Schur-Cohn Criterion Research Articles

An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications

In this study, a mathematical model describing diabetes mellitus and its complications in a population is considered. Since standard numerical methods can lead to numerical instabilities, it aims to solve the problem using a nonstandard method. Among the nonstandard methods, nonstandard finite difference (NSFD) schemes that satisfy dynamical consistency are preferred to make the model discrete. Both continuous and discrete models are analyzed to show the stability of the model at the equilibrium points. The Schur-Cohn criterion is used to perform stability analysis at the equilibrium point of the discretized model. Thus, asymptotically stability of the model is presented. Moreover, the advantages of the NSFD method are emphasized by comparing the stability for different step sizes with classical methods, such as Euler and Runge-Kutta. It has been observed that the NSFD method is convergence for larger step sizes. In addition, the numerical results obtained by NSFD schemes are compared with the Runge–Kutta–Fehlberg (RKF45) method in graphical forms. The accuracy of the NSFD method is observed.

Discretization and Stability Analysis for a Generalized Type Nonlinear Pharmacokinetic Models

Estimating the effects of drugs at different stages is directly proportional to the duration of recovery and the duration of pulling through with the disease. It is very important to estimate the effects of drugs at different stages. For this reason, solving Pharmacokinetic models which investigate these effects are very important. In this study, numerical solutions of one, two, and three-compartment nonlinear Pharmacokinetic models have been studied. Distributed order differential equations have been used for the solution. Numerical solutions have been found with the density function contained in distributed order differential equations and different values of this function. A nonstandard finite difference scheme has been used for numerical solutions. Finally, stability analyses of equilibrium points of the obtained discretized system have also been researched with the help of the Matignon criterion.

Caputo and Conformable Fractional Order Guava Model for Biological Pest Control: Discretization, Stability and Bifurcation

Abstract Two predator-prey model describing the guava borers and natural enemies are studied in this paper. Positivity, existence, and uniqueness of the solution, global and local stability analysis of the fixed points of the first model based on the Caputo fractional operator are studied. By adding piecewise constant functions to the second model including conformable fractional operator allows us to transition discrete dynamical system via discretization process. Applying Schur-Cohn criterion to the discrete system, we hold some regions where the equilibrium points in the discretized model are local asymptotically stable. We prove that discretized model displays supercritical Neimark–Sacker bifurcation at the equilibrium point. Theoretical and numerical results show that the discretized system demonstrates richer dynamic properties such as quasi-periodic solutions, bifurcation, and chaotic dynamics than the fractional order model with Caputo operator. All theoretical results are interpreted biologically and the optimum time interval for the harvesting of the guava fruit is given.

Modeling the Transmission Dynamics of Coronavirus Using Nonstandard Finite Difference Scheme

A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves as an epidemic indicator. To better understand the dynamic behavior of the continuous model, the unconditionally stable nonstandard finite difference (NSFD) scheme is constructed. The aim of developing the NSFD scheme for differential equations is its dynamic reliability, which means discretizing the continuous model that retains important dynamic properties such as positivity of solutions and its convergence to equilibria of the continuous model for all finite step sizes. The Schur–Cohn criterion is used to address the local stability of disease-free and endemic equilibria for the NSFD scheme; however, global stability is determined by using Lyapunov function theory. We perform numerical simulations using various values of some key parameters to see more characteristics of the state variables and to support our theoretical findings. The numerical simulations confirm that the discrete NSFD scheme maintains all the dynamic features of the continuous model.

The Stability Analysis for a 2 × 2 Conservation Law with PI Controller and PDP Controller

This article is concerned with the stability analysis of a 2 × 2 conservation law with PI and PDP boundary feedback control. In the PI controller case, the authors establish a sufficient and necessary stability condition for feedback parameters by Walton–Marshall criterion. Moreover, in PDP controller case, the characteristic equation is an exponential polynomial which includes two exponential arguments. Based on Ruan’s results on stability for quasi-polynomials and Schur–Cohn criterion on the roots inside the unit circle for a real coefficient polynomial, a complete characterization for system parameters and the feedback delay is presented. Also, the delay-independent and delay-dependent stability intervals are obtained in this article.

Complex dynamics of COVID-19 mathematical model on Erdős–Rényi network

In this study, a conformable fractional order Lotka–Volterra predator-prey model that describes the COVID-19 dynamics is considered. By using a piecewise constant approximation, a discretization method, which transforms the conformable fractional-order differential equation into a difference equation, is introduced. Algebraic conditions for ensuring the stability of the equilibrium points of the discrete system are determined by using Schur–Cohn criterion. Bifurcation analysis shows that the discrete system exhibits Neimark–Sacker bifurcation around the positive equilibrium point with respect to changing the parameter d and e. Maximum Lyapunov exponents show the complex dynamics of the discrete model. In addition, the COVID-19 mathematical model consisting of healthy and infected populations is also studied on the Erdős Rényi network. If the coupling strength reaches the critical value, then transition from nonchaotic to chaotic state is observed in complex dynamical networks. Finally, it has been observed that the dynamical network tends to exhibit chaotic behavior earlier when the number of nodes and edges increases. All these theoretical results are interpreted biologically and supported by numerical simulations.

