Differential Equation Model Research Articles

Advancing very short-term rainfall prediction with blended U-Net and partial differential approaches

Accurate and timely prediction of short-term rainfall is crucial for reducing the damages caused by heavy rainfall events. Therefore, various precipitation nowcasting models have been proposed. However, the performance of these models still remains limited. In particular, the current operational precipitation nowcasting method, which is based on radar echo tracking, such as the McGill Algorithm for Precipitation Nowcasting by Lagrangian Extrapolation (MAPLE), has a critical drawback when predicting newly developed or decayed precipitation fields. Recently proposed deep learning models, such as the U-Net convolutional neural network outperform the models based on radar echo tracking. However, these models are unsuitable for operational precipitation nowcasting due to their blurry predictions over longer lead times. To address these blurry effects and enhance the performance of U-Net-based rainfall prediction, we propose a blended model that combines a partial differential equation (PDE) model based on fluid dynamics with the U-Net model. The evaluation of the forecast skill, based on both qualitative and quantitative methods for 0–3-h lead times, demonstrates that the blended model provides less blurry and more accurate rainfall predictions compared with the U-Net and partial differential equation models. This indicates the potential to enhance the field of very short-term rainfall prediction. Additionally, we also evaluated the monthly-averaged forecast skills for different seasons, and confirmed the operational feasibility of the blended model, which contributes to the performance enhancement of operational nowcasting.

Modeling grazer-mediated effects of demographic and material connectivity on giant kelp metapopulation dynamics

From dispersal-based metapopulations to meta-ecosystems that arise from flows of non-living materials, spatial connectivity is a major driver of population dynamics. One potentially important process is material transport between populations also linked by individual dispersal. Here, I explored material and demographic connectivity in metapopulations of giant kelp Macrocystis pyrifera, a foundation species that produces both detritus and reproductive spores. Kelp detritus (drift) subsidizes grazers, helping maintain the kelp forest ecosystem state. Drift could potentially be exchanged among kelp patches, but this is less studied than spore dispersal. Therefore, I built an ordinary differential equation (ODE) model to investigate conditions under which drift and/or spore connectivity promotes the kelp forest state. I fit statistical models (generalized linear mixed models, GLMMs) to observational data and used the GLMM’s predictions to validate the ODE model. My results suggest kelp patch dynamics are best explained by connectivity of both drift and spores, and that the impacts of these forms of connectivity depend on local grazer (urchin) abundance. Both models predicted greater kelp persistence in well-connected patches across a range of urchin densities. These effects were largely driven by drift, which reduced grazing in recipient patches and thereby enhanced spore recruitment. While testing these predictions will require greater empirical quantification of interpatch drift transport, my findings indicate drift connectivity may be an important spatial process in kelp forest systems. More broadly, this work highlights the role of meta-ecosystem dynamics within a single ecosystem type, reinforcing the need to expand traditional metapopulation perspectives to consider multiple forms of spatial connectivity.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Bifurcation and stability analysis of atherosclerosis disease model characterizing the anti-oxidative activity of HDL during short- and long-time evolution

Abstract In this article, a partial differential equation (PDE) model for atherosclerosis disease is presented that analyzes the anti-oxidative activity of high-density lipoprotein (HDL) during the reverse cholesterol transport (RCT) process. The model thoroughly investigates the complex interplay between oxidized low-density lipoprotein (ox-LDL) and high-density lipoprotein in the context of atherosclerosis, emphasizing their combined impact on plaque formation, disease progression, and regression. In addition to this, we considered that monocytes are also attracted by the presence of ox-LDL within the intima. Detailed discussions on stability analyses of the reaction dynamical system at non-inflammatory and chronic equilibrium are provided, followed by a bifurcation analysis for the proposed system. Furthermore, stability analysis for the PDE model in the presence of diffusion is conducted. Our study reveals that the oxidation rate of LDL by monocytes (δ) and the influx rate of HDL (ϕ) due to drugs/diet are primarily responsible for the existence of bi-stability of equilibrium points. In the numerical results, we observe that non-inflammatory or chronic equilibrium points exist for either a short or a long time, and these findings are validated with existing results. The biological elucidation shows the novelty in terms of enhancing our ability to assess intervention efficacy to generate therapeutic strategies resulting in the reduction of the atherosclerotic burden and associated cardiovascular risks.

