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In this paper, we formulate a population suppression model and a population replacement model with periodic impulsive releases of Nilaparvata lugens infected with wStri. The conditions for the stability of wild- $ N.\,lugens $ N. l u g e n s -eradication periodic solution of two systems are obtained by applying the Floquet theorem and comparison theorem. And the sufficient conditions for the persistence in the mean of wild $ N.\,lugens $ N. l u g e n s are also given. In addition, the sufficient conditions for the extinction and persistence of the wild $ N.\,lugens $ N. l u g e n s in the subsystem without wLug are also obtained. Finally, we give numerical analysis which shows that increasing the release amount or decreasing the release period are beneficial for controlling the wild $ N.\,lugens $ N. l u g e n s , and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions. [ABSTRACT FROM AUTHOR]

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the global properties of a computer virus propagation model. It is, interesting to note that the classical method of Lyapunov functions combined with the Volterra-Lyapunov matrix properties, can lead to the proof of the endemic global stability of the dynamical model characterizing the spread of computer viruses over the Internet. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in computer virus propagation model. A numerical study of the model is also carried out to investigate the analytical results. [ABSTRACT FROM AUTHOR]

• A multi-scale novel model for cholera is proposed. • The epidemiological dynamics for cholera are further shown. • The connection between the within-host and between-host dynamics for cholera is discovered. • The new suggestions on possible mechanisms of the cholera prevalence are given. In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R 0 for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R 0 < 1, and the endemic equilibrium is globally asymptotically stable when R 0 > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load. [ABSTRACT FROM AUTHOR]

This paper studies a fractional‐order modelling chronic wasting disease (CWD). The basic results on existence, uniqueness, non‐negativity, and boundedness of the solutions are investigated for the considered model. The criterion for local as well as global stability of the equilibrium points is derived. A numerical analysis for Hopf‐type bifurcation is presented. Finally, numerical simulations are provided to justify the results obtained. [ABSTRACT FROM AUTHOR]

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R0) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R0 and holds great promise for lowering the number of children and youth who addict to online game in Thailand. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonstandard finite difference (NSFD) scheme to solve numerically a cholera epidemic model with spatial diffusion. Through constructing discrete Lyapunov functions, we prove the globally asymptotical stabilities of the diseasefree equilibrium and the chronic infection equilibrium, which coincide with the corresponding continuous model. Finally, numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a nonlinear mathematical model of the HIV/AIDS and screening of unaware infectives on the transmission dynamics of the disease in a homogeneous population with constant immigration of susceptibles incorporating use of condom, screening of unaware infectives and treatment of the infected. We consider constant controls and thereafter by incorporating the theory of Volterra-Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis of HIV/AIDS. The analysis and results presented in this paper make building blocks toward a comprehensive study and deeper understanding of the fundamental mechanism in HIV/AIDS. A numerical study of the model is also carried out to investigate the analytical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we revisit a diffusive virus dynamics model with general incidence function and time delay. One novelty of our model is that we introduce cell-to-cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus-to-cell infection. We justify the well-posedness of the model and identify the basic reproduction number ℜ 0 for the model to be a sharp threshold value. The global stability of equilibria is determined by constructing suitable Lyapunov functionals in the sense that: the infection-free equilibrium is globally asymptotically stable if ℜ 0 ≤ 1 , and when ℜ 0 > 1 , the global asymptotic stability of infection equilibrium implies that the infection will persist. A significant impact of the cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model that is under-evaluated. Last, we perform numerical simulation to support our theoretic results. We set the domain of the viruses to be a two-dimensional square domain with the homogeneous Neumann boundary conditions to reflect the spatial spreading. [ABSTRACT FROM AUTHOR]

