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The purpose of this paper is to discuss a nonlinear rational difference equation system with three variables. Firstly, by utilizing Jacobian theory, the stability of the system is obtained. Then, we explore the saddle point, nonhyperbolic point, and prime period‐two solution and its boundedness. Finally, we verify the results obtained in the paper via computer numerical calculations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that 'slower diffuser prevails', which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the "weak competition" case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the "strong–weak competition" case, we give the expression of critical competition intensity and the weak competitor will be wiped out. [ABSTRACT FROM AUTHOR]

In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management. [ABSTRACT FROM AUTHOR]

We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune-free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for τ ≥ 0 and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle-node-Hopf bifurcation. Numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation xn+1 = αxn-m+ηxn-k+σxn-l+δxn/β+γxn-kxn-l(xn-k+xn-l), n = 0,1,2... where the parameters α, β, γ, η, σ ∊(0,∞) while m;k; l are positive integers, such that m, k, l. The initial conditions x-m....,x-k...,x-l...,x-1...,x0 are arbitrary positive real numbers. We will give some numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the control problem of linear systems with periodic sampling period subject to actuator saturation is considered via delta operator approach. Using periodic Lyapunov function, sufficient conditions of local stabilization for periodic sampling systems are given. By solving an optimization problem, we derive the periodic feedback control laws and the estimate of the domain of attraction. As the saturation function sat. belongs to the sector OE0; 1, sufficient conditions are derived by constructing saturation-dependent Lyapunov functions to ensure that the periodic sampling system is globally asymptotically stable. A numerical example is given to illustrate the theoretical results proposed in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates the globally asymptotically stable and L2-gain of robust H∞ control for switched nonlinear systems under sampled data. By considering the relationship between the sampling period and the dwell time, the non-switching and one switching are discussed in the sampling interval, respectively. Firstly, a state feedback sampled-data controller is constructed by the back-stepping method, and the switching converts to asynchronous switching if it happens within the sampling interval. Then, under the limiting conditions of the sampling period, which are obtained by the average dwell time method, the closed-loop system is globally asymptotically stable and has L2-gain. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study a diffusive HCV infections epidemic model with nonlinear incidence rate and analyze the stability of the two kinds of equilibria. By constructing various Lyapunov functions, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R 0 < 1 and the endemic equilibrium is globally asymptotically stable when the basic reproduction number R 0 > 1 . Finally, some numerical simulations are given to confirm the theoretical analysis. The results show that when other parameters are the same, the linear infection rate and the non-linear infection rate have different effects on disease spread. [ABSTRACT FROM AUTHOR]

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion. [ABSTRACT FROM AUTHOR]

In the current era, information dissemination is more convenient, the harm of rumors is more serious than ever. At the beginning of 2020, COVID-19 is a biochemical weapon made by a laboratory, which has caused a very bad impact on the world. It is very important to control the spread of these untrue statements to reduce their impact on people’s lives. In this paper, a new rumor spreading model with comprehensive interventions (background detection, public education, official debunking, legal punishment) is proposed for qualitative and quantitative analysis. The basic reproduction number with important biological significance is calculated, and the stability of equilibria is proved. Through the optimal control theory, the expression of optimal control pairs is obtained. In the following numerical simulation, the optimal control under 11 control strategies are simulated. Through the data analysis of incremental cost-effectiveness ratio and infection averted ratio of all control strategies, if we consider the control problem from different perspectives, we will get different optimal control strategies. Our results provide a flexible control strategy for the security management department. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of the guaranteed $\mathcal{H_{\infty}}$ performance state estimation for static neural networks with interval time-varying delay. Based on a modified Lyapunov-Krasovskii functional and the linear matrix inequality technique, a novel delay-dependent criterion is presented such that the error system is globally asymptotically stable with guaranteed $\mathcal{H_{\infty}}$ performance. In order to obtain less conservative results, Wirtinger's integral inequality and reciprocally convex approach are employed. The estimator gain matrix can be achieved by solving the LMIs. Numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new online game addiction model with low and high risk exposure. With the help of the next generation matrix, the basic reproduction number R0 is obtained. By constructing a suitable Lyapunov function, the equilibria of the model are Globally Asymptotically Stable. We use the optimal control theory to study the optimal solution problem with three kinds of control measures (isolation, education and treatment) and get the expression of optimal control. In the simulation, we first verify the Globally Asymptotical Stability of Disease-Free Equilibrium and Endemic Equilibrium, and obtain that the different trajectories with different initial values converges to the equilibria. Then the simulations of nine control strategies are obtained by forward-backward sweep method, and they are compared with the situation of without control respectively. The results show that we should implement the three kinds of control measures according to the optimal control strategy at the same time, which can effectively reduce the situation of game addiction. [ABSTRACT FROM AUTHOR]

