Ordinary Differential Model Research Articles

Equilibrium points and their stability of COVID-19 in US

This study aims to develop an advanced mathematic model and investigate when and how will the COVID-19 in the US be evolved to endemic. We employed a nonlinear ordinary differential equations-based model to simulate COVID-19 transmission dynamics, factoring in vaccination efforts. Multi-stability analysis was performed on daily new infection data from January 12, 2021 to December 12, 2022 across 50 states in the US. Key indices such as eigenvalues and the basic reproduction number were utilized to evaluate stability and investigate how the pandemic COVD-19 will evolve to endemic in the US. The transmissional, recovery, vaccination rates, vaccination effectiveness, eigenvalues and reproduction numbers (R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}$$\\end{document} and R0end\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}^{end}$$\\end{document}) in the endemic equilibrium point were estimated. The stability attractor regions for these parameters were identified and ranked. Our multi-stability analysis revealed that while the endemic equilibrium points in the 50 states remain unstable, there is a significant trend towards stable endemicity in the US. The study's stability analysis, coupled with observed epidemiological waves in the US, suggested that the COVID-19 pandemic may not conclude with the virus's eradication. Nevertheless, the virus is gradually becoming endemic. Effectively strategizing vaccine distribution is pivotal for this transition.

Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization

Abstract Interest in the thermal effects of nanofluid (NF) has increased recently due to the use of nanocomposites to magnify the thermal conductivity of conventional liquids and so boost the heat transit phenomena. Based on this fundamental concept, the current study inspects the thermal advanced third-grade fluid flow with nanocomposites with an extended surface and the inclusion of stratification, non-Fourier heat flux, mass flux, and radiation. Buongiorno’s NF model is employed to observe the thermophoresis and Brownian motion properties. The gyrotactic microorganisms, which are connected to the bioconvection phenomenon that intrigues most, are also considered to be present in the nanoparticles. The governing models are composed of partial differential equations; thereafter, the relevant transformations are applied to these equations to convert the structure into an ordinary differential model. These resultant models are solved by implementing the homotopy analysis method. It is explained in detail how the pertinent parameters are affecting the motion, temperature of fluid, nanocomposite volume, dynamic microbe density, skin friction rates, local Nusselt, and local Sherwood numbers. Applications for the flow of nanoparticles carrying gyrotactic microorganisms include enzyme biosensors, microfluidic devices, microbial fuel cells, and biotechnology.

Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives

Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.

A new approach for the globally optimal design of gasketed plate heat exchangers with variable properties

In this paper, we present two novel advances in the design optimization of gasketed plate exchangers. Our first novelty is the use of an ordinary differential model that represents the heat transfer and pressure drop along the surface of the plate considering the variation of the thermophysical properties (density, heat capacity, viscosity, etc.) with temperature. Previous literature did not address this aspect of the problem or addressed it only partially. We also take into account all pass arrangements. Although there is no novelty in this, it adds to making our model comprehensive. The second novelty is the design optimization method used: Partial Set Trimming and Smart Enumeration. This method guarantees global optimality, it is robust (no convergence issues), and it is fast. A new feature added to the Set Trimming methodology is Proxy Set Trimming. The advantage of Set Trimming followed by Smart Enumeration is that there is no need to discretize the differential equations and build very large models with the resulting large number of variables and algebraic equations, rendering highly non-convex models difficult to solve using MINLP procedures. A comparison of the performance of the proposed procedure with two different metaheuristic methods showed that this approach presents better computational performance.

Global dynamics of a tumor-immune model with an immune checkpoint inhibitor

In this paper, a mathematical ordinary differential tumor-immune model is proposed based on an immune checkpoint inhibitor, which is an innovative method for tumor immunotherapies. Two important factors in tumor-immune response are the programmed cell death protein 1 (PD-1) and its ligand PD-L1. The model consists of three populations: tumor cells, activated T cells and anti-PD-1. By analyzing the dynamics of the model, it is found that there is always a unique tumor-free equilibrium and at most two tumor interior equilibria. The nonexistence of nontrivial positive periodic orbits is established by using the new Dulac function, and then a global dynamics of the model is obtained. The conclusions of our analysis show that increasing the possibility of T cells killing tumor cells ([Formula: see text]), early detection of tumor cells, or the use of PD-1 inhibitors to activate T cells are effective in eliminating tumor cells.

