Routh-Hurwitz Conditions Research Articles

Stability analysis of a nonlinear malaria transmission epidemic model using an effective numerical scheme

Malaria is a fever condition that results from Plasmodium parasites, which are transferred to humans by the attacks of infected female Anopheles mosquitos. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. More so, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The reproduction number R o\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(o\\right)$$\\end{document} was determined to be an asymptotically stable condition for disease free and endemic equilibria. In this paper, we used various schemes such as Runge–Kutta order 4 (RK-4) and non-standard finite difference (NSFD). All of the schemes produce different results, but the most appropriate scheme is NSFD. This is true for all step sizes. Various criteria are used in the NSFD scheme to assess the local and global stability of disease-free and endemic equilibrium points. The Routh–Hurwitz condition is used to validate the local stability and Lyapunov stability theorem is used to prove the global asymptotic stability. Global asymptotic stability is proven for the disease-free equilibrium when R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\le 1$$\\end{document}. The endemic equilibrium is investigated for stability when R0≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\ge 1$$\\end{document}. All of the aforementioned schemes and their effects are also numerically demonstrated. The comparative analysis demonstrates that NSFD is superior in every way for the analysis of deterministic epidemic models. The theoretical effects and numerical simulations provided in this text may be used to predict the spread of infectious diseases.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models.

Stability and optimal control analysis for studying the transmission dynamics of a fractional-order MSV epidemic model

In this research paper, we study the transmission dynamics of the maize streak virus disease in agriculture using fractional calculus. Our fractional-order model (FOM) consists of fractional differential equations within Caputo fractional derivative (CFD). Further, we investigate the mathematical analysis of this FOM such as the positivity and boundedness of the projected solutions, compute the control reproduction number (Rc) by the next-generation matrix method, examine the local stability by using Routh–Hurwitz conditions and global stability by using fractional LaSalle’s invariance principle for all possible equilibrium points. A sensitivity analysis is conducted on the parameters integrated into Rc to evaluate how accurately the parameter estimation fits. Additionally, we present further numerical results on the FOM to demonstrate the obtained analytical results. Then, we formulate the fractional optimal control problem (FOCP) depending on the suggested FOM with three control efforts u1, u2 and u3 called respectively, prevention, quarantine and insecticide chemical. Also, we study this FOCP analytically to derive the fractional necessary optimality conditions (NOCs) based on a kind of Pontryagin’s maximum principle. After that, we solve these fractional NOCs numerically where the state and co-state variables are subject to the left CFD. We present various strategies with different combinations of suggested controls to examine the best strategy for decreasing the transmission of MSV in maize plants, where each one of these strategies is able to reduce the maize streak disease (MSD) at the period of control time. We perform simulations of outcomes using MATLAB by developing the forward–backward sweep method utilizing the fractional Euler method. Moreover, some figures are presented to show the impact of both control interventions, change in weight factors and fractional order within the suggested strategies. We can observe that the fractional order plays a significant role in limiting the spread of MSD where the obtained results illustrated that the MSD can be reduced through the suggested FOM, which has a notable influence on plant epidemiology.

Optimal control of the coronavirus pandemic with both pharmaceutical and non-pharmaceutical interventions

Coronaviruses are types of viruses that are widely spread in humans, birds, and other mammals, leading to hepatic, respiratory, neurologic, and enteric diseases. The disease is presently a pandemic with great medical, economical, and political impacts, and it is mostly spread through physical contact. To extinct the virus, keeping physical distance and taking vaccine are key. In this study, a dynamical transmission compartment model for coronavirus (COVID-19) is designed and rigorously analyzed using Routh–Hurwitz condition for the stability analysis. A global dynamics of mathematical formulation was investigated with the help of a constructed Lyapunov function. We further examined parameter sensitivities (local and global) to identify terms with greater impact or influence on the dynamics of the disease. Our approach is data driven to test the efficacy of the proposed model. The formulation was incorporated with available confirmed cases from January 22, 2020, to December 20, 2021, and parameterized using real-time series data that were collected on a daily basis for the first 705 days for fourteen countries, out of which the model was simulated using four selected countries: USA, Italy, South Africa, and Nigeria. A least square technique was adopted for the estimation of parameters. The simulated solutions of the model were analyzed using MAPLE-18 with Runge–Kutta–Felberg method (RKF45 solver). The model entrenched parameters analysis revealed that there are both disease-free and endemic equilibrium points. The solutions depicted that the free equilibrium point for COVID-19 is asymptotic locally stable, when the epidemiological reproduction number condition (R_{0}<1). The simulation results unveiled that the pandemic can be controlled if other control measures, such as face mask wearing in public areas and washing of hands, are combined with high level of compliance to physical distancing. Furthermore, an autonomous derivative equation for the five-dimensional deterministic was done with two control terms and constant rates for the pharmaceutical and non-pharmaceutical strategies. The Lagrangian and Hamilton were formulated to study the model optimal control existence, using Pontryagin’s Maximum Principle describing the optimal control terms. The designed objective functional reduced the intervention costs and infections. We concluded that the COVID-19 curve can be flattened through strict compliance to both pharmaceutical and non-pharmaceutical strategies. The more the compliance level to physical distance and taking of vaccine, the earlier the curve is flattened and the earlier the economy will be bounce-back.