Modeling and Analysis of an Age-Structured Malaria Model in the Sense of Atangana–Baleanu Fractional Operators

In this paper, integer- and fractional-order models are discussed to investigate the dynamics of malaria in a human host with a varied age distribution. A system of differential equation model with five human state variables and two mosquito state variables was examined. Preschool-age (0–5) and young-age individuals make up our model’s division of the human population. We investigated the existence of an area in which the model is both mathematically and epidemiologically well posed. According to the findings of our mathematical research, the disease-free equilibrium exists whenever the fundamental reproduction number R0 is smaller than one and is asymptotically stable. The disease-free equilibrium point is unstable when R0>1. We showed that the endemic equilibrium point is unique for R0>1. Also, the most influential control parameters of the spread of malaria were identified. Numerical simulations of both classical and fractional order were conducted, and we used ODE (45) for classical part and numerical technique developed by Toufik and Atangana for fractional order. The infected population will grow because of the high biting frequency of the mosquito and the high likelihood of transmission from the infected mosquito to the susceptible human. R=1.622, which is more than one, indicating that the mosquito vector keeps on growing. This supports the stability of the endemic equilibrium point theorem, which states that the disease becomes endemic when R=1. The susceptible human population will decrease because of the presence of the infective mosquito, which has a high biting frequency for the first couple of days. Since the infective mosquito bit the susceptible human, the susceptible human became infected and went to the infected human compartments. Then, the susceptible population will decrease and the infested human population will increase. After a certain amount of time, it becomes zero due to the growth of protected classes. In this case, a disease-free equilibrium point exists and is stable. This condition exists because R0=2.827×10−5 is less than 1. This supports the theorem that the stability of the disease-free equilibrium point is obtained when R0<1. Depending on equation, we have shown that the possibility of some endemic equilibria exists when R0<1, that is, it undergoes backward bifurcation, even when the disease-free equilibrium is locally stable, and the result means that the society may misunderstand the level of malaria prevalence in the community.

Thermal error compensation for a fluid-cooling ball-screw feed system

Thermal deformation resulted from the heat generated by the motor and the frictional behavior between the moving pairs significantly reduces the positioning accuracy of feed systems and thus influences the machining quality of parts. The internal cooling ball screw takes advantage of the cooling liquid to strengthen the convection heat transfer actively and suppress the temperature rise. Owing to the intermittent starting and stopping operation modes of the cooling machine, the heat flux arising from various heat sources cannot be completely removed by the coolant; consequently, thermal errors still exist. In this study, K-means++ clustering and correlation analyses were used to select thermal key points. A difference equation model of the thermal error was established to describe the transient change relationship between the temperatures of the thermal key points and the ball-screw shaft elongation, which was separated from the thermal characteristic experimental data according to the linear superposition principle of geometric and thermal errors to constitute the positioning error. The model parameters were identified using the least-squares method, and a thermal error compensation strategy based on the origin offset method was implemented. Experiments comparing the thermal error compensation of the three models under different working conditions were conducted to confirm that the thermal error difference equation model can be applied to reduce the thermal error of the feed system effectively and maintain excellent robustness.

How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices?

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer's disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

Construction of innovation and entrepreneurship teaching capacity in colleges and universities and its optimization path based on differential equation modeling

Abstract The rapid development of information and artificial intelligence technology has brought new opportunities for innovation and transformation in higher education, leading to innovative applications of data-driven-assisted teaching decisions. This paper analyzes the equilibrium point in predicting students’ performance using nonlinear differential equations, solves the expected value of students’ performance, and constructs a prediction model for students’ innovation and entrepreneurship performance. Student achievement data and characteristics are preprocessed to improve the accuracy of the prediction model. Based on Pearson’s coefficient, students’ innovation and entrepreneurship achievement are significantly correlated with teaching ability (P<0.05). The results show that, after implementing innovation and entrepreneurship teaching based on this prediction model, the predicted achievement of students in the intervention group is significantly higher than that of students in the control group (P<0.05). In the actual achievement test, the average total grade for the innovative entrepreneurship course of students in the intervention group was about 4.68-8.78 points higher than that of students in the control group. The achievement prediction model developed in this paper, along with the proposed optimization path for building innovation and entrepreneurship teaching capacity, offers significant guidance for the teaching research department in implementing effective teaching interventions and in the development of innovation and entrepreneurship education.

