ODE Model Research Articles

An ODEs multiscale model with cell proliferation for hepatitis C virus infection treated with direct acting antiviral agents

In a recent study, a mathematically identical ODE model is derived from a multiscale PDE model of hepatitis C virus infection, which helps to overcome the limitations of the PDE model in the analysis. Here, an extended proposed model is formulated for this transformed ODE model by including the hepatocyte proliferation of both uninfected and infected cells. Unlike the transformed model, the proposed model can predict the triphasic viral decline and the virus level after therapy cessation without oscillations. Numerical simulations are performed to investigate the effect of hepatocyte proliferation and therapy with direct-acting antivirals agents (DAAs). The basic reproduction number is obtained, the equilibrium points are specified, and their stability is analysed. A bifurcation analysis is performed to specify the bifurcation points and to study the effect of varying system parameters. Various viral load profiles generated by the model are confirmed to fit with reported data in the literature.

Bounding Seed Loss from Isolated Habitat Patches

Dispersal of propagules (seeds, spores) from a geographically isolated habitat into an uninhabitable matrix can play a decisive role in driving population dynamics. ODE and integrodifference models of these dynamics commonly feature a “dispersal success” parameter representing the average proportion of dispersing propagules that remain in viable habitat. While dispersal success can be estimated by empirical measurements or by integration of dispersal kernels, one may lack resources for fieldwork or details on dispersal kernels for numerical computation. Here we derive simple upper bounds on the proportion of propagule loss—the complement of dispersal success—that require only habitat area, habitat perimeter, and the mean dispersal distance of a propagule. Using vector calculus in a probabilistic framework, we rigorously prove bounds for the cases of both symmetric and asymmetric dispersal. We compare the bounds to simulations of integral models for the population of Asclepias syriaca (common milkweed) at McKnight Prairie—a 14 hectare reserve surrounded by agricultural fields in Goodhue County, Minnesota—and identify conditions under which the bounds closely estimate propagule loss.

Hierarchical Bayesian data selection

There are many issues that can cause problems when attempting to infer model parameters from data. Data and models are both imperfect, and as such there are multiple scenarios in which standard methods of inference will lead to misleading conclusions; corrupted data, models which are only representative of subsets of the data, or multiple regions in which the model is best fit using different parameters. Methods exist for the exclusion of some anomalous types of data, but in practice, data cleaning is often undertaken by hand before attempting to fit models to data. In this work, we will employ hierarchical Bayesian data selection; the simultaneous inference of both model parameters, and parameters which represent our belief that each observation within the data should be included in the inference. The aim, within a Bayesian setting, is to find the regions of observation space for which the model can well-represent the data, and to find the corresponding model parameters for those regions. A number of approaches will be explored, and applied to test problems in linear regression, and to the problem of fitting an ODE model, approximated by a finite difference method. The approaches are simple to implement, can aid mixing of Markov chains, and are broadly applicable to many inferential problems.

A new approach to modeling transdermal ethanol kinetics.

Measurement of ethanol above the skin surface (supradermal) is used to monitor blood alcohol concentrations (BAC) in both legal and consumer settings. Previously, the relationship between supradermal alcohol concentration (SAC) and BAC was described using partial and ordinary differential equations (PDE model: J. Appl. Physiol. 100: 649-55, 2006). Using a range of BAC profiles by varying absorption times and peak concentrations, the PDE model accurately predicted experimental measures of SAC. Recently, other mathematical models have relied on the PDE model. This paper proposes a new approach to modeling transdermal ethanol kinetics using a mass transfer coefficient and only ordinary differential equations (ODE model). Using a range of BAC profiles, the ODE model performed very similarly to the PDE model. The ODE model had slightly slower washout rates and slightly slower times to peak SAC and to zero SAC. Similar to the PDE model, a sensitivity analysis on the ODE model showed changes in solubility and diffusivity within the stratum corneum, stratum corneum thickness, and the volume of gas above the skin affected model performance. This new model will streamline integration into larger physiologic models, reduce computation time, and decrease the time to transform skin alcohol measurements to blood alcohol concentrations.