In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if $\mathfrak{R}_{0}\leq1$ , then the disease-free equilibrium $P_{0}$ is globally asymptotically stable; if $\mathfrak{R}_{0}>1$ , then the infection equilibrium $P_{*}$ is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model. [ABSTRACT FROM AUTHOR]

• Modelling the predator-prey ecosystem using a 'generalized' impulsive response function for the first time. • Dynamical behaviors such as sufficient conditions for the stability of prey-eradication and the permanence have been given. • The generalized solutions under asymptotically stable conditions are validated through various specific response functions. • The correctness of the derived theoretical results are further validated using numerical simulations. • Unlike previous research this paper also models the non-simultaneous occurrence of predator stocking and prey harvesting. Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environmental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a 'generalized' impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the harvesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradication periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a 'generalized' impulsive response function. Subsequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the 'general impulsive response function' under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response function (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elimination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator parameter and the incremental stocking of the predator parameter. [ABSTRACT FROM AUTHOR]

It is widely accepted that diffusive dispersal can permit persistence in an advective environment. This paper studies in some sense the optimal diffusion rate of species in a flowing habitat with hostile downstream boundary conditions. Firstly, we study the dependence of the critical length of the habitat on the dispersal rate d. It is shown that the critical length first decreases and then increases and asymptotically tends to infinity. Then there is a unique optimal diffusion rate $d_{0}$ for a single species to evolve. Then, by using this observation, we study the competition system of two species which are the same but only with different dispersal rates. We get an open finite interval, which is a neighborhood of $d_{0}$ , such that, if one of the dispersal rates lies within the interval but the other rate falls outside, then competition exclusion occurs. If the two dispersal rates both lie within the interval, the one with an intermediate dispersal rate can always invade the other with its dispersal rate near the ends of the interval. [ABSTRACT FROM AUTHOR]

In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number R 0 and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if R 0 < 1 , and the infection equilibrium is globally asymptotically stable if R 0 > 1. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number R j 𝜖 > 1 , while virus will extinct if R j 𝜖 < 1. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

This paper focuses on the global stability of an epidemic model with vaccination, treatment and isolation. The basic reproduction number R 0 is derived. By constructing suitable Lyapunov functions, sufficient conditions for the global asymptotic stability of equilibria are obtained. Numerical simulations are performed to verify and complement the theoretical results. Furthermore, we consider the uncertainty and sensitivity analysis of the basic reproduction number R 0 . The results show that the transmission rate, the fraction of infected receives treatment, vaccination rate, the isolation rate are crucial to prevent the spread of infectious diseases. These suggest that public health workers design the control strategies of disease should consider the influence of vaccination, treatment and isolation. [ABSTRACT FROM AUTHOR]

In this paper, a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate is set up. We derive the basic reproductive number R0, the cytotoxic T lymphocytes immune response reproductive number R1, the humoral immune response reproductive number R2, humoral immune response competitive reproductive number R3, and cytotoxic T lymphocytes immune response competitive reproductive number R4, and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete susceptible-infected-removed epidemic model with vaccination and treatment and it is shown that the global dynamics is determined by the basic reproduction number. If, then the disease-free equilibrium is globally asymptotically stable and if, then the endemic equilibrium is globally asymptotically stable. [ABSTRACT FROM PUBLISHER]

In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number . In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder's fixed point theorem and the limiting arguments, we show that when , there exists a constant such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for , and when and , the model has no positive traveling wave solutions connecting them. [ABSTRACT FROM AUTHOR]

In this paper, we designed and analysed a discrete model to solve a delayed within-host viral infection model by using non-standard finite difference scheme. The original model that we considered was a delayed viral infection model with cell-to-cell transmission, cell-mediated immune response and general nonlinear incidence. We show that the discrete model has equilibria which are exactly the same as those of the original continuous model and the conditions for those equilibria to be globally asymptotically stable are consistent with the original continuous model with no restriction on the time step size. The results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for corresponding continuous model. [ABSTRACT FROM PUBLISHER]

A delayed multi-group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number [ABSTRACT FROM AUTHOR]

In this paper, an inverse-free dynamical system is built to solve the absolute value equations (AVEs), whose equilibrium points coincide with the solutions of the AVEs. Under proper assumptions, the equilibrium points of the dynamical system exist and could be (globally) asymptotically stable. In addition, with strongly monotone property, a global projection-type error bound is provided to estimate the distance between any trajectories and the unique equilibrium point. Compared with four existing dynamical systems for solving the AVEs, our method is inverse-free and is still valid even if 1 is an eigenvalue of the coefficient matrix. Some numerical simulations are given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we provide a criterion to guarantee the existence of a globally attracting positive periodic solution for the nonautonomous Ricker system whose coefficients are all periodic functions of a common period, that is, under certain circumstances, all species will coexist in a limit cycle fashion. [ABSTRACT FROM AUTHOR]

