Mathematical Stability Analysis Research Articles

Fractional-Order Mathematical Modeling and Stability Analysis of Tumor-Immune System Relation Including Chemotherapy Drug Effect

Cancer diseases rank second in global mortality rates. Among the prevalent approaches for treating tumors, chemotherapy stands out as a prominent method capable of reducing tumor size and managing the advancement of cancer ailments. To gain deeper insights into the intricacies of chemotherapy mechanisms, we devised a Fractional-order mathematical model, depicting tumor growth in the presence of chemotherapy. This comprehensive model encompasses both the immune system's response and the effects of drug therapy. To demonstrate that the system is biologically meaningful, we examined the existence and uniqueness of solutions and proved the positivity and boundedness of solutions. We characterize the dynamics properties of this differential equation model by finding the equilibrium points and exploring the stability conditions in a range of model parameters. In addition, we have performed numerical simulations considering different parameter values. Furthermore, to demonstrate the memory effect of the fractional derivative, we have simulated the dynamic behavior of the system for different orders of fractional derivatives. So, we have concluded that the memory effect occurs as the α decreases from 1 and the chemotherapy drug is quite effective on the populations. We hope that this work will contribute to helping medical scientists take the necessary measures during the screening process and treatment.

Comprehensive analysis of COVID-19 transmission dynamics: mathematical modeling, stability analysis, and optimal control strategies

This paper presents a mathematical model for comprehensively analyzing the transmission dynamics of COVID-19. We investigate the model’s various properties, such as positivity, boundedness, and the existence and uniqueness of solutions. Additionally, we calculate the basic reproductive number, denoted as R 0, to gauge the epidemic’s potential spread. Furthermore, we conduct a stability analysis to understand the long-term behavior of the model. Furthermore, we devised an optimal control strategy to effectively curb disease transmission. Using graphical analysis, we assess the impact of secondary infection rates and quarantine rates across different population groups. Finally, we compare our proposed numerical scheme with the well-known RK-4 scheme, emphasizing the NSFD scheme’s ability to maintain positivity, unlike the RK-4 scheme. Our numerical simulations offer strong evidence supporting the theoretical findings, demonstrating the effectiveness of our results.

Mathematical Model and Stability Analysis for Dusty-hybrid Nanofluid over a Curved Surface with Cattaneo–Christov Heat Flux and Fourier law with Slip Effect

Mathematical Model and Stability Analysis for Dusty-hybrid Nanofluid over a Curved Surface with Cattaneo–Christov Heat Flux and Fourier law with Slip Effect

Mathematical analysis of stability and Hopf bifurcation in a delayed HIV infection model with saturated immune response

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results.

Propagation of solitary wave solutions to (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation arise in mathematical physics and stability analysis

This innovative study simulates the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation, which has anomalous applications in the field of mathematical physics. The current research has two basic pillars. Firstly, numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, periodic, combo, W-shape, mixed trigonometric, exponential, and rational, based on the modified sardar sub-equation (MSSE) method and the improved F-expansion method. Secondly, the stability analysis of the selected model is manifested to study modulation instability (MI) gain. Furthermore, for the physical demonstration of the acquired solutions in 3D and 2D, contour and density plots are depicted. The discovered results have a distinctive feature that has not been developed before and indicate a good balance between the nonlinear physical components. Also, the resulting structure of the acquired results can enrich the nonlinear dynamical behaviours of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

A bridgeless configured asymmetrical alternating current–direct current converter‐based isolated single‐stage electric vehicle battery charger with supply side power factor enhancement