A Novel Ergodic Cellular Automaton Model of Gene-Protein Network: Theoretical Nonlinear Analyses and Efficient FPGA Implementation

A novel ergodic cellular automaton model of gene-protein network is presented. It is shown that the presented model can predict occurrences of typical nonlinear phenomena of a conventional ordinary differential equation gene-protein network model. In addition, theoretical analysis methods of the presented model are proposed. Using the analysis methods, an important advantage of the presented model is revealed: the ergodic cellular automaton is better suited to predict the occurrences of the nonlinear phenomena of the differential equation gene-protein network model compared to a regular (standard) cellular automaton. Furthermore, the presented model is implemented by a field programmable gate array and experiments validate its operations. It is then revealed that the presented model is much more hardware-efficient compared to a standard numerical integration formula of the differential equation model.

Mathematical Modeling of MHD Flow of CNTs/Ag Nanoparticles Past a Heated Stretchy Surface with Nonuniform Heat Sink/Source and Thermal Radiation Impacts

This research communication intends to evaluate the impact of time‐dependent MHD Darcy–Forchheimer flow of CNTs/Ag nanoparticles on a heated stretchy surface. Water is employed as a base fluid, and two types of CNTs such as single‐ and multiwall carbon nanotubes are considered. The significance of nonlinear radiation and heat sink/source is added to our analysis. To accommodate the suitable variables, the governing nonlinear partial differential models are transformed into a set of ordinary differential models. These resulting models are solved analytically and numerically by utilizing the homotopy analysis technique and the bvp4c procedure in MATLAB. The distinctive behavior of pertinent physical parameters on the dimensionless profiles are displayed and discussed through diagrams, tables, and charts. It is discovered that the velocity profile decrepitude whenever there is a change in the unsteady, porosity, and injection/suction parameters. The space and temperature‐dependent heat sink/source parameter cause to elevate the thermal profile. The Ag nanoparticles have a lesser surface shear stress compared to both CNTs. The heat transfer gradient develops for larger quantities of radiation and temperature ratio parameters. This research has significant applications in many industrial sectors, such as thermal exchangers, chemical reactors, microelectronics, biomedical engineering, aerodynamics, and industrial production processes.

Designing Meyer wavelet neural networks for the three-species food chain model

<abstract> <p>The current research work is related to present the numerical solutions of three-species food chain model (TS-FCM) by exploiting the strength of Meyer wavelet neural networks (MWNNs) along with the global and local search competencies. The particle swarm optimization technique works as a global operator, while the sequential quadratic programming scheme is applied as a local operator for the TS-FCM. The nonlinear TS-FCM is dependent upon three categories, called consistent of prey populations, specialist predator and top predator. The optimization of an error-based fitness function is presented by using the hybrid computing efficiency of the global and local search schemes, which is designed through the differential form of the designed ordinary differential model and its initial conditions. The proposed results of the TS-FCM are calculated through the stochastic numerical techniques and further comparison is performed by the Adams method to check the exactness of the scheme. The absolute error in good ranges is performed, which shows the competency of the proposed solver. Moreover, different statistical procedures have also been used to check the reliability of the proposed stochastic procedure along with forty numbers of independent trials and 10 numbers of neurons.</p> </abstract>

A combination therapy of oncolytic viruses and chimeric antigen receptor T cells: a mathematical model proof-of-concept

<abstract><p>Combining chimeric antigen receptor T (CAR-T) cells with oncolytic viruses (OVs) has recently emerged as a promising treatment approach in preclinical studies that aim to alleviate some of the barriers faced by CAR-T cell therapy. In this study, we address by means of mathematical modeling the main question of whether a single dose or multiple sequential doses of CAR-T cells during the OVs therapy can have a synergetic effect on tumor reduction. To that end, we propose an ordinary differential equations-based model with virus-induced synergism to investigate potential effects of different regimes that could result in efficacious combination therapy against tumor cell populations. Model simulations show that, while the treatment with a single dose of CAR-T cells is inadequate to eliminate all tumor cells, combining the same dose with a single dose of OVs can successfully eliminate the tumor in the absence of virus-induced synergism. However, in the presence of virus-induced synergism, the same combination therapy fails to eliminate the tumor. Furthermore, it is shown that if the intensity of virus-induced synergy and/or virus oncolytic potency is high, then the induced CAR-T cell response can inhibit virus oncolysis. Additionally, the simulations show a more robust synergistic effect on tumor cell reduction when OVs and CAR-T cells are administered simultaneously compared to the combination treatment where CAR-T cells are administered first or after OV injection. Our findings suggest that the combination therapy of CAR-T cells and OVs seems unlikely to be effective if the virus-induced synergistic effects are included when genetically engineering oncolytic viral vectors.</p></abstract>

