. In particular, we derive a critical threshold for the diffusion coefficient below which disease transmission dynamics exhibit spatial heterogeneity. We validate our analytical results with individual-based simulations of disease transmission across a two-dimensional continuous landscape. Using this framework, we further explore how key epidemiological parameters such as the probability of disease establishment, its maximum incidence, and its final epidemic size are affected by incorporating spatial structure into SI, SIS, and SIR models. We discuss the implications of our findings for epidemiological modeling and identify design considerations and limitations for spatial simulation models of disease dynamics.

Stochastic Models of Zoonotic Avian Influenza with Multiple Hosts, Environmental Transmission, and Migration in the Natural Reservoir.

Avian influenza virus type A causes an infectious disease that circulates among wild bird populations and regularly spills over into domesticated animals, such as poultry and swine. As the virus replicates in these intermediate hosts, mutations occur, increasing the likelihood of emergence of a new variant with greater transmission to humans and a potential threat to public health. Prior models for spread of avian influenza have included some combinations of the following components: multi-host populations, spillover into humans, environmental transmission, seasonality, and migration. We develop an ordinary differential equation (ODE) model for spread of a low pathogenic avian influenza virus that combines all of these factors, and we translate this into a stochastic continuous-time Markov chain model. Linearization of the ODE near the disease-free solution leads to the basic reproduction number , a threshold for disease extinction in both the ODE and Markov chain. The linearized Markov chain leads to a branching process approximation which provides an estimate for probability of disease extinction, i.e., probability no major disease outbreak in the multi-host system. The probability of disease extinction depends on the time and the population into which infection is introduced and reflects the seasonality inherent in the system. Some of the most sensitive parameters to model outcomes include wild bird recovery and environmental transmission. We find that migratory wild birds can drive infection numbers in other populations even when transmission parameters for those populations are low, and that environmental transmission can be a significant driver of infections.

Developing, Characterizing, and Modeling CRISPR-Based Point-of-Use Pathogen Diagnostics.

Recent years have seen intense interest in the development of point-of-care nucleic acid diagnostic technologies to address the scaling limitations of laboratory-based approaches. Chief among these are combinations of isothermal amplification approaches with CRISPR-based detection and readouts of target products. Here, we contribute to the growing body of rapid, programmable point-of-care pathogen tests by developing and optimizing a one-pot NASBA-Cas13a nucleic acid detection assay. This test uses the isothermal amplification technique NASBA to amplify target viral nucleic acids, followed by the Cas13a-based detection of amplified sequences. We first demonstrate an in-house formulation of NASBA that enables the optimization of individual NASBA components. We then present design rules for NASBA primer sets and LbuCas13a guide RNAs for the fast and sensitive detection of SARS-CoV-2 viral RNA fragments, resulting in 20-200 aM sensitivity. Finally, we explore the combination of high-throughput assay condition screening with mechanistic ordinary differential equation modeling of the reaction scheme to gain a deeper understanding of the NASBA-Cas13a system. This work presents a framework for developing a mechanistic understanding of reaction performance and optimization that uses both experiments and modeling, which we anticipate will be useful in developing future nucleic acid detection technologies.

Model Selection for Ordinary Differential Equations: A Statistical Testing Approach.

Ordinary differential equations(ODEs) are foundational tools in modeling intricate dynamics across a gamut of scientific disciplines. Yet, a possibility to represent a single phenomenon through multiple ODE models, driven by different understandings of nuances in internal mechanisms or abstraction levels, presents a model selection challenge. This study introduces a testing-based approach for ODE model selection amidst statistical noise. Rooted in the model misspecification framework, we adapt classical statistical paradigms (Vuong and Hotelling) to the ODE context, allowing for the comparison and ranking of diverse causal explanations without the constraints of nested models. Our simulation studies numerically investigate the statistical properties of the test, demonstrating its attainment of the nominal size and power across various settings. Real-world data examples further underscore the algorithm's applicability in practice. To foster accessibility and encourage real-world applications, we provide a user-friendly Python implementation of our model selection algorithm, bridging theoretical advancements with hands-on tools for the scientificcommunity.