We establish an SAIQR epidemic network model, in which asymptomatic infected people (A) are as contagious as infected people (I). The basic reproductive number R 0 is calculated, and the globally asymptotically stable of the disease-free equilibrium, the globally attractive and globally asymptotically stable of the endemic equilibrium are obtained. For the control of epidemic transmission, we take into account the non-instantaneous impulsive vaccination in the model, calculate the basic reproduction number R 0 ⁎ of the model, and demonstrate that the disease-free T -periodic solution is globally attractive and the model is permanent. Finally, we choose scale-free network to simulate numerically and validate the results of this paper. • An SAIQR epidemic network model is established, in which asymptomatic infected people are as contagious as infected people. • The globally asymptotically stable of the disease-free equilibrium and the endemic equilibrium are obtained. • Non-instantaneous impulsive vaccination is considered in the system, and the permanence of the system are investigated. [ABSTRACT FROM AUTHOR]

The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates. As an application, we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration. [ABSTRACT FROM AUTHOR]

In this paper, a model of Beddington-DeAngelies chemostat involving two species of micro-organism competing for two perfectly complementary growth-limiting nutrients and pulsed input of toxicant in the polluted environment was studied. Using Floquet theory and small amplitude perturbation method, a conclusion was that there exists two-micro-organism eradication periodic solution and which is global asymptotical stability. At the same time, the condition of the permanence for system was obtained. From the biological point of view, the method for protecting species is to improve the amount of impulsive period, and control the amount of toxicant input to the chemostat. Finally, our results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

Acyclic networks are dynamical systems whose dependency graph (or wiring diagram) is an acyclic graph. It is known that such systems will have a unique steady state and that it will be globally asymptotically stable. Such result is independent of the mathematical framework used. More precisely, this result is true for discrete-time finite-space, discrete-time discrete-space, discrete-time continuous-space and continuous-time continuous-space dynamical systems; however, the proof of this result is dependent on the framework used. In this paper we present a novel and simple argument that works for all of these frameworks. Our arguments support the importance of the connection between structure and dynamics. [ABSTRACT FROM AUTHOR]

Dengue presents over 390 million cases worldwide yearly. Releasing Wolbachiainfected male mosquitoes to suppress wild mosquitoes via cytoplasmic incompatibility has proven to be a promising method for combating the disease. As cytoplasmic incompatibility causes early developmental arrest of the embryo during the larval stage, we introduce the Ricker-type survival probability to assess the resulting effects. For periodic and impulsive release strategies, our model switches between two ordinary differential equations. Owing to a Poincar'e map and rigorous dynamical analyses, we give thresholds T*; c* and c**(> c*) for the release period T and the release amount c. Then, we assume c > c* and prove that our model admits a globally asymptotically stable periodic solution, provided T > T*, and it admits at most two periodic solutions when T < T*. Moreover, for the latter case, we assert that the origin is globally asymptotically stable if c ≥ c**, and there exist two positive numbers such that whenever there is a periodic solution, it must initiate in an interval composed of the aforementioned two numbers, once c* < c < c**. We also offer numerical examples to support the results. Finally, a brief discussion is given to evoke deeper insights into the Ricker-type model and to present our next research directions. [ABSTRACT FROM AUTHOR]

In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management.

This paper studies the stabilization problem for controllable and observable linear systems subject to output saturation. An efficient approach to global asymptotical stabilization is proposed, where multiple output components can be collectively steered to escape from saturation. This differs from previous approaches in which each output component is driven one by one. Sufficient conditions for collectively driving all output elements are presented in single-batch driven strategy. In the case of the above sufficient conditions not being satisfied, a multi-batch driven strategy is constructed to guarantee that each component is steered away from saturation at least once. Moreover, an algorithm is put forward for selecting controller parameters in the multi-batch driven case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper studies the global dynamics of a discontinuous delayed model of malicious signal transmission in wireless sensor networks under the framework of differential inclusion. The local stability of two types of steady states are investigated for the discontinuous system by studying the corresponding characteristic equation. The sufficient conditions for the existence of two types of globally asymptotically stable steady states are obtained for the discontinuous system by using the comparison arguments method. Furthermore, the optimal control of the discontinuous system is investigated by using Pontryagin's maximum principle. Numerical simulations of two examples are carried out to illustrate the main theoretical results. The obtained results can help us to better control and predict the spread of malicious signal transmission in wireless sensor networks. [ABSTRACT FROM AUTHOR]