Modeling of zirconium alloy cladding corrosion behavior based on neural ordinary differential equation

Current zirconium alloy cladding corrosion models are mainly semi-empirical and show significant dispersion when compared to measured data. This study introduces neural ordinary differential equation (neural ODE) to model corrosion behavior, utilizing data-model fusion approach for network training. Initially, a semi-empirical model for zirconium alloy cladding corrosion is established through differential evolution algorithm, generating a large dataset for pre-training the neural network. The network is then fine-tuned using measured data. These methods effectively address the challenges of sparse cladding corrosion data and data available only at fixed time points, resulting in a more accurate model. The results shows that the differential evolution algorithm can identify a set of appropriate parameters for the semi-empirical model, achieving a standard deviation of 0.040. The neural ODE model demonstrates even higher accuracy, reducing the standard deviation to 0.031 and improving accuracy by approximately 25%. Additionally, the model demonstrates excellent generalization capacity on other time points and new power histories.

A system of ODEs for representing trends of CGM signals

Diabetes Mellitus is a metabolic disorder which may result in severe and potentially fatal complications if not well-treated and monitored. In this study, a quantitative analysis of the data collected using CGM (Continuous Glucose Monitoring) devices from eight subjects with type 2 diabetes in good metabolic control at the University Polyclinic Agostino Gemelli, Catholic University of the Sacred Heart, was carried out. In particular, a system of ordinary differential equations whose state variables are affected by a sequence of stochastic perturbations was proposed and used to extract more informative inferences from the patients’ data. For this work, Matlab and R programs were used to find the most appropriate values of the parameters (according to the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC)) for each patient. Fitting was carried out by Particle Swarm Optimization to minimize the ordinary least squares error between the observed CGM data and the data from the ODE model. Goodness of fit tests were made in order to assess which probability distribution was best suitable for representing the waiting times computed from the model parameters. Finally, both parametric and non-parametric density estimation of the frequency histograms associated with the variability of the glucose elimination rate from blood were conducted and their representative parameters assessed from the data. The results show that the chosen models succeed in capturing most of the glucose fluctuations for almost every patient.

Optimization of SOX2 Expression for Enhanced Glioblastoma Stem Cell Virotherapy

The Zika virus has been shown to infect glioblastoma stem cells via the membrane receptor αvβ5, which is activated by the stem-specific transcription factor SOX2. Since the expression level of SOX2 is an important predictive marker for successful virotherapy, it is important to understand the fundamental mechanisms of the role of SOX2 in the dynamics of cancer stem cells and Zika viruses. In this paper, we develop a mathematical ODE model to investigate the effects of SOX2 expression levels on Zika virotherapy against glioblastoma stem cells. Our study aimed to identify the conditions under which SOX2 expression level, viral infection, and replication can reduce or eradicate the glioblastoma stem cells. Analytic work on the existence and stability conditions of equilibrium points with respect to the basic reproduction number are provided. Numerical results were in good agreement with analytic solutions. Our results show that critical threshold levels of both SOX2 and viral replication, which change the stability of equilibrium points through population dynamics such as transcritical and Hopf bifurcations, were observed. These critical thresholds provide the optimal conditions for SOX2 expression levels and viral bursting sizes to enhance therapeutic efficacy of Zika virotherapy against glioblastoma stem cells. This study provides critical insights into optimizing Zika virus-based treatment for glioblastoma by highlighting the essential role of SOX2 in viral infection and replication.

Algorithm for globally identifiable reparametrizations of ODEs

Structural global parameter identifiability indicates whether one can determine a parameter's value in an ODE model from given inputs and outputs. If a given model has parameters for which there is exactly one value, such parameters are called globally identifiable. Given an ODE model involving not globally identifiable parameters, first we transform the system into one with locally identifiable parameters. As a main contribution of this paper, then we present a procedure for replacing, if possible, the ODE model with an equivalent one that has globally identifiable parameters. We first derive this as an algorithm for one-dimensional ODE models and then reuse this approach for higher-dimensional models.