Parameter Estimation and Identifiability in Kinetic Flux Profiling Models of Metabolism

Metabolic fluxes are the rates of life-sustaining chemical reactions within a cell and metabolites are the components. Determining the changes in these fluxes is crucial to understanding diseases with metabolic causes and consequences. Kinetic flux profiling (KFP) is a method for estimating flux that utilizes data from isotope tracing experiments. In these experiments, the isotope-labeled nutrient is metabolized through a pathway and integrated into the downstream metabolite pools. Measurements of proportion labeled for each metabolite in the pathway are taken at multiple time points and used to fit an ordinary differential equations model with fluxes as parameters. We begin by generalizing the process of converting diagrams of metabolic pathways into mathematical models composed of differential equations and algebraic constraints. The scaled differential equations for proportions of unlabeled metabolite contain parameters related to the metabolic fluxes in the pathway. We investigate flux parameter identifiability given data collected only at the steady state of the differential equation. Next, we give criteria for valid parameter estimations in the case of a large separation of timescales with fast–slow analysis. Bayesian parameter estimation on simulated data from KFP experiments containing both irreversible and reversible reactions illustrates the accuracy and reliability of flux estimations. These analyses provide constraints that serve as guidelines for the design of KFP experiments to estimate metabolic fluxes.

Dynamic modelling of signalling pathways when ordinary differential equations are not feasible.

Mathematical modelling plays a crucial role in understanding inter- and intracellular signalling processes. Currently, ordinary differential equations (ODEs) are the predominant approach in systems biology for modelling such pathways. While ODE models offer mechanistic interpretability, they also suffer from limitations, including the need to consider all relevant compounds, resulting in large models difficult to handle numerically and requiring extensive data. In previous work, we introduced the retarded transient function (RTF) as an alternative method for modelling temporal responses of signalling pathways. Here, we extend the RTF approach to integrate concentration or dose-dependencies into the modelling of dynamics. With this advancement, RTF modelling now fully encompasses the application range of ODE models, which comprises predictions in both time and concentration domains. Moreover, characterizing dose-dependencies provides an intuitive way to investigate and characterize signalling differences between biological conditions or cell types based on their response to stimulating inputs. To demonstrate the applicability of our extended approach, we employ data from time- and dose-dependent inflammasome activation in bone marrow-derived macrophages treated with nigericin sodium salt. Our results show the effectiveness of the extended RTF approach as a generic framework for modelling dose-dependent kinetics in cellular signalling. The approach results in intuitively interpretable parameters that describe signal dynamics and enables predictive modelling of time- and dose-dependencies even if only individual cellular components are quantified. The presented approach is available within the MATLAB-based Data2Dynamics modelling toolbox at https://github.com/Data2Dynamics and https://zenodo.org/records/14008247 and as R code at https://github.com/kreutz-lab/RTF.

Parameter estimation for allometric trophic network models: A variational Bayesian inverse problem approach

Abstract Differential equation models are powerful tools for predicting biological systems, capable of projecting far into the future and incorporating data recorded at arbitrary times. However, estimating these models' parameters from observations can be challenging because numerical methods are often required to approximate their solution. An example of such a model is the allometric trophic network model, for which studies considering its inverse problem are limited, particularly in the Bayesian framework. Here we develop a variational Bayesian method for parameter inference of the allometric trophic network model and explore how accurately we can recover its parameter values. We represent the model as a Bayesian neural network, which combines an artificial neural network with Bayesian inference, using a surrogate for the posterior distribution of model parameters, and train this model by evolutionary optimization to avoid potentially costly computation of the gradient with respect to the model parameters. Using synthetic data, we compare the accuracy of this variational inference to ordinary least squares estimation. To reduce the number of estimated parameters, we focus on the inference of functional response parameters. Our variational Bayesian method yields parameter estimates that are comparable to the ordinary least squares results in terms of accuracy. The method provides a promising approach for including uncertainty quantification in parameter estimation, which the simple ordinary least squares approach as it is does not address. Regardless of the method, potential multimodality of the inference problem is nonetheless important to keep in mind. The present study suggests a technique for parameter inference of ordinary differential equation models in the Bayesian context. We propose the method especially for validation of the allometric trophic network model against empirical data.

Opinion response functions are key to understanding the tipping of social conventions

Abstract The extent to which committed minorities can overturn social conventions is an active area of research in the mathematical modeling of opinion dynamics. Researchers generally use simulations of agent-based models (ABMs) to compute approximate values for the minimum committed minority size needed to overturn a social convention. In this manuscript, we expand on previous work by studying an ABM’s mean-field behavior using ordinary differential equation models and a new tool, namely opinion response functions (ORFs). Using this method allows for formal analysis of the deterministic model, which can provide a theoretical explanation for observed behaviors, e.g. coexistence or overturning of opinions. In particular, ORFs are a method for characterizing equilibria in our social model. Our analysis confirms earlier numerical results and supplements them with a precise formula for computing the minimum committed minority size required to overturn a social convention.