Virus can disseminate between uninfected target cells via two modes, namely, the diffusion-limited cell-free viral spread and the direct cell-to-cell transfer using virological synapses. To examine how these two viral infection modes impact the viral dynamics, in this paper, we propose and analyze a general viral infection model that incorporates these two viral infection modes. The model also includes nonlinear target-cell dynamics, infinitely distributed intracellular delays, nonlinear incidences, and concentration-dependent clearance rates. It is shown that the numbers of secondly infected cells through the cell-free infection mode and the cell-to-cell infection mode both contribute to the basic reproduction number. Under some reasonable assumptions, the model exhibits a global threshold dynamics: the infection is cleared out if the basic reproduction number is less than one and the infection persists if the basic reproduction number is larger than one. Two specific examples are provided to illustrate that our theoretical results cover and improve some existing ones. When the underlying assumptions are not satisfied, oscillations via global Hopf bifurcation can be observed. A brief simulation of two-parameter bifurcation analysis is given to explore the joint impacts on viral dynamics for the interplay between nonlinear target-cell dynamics and intracellular delays. [ABSTRACT FROM AUTHOR]

This paper describes a prey–predator type ecological model with infection in the prey populations. We also consider here a nonlinear functional response for disease transmission and a constant amount of refuge for the sound prey populations. The dynamical behavior of the mathematical model is described from the point of view of stability and bifurcation. A geometric method is also applied to establish the global asymptotic stability at the co-existence equilibrium point. Some computer simulation works have been presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates an SIS epidemic model with variable population size including a vaccination program. Dynamics of the endemic equilibrium of the model are obtained, and it will be shown that this equilibrium exists and is locally asymptotically stable when R0>1. In this case, the disease uniformly persists, and moreover, using a geometric approach we conclude that the model is globally asymptotically stable under some conditions. Also, a numerical discussion is given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed and analyzed an agricultural pest control system. For this purpose, an eco-epidemiological type predator–prey model has been proposed with the consideration of a sound predator population and two classes of pest populations namely susceptible pest and infected pest. Further to consider uncertainty, we modify our model and transform it into a fuzzy system with incorporation of imprecise parameters. The dynamical behavior of the proposed model has been investigated by examining the existence and stability criteria of all feasible equilibria. An optimal control problem is formed by considering the pesticide control as the control parameter and then the problem is solved both theoretically and numerically with the help of some computer simulation works. [ABSTRACT FROM AUTHOR]

This paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non-convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S-Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = αxn + βxn-l + γxn-k + axn-l + bxn-k/cxn-l + dxn-k, n + 0, 1, ..., where the parameters α,β,γ,a,b,c,d ∊(0,∞) and the initial conditions x-s, x-s+1 ..., x-l and x0 are positive real numbers where s = max{l,k}. Examples to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a Lotka-Volterra competition system which describes the dynamics of two aquatic species in a river with unidirectional water flow. It is assumed that two species are identical except their movement strategies distinguished by diffusion and advection rates. Assume further that the downstream end of the river is open, then at this boundary there is always a net loss of individuals incurred by water flow. Our final results suggest that when the downstream loss is small, the strategy of a combination of faster diffusion but smaller advection is always beneficial for species to win the competition; while the movement of both faster diffusion and advection can be favorable or unfavorable depending on the gap between two advection rates. Some different dynamics are illustrated when the downstream loss becomes severe. [ABSTRACT FROM AUTHOR]

Nowadays, the dynamical modeling has been an important approach to macroscopically study the propagation behaviors of computer virus on the Internet, and a large number of models have been established in this field. To our knowledge, however, the existing models did not take into account the difference in the ability of computer security defense. As it is known, in reality, due to the different nature and usefulness, not all computers in a system have the same security defense ability. In this paper, a new computer virus propagation model, which addresses the impact of different security defense abilities on computer virus spreading, is proposed and analyzed. In this context, a threshold is given to determine when the computer security defense level is need to be upgraded. Then, three potential equilibria are obtained, and their local and global asymptotic stability are fully studied. Furthermore, the optimal control problem of proposed model is formulated. On this basis, some numerical experiments are made to justify our results. [ABSTRACT FROM AUTHOR]