Overcompensation of transient and permanent death rate increases in age-structured models with cannibalistic interactions

There has been renewed interest in understanding the mathematical structure of ecological population models that lead to overcompensation, the process by which a population recovers to a higher level after suffering a permanent increase in predation or harvesting. Here, we apply a recently formulated kinetic population theory to formally construct an age-structured single-species population model that includes a cannibalistic interaction in which older individuals prey on younger ones. Depending on the age-dependent structure of this interaction, our model can exhibit transient or steady-state overcompensation of an increased death rate as well as oscillations of the total population, both phenomena that have been observed in ecological systems. Analytic and numerical analysis of our model reveals sufficient conditions for overcompensation and oscillations. We also show how our structured population partial integrodifferential equation (PIDE) model can be reduced to coupled ODE models representing piecewise constant parameter domains, providing additional mathematical insight into the emergence of overcompensation.

Boundedness in a Chemotaxis-May-Nowak model for virus dynamics with gradient-dependent flux limitation

This paper investigates an extension of the May-Nowak ODE model for virus dynamics with gradient-dependent flux limitation of cross diffusion. In particular, we consider the associated no-flux initial–boundary value problem (0.1)ut=Δu−∇⋅(uf(|∇v|2)∇v)+κ−u−uw,vt=Δv−v+uw,wt=Δw−w+vin a smoothly bounded domain Ω⊂Rn(n≤3), where the parameter κ≥0. The prototypical chemotactic sensitivity function f∈C2([0,∞)) is given by f(ξ)=(1+ξ)−α,ξ≥0 with some α∈R. It is proved that whenever α∈R,ifn=1,α>n−22(n−1),ifn={2,3},global classical solutions to (0.1) exist and are uniformly bounded. Such result consists with that in [Winkler (2022), Proposition 1.2] when n≤3, which shows that the effect of gradient-dependent flux limitation in weakening the cross-diffusion term remains unchanged even in the context of nonlinear signal production mechanism.

SPATIOTEMPORAL DYNAMICS IN A MODEL OF IMMUNODOMINANCE

Adapative immune response to viral infection is mediated by cytotoxic T-lymphocytes (CTLs) eliminating infected cells and virus-neutralizing antibodies produced by B-lymphocytes. Different epitopes, that is, some parts of antigens, stimulate clonal expansion of the corresponding lymphocytes. These different types of CTLs and antibodies begin to compete with each other leading to immunodominance of some of them. As a result, some types of CTLs and antibodies will persist and increase their number, while some others will become extinct. In this work, we study spatiotemporal dynamics in a previously developed mathematical model of immunodominance. We begin with the investigation of complex oscillations exhibited by the concentrations of epitopes and corresponding cytotoxic T-lymphocytes in the ODE model and determine the mechanism of these oscillations. After that, we study the reaction–diffusion system of equations describing their spatiotemporal dynamics and analyze how these dynamics depend on the interaction of different virus variants and CTL types.

Enhanced Trajectory Forecasting for Hypersonic Glide Vehicle via Physics-Embedded Neural ODE

Forecasting hypersonic glide vehicle (HGV) trajectories accurately is crucial for defense, but traditional methods face challenges due to the scarce real-world data and the intricate dynamics of these vehicles. Data-driven approaches based on deep learning, while having emerged in recent years, often exhibit limitations in predictive accuracy and long-term forecasting. Whereas, physics-informed neural networks (PINNs) offer a solution by incorporating physical laws, but they treat these laws as constraints rather than fully integrating them into the learning process. This paper presents PhysNODE, a novel physics-embedded neural ODE model for the precise forecasting of HGV trajectories, which directly integrates the equations of HGV motion into a neural ODE. PhysNODE leverages a neural network to estimate the hidden aerodynamic parameters within these equations. These parameters are then combined with observable physical quantities to form a derivative function, which is fed into an ODE solver to predict the future trajectory. Comprehensive experiments using simulated datasets of HGV trajectories demonstrate that PhysNODE outperforms the state-of-the-art data-driven and physics-informed methods, particularly when training data is limited. The results highlight the benefit of embedding the physics of the HGV motion into the neural ODE for improved accuracy and stability in trajectory predicting.