Effects of delay in a biological environment subject to tumor dynamics

In the present work, we perform a study investigating a generic model of tumor growth with a delay distribution in the proliferation of tumor-stimulating effectors using combinations of analytical and numerical methods. We examine two borderline cases of the distribution: the first limit case is the Dirac distribution, leading to a model with constant delay and the second limit case is the exponential distribution leading to a model with an additional equation. The main objective is to assess the effect of delays in the response of the immune system on the dynamic stability of interaction between tumor, immune and host cells. Analytical and numerical investigations reveal that in the absence of delay, the stationary states of the two models can be stable or unstable for all the parameters used. In the case of constant delay, the analysis focuses on the stability switch with increasing delay. We show using the generalized Sturm criterion that the space of the parameters of concern is divided into four regions determined by a sequence of discrimination and the Routh-Hurwitz conditions: the system can undergo no stability switch and remain unstable regardless of the delay or undergo exactly a stability switch causing the coexisting equilibrium to pass from stable to unstable when the parameters are chosen in a well-defined region. This shows that the delay plays the role of destabilizer and not of stabilizer. We also show in this case that the destabilization of the system by the delay induces a chaotic behavior in the a priori non-chaotic system in the absence of delay. In the case of exponential distribution, we show that the delay induces certain phenomena such as the Hopf bifurcation, the doubling of periods, the intermittence by saddle-node bifurcation and chaos. We show the importance of characterizing the delay-induced chaos and dynamic states of the system by examining the maximum tumor size for each dynamic state. In both cases of study, it is observed that small delays guarantee stability at the stable equilibrium level, but delays greater than a critical value can produce periodic solutions by Hopf bifurcation and larger delays can even lead to chaotic attractors. The implications of these results are discussed. We examined the other scenarios by showing the influence of the probability density parameters on the behavior of the solutions as well as the dynamics of the model. It is shown that, the region of stability for distributed delays is relatively larger than that of the presence of any discrete delay.

Mathematical Analysis and Optimal Control of Giving up the Smoking Model

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.

Computation of Stability Criterion for Fractional Ocean Circulation Box Model Using Optimal Routh—Hurwitz Conditions

Atlantic ocean thermohaline circulation is a deep ocean circulation occur in the Atlantic ocean which shows mixed of salt and freshwater transportation. The ocean circulation box model is defined to cover the large-scale behavior of the thermohaline circulation. On the other hand, fractional order dynamical systems are more flexible and realistic for real-life problems if compare with integer order dynamical systems. Hence, research on the stability for fractional dynamical systems is still infant and more difficult to analyze analytically. In this paper, we will extend the ocean circulation 3-box model into fractional order and investigate stability criterion for this fractional model by applying fractional Routh-Hurwitz conditions. Routh-Hurwitz conditions allow us to find the range of adjustable control parameter F1 which can detect the stability criterion for the fractional ocean circulation model.

DYNAMICAL ANALYSIS OF FRACTIONAL ORDER ZIKV MODEL

The ZIKV model presented in this article is developed by modifying \cite{Bonyah2016}’s model. The classical order is changed into fractional order model. The equilibrium points of the model are determined and the stability conditions of each equilibrium point have been done using Routh-Hurwitz conditions. Numerical simulation is presented to verify the result of stability analysis result. Numerical simulation is also used to shows the effect of the order $\alpha$ to the stability of the model’s equilibrium point.

