Routh-Hurwitz Criterion Research Articles

Mathematical Modeling of COVID‐19 Disease Dynamics With Contact Tracing: A Case Study in Ethiopia

In this paper, we developed a mathematical model for the dynamics of coronavirus disease (COVID‐19) transmission. The model embraces the notion of contact tracing and contaminated surfaces which are vital for disease control and contribute to disease transmission, respectively. We analyzed the model properties such as the positivity of the solution, invariant region, existence, and stability nature of equilibria. Besides, we computed the basic reproduction number R0. The local stability and the global stability of disease‐free equilibrium (DFE) points are proved by using the Routh–Hurwitz criteria and the Castillo‐Chavez and Song approach, respectively. LaSalle’s invariant principle is applied to prove the stability of an endemic equilibrium (EE) point. The possibility of bifurcation is discussed using the center manifold theory. We used real data on the spread and control of COVID‐19 disease in Ethiopia. Based on the data reported, we estimated the values of the parameters using the least squares method together with the fmin function in the MATLAB optimization toolbox. The sensitivity analysis of the model is explored numerically to illustrate the impact of the parameters on disease transmission. The study addressed that contact tracing is especially important because COVID‐19 often has asymptomatic carriers, and there are many asymptomatic individuals unaware in Ethiopia. The new infections would decrease in the communities by detecting and isolating COVID‐19 cases before they could spread the virus to others. Moreover, the study endorsed that the contaminated surface has contributed to disease transmission. The sensitivity analysis shows that if the rate of disinfected contaminated objects (ϕ) rises, then the transmission of the disease is reduced. Consequently, this study will aid in the fight against COVID‐19 policymakers and NGOs. It can also be used as a policy input for different countries under this crisis. Because of the mathematical modeling of this global pandemic, there is another point of view rather than public health research outputs. Additionally, with the concept of contact tracing and contaminated surfaces incorporated into the model, the result provides insight for disease prevention.

A Stability Study by Routh-Hurwitz Criterion and Gershgorin Circles for Covid-19

A Stability Study by Routh-Hurwitz Criterion and Gershgorin Circles for Covid-19

Resilient Frequency-shifted IMC-PID Scheme for Mitigating Scaling and Random Attacks in Thermal Plants Amid Delay

In recent times, several network-related scenarios like Communication Delays (CDs) and malicious Cyber Attacks (CAs) have posed a major challenge in keeping the operational frequency of automatic generation systems within admissible levels. Hence, the proposed work focuses on investigating the applicability of an Frequency-Shifted Internal Model Control-Proportional Integral Derivative (FIMC-PID) controller for Load Frequency Control (LFC) of a non-identical dual-area Power System (PS) in the presence of inherent CDs and CAs. The FIMC approach is based on the integration of concepts like model order reduction and pole-zero shifting of the reduced order transfer function by the constant (ψ) which indicates the amount by which system robustness can be manipulated. In addition to that, an analytical approach for obtaining a stable numerical range of the tunable parameter ψ is introduced using Routh-Hurwitz criteria. This range serves as the input for the Golden Jackal Optimizer (GJO) for finding the optimal value of ψ by minimizing the Integrated Time-Squared Error (ITSE). False Data Injection (FDI) threats to the cyber-physical systems like scaling and random attacks have been modeled and the efficacy of GJO-tuned FIMC strategy in mitigating the same is vindicated. Finally, hardware-in-loop real-time verification of the suggested strategy in mitigating frequency fluctuations is carried out in the OPAL-RT platform.