This paper investigates the impacts of state-dependent impulses on the stability of switching Cohen-Grossberg neural networks (CGNN) by means of B-equivalence method. Under certain conditions, the state-dependent impulsive switching systems can be reduced to the fixed-time ones. Furthermore, a stability criterion for the considered CGNN using the proposed comparison system is established. Finally, two numerical examples are provided to illustrate the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a dynamical system of a SEIQV mathematical model with nonlinear generalized incidence arising in biology is investigated. The stability of the disease-free and endemic equilibrium is discussed. The basic reproduction number of the model is obtained. We found that the disease-free and endemic equilibrium is stable locally as well as globally asymptotically stable. For , the disease-free equilibrium is stable both locally and globally and for , the endemic equilibrium is stable globally asymptotically. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

Virus patches can be disseminated rapidly through computer networks and take effect as soon as they have been installed, which significantly enhances their virus-containing capability. This paper aims to theoretically assess the impact of patch forwarding on the prevalence of computer virus. For that purpose, a new malware epidemic model, which takes into full account the influence of patch forwarding, is proposed. The dynamics of the model is revealed. Specifically, besides the permanent susceptible equilibrium, this model may admit an infected or a patched or a mixed equilibrium. Criteria for the global stability of the four equilibria are given, respectively, accompanied with numerical examples. The obtained results show that the spectral radii of the patch-forwarding network and the virus-spreading network both have a marked impact on the prevalence of computer virus. The influence of some key factors on the prevalence of virus is also revealed. Based on these findings, some strategies of containing electronic virus are recommended. [ABSTRACT FROM AUTHOR]

In this paper, we study the global stability and attractivity of the endemic equilibrium for a network-based SIS epidemic model with nonmonotone incidence rate. The model was introduced in Li (2015). We prove that the endemic equilibrium is globally asymptotically stable if α (a parameter of this model) is sufficiently large, and is globally attractive if the transmission rate λ satisfies λ λ c ∈ ( 1 , 2 ] , where λ c is the epidemic threshold. Some numerical experiments are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = axn + αxnxn-l/β+γn-k,n = 0, 1, ..., where the parameters αβγ and a are positive real numbers and the initial conditions x-t,x-t+1 ..., x-1 and x0 are positive real numbers where t = {l,k}. Numerical examples to the difference equation are given to explain our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a delayed HIV-1 infection model with nonlinear incidence of infection is reinvestigated. It is shown that if the reproduction number , then the system is permanent, and the infective equilibrium of the system is globally asymptotically stable. Thus, the global dynamics of the system is completely determined by the reproduction number . The results obtained enrich and improve the corresponding results given by Wang et al. [X. Wang, Y. Tao, X. Song, A delayed HIV-1 infection model with Beddington–DeAngelis functional response, Nonlinear Dynamics 62 (2010) 67–72]. The conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper [D. Li, W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007) 683–691]. [Copyright &y& Elsevier]

In this paper, we propose a numerical method for delayed partial differential equations that describe the dynamics of viral infections such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). We first prove that the proposed numerical method preserves the positivity and boundedness of solutions in order to ensure the well-posedness of the problem. By constructing appropriate discrete Lyapunov functionals, we show that the proposed method also preserves the global stability of equilibria of the corresponding continuous system with no restriction on the space and time step sizes. Moreover, the discrete model and main results presented in Qin et al. (2014) are extended and generalized. [ABSTRACT FROM AUTHOR]

In the paper we focus on modelling immune system–cancer cells interactions with time delays included into the model. The first model we consider, is the (Feyissa and Banerjee, 2013) model of humoral immune response, for which we show that the non-negativity property is not preserved. This is the reason we propose a modification, which under the assumption that the tumour is non-immunogenic can be simplified into the system of three equations without delays, that is the second model we consider. For the simplified model we have made mathematical analysis, including asymptotic stability, as well as numerical analysis, showing wide spectrum of possible model dynamics, including globally stable steady state, bistability and oscillatory behaviour. The dynamics of the simplified model is numerically compared with the dynamics of the full model with time delays for the parameter values from the literature. It occurs that if tumour immunogenicity is low time delays have no significant influence on the full system and both systems behave similarly. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions. Then, we consider the discretization of the model by using nonstandard finite difference scheme. It is then followed by the investigation of the global stability of the equilibria of the discrete model. Our study shows that if the basic reproduction number R 0 ≤ 1 , then the infection-free equilibrium is globally asymptotically stable; however if R 0 > 1 , then the infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors. [Copyright &y& Elsevier]