Spatio-temporal envolutional graph neural network for traffic flow prediction in UAV-based urban traffic monitoring system

With the advancement of modern UAV technology, UAVs have become integral to creating traffic management monitoring systems. Additionally, UAV-based traffic monitoring systems can predict traffic flow by integrating machine learning methods. Specifically, traffic flow data contains both spatial and temporal information, which can be effectively processed by graph neural networks (GNNs). However, GNNs often face the challenge of oversmoothing, which hinders their ability to capture complex structures in the data. The Spatio-Temporal Graph Ordinary Differential Equations (STGODE) model addresses this issue by introducing Neural Ordinary Differential Equations (NODEs) to construct deeper GNNs. Despite this, STGODE relies on initially predefined semantic neighborhood matrices, which do not adapt well to the dynamic nature of traffic information. To overcome this limitation, we propose an evolutionary graph neural network for traffic prediction, capable of continuously updating the semantic adjacency matrix throughout the training process. This dynamic evolution of the semantic adjacency matrix allows it to adapt to the features and semantic relations of the current data, enhancing its ability to capture the complexity and variability of traffic patterns. We validate our approach through experiments on several real-world datasets, demonstrating that our method outperforms state-of-the-art benchmarks.

Leveraging mathematical modeling framework to guide regimen strategy for phage therapy

Bacteriophage (phage) cocktail therapy has been relied upon more and more to treat antibiotic-resistant infections. Understanding of the complex kinetics between phages, target bacteria, and the emergence of phage resistance remain hurdles to successful clinical outcomes. Building upon previous mathematical concepts, we develop biologically-motivated nonlinear ordinary differential equation models to explore single, cocktail, and sequential phage treatment modalities. While the optimal pairwise phage treatment strategy was the double simultaneous administration of two highly potent and asymmetrically binding phage strains, it appears unable to prevent the evolution of resistance. This treatment regimen did have a greater lysis efficiency, promoted higher phage population sizes, reduced bacterial density the most, and suppressed the evolution of resistance the longest compared to all other treatments strategies tested. Conversely, the combination of phages with polar potencies allows the more efficiently replicating phages to monopolize susceptible host cells, thereby quickly negating the intended compounding effect of cocktails. Together, we demonstrate that a biologically-motivated modeling-based framework can be leveraged to quantify the effects of each phage’s properties to more precisely predict treatment responses.

Influence of Human Hunting Strategies and Large Carnivore Presence on Population Dynamics of European Facultative Scavengers.

Ungulates serve as the primary carrion source for facultative scavengers in European ecosystems. In the absence of large carnivores, such as wolves (Canis lupus), human hunting leftovers are the main source of carrion for these scavengers. Additionally, wild boars (Sus scrofa) are heavily culled in many ecosystems and are both a significant prey species for wolves as well as a key scavenger. Nowadays, wolves and wild boars are re-establishing their historical home ranges. However, it remains unclear how their presence influences the population dynamics of facultative scavengers under different scenarios of human hunting strategies. We simulated the biomass densities of all states in the trophic web including European scavengers and wolves using an ordinary differential equations (ODE) model. The presence of wolves led to a positive trend in scavenger biomass in general. However, in general, we found that plant-based resources were more important for scavenger dynamics than carrion, regardless of whether the carrion originated from human hunting or wolf predation. Only when wolves were absent but boars present, the human hunting strategy became important in determining scavenger dynamics via carrion supply. In conclusion, our model indicates that population dynamics of facultative scavengers are not mainly driven by the availability of carrion, but rather by the presence of and competition for vegetation. Furthermore, our simulations highlight the importance of adapting human hunting strategies in accordance with the re-establishment of wolf and boar as these can cause fluctuating population patterns over the years.