Unlike conventional methods of pests control, introducing in an appropriate mathematical model can contribute a batter performance on pests control with higher efficiency while lest damage to ecosystem. To fill the research gap on plant root pest control, we propose a plant root pest management model with state pulse feedback control. Firstly, the stability of the equilibrium point of the model (1.3) is analyzed by using the linear approximate equation, given that the only positive equilibrium point of model (1.3) is globally asymptotically stable. Moreover, the existence and uniqueness of order 1 periodic solutions of model (1.3) are discussed in detail according to the geometric theory of semi-continuous dynamical systems, successor functions method and the qualitative theory of differential equations. Finally, with further analysis in different methods, the asymptotic stability of the order 1 periodic solution of model (1.3) is obtained by using Similar Poincare Criterion or interval set theorem. The results show that this model can effectively control the number of pests below the economic level of damage. [ABSTRACT FROM AUTHOR]

In this article, we discuss the global asymptotic stability of following system of difference equations with quadratic terms:... where α, β are positive numbers and the initial values are positive numbers. We also study the rate of convergence and oscillation behaviour of the solutions of related system. We will give also, some numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

• Studied an Alzheimer's disease model with discrete sizes and two types of A β , the interaction with P r p c as well as the treatment. • Proved the global asymptotic stability. • Presented numerical simulations and investigate the contributions from two types of A β. • Explained A β 42 paranuclei can be used as a therapeutic target. The accumulation of β -amyloid (A β) is one of the most important pathogenic factors in the occurrence of Alzheimer's disease (AD). Studies have shown that oligomers are more toxic in the process of A β aggregation because oligomers can interact with receptors such as prion proteins (P r p c) and the interaction causes P r p c to be misfolded into pathogenic oligomeric prion proteins (P r p o l ). In this paper, we propose an AD model including two types of A β , the interaction of oligomers with P r p c and anti- A β drugs treatment. The existence, uniqueness, and non-negativity of the solutions are analyzed. Furthermore, we prove that the model admits a unique globally asymptotically stable equilibrium, which means the drug cannot cure AD completely. Finally, we present some numerical simulations and investigate the relative contributions to AD with two types of A β , A β 40 and A β 42. In addition, elastic analysis description A β 42 paranuclei can be used as a therapeutic target. [ABSTRACT FROM AUTHOR]

This paper considers a three species eco-epidemiological model perturbed by white noise. Stochastic stability and long time behavior around the equilibrium of deterministic eco-epidemiological model are analyzed by Lyapunov methods. Numerical simulations for a set of parameter values are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

In this paper the application of projection neural network for solving bilinear programming problems (BLPs) is obtained. So far as we know, no study has yet been attempted for these problems via neural network. In fact, some interesting reformulations of BLP and mixed-integer bilinear programming problem (MIBLP) with a binary vector to linear complementarity problem (LCP) are given. Additionally, we show that the special type of MIBLP with a binary vector is equal to a quadratic program and on the other hand, it is equal to a mixed-integer linear program (MILP). Finally, we use a neural network to solve projection equation which has the same solution with LCP. Then, by presenting a Lyapunov function, we show that the proposed neural network is globally asymptotically stable. Illustrative examples are given to show the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $, $ n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) $, while $ m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Influenza remains one of the major infectious diseases that target humankind, therefore, understand transmission mechanisms and control strategies can help us obtain more accurate predictions. There are many control strategies, one of them is vaccination. In this paper, our purpose is to extend the incidence rate of a two-strain flu model with a single vaccination, which includes a wide range of incidence rates among them, some cases are not monotonic nor concave, which may be used to reflect media education or psychological effect. Our main aim is to mathematically analyze the effect of the vaccine for strain 1, the general incidence rate of strain 1 and the general incidence rate of strain 2 on the dynamics of the model. Four equilibrium points were obtained and the global dynamics of the model are completely determined via suitable Lyapunov functions. We illustrate our results by some numerical simulations. Our results showed that the vaccination is always beneficial for controlling strain 1, its impact on strain 2 depends on the force of infection of strain 2. Also, the psychological effect is always beneficial for controlling the disease.

Abstract In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if R 0 ≤ 1 $\mathfrak{R}_{0}\leq1$ , then the disease-free equilibrium P 0 $P_{0}$ is globally asymptotically stable; if R 0 > 1 $\mathfrak{R}_{0}>1$ , then the infection equilibrium P ∗ $P_{*}$ is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model.