Local Asymptotic Stability Of Equilibria Research Articles

Host vector dynamics of a nonlinear pine wilt disease model in deterministic and stochastic environments

In this study, we develop a vector-host transmission model with general incidence rates for the dynamics of pine wilt disease in deterministic and stochastic environments. The existence and local asymptotic stability of equilibria are investigated in the deterministic case. We show the required conditions for the ergodic stationary distribution and extinction of the model in the stochastic case by constructing appropriate Lyapunov functions. Furthermore, by solving the corresponding Fokker-Planck equation, we obtain exact expressions of probability density function around the quasi-equilibrium of the stochastic model. Finally, we employ comprehensive numerical simulations to support our results and compare deterministic and stochastic situations.

Effects of additional food availability and pulse control on the dynamics of a Holling-($ p $+1) type pest-natural enemy model

<abstract><p>In this paper, a novel pest-natural enemy model with additional food source and Holling-($ p $+1) type functional response is put forward for plant pest management by considering multiple food sources for predators. The dynamical properties of the model are investigated, including existence and local asymptotic stability of equilibria, as well as the existence of limit cycles. The inhibition of natural enemy on pest dispersal and the impact of additional food sources on system dynamics are elucidated. In view of the fact that the inhibitory effect of the natural enemy on pest dispersal is slow and in general deviated from the expected target, an integrated pest management model is established by regularly releasing natural enemies and spraying insecticide to improve the control effect. The influence of the control period on the global stability and system persistence of the pest extinction periodic solution is discussed. It is shown that there exists a time threshold, and as long as the control period does not exceed that threshold, pests can be completely eliminated. When the control period exceeds that threshold, the system can bifurcate the supercritical coexistence periodic solution from the pest extinction one. To illustrate the main results and verify the effectiveness of the control method, numerical simulations are implemented in MATLAB programs. This study not only enriched the related content of population dynamics, but also provided certain reference for the management of plant pest.</p></abstract>

Deciphering the transmission dynamics of COVID-19 in India: optimal control and cost effective analysis

In this paper we assess the effectiveness of different non-pharmaceutical interventions (NPIs) against COVID-19 utilizing a compartmental model. The local asymptotic stability of equilibria (disease-free and endemic) in terms of the basic reproduction number have been determined. We find that the system undergoes a backward bifurcation in the case of imperfect quarantine. The parameters of the model have been estimated from the total confirmed cases of COVID-19 in India. Sensitivity analysis of the basic reproduction number has been performed. The findings also suggest that effectiveness of face masks plays a significant role in reducing the COVID-19 prevalence in India. Optimal control problem with several control strategies has been investigated. We find that the intervention strategies including implementation of lockdown, social distancing, and awareness only, has the highest cost-effectiveness in controlling the infection. This combined strategy also has the least value of average cost-effectiveness ratio (ACER) and associated cost.

Qualitative investigation of cytolytic and noncytolytic immune response in an HBV model

We investigate a mathematical model of infection by the hepatitis B virus (HBV) that includes cytolytic and noncytolytic immune response. The model exhibits a variety of steady-state solutions depending on parameter values, including nonunique and unique equilibrium solutions and periodic behavior. The disease-free equilibrium Ef and the positive-disease equilibrium Ed are examined. The basic reproduction ratio R0 is computed in order to examine the uniqueness and local asymptotic stability of equilibria and to understand the model’s biological implications for HBV dynamics.

Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations

Recently, systems of coupled renewal and retarded functional differential equations have begun to play a central role in complex and realistic models of population dynamics. In view of studying the local asymptotic stability of equilibria and (mainly) periodic solutions, we propose a pseudospectral collocation method to approximate the eigenvalues of the evolution operators of linear coupled equations, providing rigorous error and convergence analyses and numerical tests. The method combines the ideas of the analogous techniques developed separately for renewal equations and for retarded functional differential equations. Coupling them is not trivial, due to the different state spaces of the two classes of equations, as well as to their different regularization properties.

Analysis of Eco-epidemiological SIS Model under Sparse Effect

In this paper we formulated and analyzed the eco-epidemiological SIS model under sparse effect, and the sufficient condition of the local asymptotical stability of the equilibrium was studied with the method of latent root, analyze the global asymptotical stability of the boundary equilibriums, and also discuss the local asymptotical stability of the positive equilibrium, and established the sufficient condition of their stability.

Dynamics of an avian influenza model with half-saturated incidence

In this paper, an avian influenza epidemic model with half saturated incidence is investigated. Firstly we study the local asymptotical stability of equilibria of the deterministic model. Then for the stochastic model, the existence of the global positive solution is proved and different thresholds are obtained. In addition, we get sufficient conditions for the existence of an ergodic stationary distribution by using Has’minskii theory and Lyapunov function. Finally, some numerical simulations illustrate the analytical results.

Bifurcation analysis and chaos control in discrete-time eco–epidemiological models of pelicans at risk in the Salton Sea

Parasites play vital role in dynamics of predator–prey interaction and regulating bio-diversity. We study qualitative behavior of two 3-dimensional discrete-time predator–prey-parasite models. Bifurcation analysis and chaos control are discussed by taking into account the study of an eco–epidemiological model of pelicans at risk in the Salton Sea. Discrete-time models are obtained with implementations of Euler’s forward scheme and piecewise constant argument for differential equations. Local asymptotic stability of equilibria is investigated, and explicit Hopf bifurcation and period-doubling bifurcation criteria are implemented to discuss emergence of both type of bifurcations at positive steady-states of discrete-time models. Moreover, some chaos control techniques are implemented for controlling chaotic behavior under the influence of bifurcations. Numerical simulations are provided to illustrate theoretical discussion.