Mathematical Model of Celebrity Worship Tendency Among K-Pop Fans in South Sulawesi

Celebrity worship is a form of almost obsessional participation in which people idolize and even start to “worship” their favorite celebrities. Twitter reveals that in 2021, Indonesia was ranked fourth as the country with the most number of Korean Pop (K-Pop) fans. In this study, we constructed a SEIRS mathematical model of celebrity worship tendency among K-Pop fans (CWKF) because there have not been specific mathematical models built on the problem of celebrity worship tendency among K-Pop fans. The CWKF free equilibrium point (E0), endemic equilibrium point (E1), and using the Next Generation Matrix for the Basic Reproduction Number (R0) are calculated through analysis. Using the Routh-Hurwitz criteria for equilibrium analysis, we found that E0 is locally asymptotically stable if R0 < 1, and E1 is locally asymptotically stable if R0 > 1. Secondly, we conduct numerical simulations using Maple software, and research data consisting of 204 individuals from various regions in South Sulawesi serves as the basis for this mathematical analysis. The numerical simulation obtained supports the dynamic analysis results that the number of individuals who are K-Pop fans will decrease in the population and disappear within a certain period, even without any special treatment.

Tritrophic fractional model with Holling III functional response

<abstract><p>In this paper, we analyzed the local stability of three species in two fractional tritrophic systems, with Caputo's fractional derivative and Holling type Ⅱ and Ⅲ functional responses, when the prey density has a linear growth. To begin, we obtained the equilibria in the first octant under certain conditions for the parameters. Subsequently, through linearization and applying the Routh-Hurwitz Criterion, we concluded that only the system with Holling type Ⅲ exhibits an asymptotically stable equilibrium point, where the fractional derivative order belongs to the interval $ (0, 1] $. Finally, we obtained the solution of the system with the Holling type Ⅲ functional response, using the multistage homotopic perturbation method, and presented an example that shows the dynamics of the solutions around the stable equilibrium point.</p></abstract>

Dynamics of a diffusive two predators- one prey system

This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

Analytical study on nonlinear frequency response and vibration isolation of multistage clutch damper in automotive powertrain

This paper studies the nonlinear frequency response and vibration isolation performance of the multistage clutch damper at resonance. First, a 2-DOF drivetrain model is established considering the piecewise-linear stiffness, piecewise-linear damping and asymmetric transition angles of the multistage clutch damper. Then, the averaging method is employed to deduce the analytical solution of the nonlinear motion equation, and the stability of the periodic solution is studied using Routh-Hurwitz criterion. The nonlinear behaviors of the oscillator are discussed based on the theoretical results, and the numerical simulation is carried out to verify the credibility of the proposed method. Finally, the sensitivity analysis is conducted to investigate the influence of the system parameters on the amplitude-frequency response of the multistage clutch damper. The results indicate that the isolation performance of the multistage clutch damper is sensitive to the first and second stages of damping and the first-stage transition angles. By increasing the first and second stages of viscous damping by 50% and simultaneously reducing the first-stage angle range by 3° with 1° asymmetry, the isolation performance of the clutch damper is effectively improved, reducing the vibration amplitude at resonance by 28.8%.

Piecewise nonlinear resistor-capacitor shunted josephson junction circuit: dynamical and microcontroller implementation probing

This paper exposes the theoretical and microcontroller implementation probing of the piecewise nonlinear resistor-capacitor shunted Josephson junction circuit (PNRCSJJC). The PNRCSJJC is characterized by no steady state when the applied current is greater than one and exhibits two steady states in which one is a focus and its counterpart a saddle-node for excitation current less than or equal to one with credit to the Routh–Hurwitz criterion. The PNRCSJJC exhibits periodic characteristics, quasi-periodic characteristics, varying structures of chaotic characteristics, and coexisting behaviors which is proved qualitatively by the microcontroller execution method. The polarity of the chaotic signal in the voltage state variable is flexibly altered by varying a constant parameter included in the rate equations of PNRCSJJC.