AbstractAn approach is presented to employ two different types of converters in bridgeless configuration for supply side power factor enhancement of the system. The isolated single‐stage electric vehicle battery charger uses two different converters in a bridgeless configuration to extract the advantages of both converters for supply‐side power factor enhancement. For the negative and positive semi‐cycles of the supply voltage, the power factor‐enhanced asymmetrical alternating current–direct current converter utilises a fourth order single‐ended primary‐inductor converter and a second order buck‐boost converter, respectively. The use of single‐ended primary‐inductor converter and buck‐boost converter in bridgeless configuration reduces the net order of the system with respect to conventional bridgeless‐single‐ended primary‐inductor converter schemes. The buck‐boost converter also needs the supply‐side filter to eradicate the unwanted harmonics in the supply current which increases the order of the system. The usage of both converters presents many benefits like input inductance of the single‐ended primary‐inductor converter can be utilised as a filtering element with a capacitor for the buck‐boost converter. The anti‐parallel diode conduction operation of both switches facilitates the elimination of extra reverse feed diodes (generally used in bridgeless schemes). The single‐stage charger itself comes with the benefit of elimination of extra stages and thus the losses associated with it. The presented charger also witnesses the elimination of the rectifier due to usage of bridgeless configuration. The isolated single‐stage electric vehicle battery charger is also garnished with electrical isolation which adds to the safety standard of the system. To attain power factor enhancement, the asymmetrical alternating current–direct current converter functions in discontinuous current conduction mode in the present work. The elimination of extra‐stages (with respect to two stage charger), a filter, a rectifier, two extra reverse‐feeding diodes, one voltage sensor, one current sensor (with respect to continuous current conduction mode), and electrical isolation not only makes the system compact and safer but also makes the system cheaper. Elaborated mathematical modelling and stability analysis of the presented alternating current–direct current converter using a pole‐zero map and bode plot have been included in the article. The prototype and MATLAB/Simulink model of isolated single‐stage electric vehicle battery charger system with discontinuous current conduction mode control has been built and results of both prototype and MATLAB/Simulink are deployed to verify isolated single‐stage electric vehicle battery charger system's performance during dynamic and steady‐state conditions.

Mathematical Modeling and Stability Analysis of Systemic Risk in the Banking Ecosystem

This paper investigates the dynamics of systemic risk in banking networks by analyzing equilibrium points and stability conditions. The focus is on a model that incorporates interactions among distressed and undistressed banks. The equilibrium points are determined by solving a reduced system of equations, considering both homogeneous and heterogeneous scenarios. Local and global stability analyses reveal conditions under which equilibrium points are stable or unstable. Numerical simulations further illustrate the dynamics of systemic risk, while the theoretical findings offer insights into the behavior of distressed banks under varying conditions. Overall, the model enhances our understanding of systemic financial risk and offers valuable insights for risk management and policymaking in the banking sector.

Mathematical Modeling and Stability Analysis of the Delayed Pine Wilt Disease Model Related to Prevention and Control

Forest pests and diseases have been seriously threatening ecological security. Effective prevention and control of such threats can extend the growth cycle of forest trees and increase the amount of forest carbon sink, which makes a contribution to achieving China’s goal of “emission peak and carbon neutrality”. In this paper, based on the insect-vector populations (this refers to Monochamus alternatus, which is the main vector in Asia) in pine wilt disease, we establish a two-dimensional delay differential equation model to investigate disease control and the impact of time delay on the effectiveness of it. Then, we analyze the existence and stability of the equilibrium of the system and the existence of Hopf bifurcation, derive the normal form of Hopf bifurcation by using a multiple time scales method, and conduct numerical simulations with realistic parameters to verify the correctness of the theoretical analysis. Eventually, according to theoretical analysis and numerical simulations, some specific suggestions are put forward for prevention and control of pine wilt disease.

Multiple solutions in magnetohydrodynamic stagnation flow of hybrid nanofluid past a sheet with mathematical chemical reactions model and stability analysis

The homogeneous heterogeneous reactions (H–H reactions) in the magnetohydrodynamic (MHD) boundary layer stagnated flow of an Al2O3–Cu–water base hybrid nanofluid past a stretching shrinking sheet are studied. A newly developed two-phase hybrid nanofluid model based on Buongiorno's model is used to understand the nanofluids behaviors. Multiple solutions are observed for specific ranges of various parameters, whose stabilities are checked and discussed, which seem to have been neglected in previously published articles about studies of MHD-stagnation flow and mathematical chemical reactions models using hybrid nanofluid. The heterogeneous reaction considered in this case is isothermal and first order, whereas the homogeneous chemical reaction is isothermal cubic autocatalytic. The solutions so captured are examined using various graphs to demonstrate the impact of different physical parameters, and their physical insights are also given. The results show that hybrid nanofluids, which have distinct functions in the processes of homogeneous and heterogeneous reactions, play a key role in the homogeneous–heterogeneous reactions' transport mechanism. The quadratic multiple regression analysis evaluations of the local Nusselt number demonstrate that the thermophoretic impact predominates over Brownian motion for both magnetic and non-magnetic effects.