Performance analysis of passive heave compensator operated with inclined piston using analytical method

When performing operations such as drilling in the ocean, floating platforms undergo vertical motion due to waves, which negatively impacts these operations. So passive heave compensator(PHC) is used to counteract it. In many studies, the analysis of PHC with pneumatic springs has been done. Some of these papers analyzed it using simple second-order differential equations, but there was no case of modeling considering the pneumatic spring installed tilted in the PHC. On the other hand, in some papers, the characteristics were analyzed by numerical methods using multi-body dynamics equations, and it is time consuming. Therefore, in this paper, a second-order ordinary differential equation modeling method for PHC with an inclined pneumatic spring was presented and verified by comparison with the numerical method. Using the model presented in this paper, it is possible to analyze as accurate as conventional analytical methods, and to analyze characteristics faster than numerical methods.

Benchmarking methods for computing local sensitivities in ordinary differential equation models at dynamic and steady states.

Estimating parameters of dynamic models from experimental data is a challenging, and often computationally-demanding task. It requires a large number of model simulations and objective function gradient computations, if gradient-based optimization is used. In many cases, steady-state computation is a part of model simulation, either due to steady-state data or an assumption that the system is at steady state at the initial time point. Various methods are available for steady-state and gradient computation. Yet, the most efficient pair of methods (one for steady states, one for gradients) for a particular model is often not clear. In order to facilitate the selection of methods, we explore six method pairs for computing the steady state and sensitivities at steady state using six real-world problems. The method pairs involve numerical integration or Newton's method to compute the steady-state, and-for both forward and adjoint sensitivity analysis-numerical integration or a tailored method to compute the sensitivities at steady-state. Our evaluation shows that all method pairs provide accurate steady-state and gradient values, and that the two method pairs that combine numerical integration for the steady-state with a tailored method for the sensitivities at steady-state were the most robust, and amongst the most computationally-efficient. We also observed that while Newton's method for steady-state computation yields a substantial speedup compared to numerical integration, it may lead to a large number of simulation failures. Overall, our study provides a concise overview across current methods for computing sensitivities at steady state. While our study shows that there is no universally-best method pair, it also provides guidance to modelers in choosing the right methods for a problem at hand.

Mathematical Modeling of Dengue Disease Transmission Dynamics

Objectives: This study developed a compartmental ordinary differential equation model to investigate dengue transmission dynamics within a human population. The model stratified the population into susceptible, exposed, infected, and recovered classes, incorporating key epidemiological factors. Methods: Model equilibrium analysis was conducted to determine the stability of disease-free and endemic states. The basic reproduction number (R₀) was calculated to quantify the potential for disease spread. Additionally, sensitivity analysis was performed to assess the impact of key parameters on model outcomes. Findings: Results indicate that reducing mosquito biting rates and increasing human recovery rates are effective strategies for controlling dengue transmission. The model exhibits backward bifurcation, suggesting that even when R₀ is less than one, the disease can persist in the population under certain conditions. Novelty: This study presents a novel modeling framework that can inform the development of targeted prevention and management strategies for dengue. Keywords: Dengue, Mathematical Model, Ordinary differential equations, Basic reproduction number, Sensitivity analysis

Choice of landscape discretisation method affects the inferred rate of spread in wildlife disease spread models

Disease modelling at the livestock-wildlife interface is an important topic for which discrete-space models are used for the wildlife component. One prominent example is African Swine Fever, where wild boar play an influential role as reservoirs of disease spillover into domestic pig farms. In this paper, we present a simulation study that demonstrates the impact of seemingly arbitrary choices of landscape discretisation method on the inferred rate of spread within the model. We use an ordinary differential equation model to implement a simplified model of disease transmission between discrete groups of wild boar with spillover into domestic pig farms contained within a homogeneous landscape. We examine a range of scenarios whereby the landscape is discretised into wild boar patches of varying size and shape, and compare the rate of spread between domestic pig farms placed at fixed points on the landscape. Our results demonstrate a non-monotonic relationship between patch size and rate of spread, which is particularly unstable and unpredictable for square and triangular shaped patches. Discretisation of the landscape into hexagons appears to produce a more stable relationship between patch size and rate of spread for the three types of transmission kernel we investigated. Although the rate of disease spread does converge to a stable value, this occurs at patch sizes that are much smaller than would be used in practice for wild boar. We conclude that outputs of disease models containing a wildlife component should not be considered to be robust to arbitrary choices for patch size and placement, but rather as a source of uncertainty to be examined using sensitivity analysis. Furthermore, we strongly recommend the use of hexagons rather than squares or right triangles for landscape discretisation.