TRAFFIC CONGESTION ANALYSIS USING SIR EPIDEMIC MODEL

In this work, we propose a mathematical model to represent traffic congestion in the street under some consideration. A congestion problem in a city highway becomes a critical issue since congestion at one point affected congestion propagation on the other points. We focus on the propagation of traffic propagation by adopting the concept of disease spread using the SIR model. We consider that the disease in traffic problems is congestion. Meanwhile, vehicles that enter the highway are susceptible to congestion. In contrast, vehicles free from traffic jams represent individuals free from disease. The SIR model can explain the spread of congestion by looking at the congestion variable as an infected variable. We discuss and analyze the existence and stability of the equilibrium points. The local stability equilibrium point is verified using the Routh-Hurwitz criteria. At the same time, the global stability is analyzed using Lyapunov function. The numerical simulation is provided in the last section to validate the discussion results.

Dynamical analysis for the motion of a 2DOF spring pendulum on a Lissajous curve

This study examines the motion of a spring pendulum with two degrees-of-freedom (DOF) in a plane as a vibrating system, in which its pivot point is constrained to move along a Lissajous curve. In light of the system’s coordinates, the governing equations of motion (EOM) are obtained utilizing the equations of Lagrange’s. The novelty of this work is to use the approach of multiple scales (AMS), as a traditional method, to obtain novel approximate solutions (AS) of the EOM with a higher degree of approximation. These solutions have been compared with the numerical ones that have been obtained using the fourth-order Runge–Kutta algorithm (4RKA) to reveal the accuracy of the analytic solutions. According to the requirements of solvability, the emergent resonance cases are grouped and the modulation equations (ME) are established. Therefore, the solutions at the steady-state case are confirmed. The stability/instability regions are inspected using Routh–Hurwitz criteria (RHC), and examined in accordance with the steady-state solutions. The achieved outcomes, resonance responses, and stability areas are demonstrated and graphically displayed, to evaluate the positive effects of different values of the physical parameters on the behavior of the examined system. Investigating zones of stability/instability reveals that the system’s behavior is stable for a significant portion of its parameters. A better knowledge of the vibrational movements that are closely related to resonance is crucial in many engineering applications because it enables the avoidance of on-going exposure to potentially harmful occurrences.

Parametric resonance of an axially accelerating viscoelastic membrane with a fractional model

In this paper, the parametric resonance of an axially accelerating viscoelastic membrane is examined with a fractional model. The membrane is transported with a time-dependent velocity. The viscoelastic model incorporating fractional derivative into the Kelvin-Voigt relation is considered. The governing equation is formulated from Newton's second law and von-Karman theory, and solved via Galerkin truncation and the multiple-scale method. Furthermore, the frequency-responses of the nonlinear resonance are estimated by eliminating the secular terms, and the Routh-Hurwitz criterion is employed to identify the stability and bifurcation of the steady-state responses. Finally, the numerical results are used to analyze the effects of the fractional order, viscoelastic coefficient and time-dependent velocity on the frequency- responses, solution stability and bifurcation points. The novelty of this study is to introduce a fractional viscoelastic model and time-dependent velocity to analyze the resonance stability of axially accelerating viscoelastic membranes in a roll-to-roll system. The results of this investigation are expected to be useful for quantifying the relative influence of the system parameters in the roll-to-roll manufacturing process.

A generalized Routh–Hurwitz criterion for the stability analysis of polynomials with complex coefficients: Application to the PI-control of vibrating structures

The Routh–Hurwitz criterion is one of the most popular methods to study the stability of polynomials with real coefficients, given its simplicity and ductility. However, when moving to polynomials with complex coefficients, some generalization exist but are either incorrect or inapplicable to most practical cases. To fill this gap, we hereby propose a directed generalization of the criterion to the case of complex polynomials, broken down in an algorithmic form, so that the method is now easily accessible and ready to be applied. Then, we demonstrate its use to determine the external stability of a system consisting of the interconnection between a rotating shaft and a PI-regulator, obtaining the necessary and sufficient conditions to achieve stabilization of the system.

Traveling waves in nonlocal delayed reaction–diffusion bistable equations and applications

In this paper, we are concerned with traveling waves in a class of reaction–diffusion equations with nonlocal delays. By introducing new variables, we transform equations with nonlocal delays to non‐delayed system; the existence of traveling waves is obtained by means of Routh–Hurwitz criterion, and by considering the regularity of the Cauchy problem and using spectral theory, we investigate the global stability of traveling waves and then the uniqueness of wave speeds with the help of upper and lower solution method. Finally, our results are applied to two nonlocal delayed reaction–diffusion equations.