An Evaluation of Mathematical Models and Stability Analysis of Learning Based on Reaction Kinetics

Human cognition and consciousness are perhaps the most confounding mystery. Somehow it has a linkage to the process of learning and storage of short-term and long-term memory in the form of knowledge. This paper examines a brief background of early models in learning presented by Atkinson and Shriffrin (1965) and related stochastic models utilizing probability functions. Each of these learning models capture certain facets of the learning process but are ineffective in describing the physical basis in which learning occurs. For this reason, this paper explores analogous mathematical models based on reaction kinetics that have been shown to represent chemical reactions found in nature. Six learning models are presented of unitary, binary, reversible binary, reversible binary with mass action, and enzyme learning model reactions with and without decay. Preliminary analysis of time series plots, phase line diagrams, and phase plane plots were conducted to illustrate equilibrium conditions and stability of the models. Each model is examined in terms of its limitations in the philosophy and inability to capture certain elements that are understood about the learning process. Finally, this paper concludes that the feasibility of understanding behavior such as stability through the tools of applied mathematics and thereby illuminating certain layers of human cognition and learning is a useful tool in examining the suitability of a possible deterministic model that could describe the learning process. Further analysis with empirical data would validate the suitability of the presented models.

A Bridgeless Cuk-BB-Converter-Based BLDCM Drive for MEV Applications

This article presents a brushless DC motor (BLDCM) drive for a maritime electric vehicle (MEV) application. The presented BLDCM drive uses a bridgeless Cuk-buckboost (BL-Cuk-BB) converter for input-side power factor (PF) improvement. The BL-Cuk-BB converter uses the buckboost converter for the negative half-cycles of the input AC voltages and the Cuk converter for the positive half-cycles. In the case of MEVs, the drive systems are generally fed by diesel engine generators (DEGs). The asymmetric BL-Cuk-BB converter is operated in a discontinuous inductor current mode (DICM) in the present work to attain better power quality. The usage of a second-order buckboost converter with a fourth-order Cuk converter results in a decrement in the net order of the system. Additionally, the input inductor of the Cuk converter also participates as the filter component along with capacitor C2 during buckboost converter operation to enhance the power quality. The total component count reduction in the BL-Cuk-BB converter is also achieved by eliminating the usage of extra/external back-feeding diodes, which are generally used in bridgeless schemes. The present scheme uses the inbuilt anti-parallel diodes for the same purpose. The lesser components requirement in the BL-Cuk-BB-converter-based BLDCM drive implies lesser cost and volume, along with greater reliability, lower conduction losses, and lower weight of the BLDCM drive, which adds to the merits of the model. The paper includes a detailed mathematical model and stability analysis using pole-zero maps and bode plots of the BL-Cuk-BB converter for each half-supply AC voltage cycle. The BL-Cuk-BB-converter-based BLDCM drive for an EV application has been developed on the MATLAB/Simulink platform for a DICM operation, and the MATLAB simulation results have been presented for validation of the BL-Cuk-BB-converter-based BLDCM drive.

Mathematical model and stability analysis of university-PhD-postdoc-industry migration and unemployment in south Africa

Mathematical model and stability analysis of university-PhD-postdoc-industry migration and unemployment in south Africa

Changing the paradigm in modelling the Bray-Liebhafsky oscillatory chemical reaction.