Implementasi Metode Beda Hingga Tak Standar untuk Model Penyebaran Campak

The measles distribution model is a system of differential equations that is included in a continuous dynamic system. This research focuses on transforming the continuous form into discrete form by discretization using non-standard finite difference and stability analysis which is then carried out by numerical simulations to prove its stability graphically. Based on the analysis, it is found that the measles distribution model which is assumed to have two fixed points, namely the disease-free fixed point (R_01) and the endemic fixed point (R_01), is stable. The stability of the two fixed points is proven by the Schur-Cohn criteria and is obtained stable with the condition 0ϕ(h)≤5 which meets the value of h0. The results of the numerical simulation show that the measles distribution model is dynamically consistent and tends to the fixed point. In addition, numerical simulations show that the larger the value of h, the more the graph tends to the fixed point.

Construction a distributed order smoking model and its nonstandard finite difference discretization

<abstract><p>Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.</p></abstract>

Comparison of dynamical behavior between fractional order delayed and discrete conformable fractional order tumor-immune system

In this paper, we analyze the dynamical behavior of the delayed fractional-order tumor model with Caputo sense and discretized conformable fractional-order tumor model. The model is constituted with the group of nonlinear differential equations having effector and tumor cells. First of all, stability and bifurcation analysis of the delayed fractional-order tumor model in the sense of Caputo fractional derivative is studied, and the existence of Hopf bifurcation depending on the time delay parameter is proved by using center manifold and bifurcation theory. Applying the discretization process based on using the piecewise constant arguments to the conformable version of the model gives a two-dimensional discrete system. Stability and Neimark–Sacker bifurcation analysis of the discrete system are demonstrated using the Schur-Cohn criterion and projection method. This study reveals that the delay parameterτin the model with Caputo fractional derivative and the discretization parameterhin the discrete-time conformable fractional-order model have similar effects on the dynamical behavior of corresponding systems. Moreover, the effect of the order of fractional derivative on the dynamical behavior of the systems is discussed. Finally, all results obtained are interpreted biologically, and numerical simulations are presented to illustrate and support theoretical results.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].

Multiple bifurcations in an early brain tumor model with piecewise constant arguments

In this paper, a differential equation with piecewise constant arguments modeling an early brain tumor growth is considered. The discretization process in the interval [Formula: see text] leads to two-dimensional discrete dynamical system. By using the Schur–Cohn criterion, stability conditions of the positive equilibrium point of the system are obtained. Choosing appropriate bifurcation parameter, the existence of Neimark–Sacker and flip bifurcations is verified. In addition, the direction and stability of the Neimark–Sacker and flip bifurcations are determined by using the normal form and center manifold theory. Finally, the Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors of the system.

Stability criterion for difference equations involving generalized difference operator

In this study, some necessary and su¢ cient conditions are givenfor the stability of some class of diğerence equations including generalizeddiğerence operator. For this, Schur-Cohn criteria is used and some examplesare given to verify the results obtained

Quantity and price competition in a differentiated triopoly: static and dynamic investigations

The game among three competing firms, which is called a triopolistic game, is formulated and studied in this paper. The adopted prices by those firms are derived from a quadratic form of utility that is quasi-concave and has a strict convexity to the origin level curves. The paper is divided into three parts: the first part analyzes a static model on which firms use different strategic variables. The stability condition of the equilibrium points of that model is investigated. In the second part, it is assumed that the three firms may conjecture that their opponents’ current step will remain as it is in the next time step and therefore an instantaneous adjustment model is considered. Schur–Cohn criterion of stability is adopted to detect the stability condition of the equilibrium points of that model. The fractional bounded rationality is introduced in the third part. We introduce in this part a dynamical model of three competing firms whose strategic variables are quantities. As in the previous models, the stability condition of the equilibrium points of the model is studied. The main interesting observation we obtain in this part is that for the firms to stay stable for a long time in the market they should play with fractional bounded rationality rather than the traditional bounded rationality. This important observation is confirmed using numerical simulation for some cases under the same values of parameters. Furthermore, for the models considered in this paper some numerical experiments are carried out to satisfy our theoretical results. Moreover, Karush–Kuhn–Tucker (KKT) proximity measure based on evolutionary algorithm is performed in order to confirm the obtained results.