Curtailing the spread of typhoid fever: An optimal control approach

In this paper, a deterministic compartmental model for the control of typhoid fever, which takes into account a partially effective vaccine and drug resistance effect was proposed. We incorporated five control strategies which consist of mass public health education, vaccination, treatment, second-line of treatment, and protection or environmental sanitation to curtail the spread of the disease in the population. The model consists of eight (8) compartments that include: Vaccinated population, susceptible population, exposed population, asymptomatic infected human population, symptomatic infected human population, resistant human population, treated human population, and the bacterial population. We developed a non-linear differential equation to study the dynamics of the model. We computed the basic reproduction number for a case of constant control which can be used to control the transmission dynamics of the disease and proved the local and global stability of the disease-free equilibrium, the result of stability analysis revealed that the disease-free equilibrium (DFE) is locally asymptotically stable if the basic reproduction number is less than one (1) using Routh–Hurwitz Criterion (RHC) and globally asymptotically stable if the basic reproduction number is less than one through the method of Castillo-Chavez. The stability analysis of the disease-free equilibrium result indicates that typhoid can be completely wiped out if the average number of secondary infection is kept at a number less than unity. We carry out sensitivity analyses on the reproduction number to ascertain the parameters that affect the reproduction number. Based on the results, an optimal control problem was formulated and analyzed using Pontryagin’s Maximum Principle (PMP) to determine the optimality system. We solved the optimality system using the forward and backward sweep method and the results revealed that the combination of vaccination, public health education, treatment, the second line of treatment, and environmental sanitation is the best strategy for controlling the spread of the disease.

Nonlinear dynamics and passive control of GLYCOLYTIC oscillations

This paper addresses the issues of nonlinear dynamics and passive control of the main first stage of glycolytic oscillations. The Routh–Hurwitz criterion, the Whittaker method and the Floquet theory are utilized to analytically determine the stability boundaries of linear and nonlinear oscillations. Routes to chaos are investigated through bifurcation diagram, Lyapunov exponant, times stories and phase portraits. The passive control scheme is considered to get rid of chaotic oscillations. Results of analytical investigations are validated and complemented by numerical simulations.

The Impact of Peer Pressure Mathematical Models Armed Criminal Groups with Criminal Mapping Area

Model Armed Criminal Groups is mathematically realistic to be considered in the study of mathematical science. The aim of this research is to form a mathematical model of social cases of criminal acts. The given model is a criminal form that adopts the conformity of the conditions in the susceptible, exposed, infected, and recovered (SEIR) disease distribution model. The research method used is literature study and analysis. The research results show that there are 2 non-negative equilibrium, and one of them is stability analysis. Stability analysis is only carried out at equilibrium that does not contain a zero value with the Routh-Hurwitz criteria. In the results of other research the trajectories show that population growth tends not to experience fluctuations, this indicates that the population is growing towards stability rapidly. In case studies in the field, this marks a cycle of crime that quickly subsides or only occurs in a short period of time and does not occur in a sustainable manner. Overall the susceptible population, the exposed population, the infected population, and the recovered population experience the same conditions.

Routes to chaos and generation of cnoidal wave envelopes in exciton-polariton microcavities

We show the presence of intermittent chaos as a function of the variation of the chemical potential during the condensation of exciton-polaritons using the bifurcation diagram and the Lyapunov Exponent. We then show the possibility of controlling the chaos in the homogeneous solution by the variation of the chemical potential when the pump increases. The influence of the chemical potential is demonstrated in stationary state by studying stability using the Routh-Hurwitz criterion. We also prove the existence of cnoidal waves and through phase portraits, their stabilities in propagation are established using the notion of symmetry. For this, we use a dissipative Gross-Pitaevskii equation for the polariton field, coupled to a reservoir dynamics equation, commonly used to describe polariton condensates under non-resonant pumping.