Optimal control for an ordinary differential equation online social network model

Optimal control for an ordinary differential equation online social network model

Computational Insights of Bioconvective Third Grade Nanofluid Flow past a Riga Plate with Triple Stratification and Swimming Microorganisms

The goal of this study is to examine the heat‐mass effects of a third grade nanofluid flow through a triply stratified medium containing nanoparticles and gyrostatic microorganisms swimming in the flow. The heat and mass fluxes are considered as a non‐Fourier model. The governing models are constructed as a partial differential system. Using correct transformations, these systems are converted to an ordinary differential model. Ordinary systems are solved using convergent series solutions. The effects of physical parameters for fluid velocity, fluid temperature, nanoparticle volume percentage, motile microbe density, skin friction coefficients, local Nusselt number, and local Sherwood number are all illustrated in detail. When the values of the bioconvection Lewis number increase, the entropy rate also rises. The porosity parameter and modified Hartmann number show the opposite behaviour in the velocity profile.

A two diffusion stochastic model for the spread of the new corona virus SARS-CoV-2.

We propose a refined version of the stochastic SEIR model for epidemic of the new corona virus SARS-Cov-2, causing the COVID-19 disease, taking into account the spread of the virus due to the regular infected individuals (transmission coefficient ), hospitalized individuals (transmission coefficient , ) and superspreaders (transmission coefficient ). The model is constructed from the corresponding ordinary differential model by introducing two independent environmental white noises in transmission coefficients for above mentioned classes - one noise for infected and hospitalized individuals and the other for superspreaders. Therefore, the model is defined as a system of stochastic differential equations driven by two independent standard Brownian motions. Existence and uniqueness of the global positive solution is proven, and conditions under which extinction and persistence in mean hold are given. The theoretical results are illustrated via numerical simulations.

An Advanced Susceptible-Exposed-Infectious-Recovered model for quantitative analysis of COVID-19

The infectious diseases with a viral origin are of significant worldwide concern. In recent times, pandemics are creating havoc across the entire globe. This paper presents a constructive analysis of a new mathematical concept that will help the medical authorities to predict and to take controlling measures. In this work, we use ordinary first-order differential equations and compartmental model analysis for calculation of infection rate, transmission rate, and reproduction number of the patients. A new Advanced Susceptible-Exposed-Infectious-Recovered model has been introduced, which has greater accuracy of the reproduction number. The prediction of mode of disease transmission demonstrates the performance characteristics of the proposed model.

Neural ordinary differential grey model and its applications

Due to the efficiency of grey models in predicting the time series of small samples, grey system theory has been well studied since it was first proposed and has now become an important method for small sample prediction. Inspired by the neural ordinary differential equations (NODE), this paper proposes a novel grey forecasting model called the neural ordinary differential grey model (NODGM). The NODGM model includes a novel whitening equation that allows the prediction model to be learned through a training procedure. Therefore, compared with other models whose structures and terms need to be artificially predefined based on the laws of real samples, the NODGM model has a wider application range and can learn the characteristics of different data samples. Then, to obtain a model with better prediction performance, we apply NODE to train the model. Finally, the predicting sequence is obtained by using the Runge–Kutta method to solve the model. In the experiments, we apply NODGM model to two energy samples and then contrast the experimental results with the results of some classical grey models to validate the effectiveness of NODGM model. The comparison results from predicting China’s annual crude oil consumption and forecasting oilfield production in northern China show that the prediction accuracies achieved by NODGM model are 28% and 8% higher, respectively, than those achieved by the state-of-the-art grey forecasting models.

A fractional model of cancer-immune system with Caputo and Caputo-Fabrizio derivatives.