Mathematical analysis of a two-strain disease model with amplification

We investigate a two-strain disease model with amplification to simulate the prevalence of drug-susceptible (s) and drug-resistant (m) disease strains. Drug resistance first emerges when drug-susceptible strains mutate and become drug-resistant, possibly as a consequence of inadequate treatment, i.e. amplification. In this case, the drug-susceptible and drug-resistant strains are coupled. We perform a dynamical analysis of the resulting system and find that the model contains three equilibrium points: a disease-free equilibrium; a mono-existent disease-endemic equilibrium at which only the drug-resistant strain persists; and a co-existent disease-endemic equilibrium where both the drug-susceptible and drug-resistant strains persist. We found two basic reproduction numbers: one associated with the drug-susceptible strain (R0s); the other with the drug-resistant strain (R0m), and showed that at least one of the strains can spread in a population if max[R0s,R0m]>1. Furthermore, we also showed that if R0m>max[R0s,1], the drug-susceptible strain dies out but the drug-resistant strain persists in the population (mono-existent equilibrium); however if R0s>max[R0m,1], then both the drug-susceptible and drug-resistant strains persist in the population (co-existent equilibrium). We conducted a local stability analysis of the system equilibrium points using the Routh-Hurwitz conditions and a global stability analysis using appropriate Lyapunov functions. Sensitivity analysis was used to identify the key model parameters that drive transmission through calculation of the partial rank correlation coefficients (PRCCs). We found that the contact rate of both strains had the largest influence on prevalence. We also investigated the impact of amplification and treatment/recovery rates of both strains on the equilibrium prevalence of infection; results suggest that poor quality treatment/recovery makes coexistence more likely and increases the relative abundance of resistant infections.

Complex dynamics in susceptible-infected models for COVID-19 with multi-drug resistance.

Nowadays, exploring complex dynamic of epidemic models becomes a focal point for research after the outbreak of COVID-19 pandemic which has no vaccine or fully approved drug treatment up till now. Hence, complex dynamics in a susceptible-infected (SI) model for COVID-19 with multi-drug resistance (MDR) and its fractional-order counterpart are investigated. Existence of positive solution in fractional-order model is discussed. Local stability based on the fractional Routh-Hurwitz (FRH) conditions is considered. Also, new FRH conditions are introduced and proved for the fractional case (0,2]. All these FRH conditions are also applied to discuss local stability of the multi-drug resistance steady states. Chaotic attractors are also found in this model for both integer-order and fractional-order cases. Numerical tools such as Lyapunov exponents, Lyapunov spectrum and bifurcation diagrams are employed to confirm existence of these complex dynamics. This study helps to understand complex behaviors and predict spread of severe infectious diseases such as COVID-19.

Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel

Fractional calculus (FC) is widely used in many interdisciplinary branches of science due to its effectiveness in describing and investigating complicated phenomena. In this work, nonlinear dynamics for a new physical model using nonlocal fractional differential operator with singular kernel is introduced. New Routh-Hurwitz stability conditions are derived for the fractional case as the order lies in [0,2). The new and basic Routh-Hurwitz conditions are applied to the commensurate case. The local stability of the incommensurate orders is also discussed. A sufficient condition is used to prove that the solution of the proposed system exists and is unique in a specific region. Conditions for the approximating periodic solution in this model via Hopf bifurcation theory are discussed. Chaotic dynamics are found in the commensurate system for a wide range of fractional orders. The Lyapunov exponents and Lyapunov spectrum of the model are provided. Suppressing chaos in this system is also achieved via two different methods.

Computational modelling and bifurcation analysis of reaction diffusion epidemic system with modified nonlinear incidence rate

ABSTRACT The aim of this work is to design two novel implicit and explicit finite difference (FD) schemes to solve SIR (susceptible, infected and recovered) epidemic reaction–diffusion system with modified saturated incidence rate. Since this model is based on population dynamics, therefore solution of the continuous system possesses the positivity property. The proposed finite difference schemes retain the positivity property of sub population which is an essential feature in population dynamics. Von Neumann stability analysis reveals that proposed FD schemes are unconditionally stable. It is verified with the help of Taylor's series expansion that proposed FD schemes are consistent. The proposed implicit scheme is unconditionally consistent, i.e. for . On the other hand the proposed explicit scheme gives conditional consistency for . The proposed FD schemes are compared with two other FD schemes, i.e. forward Euler and Crank Nicolson scheme. Simulations are performed for the verification of all the attributes for the underlying FD schemes. Furthermore, stability of the reaction diffusion system is discussed by applying Routh–Hurwitz criteria. Bifurcation values of infection coefficient are also obtained from Routh–Hurwitz condition.

On some extended Routh–Hurwitz conditions for fractional-order autonomous systems of order α ∈ (0, 2) and their applications to some population dynamic models

Abstract The Routh–Hurwitz stability criterion is a useful tool for investigating the stability property of linear and nonlinear dynamical systems by analyzing the coefficients of the corresponding characteristic polynomial without calculating the eigenvalues of its Jacobian matrix. Recently some of these conditions have been generalized to fractional systems of order α ∈ [0, 1). In this paper we extend these results to fractional systems of order α ∈ [0, 2). Stability diagram and phase portraits classification in the (τ, Δ)-plane for planer fractional-order system are reported. Finally some numerical examples from population dynamics are employed to illustrate our theoretical results.