Role of Allee Effect, Hunting Cooperation, and Dispersal to Prey–Predator Model

In prey–predator interaction, many factors, such as the fear effect, Allee effect, cooperative hunting, and group behavior, can influence the population dynamics. Hence, studying these factors in prey–predator makes the model more realistic. In this paper, we have proposed the prey–predator model having herd and Allee effect in prey population, where predators follow hunting cooperation. We have employed temporal analysis to examine the role of the Allee effect and hunting cooperation. Furthermore, we have extended the analysis to spatiotemporal analysis to examine the role of dispersal and the type of spatial structure formed by the population due to random movement. We first discuss the proposed model’s existence and positivity, then the stability of the existing equilibrium points through Routh–Hurwitz criteria. The temporal analysis is carried out through Hopf-bifurcation at the coexistence equilibrium point by considering the Allee threshold ([Formula: see text]), hunting cooperation ([Formula: see text]), and attack rate ([Formula: see text]) as controlled parameters. With the addition of diffusion to the model, we examine the spatial model stability and derive the Turing instability condition, which will give rise to various Turing patterns. Finally, numerical simulations are performed to validate the analytical results. The theoretical study and numerical simulation results demonstrate that the Allee effect, hunting cooperation, and diffusion coefficient are sensitive parameters to the model’s stability.

THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, AND RECOVERED (SIR) MODEL WITH VACCINATION

The Covid-19 pandemic in 2020 has caused severe problems in Indonesia. The Covid-19 virus epidemic can be modeled using the Susceptible, Infected, and Recovered (SIR) model. This modeling aims to look at the dynamics of Covid-19 to predict when disease-free and endemic disease occurs and to find the basic reproduction number ( ) for policy making in suppressing the spread of Covid-19. In this article, we describe and solve a research result on the SIR model with an assumption. The assumption in the model is that there is vaccination for the population. There are live stages of research conducted. The first is creating the SIR model and determining the equilibrium points on disease-free and disease-endemic. The Second is getting the basic reproduction number. The third is determining the stability around the equilibrium points using the Routh-Hurwitz criteria. Fourth, create a diagram for the subpopulations state at a specific time using Wolfram Mathematica software. As an implementation of the model created, COVID-19 data at the Batanghari Community Health Center Inpatient UPTD was used. Finally, determine the model error percentage with MAPE. The SIR Covid-19 model was made using eight parameters, namely , which are all positive. The results showed that the disease-free and disease-endemic equilibrium points were locally asymptotically stable after being analyzed using the Routh-Hurwitz stability criteria. The model trial using data from UPTD Puskesmas Batanghari obtained a stable condition for up to 100 months with a MAPE of 2.8%. From this study, obtained an . This means that if you want to reduce the rate of spread, then reduce the number of people who are easily infected ( ) and reduce contacts ( ), and increase the healing rate ( ).

A next-generation matrix approach using Routh–Hurwitz criterion and quadratic Lyapunov function for modeling animal rabies with infective immigrants

Rabies remains a major public health concern in many regions of the world, particularly in poorer nations. This study proposes a mathematical model for the animal rabies transmission dynamics, considering infective immigrants and vaccination as potential control measures to address this issue. The effective reproduction number (Re) was calculated by using the next-generation matrix approach. By using Routh–Hurwitz Criterion, a disease-free equilibrium point (DFE) was obtained and proved to be locally asymptotically stable if Re<1 and unstable otherwise, and also, by using the quadratic Lyapunov function DFE was determined to be globally asymptotically stable. Moreover, sensitivity analysis of the model parameters on the (Re) was performed using the normalized forward sensitivity index method. For simulation analysis, the numerical analysis was performed with the help of the MATLAB software. The simulated data results revealed that the increase in vaccination rate and decreasing infective immigrants will reduce the spread of the decrease, which was depicted graphically.