An Efficient Platform for Numerical Modeling of Partial Differential Equations

An Efficient Platform for Numerical Modeling of Partial Differential Equations

Solution of logistic differential equation in an uncertain environment using neutrosophic numbers

The modeling and forecasting of population dynamics, as well as growth in biological systems more generally, have required the construction of various growth models. This paper presents the logistic growth model, which is a modified version of the exponential growth model and offers a more flexible framework to capture fluctuations and potential deviations from deterministic predictions. This paper aims to advance the model of a logistic growth differential equation within a neutrosophic environment, with a focus on validating the proposed framework by applying it to predict the projected population of India from the year 2001 to 2050. We investigated the neutrosophic solutions and validated them with the exact solution with available projected data on the portal of UNITED NATION. The developed logistic growth differential equation in a neutrosophic environment is visually elucidated through graphical representation, enhancing the clarity and accessibility of the presented model.

Comparative Study of Tumor Growth Based on Single Species Models

Although the ideal way to simulate tumour development is still unknown, mathematical models are often employed to forecast how cancer will advance and how treatments will be developed. Five alternative ordinary differential Equation (ODE) models of tumour development have been proposed: exponential, logistic, Allee effect, Gompertz, and Bertalanffy. The problem is no consensus exists on the best model for a given cancer type. Using a sample dataset, we compared how these quantities differ depending on the growth model chosen. Equations were developed to determine the largest possible size of the tumour and the least level of chemotherapy needed for suppressing the tumour.

Modeling disease awareness and variable susceptibility with a structured epidemic model

<abstract><p>We developed an epidemic model with disease awareness and variable susceptibility, consisting of a two-dimensional, nonlocal, transport equation. From this model, we deduced a 3D ordinary differential equation (ODE) model, which is reminiscent of (but not reducible to) more traditional susceptible-infectious-susceptible (SIS)-type models, where the dynamical variables are the infected population proportion, the mean awareness of the population, and the mean susceptibility to reinfection. We show that a reproduction number $ R_0 $ exists whose threshold value determines the stability of the disease-free equilibrium, alongside the existence of an endemic one. We deduced conditions on the model parameters and ensured the stability and uniqueness of the endemic equilibrium. The transport equation was studied, and we showed some numerical experiments. Our results suggest that disease awareness dynamics can have a major role in epidemiological outcomes: we showed that even for high $ R_0 $, the infection prevalence could be made as small as desired, as long as the awareness decay was small. On the other hand, numerical evidence suggested that the relation between epidemiological outcomes and awareness levels was not straightforward, in the sense that sustained high awareness may not always lead to better outcomes, as compared to time-limited awareness peaks in response to outbreaks.</p></abstract>

Finite-Time Consensus Adaptive Neural Network Control for Nonlinear Multiagent Systems Under PDE Models.

In this article, a novel adaptive control method based on neural networks is proposed for a class of multiagent systems (MASs) with nonlinear functions and external disturbances. First, the approximation properties of neural networks are used to approximate the MAS partial differential equation (PDE) model with nonlinear terms containing two variables, time t, and spatial variable x. Second, an adaptive controller is constructed to actuate the parabolic MAS to reach consensus under external disturbances. Based on this, the finite-time theorem and special inequalities are applied to prove the stability of the closed-loop system. Thus, MAS that have nonlinear functions and external disturbances are enabled with finite-time consensus. Finally, the effectiveness of the proposed control method is demonstrated by numerical simulations.

Hybrid data-mechanism-driven model of the unsteady soil temperature field for long-buried crude oil pipelines with non-isothermal batch transportation

The thermal simulation of oil pipeline transportation is crucial for ensuring safe transportation of pipelines and optimizing energy consumption. The prediction of the soil temperature field is the key to the thermal calculation for the non-isothermal batch transportation of the buried pipeline, while the standard numerical simulation of the soil temperature field is time-consuming. Coupling with a data-driven Bayesian neural network and mechanism-informed partial differential equation, an efficient and robust prediction model of soil temperature field is proposed to dynamically adapt the spatio-temporal changes of boundary conditions. Based on the soil temperature field predicted by the proposed model, the oil temperature at the outlet of the pipeline is further obtained, which is compared with that from the field data and the standard numerical simulation. It is found that the former is in good agreement with the latter two, verifying the proposed model. However, the calculation of the proposed model only takes 10.59 s, which is 29.53 times faster than the standard numerical simulation. Moreover, the predicted error of the proposed model only changes by 0.12 % (from 3.05 % to 3.17 %) when the training data decreases from 100 % to 2.2 %, which is lower than that of two data-driven surrogate models.