The importance of the incubation time distribution in compartmental epidemiological models

This study investigates the utilization of various mathematical models for comprehending and managing outbreaks of infectious diseases, with a specific focus on how different distributions of incubation times influence predictions regarding epidemics. Two methodologies are examined: a compartmental SEnIR ODE model, which represents an enhanced version of the mean-field SEIR model, and a stochastic agent-based complex network model. Our findings demonstrate that the selection of diverse incubation time distributions can result in noteworthy discrepancies in critical epidemic forecasts, highlighting the crucial role of precise modeling in shaping effective public health interventions. The research underscores the necessity of integrating authentic distribution patterns into epidemic modeling to increase its reliability and applicability.

Analyzing the Stability of Lanchester Warfare Models for Symmetric Warfare Scenarios

This study analyzes the stability of Lancaster-type ODE models in symmetric warfare situations. In symmetric warfare situations, when the lethality coefficients (K) are equal for both forces in battle, the system exhibits marginal stability, characterized by poles at ± K, indicating that the model is stable in some regions and unstable in others. The present research illustrates a controlled rhythmicity, and these forces show a harmonized balance between the two battling forces in order to develop a strong strategy and decision for military.

End-to-End Statistical Model Checking for Parameterization and Stability Analysis of ODE Models

We propose a simulation-based technique for the parameterization and the stability analysis of parametric Ordinary Differential Equations. This technique is an adaptation of Statistical Model Checking, often used to verify the validity of biological models, to the setting of Ordinary Differential Equations systems. The aim of our technique is to estimate the probability of satisfying a given property under the variability of the parameter or initial condition of the ODE, with any metrics of choice. To do so, we discretize the values space and use statistical model checking to evaluate each individual value w.r.t. provided data. Contrary to other existing methods, we provide statistical guarantees regarding our results that take into account the unavoidable approximation errors introduced through the numerical integration of the ODE system performed while simulating. In order to show the potential of our technique, we present its application to two case studies taken from the literature, one relative to the growth of a jellyfish population, and the other concerning a well-known oscillator model.

Revealing endogenous conditions for Peto’s paradox via an ordinary differential equation model

Cancer, a disease intimately linked to cellular mutations, is commonly believed to exhibit a positive association with the cell count and lifespan of a species. Despite this assumption, the observed uniformity in cancer rates across species, referred to as the Peto’s paradox, presents a conundrum. Recognizing that tumour progression is not solely dependent on cancer cells but involves intricate interactions among various cell types, this study employed a Lotka-Volterra (LV) ordinary differential equation model to analyze the evolution of cancerous cells and the cancer incidence in an immune environment. As a result, this study uncovered the sufficient conditions underlying the absence of correlation in Peto’s paradox and provide insights into the reasons for the equitable distribution of cancer incidence across diverse species by applying nondimensionalization and drawing an analogy between the characteristic time interval for the variation of cell populations in the ODE model and that of cell cycles of a species.