The Bray-Liebhafsky (BL) reaction is one of the simplest chemical oscillators consisting initially of only three components. Despite this, its mechanism is unknown for more than 100 years due to the absence of selective, sensitive, and fast experimental techniques for following all of the involved intermediates. The modeling of the BL mechanism assumes presumably mass action kinetics "adjustable" to oscillatory solutions by the application of mathematical stability analysis and treating the system as homogeneous. Such a basically mathematical approach need not suggest physically realistic kinetic parameters and is not unique since a number of models can be proposed. Based on recent experimental and computational results, a new model of the BL oscillatory reaction mechanism is constructed by including heterogeneous processes occurring in the system. The same set of equations is able to demonstrate not only the oscillatory evolution but also mixing effects on the oscillatory dynamics, and non-oscillatory stepwise-iodine oxidation and can rationalize other effects described in literature. Thus, the paradigm of treating the BL oscillatory system as a homogeneous one, described by formal kinetics only, is extended for a better understanding of the chemistry of this apparently simple system. The introduced ideas of energy redistribution may contribute to establishing an improved conceptual base for considering other complex oscillators in various fields of science.

Mathematical Modeling and Stability Analysis of the COVID-19 Spread by Considering Quarantine and Hospitalize

A virus that attacks the human respiratory system that first appeared in the province of Wuhan, China is known as COVID-19 (SARS COV2 n-corona virus). In order to anticipate the increasing of the cases, a strategy is needed to inhibit its growth and spread. Seeing the projection of future situations becomes very important, so we can anticipate with a selection of policy scenarios. The dynamical system model approach is very important for predicting future situations as well as selectable scenarios based on simulation results. In this study, we develop a COVID-19 transmission model which can be used to predict the epidemiological outcome and simultaneously to evaluate the effect of quarantine and hospitalization to COVID-19 spread. The mathematical model of the transmission of COVID-19 was developed in the form of the non-linear differential equation system, with seven variables, namely susceptible, exposed, infected, quarantined-1 (exposed individuals who were quarantined), quarantined-2 (infected individuals who were quarantined), hospitalized and recovered. The proposed model has a non-endemic and endemic equilibrium point. Local stability analysis of the non-endemic equilibrium point was investigated using the Routh-Hurwitz criterion, while global stability of the endemic equilibrium point was analyzed by using the Lyapunov method. If the basic reproduction number is less than one, then non-endemic equilibrium point is stable. On the other hand, if the basic reproduction number is greater than one, then endemic equilibrium point is stable. Verification of the developed model was carried out through numerical simulations using data from Central Java Province, Indonesia. We have investigated that parameter related to quarantine and hospitalize affect the number of new infections COVID-19 and the basic reproduction number. From the simulation results, it was found that strict quarantine and hospitalize have the potential to succeed in reducing and inhibiting the transmission of the COVID-19. It can be used by the government in making policies to increase the implementation of quarantine and hospitalize in the community.

Chewing Khat Transmission Dynamics: A Mathematical Model and Stability Analysis

In this study, the authors proposed a nonlinear deterministic model and stability analysis for the transmission dynamics of chewing khat. The model’s solution is proved to be positive and bounded, and the basic reproduction number ( R 0 ) is calculated using the next-generation matrix method. Following that, the authors have looked at the local and global stability of the model’s khat-free and endemic equilibrium points. When R 0 < 1 , the chewing khat-free equilibrium point is locally and globally asymptotically stable, whereas when R 0 > 1 , the endemic equilibrium point is locally and globally asymptotically stable. The simulation results demonstrate the analytical results.

