Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.     

Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.     

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.     

Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment

In this paper, the dynamics behavior of a delayed viral infection model with logistic growth and immune impairment is studied. It is shown that there exist three equilibria. By analyzing the characteristic equations, the local stability of the infection-free equilibrium and the immune-exhausted equilibrium of the model are established. By using suitable Lyapunov functional and LaSalle invariant principle, it is proved that the two equilibria are globally asymptotically stable. In the following, the stability of the positive equilibrium is investigated. Furthermore, we investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Stability preserving NSFD scheme for a multi‐group SVIR epidemic model

A discrete multi‐group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Global properties of a discrete viral infection model with general incidence rate

In this paper, we propose a discrete viral infection model with a general incidence rate. The discrete model is derived from a continuous case by using a 'mixed' Euler method, which is a mixture of both forward and backward Euler methods. We prove that the mixed Euler method preserves the qualitative properties of the corresponding continuous system, such as positivity, boundedness, and global behaviors of solutions. Furthermore, the model and mathematical results presented in another previous study are extended and generalized. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response

A class of more general delayed viral infection model with lytic immune response is proposed based on some important biological meanings. The effect of time delay on stabilities of the equilibria is given. The sufficient criteria for local and global asymptotic stabilities of the viral free equilibrium and the local asymptotic stabilities of the no-immune response equilibrium are given. We also get the sufficient criteria for stability switch of the positive equilibrium. Numerical simulations are carried out to explain the mathematical conclusions.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013     

A finite difference scheme for the equations of population dynamics

In this paper, we develop a discrete age-dependent population model by applying a finite difference scheme to the McKendrick type population equation. The properties of the system are analyzed. The convergence of the sequence of solutions and the critical fertility to those of the continuous model are proved (as the age step approaches zero).     

Algorithms for generalized potential games with mixed-integer variables

We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches.     

Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model

A non-standard finite difference scheme for a harvesting Leslie–Gower equations is constructed. It is shown that the obtained difference system has the same dynamics as the original continuous system, such as positivity of solutions, equilibria and their local stability properties, irrespective of the size of numerical time step. To illustrate the analytical results, we present some numerical simulations.     

一类具有饱和发生率和CTL免疫反应的HIV-1感染时滞模型的全局稳定性分析

通过构造合适的Lyapunov函数证明了一类具有饱和发生率和CTL免疫反应的HIV-1感染时滞模型各可能平衡点的全局稳定性.     

THE UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR TWO DIMENSIONAL SEMILINEAR PARABOLIC SYSTEMS

In this paper we are going to discuss the difference schemes with intrinsic parallelism for the boundary value problem of the two dimesional semilinear parabolic systems.The unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W2^(2,1) norms.Then the uniqueness of the discrete vector solution of this difference scheme follows as the consequence of the stability.     

Dynamics of a continuous Hénon model

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular, high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.     

HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission

HIV infection persists despite long-term administration of antiretroviral therapy. The mechanisms underlying HIV persistence are not fully understood. Direct viral transmission from infected to uninfected cells (cell-to-cell transmission) may be one of them. During cell-to-cell transmission, multiple virions are delivered to an uninfected cell, making it possible that at least one virion can escape HIV drugs and establish infection. In this paper, we develop a mathematical model that includes cell-to-cell viral transmission to study HIV persistence. During cell-to-cell transmission, it is assumed that various number of virus particles are transmitted with different probabilities and antiretroviral therapy has different effectiveness in blocking their infection. We analyze the model by deriving the basic reproduction number and investigating the stability of equilibria. Sensitivity analysis and numerical simulation show that the viral load is still sensitive to the change of the treatment effectiveness in blocking cell-free virus infection. To reduce this sensitivity, we modify the model by including density-dependent infected cell death or HIV latent infection. The model results suggest that although cell-to-cell transmission may have reduced susceptibility to HIV drugs, HIV latency represents a major reason for HIV persistence in patients on suppressive treatment.     

Equilibrium strategies for multiple interdictors on a common network

In this work, we introduce multi-interdictor games, which model interactions among multiple interdictors with differing objectives operating on a common network. As a starting point, we focus on shortest path multi-interdictor (SPMI) games, where multiple interdictors try to increase the shortest path lengths of their own adversaries attempting to traverse a common network. We first establish results regarding the existence of equilibria for SPMI games under both discrete and continuous interdiction strategies. To compute such an equilibrium, we present a reformulation of the SPMI game, which leads to a generalized Nash equilibrium problem (GNEP) with non-shared constraints. While such a problem is computationally challenging in general, we show that under continuous interdiction actions, an SPMI game can be formulated as a linear complementarity problem and solved by Lemke’s algorithm. In addition, we present decentralized heuristic algorithms based on best response dynamics for games under both continuous and discrete interdiction strategies. Finally, we establish theoretical lower bounds on the worst-case efficiency loss of equilibria in SPMI games, with such loss caused by the lack of coordination among noncooperative interdictors, and use the decentralized algorithms to numerically study the average-case efficiency loss.     

Matching and stabilization of discrete mechanical systems

Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)