Federated Learning of Neural ODE Models with Different Iteration Counts

Federated Learning of Neural ODE Models with Different Iteration Counts

Mathematical modeling and analysis of cancer treatment with radiation and anti-PD-L1

In cancer treatment, radiation therapy (RT) induces direct tumor cell death due to DNA damage, but it also enhances the deaths of radiosensitive immune cells and is followed by local relapse and up-regulation of immune checkpoint ligand PD-L1. Since the binding between PD-1 and PD-L1 curtails anti-tumor immunities, combining RT and PD-L1 inhibitor, anti-PD-L1, is a potential method to improve the treatment efficacy by RT. Some experiments support this hypothesis by showing that the combination of ionizing irradiation (IR) and anti-PD-L1 improves tumor reduction comparing to the monotherapy of IR or anti-PD-L1. In this work, we create a simplified ODE model to study the order of tumor growths under treatments of IR and anti-PD-L1. Our synergy analysis indicates that both IR and anti-PD-L1 improve the tumor reduction of each other, when IR and anti-PD-L1 are given simultaneously. When giving IR and anti-PD-L1 separately, a high dosage of IR should be given first to efficiently reduce tumor load and then followed by anti-PD-L1 with strong efficacy to maintain the tumor reduction and slow down the relapse. Increasing the duration of anti-PD-L1 improves the tumor reduction, but it cannot prolong the duration that tumor relapses to the level of the control case. Under some simplification, we also prove that the model has an unstable tumor free equilibrium and a locally asymptotically stable tumor persistent equilibrium. Our bifurcation diagram reveals a transition from tumor elimination to tumor persistence, as the tumor growth rate increases. In the tumor persistent case, both anti-PD-L1 and IR can reduce tumor amount in the long term.

Optimal vaccination and treatment policies with constrained inequalities to study limited vaccination resources for a multistrain reaction–diffusion [formula omitted] model of COVID-19

The availability of an effective vaccine is a major challenge in COVID-19 vaccination campaigns. In this article, we propose a comprehensive strategy incorporating distributed controls via vaccination and treatment while considering the limitations of available resources expressed as inequalities constrained. Our approach utilizes a spatiotemporal model that includes multiple strains of the virus and considers spatial heterogeneity, providing a more accurate representation of disease transmission. By using a generalized two-strain ODE model, we effectively capture the evolution of COVID-19, including various strains and variants. To validate our model, we establish the positivity and boundedness of the system’s solution. Through the exploration of vaccination and treatment controls. Our study aims to reduce the spread of the disease between strains and identify optimal values that minimize the number of infected individuals and associated costs. This work provides useful insights into effective strategies for controlling the COVID-19 epidemic, even with limited vaccination resources. Furthermore, we demonstrate the existence of controls and describe their characterizations in terms of state and adjoint functions. The penalty method is used to treat the inequalities constrained. More importantly, our numerical simulation shows that adequate vaccination resources and simultaneous treatment can effectively reduce and even eliminate the spread of COVID-19.

Enhancing continuous time series modelling with a latent ODE-LSTM approach

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent Neural Networks (RNNs), RNNs have been generalised to have continuous-time hidden dynamics defined by a Neural Ordinary Differential Equation (Neural ODE), leading to the ODE-RNN model. Another approach that provides a better modelling is that of the Latent ODE model, which constructs a continuous-time model where a latent state is defined at all times. The Latent ODE model uses a standard RNN as the encoder and a Neural ODE as the decoder. However, since the RNN encoder leads to difficulties with missing data and ill-defined latent variables, a Latent ODE-RNN model has recently been proposed that uses a ODE-RNN model as the encoder instead.Both the Latent ODE and Latent ODE-RNN models are difficult to train due to the vanishing and exploding gradients problem. To overcome this problem, the main contribution of this paper is to propose and illustrate a new model based on a new Latent ODE using an ODE-LSTM (Long Short-Term Memory) network as an encoder - the Latent ODE-LSTM model. To limit the growth of the gradients, the Norm Gradient Clipping strategy was embedded on the Latent ODE-LSTM model.The performance evaluation of the new Latent ODE-LSTM (with and without Norm Gradient Clipping) for modelling CTS with regular and irregular sampling rates is then demonstrated. Numerical experiments show that the new Latent ODE-LSTM performs better than Latent ODE-RNNs and can avoid the vanishing and exploding gradients during training.Code implementations developed in this work are available at github.com/CeciliaCoelho/LatentODELSTM.