Modified Schur-Cohn Criterion for Stability of Delayed Systems

A modified Schur-Cohn criterion for time-delay linear time-invariant systems is derived. The classical Schur-Cohn criterion has two main drawbacks; namely, (i) the dimension of the Schur-Cohn matrix generates some round-off errors eventually resulting in a polynomial ofswith erroneous coefficients and (ii) imaginary roots are very hard to detect when numerical errors creep in. In contrast to the classical Schur-Cohn criterion an alternative approach is proposed in this paper which is based on the application of triangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systems. The advantages of the proposed approach are that it halves the dimension of the polynomial and it only requires seeking real roots, making this modified criterion comparable to the Rekasius substitution criterion.

Global behaviour of a predator–prey like model with piecewise constant arguments

The present study deals with the analysis of a predator–prey like model consisting of system of differential equations with piecewise constant arguments. A solution of the system with piecewise constant arguments leads to a system of difference equations which is examined to study boundedness, local and global asymptotic behaviour of the positive solutions. Using Schur–Cohn criterion and a Lyapunov function, we derive sufficient conditions under which the positive equilibrium point is local and global asymptotically stable. Moreover, we show numerically that periodic solutions arise as a consequence of Neimark-Sacker bifurcation of a limit cycle.

Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay

In this paper, we propose and analyze a Lotka–Volterra competition like model which consists of system of differential equations with piecewise constant arguments of delay to study of interaction between tumor cells and Cytotoxic T lymphocytes (CTLs). In order to get local and global behaviors of the system, we use Schur–Cohn criterion and constructed a Lyapunov function. Some algebraic conditions which satisfy local and global stability of the system are obtained. In addition, we investigate the possible bifurcation types for the system and observe that the system may undergo Neimark–Sacker bifurcation. Moreover, it is predicted a threshold value above which there is uncontrollable tumor growth, and below periodic solutions that leading to tumor dormant state occur.

Bağışıklık Sistemi Etkisi Altında Tümör Büyümesinin Modellenmesi

In this paper, a mathematical model which consists of system of differential equations with piecewise constant argument is constructed to describe tumor-immune system interaction. The system is based on the tumor growth model constructed by Kuznetsov et all. A solution of the system with piecewise constant arguments leads to a system of difference equations. Using Schur-Cohn criterion and a Lyapunov function, sufficient conditions are obtained for the local and global asymptotic stability of a positive equilibrium point of the system of difference equations. Neimark-Sacker bifurcations analysis shows that stable limit cycle occurs at the bifurcation point, thus resulting oscillations for tumor and immune system.

Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments

In this study, we present a Lotka-Volterra predator-prey like model for the interaction dynamics of tumor-immune system. The model consists of system of differential equations with piecewise constant arguments and based on the model of tumor growth constructed by Sarkar and Banerjee. The solutions of differential equations with piecewise constant arguments leads to system of difference equations. Sufficient conditions are obtained for the local and global asymptotic stability of a positive equilibrium point of the discrete system by using Schur-Cohn criterion and a Lyapunov function . In addition, we investigate periodic solutions of discrete system through Neimark-Sacker bifurcation and obtain a stable limit cycle which implies that tumor and immune system undergo oscillation.

Dynamical Cournot game with bounded rationality and time delay for marginal profit

In this work, a kind of delayed structure on marginal profit is introduced in a dynamical Cournot game with bounded rationality. Time delay is considered for producers’ marginal profits so that each producer follows a local adjustment process to adjust its output via a smoothed marginal profit, which averages previous marginal profits with different weights. Delayed dynamics is built for such a process and analysis of local stability is mathematically done for it. Its boundary equilibria are proved to be unstable and the conditions for local stability of its unique interior equilibrium are obtained by Schur–Cohn Criterion. To show how the delayed system evolves and what influence the model parameters including the delay weight (a memory parameter) have on the system stability, numerical simulations are done for different kinds of dynamical behaviors such as bifurcation diagram, phase portrait, chaotic attractor, convergence speed and stability region. It is demonstrated that a proper delay weight to the memory plays an important role in expanding the stability region and delaying the occurrence of complex behaviors such as bifurcation and chaos. It is also demonstrated that properly medium delay weights and properly medium adjustment rates may speed up the convergence to equilibrium.

A Numerical Comparison for a Discrete HIV Infection of CD4+T-Cell Model Derived from Nonstandard Numerical Scheme

A nonstandard numerical scheme has been constructed and analyzed for a mathematical model that describes HIV infection of CD4+T cells. This new discrete system has the same stability properties as the continuous model and, particularly, it preserves the same local asymptotic stability properties. Linearized Stability Theory and Schur-Cohn criteria are used for local asymptotic stability of this discrete time model. This proposed nonstandard numerical scheme is compared with the classical explicit Euler and fourth order Runge-Kutta methods. To show the efficiency of this numerical scheme, the simulated results are given in tables and figures.