Predicting and analysis of urban wastewater quantity to be treated by using Neural Network and ODE model: A case study

Wastewater data, including Population of the Municipality Served by the Wastewater Treatment Plants (WWTPs), Capacity, Number of WWTPs, Amount of Daily Wastewater Discharged per Person and Wastewater Treated in WWTPs (y(t)) obtained from Turkish Statistical Institute (TUIK) for Kayseri province for 2003-2020 were mathematically modelled and analysed with Artificial Neural Network (ANN) and Differential equations. Firstly, the data were augmented through ARIMA since TUIK data numbers are insufficient to ANN training and later the data were normalized. The augmented normalized data was trained with ANN twice, thus the effect of other variables on the y(t) variable was shown and mathematical ANN activation functions in the form of a tangent hyperbolic function was proposed for this variable. Then, arbitrary parameters used for a linear system consisting of differential equations representing the five variables mentioned above were estimated using normalized original data and thus the ODE (ordinary differential equation) model was proposed. Two ANN models and ODE model were evaluated on normalized real TUIK data and the performances of these three models were compared. Among these mathematical models, the model that gave the minimum MSE (mean squared error) has been determined as the ODE model. Finally, future predictions were made for the y(t) variable with the ODE model.

Interpretable Spatio-Temporal Embedding for Brain Structural-Effective Network with Ordinary Differential Equation.

The MRI-derived brain network serves as a pivotal instrument in elucidating both the structural and functional aspects of the brain, encompassing the ramifications of diseases and developmental processes. However, prevailing methodologies, often focusing on synchronous BOLD signals from functional MRI (fMRI), may not capture directional influences among brain regions and rarely tackle temporal functional dynamics. In this study, we first construct the brain-effective network via the dynamic causal model. Subsequently, we introduce an interpretable graph learning framework termed Spatio-Temporal Embedding ODE (STE-ODE). This framework incorporates specifically designed directed node embedding layers, aiming at capturing the dynamic inter-play between structural and effective networks via an ordinary differential equation (ODE) model, which characterizes spatial-temporal brain dynamics. Our framework is validated on several clinical phenotype prediction tasks using two independent publicly available datasets (HCP and OASIS). The experimental results clearly demonstrate the advantages of our model compared to several state-of-the-art methods.

RTF: an R package for modelling time course data.

The retarded transient function (RTF) approach serves as a complementary method to ordinary differential equations (ODEs) for modelling dynamics typically observed in cellular signalling processes. We introduce an R package that implements the RTF approach, originally implemented within the MATLAB-based Data2Dynamics modelling framework. This package facilitates the modelling of time and dose dependencies, and it includes the possibility of model reduction to minimize overfitting. It can be applied to experimental data or trajectories of ODE models to characterize their dynamics. Additionally, it can generate a low-dimensional representation based on the fitted RTF parameters of a set of time-resolved data, aiding in the identification of key targets of experimental perturbations. The R package RTF is available at https://github.com/kreutz-lab/RTF.

Utilising an in silico model to predict outcomes in senescence-driven acute liver injury

Currently liver transplantation is the only treatment option for liver disease, but organ availability cannot meet patient demand. Alternative regenerative therapies, including cell transplantation, aim to modulate the injured microenvironment from inflammation and scarring towards regeneration. The complexity of the liver injury response makes it challenging to identify suitable therapeutic targets when relying on experimental approaches alone. Therefore, we adopted a combined in vivo-in silico approach and developed an ordinary differential equation model of acute liver disease able to predict the host response to injury and potential interventions. The Mdm2fl/fl mouse model of senescence-driven liver injury was used to generate a quantitative dynamic characterisation of the key cellular players (macrophages, endothelial cells, myofibroblasts) and extra cellular matrix involved in liver injury. This was qualitatively captured by the mathematical model. The mathematical model was then used to predict injury outcomes in response to milder and more severe levels of senescence-induced liver injury and validated with experimental in vivo data. In silico experiments using the validated model were then performed to interrogate potential approaches to enhance regeneration. These predicted that increasing the rate of macrophage phenotypic switch or increasing the number of pro-regenerative macrophages in the system will accelerate the rate of senescent cell clearance and resolution. These results showcase the potential benefits of mechanistic mathematical modelling for capturing the dynamics of complex biological systems and identifying therapeutic interventions that may enhance our understanding of injury-repair mechanisms and reduce translational bottlenecks.