Modern methods for solving partial differential equations of parabolic type and their mathematical theoretical foundations

Abstract In this paper, the existence problem of weak solutions in partial differential equations is first discussed. Secondly, the parabolic-type partial differential equation (PDE) model with non-chiral terms is used to portray the coupling principle of microwave heating dielectric medium, and the approximate models of TEM and TE10-wave microwave heating are constructed and verified by comparing with the COMSOL model in simulation. An open-loop type iterative learning control algorithm is proposed for the distributed parameter system, and the effectiveness of the control algorithm proposed in this paper is theoretically demonstrated. It is concluded that the TEM and TE10-wave microwave heating approximation models are more successful in describing the microwave heating process than the COMSOL model, and the consistency of the simulation results fully proves the correctness and effectiveness of the proposed model. The paradigm of the system output error decreases with the increase in the number of system runs, and the paradigm of the system output error tends to be close to zero.

Differential equations frameworks and models for the physics of biological systems

Differential equations frameworks and models for the physics of biological systems

Stability of the vector-borne disease model with direct transmission using Boubaker polynomials approach

In this paper, we delve into the analysis of an epidemic model for a vector-borne disease. Our study focuses on utilizing a baseline version of the ordinary differential equations (ODE) model to capture the dynamics of the disease transmission. Specifically, we aim to study the long-term behavior and properties of the model's solutions using a novel analytical approach known as the Boubaker polynomial Expansion Scheme (BPES). Furthermore, to complement our theoretical analysis, we conduct numerical simulations to provide a more practical perspective on the epidemic.

A mathematical model of visceral leishmaniasis transmission and control: Impact of ITNs on VL prevention and elimination in the Indian subcontinent.

Visceral Leishmaniasis (VL) is a deadly, vector-borne, parasitic, neglected tropical disease, particularly prevalent on the Indian subcontinent. Sleeping under the long-lasting insecticide-treated nets (ITNs) was considered an effective VL prevention and control measures, until KalaNet, a large trial in Nepal and India, did not show enough supporting evidence. In this paper, we adapt a biologically accurate, yet relatively simple compartmental ordinary differential equations (ODE) model of VL transmission and explicitly model the use of ITNs and their role in VL prevention and elimination. We also include a game-theoretic analysis in order to determine an optimal use of ITNs from the individuals' perspective. In agreement with the previous more detailed and complex model, we show that the ITNs coverage amongst the susceptible population has to be unrealistically high (over 96%) in order for VL to be eliminated. However, we also show that if the whole population, including symptomatic and asymptomatic VL cases adopt about 90% ITN usage, then VL can be eliminated. Our model also suggests that ITN usage should be accompanied with other interventions such as vector control.

Modeling disease awareness and variable susceptibility with a structured epidemic model

<abstract><p>We developed an epidemic model with disease awareness and variable susceptibility, consisting of a two-dimensional, nonlocal, transport equation. From this model, we deduced a 3D ordinary differential equation (ODE) model, which is reminiscent of (but not reducible to) more traditional susceptible-infectious-susceptible (SIS)-type models, where the dynamical variables are the infected population proportion, the mean awareness of the population, and the mean susceptibility to reinfection. We show that a reproduction number $ R_0 $ exists whose threshold value determines the stability of the disease-free equilibrium, alongside the existence of an endemic one. We deduced conditions on the model parameters and ensured the stability and uniqueness of the endemic equilibrium. The transport equation was studied, and we showed some numerical experiments. Our results suggest that disease awareness dynamics can have a major role in epidemiological outcomes: we showed that even for high $ R_0 $, the infection prevalence could be made as small as desired, as long as the awareness decay was small. On the other hand, numerical evidence suggested that the relation between epidemiological outcomes and awareness levels was not straightforward, in the sense that sustained high awareness may not always lead to better outcomes, as compared to time-limited awareness peaks in response to outbreaks.</p></abstract>

A study of stochastic epidemic model driven by liouville fractional brownian motion coupled with seasonal air pollution

Air pollution can cause and provoke respiratory diseases. It is an important topic to the public, particularly in developing countries. Since there are many uncertain factors in the environment, stochastic differential equation model is a powerful tool to study the changes of air pollution and the transmission of infectious diseases. The removal of air pollutants as well as the transmission of diseases can be influenced by random perturbations with memories. In this research, we develop a mathematical model in the form of a system of stochastic differential equations driven by fractional Brownian motion of Liouville-type, coupled with seasonal air pollution, to study the dynamics of infectious respiratory disease spread. As a result , by using stochastic calculus techniques, we derive the equation for the level of air pollution.