Post-critical behavior of an auto-parametric aero-elastic system with two degrees of freedom

A generalized two-degree-of-freedom non-linear mathematical system with non-linear gyroscopic and non-conservative terms was used to set up a model for heave and pitch self-excited motion. The stability conditions for a girder with a bluff cross-section in wind flow were identified and presented. The analysis, based on the Routh–Hurwitz conditions and a numerical evaluation of the model, comprises both the self-excited case and an assumption of a harmonic load. The boundaries between different response types that depend on the frequencies of two principal aero-elastic modes are shown. The properties of the response located at these limits and the tendencies of the response in their neighborhood are discussed. The results can be used to explain several experimentally observed effects.

On the Aizerman Problem: Coefficient Conditions for the Existence of a Four-Period Cycle in a Second-Order Discrete-Time System

We consider in this paper an automatic control second-order discrete-time system whose nonlinearity satisfies the generalized Routh–Hurwitz conditions. Systems of this type are widely used in solving modern application problems that arise in engineering, theory of motion control, mechanics, physics, and robotics. Two constructed examples of discrete-time systems with nonlinearities that lie in a Hurwitz angle were presented in recent papers by W. Heath, J. Carrasco, and M. de la Sen. These examples demonstrate that in the discrete case, the Aizerman and Kalman conjectures are untrue even for second-order systems. One such system in these examples has a three-period cycle and the other system, a four-period cycle. We assume in the present paper that the nonlinearity is two-periodic and lies in a Hurwitz angle; here, we study a system for all possible parameter values. We explicitly present the conditions (for the parameters) under which it is possible to construct a two-periodic nonlinearity in such a way that a system with it is not globally asymptotically stable. Such a nonlinearity can be constructed in more than one way. We propose a method for constructing the nonlinearity in such a way that a family of four-period cycles is found in the system. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.

Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model

In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh–Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

Stability Analysis for Routh-Hurwitz Conditions Using Partial Pivot

Stability analysis of the polynomial characteristic equation is not easy to applied to a high-order system, in this case, it relates to search the roots of the degree n polynomial equation. It is difficult to find the roots of the equation, so stability analysis will be carried out using the stability of Routh-Hurwitz. In this study, a polynomial characteristic equation is given with a degree n with a real coefficient, then applied to the Hurwitz Matrix and a Gauss elimination procedure with a partial pivot is performed, then the growth factor is calculated. The results showed that the polynomial characteristic equation of degree n for Routh-Hurwitz conditions is said to be stable if each zero of the polynomial is located in the half left open field if and only if the elimination procedure can be performed and the optimal value of growth factor is 1.

Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment

The key objective of this paper is to study the imprecise biological complexities in the interaction of two species pertaining to harvesting threshold. It is explained by taking the prey–predator model with imprecise biological parameters and fractional order generalized Hukuhara (fgH) differentiability. In this vain, different possible systems of the model are constructed, according to the increasing and decreasing behavior of population growth. Feasibility and stability analyses of equilibrium points of the stated models are also discussed by means of variational matrix with Routh–Hurwitz conditions. In addition, the numerical elaborations are carried out by taking parametric expansion of fuzzy fractional Laplace transform (FFLT). This significantly helps the researchers in using a novel approach to analyze the constant solutions of the dynamical systems in the presence of fractional index. This would allow the avoidance of any intricacy that occurs while solving fractional order derivatives. Furthermore, this attempt also provides numerical and pictorial results, obtained through some well-known methods, namely fifth-forth Runge–Kutta method (FFRK), Grunwald–Letnikov’s definition (GL) and Adams–Bashforth method (ABM) that are deemed appropriate to scrutinize the dynamics of the system of equations.

Difference synchronization among three chaotic systems with exponential term and its chaos control

Abstract In this article, the difference synchronization and chaos control of chaotic systems with nonlinear exponential terms have been studied by using the feedback control method. The chaotic systems in the presence of an exponential terms behave differently from the polynomial chaotic systems, whose dynamics will also be different. The Routh-Hurwitz condition is used during chaos control and synchronization. The nonlinear ten-ring chaotic system, 3D chaotic system, new 3D chaotic system are considered to simulate the difference synchronization scheme for continuous case, and Wang, 3D Henon map and Rossler systems are considered during simulation of discrete time chaotic systems. The numerical simulations and the graphical results are presented to show the effectiveness and reliability of difference synchronization for continuous and discrete time chaotic systems.