Stability, bifurcation analysis and chaos control for a predator-prey system

We study qualitative behavior of a modified prey–predator model by introducing density-dependent per capita growth rates and a Holling type II functional response. Positivity of solutions, boundedness and local asymptotic stability of equilibria were investigated for continuous type of the prey–predator system. In order to discuss the rich dynamics of the proposed model, a piecewise constant argument was implemented to obtain a discrete counterpart of the continuous system. Moreover, in the case of a discrete-time prey–predator model, the boundedness of solutions and local asymptotic stability of equilibria were investigated. With the help of the center manifold theorem and bifurcation theory, we investigated whether a discrete-time model undergoes period-doubling and Neimark–Sacker bifurcation at its positive steady-state. Finally, two novel generalized hybrid feedback control methods are presented for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. In order to illustrate the effectiveness of the proposed control strategies, numerical simulations are presented.

Bifurcation Analysis and Chaos Control in a Second-Order Rational Difference Equation

AbstractThis work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

A cholera mathematical model with vaccination and the biggest outbreak of world’s history

We propose and analyse a mathematical model for cholera considering vaccination. We show that the model is epidemiologically and mathematically well posed and prove the existence and uniqueness of disease-free and endemic equilibrium points. The basic reproduction number is determined and the local asymptotic stability of equilibria is studied. The biggest cholera outbreak of world's history began on 27th April 2017, in Yemen. Between 27th April 2017 and 15th April 2018 there were 2275 deaths due to this epidemic. A vaccination campaign began on 6th May 2018 and ended on 15th May 2018. We show that our model is able to describe well this outbreak. Moreover, we prove that the number of infected individuals would have been much lower provided the vaccination campaign had begun earlier.

Approximation of Eigenvalues of Evolution Operators for Linear Renewal Equations

A numerical method based on pseudospectral collocation is proposed to approximate the eigenvalues of evolution operators for linear renewal equations, which are retarded functional equations of Volterra type. Rigorous error and convergence analyses are provided, together with numerical tests. The outcome is an efficient and reliable tool which can be used, for instance, to study the local asymptotic stability of equilibria and periodic solutions of nonlinear autonomous renewal equations. Fundamental applications can be found in population dynamics, where renewal equations play a central role.

On a fractional-order model for HBV infection with cure of infected cells

Abstract In this work, a fractional-order model for HBV infection with cure of infected cells is considered. Local asymptotic stability of equilibria is discussed. Numerical simulation using PECE method is presented to illustrate the theoretical analysis. It has been noticed that when introducing a fractional-order derivative to the model, the peak of the infection is reduced, however, the disease takes a longer time to be eradicated.

Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods

This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.

Analysis of a Eco-Epidemiological Model with Disease in the Predator

In this paper we formulated and analyzed a eco-epidemiological model with disease in the predator, analysis of the existing conditions of equilibrium point, the sufficient condition of the local asymptotical stability of the equilibrium was studied with the method of latent root, the global asymptotical stability of two of the boundary equilibriums and the local asymptotical stability of the positive equilibrium is proved by using the Lyapunov function.

Analysis of an SEIR Epidemic Model with Saturated Incidence and Saturated Treatment Function

The dynamics of SEIR epidemic model with saturated incidence rate and saturated treatment function are explored in this paper. The basic reproduction number that determines disease extinction and disease survival is given. The existing threshold conditions of all kinds of the equilibrium points are obtained. Sufficient conditions are established for the existence of backward bifurcation. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease. Numerical simulations are presented to support and complement the theoretical findings.

Dynamics of an SIR Epidemic Model with Information Variable and Limited Medical Resources Revisited

The stability of the SIR epidemic model with information variable and limited medical resources was studied. When the basic reproduction ratioℛ0<1, there exists the disease-free equilibrium and when the basic reproduction ratioℛ0>1, we obtain the sufficient conditions of the existence of the endemic equilibrium. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.

A necessary condition for local asymptotic stability of discrete-time nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in the control systems literature. In this paper, we investigate the problem of asymptotic stability for discrete-time nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Sundarapandian [1] for discrete-time nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Lin and Byrnes [2] for discrete-time nonlinear autonomous systems. We illustrate our result with several examples.

A necessary condition for local asymptotic stability of nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in control systems literature. In this paper, we investigate this problem for nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Byrnes and Sundarapandian [1] for nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Brockett [2] for nonlinear autonomous systems. We illustrate our result with several examples.

A necessary condition for local asymptotic stability of discrete-time nonlinear systems with parameters

In this paper, we derive a necessary condition for local asymptotic stability of equilibria of discrete-time nonlinear systems with parameters. Our necessary condition is analogous to the necessary condition obtained by Byrnes and Sundarapandian [1] for local asymptotic stability of equilibria of continuous-time nonlinear systems with parameters. As a corollary of our general result, we deduce the necessary condition obtained by Lin and Byrnes [2] for local asymptotic stability of equilibria of discrete-time nonlinear systems. We also illustrate our result with some examples.