Recently, it is important to try to understand diseases with large mortality rates worldwide, such as infectious disease and cancer. For this reason, mathematical modeling can be used to comment on diseases that adversely affect all people. So, this paper discuss mathematical model presented for the first time that examines the interaction between immune system and cancer cells by adding IL-12 cytokine and anti-PD-L1 inhibitor. The proposed ordinary differential new mathematical model is studied by considering in term of Caputo and Caputo–Fabrizio (CF) derivative. Stability analysis, existence, and uniqueness of the solution is examined for Caputo fractional derivative. Then numerical simulations of ordinary and fractional differential new mathematical model are given. It is obtained that a reduction (20%–80%) of the number of cancer cells for Caputo derivative and (100%) of the number of cancer cells for CF derivative. The reduction is one of the most important aspects of the new fractional model for the order discussed especially obtained for CF derivative.

Insight into 2-step continuous block method for solving mixture model and SIR model

Understanding of the solutions of first-order ordinary differential equations (ODEs), mixture model and SIR model in order to develop deep insight and exploration are major problems before the experts, biologists, scientists, and mathematicians. In all these problems, the governing equations are either single first-order or coupled ODEs kind of initial value problem (IVP). In this paper, a polynomial function q(x) that passes through the points (xn, yn), (xn+1, yn+1), . . . , (xn+2, yn+2) was adopted as the basis function that leads to third derivatives continuous 2-step block method suitable to solve first order initial value problems of ODEs. Upon using the newly proposed scheme to solve linear ODEs (i.e., mixture theory) and nonlinear ODE (i.e., SIR model), it is worth concluding that the algorithm is not only efficient but minimises error.

Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical linear ordinary differential equations linear fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of ±20 dB per frequency decade and, respectively, deviate from integer multiples of ±π/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.

A mathematical model of asynchronous data flow in parallel computers*

AbstractWe present a simplified model of data flow on processors in a high-performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton–Jacobi equation associated with this conservation law for which the existence and uniqueness of solutions can be proven. We then present the results of numerical experiments for both discrete and continuum models; these show a qualitative agreement between the two and the effect of variations in the computing environment’s processing capabilities on the progress of the modelled computation.

Epidemic dynamics of influenza-like diseases spreading in complex networks

Population contact pattern plays an important role in the spread of an infectious disease. This can be described in the framework of a complex network approach. In this paper network epidemic models for influenza-like diseases that may have infectious force in incubative or asymptomatic stage are formulated and studied. Two general types of network models are considered: the annealed and the quenched networks. The next-generation matrix approach is employed to compute the basic reproduction number of our network-based models. The implicit equations for the final epidemic size are derived, and the existence and uniqueness of solutions for implicit equations are studied by rewriting implicit equations as suitable fixed-point problems. In particular, for networks with no degree correlation, low-dimensional systems of nonlinear ordinary differential model are derived by employing an edge-based compartmental approach. Due to their low dimension, a gap between the parameter identification problem for influenza-like diseases or network inference and network epidemic models may be built through our results. The analysis is applied to an example of influenza epidemic based on the final epidemic size, from which the transmission rate and the basic reproduction number can be estimated.

Dynamics of bioconvection flow of micropolar nanoparticles with Cattaneo-Christov expressions

A numerical analysis is performed to analyze the bioconvective double diffusive micropolar non-Newtonian nanofluid flow caused by stationary porous disks. The consequences of the current flow problem are further extended by incorporating the Brownian and thermophoresis aspects. The energy and mass species equations are developed by utilizing the Cattaneo and Christov model of heat-mass fluxes. The flow equations are converted into an ordinary differential model by employing the appropriate variables. The numerical solution is reported by using the MATLAB builtin bvp4c method. The consequences of engineering parameters on the flow velocity, the concentration, the microorganisms, and the temperature profiles are evaluated graphically. The numerical data for fascinating physical quantities, namely, the motile density number, the local Sherwood number, and the local Nusselt number, are calculated and executed against various parametric values. The microrotation magnitude reduces for increasing magnetic parameters. The intensity of the applied magnetic field may be utilized to reduce the angular rotation which occurs in the lubrication processes, especially in the suspension of flows. On the account of industrial applications, the constituted output can be useful to enhance the energy transport efficacy and microbial fuel cells.