Mathematical Modeling and Stability Analysis of an Effective Design of Biomimetic AUV

Mathematical Modeling and Stability Analysis of an Effective Design of Biomimetic AUV

Mathematical Modeling and Stability Analysis of a Delayed Carbon Absorption-Emission Model Associated with China’s Adjustment of Industrial Structure

Global warming has brought about enormous damage, therefore, some scholars have begun to conduct in-depth research on peak carbon dioxide emissions and carbon neutrality. In this paper, based on the background of China’s upgrading industrial structure and energy structure, we establish a delayed two-dimensional differential equation model associated with China’s adjustment of industrial structure. Firstly, we analyze the existence of the equilibrium for the model. We also analyze the characteristic roots of the characteristic equation at each equilibrium point for the model, then, we analyze the stability of the equilibrium point for the model according to the characteristic root, and discuss the existence of Hopf bifurcation of the system by using bifurcation theory. Secondly, we derive the normal form of Hopf bifurcation by using the multiple time scales method. Then, through the official real data, we present the range of some parameters in the model, and determine a set of parameters by reasonable analysis. The validity of the theoretical results is verified by numerical simulations. Finally, we use the real data to forecast the time of peak carbon dioxide emissions and carbon neutralization. Eventually, we put forward some suggestions based on the current situation of carbon emission and absorption in China, such as planting trees to increase the growth rate of carbon absorption, deepening industrial reform and optimizing energy structure to reduce carbon emissions.

A BL-CC Converter-Based BLDC Motor Drive for Marine Electric Vehicle Applications

This article presents a BLDC (permanent magnet brushless DC) motor drive for marine electric vehicle (MEV) application. The presented scheme uses bridgeless canonical cell (BL-CC) converter with center-tapped inductor (shown as two separate inductors for analysis purpose) for source-side power factor correction (PFC); however, the BL-CC converter requires two inductors. In the presented scheme use of one center-tapped inductor eliminates the requirement of one extra inductor. Thus, center-tapped inductor usage in BLDC motor drive results in decrement in components count. The BL-CC converter-based BLDC motor drive scheme do not require extra back-feeding diodes like other bridgeless (BL) schemes, but it uses inbuilt antiparallel diodes of insulated gate bipolar transistor (IGBT) switches for the same purpose this again leads to decrement in required component count. In this work, PFC BL-CC converter is operated in discontinuous inductor current (DIC) mode to attain better power quality. The DIC mode operation requires only one voltage sensor to sense DC-link voltage, whereas in continuous conduction mode (CCM), the sensor requirement increases to three (two voltage sensors and one current sensor). The PFC BL-CC converter also eliminates the diode bridge rectifier stage. The elimination of one extra inductor, two extra back-feeding diodes, DBR stage and also requirement of only one voltage sensor in DIC mode operation instead of three sensors for CCM implies reduction in components count which implies the reduction in cost and also the volume and weight of the BLDC motor drive. The weight reduction for marine (on-board) electric vehicle is very important as this enhances the vehicle performance. This paper also presents the detailed mathematical modeling and stability analysis of presented BL-CC converter using pole zero map and Bode plot. The BL-CC converter-based BLDC motor driving system for MEV application for DIC mode operation has been developed in the laboratory as well as on MATLAB/Simulink platform and simulated MATLAB and real-time experimental results have been presented to validate the presented BLDC motor drive under steady-state and dynamic operating conditions.

Application of Neural Network in the Stability of Biped Robot and Embedded Control of Walking Mode

The biped robot adopts the human movement mode. Compared with other movement modes, the gait has good flexibility and adaptability. It is very important in the research of robotics, so it has become a hot spot of robotics research. This article aims to study the application of neural networks in the stability of biped robots and the embedded control of walking mode. A method of establishing precise mathematical modeling and stability analysis is proposed. Based on this model, the motion characteristics of the biped robot’s walking mode and the local stability of joints are studied, and the motion mode of passive walking under the control of the neural network is deeply analyzed, using a neural network to control the stability of biped robot motion and adopting the research method of the embedded control system in walking mode. Essentially, the output value of the physical network is used to judge whether the robot is in a stable position so as to perform appropriate actions and control the robot’s stability of walking. The experimental results show that the biped robot can detect movement and overcome obstacles through related networks and embedded control systems. Through the control of the embedded system, the errors of each joint of the biped robot on flat ground, stairs, and obstacles are greatly reduced. The most obvious reduction of the deviation is that the ankle joint decreases from 2.5 to 0.07 when rotating, and the knee joint angle deviation is reduced from 3.8 to 2, which greatly improves the stability of the biped robot’s walking mode.