In this paper, a delayed eco-epidemiological model with Holling type II functional response is investigated. By analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium, the susceptible predator-free equilibrium and the endemic-coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are derived for the global stability of the endemic-coexistence equilibrium, the disease-free equilibrium, the susceptible predator-free equilibrium and the predator-extinction equilibrium of the system, respectively. Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study the boundedness, investigate the global convergence and the periodicity of the solutions to the following recursive sequence xn+1 = axn + bxn-1² + cxn-2xn-3/dxn-1² + exn-2xn-3, n = 0, 1, ..., where the parameters a, b, c, d and e are positive real numbers and the initial conditions x-3, x-2, x-1 and x0 ∈ (0, ∞). Also we give some numerical examples of some special cases of considered equation and presented some related graphs and figures using Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra-type functions, composite quadratic functions and Volterra-type functionals, we provide the global stability for this model. If R0, the basic reproductive number, satisfies R0 ≤ 1, then the infection-free equilibrium state is globally asymptotically stable. Our system is persistent if R0 > 1. On the other hand, if R0 > 1, then infection-free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R0 > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper studies the existence and stability of periodic solution of the high-order discrete-time Cohen–Grossberg neural networks with varying delays. The properties of M-matrix and the contracting mapping principle are used to obtain a sufficient condition that guarantees the uniqueness and global exponential stability of the periodic solution. In addition, a numerical example is given that demonstrates the effectiveness of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the linear destabilization of three-dimensional steady wake flows developing behind rectangular plates, which are oriented perpendicularly to the uniform incoming flow. The thickness-to-width ratio of plates is fixed to 1 / 6 , while the length-to-width ratio L is varied in the range 1 ≤ L ≤ 6 . For sufficiently low Reynolds numbers, three-dimensional steady wake flows are stable and exhibit two reflectional planar symmetries. Using extensive parallel numerical computations, global stability analysis of the fully three-dimensional steady base flows is systematically performed to determine the neutral curves of least stable global modes in the control parameters space. Three destabilization scenarios are thus identified. For very low aspect ratios 1 ≤ L ≤ 2 , the flow is first destabilized, when increasing the Reynolds number, by a steady global mode which breaks the top/bottom planar reflectional symmetry of the steady base flow. The first three-dimensional wake flow bifurcation is thus a (planar) pitchfork bifurcation, similar to the (circle) pitchfork bifurcation of axisymmetric wakes developing behind sphere and disks. For long aspect ratios 2.5 ≤ L ≤ 6 , the flow is first destabilized by an unsteady global mode, which also breaks the top/bottom symmetry. The first three-dimensional wake flow bifurcation is then a Hopf bifurcation, as in the case of two-dimensional wake flows developing behind infinitely elongated cylinders. For intermediate values of the aspect ratio 2 ≤ L ≤ 2.5 , a novel wake destabilization scenario is found: a Hopf bifurcation which breaks the left/right planar reflectional symmetry. Finally, the number of global modes getting unstable in a narrow range of Reynolds number gives an insight of the rich and complex nonlinear dynamics of those wake flows. [ABSTRACT FROM AUTHOR]

In this paper we consider a four-dimensional bioreactor model, describing an anaerobic digestion process of wastewater treatment and biogas production. We propose a simple and practically realizable approach for global asymptotic stabilization of the model and for maximizing the biogas flow rate; the latter is achieved by using a numerical extremum seeking algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a multi-group SEI model with the bilinear incidence and variable susceptibility and infectivity is analyzed. The susceptible individuals are divided into n groups according to their susceptibility, and infected individuals are divided into m groups according to infectivities. Each group of infected individuals has an exposed period. The basic reproduction number is obtained, and globally asymptotic stability of disease-free equilibrium is proved if . If , the disease-free equilibrium is unstable and the endemic equilibrium exists and is unique, which is globally asymptotically stable. Finally, some numerical examples are given to illustrate the above analysis. [Copyright &y& Elsevier]

Abstract: The present paper investigates the global stability and stabilization of switched Boolean network with impulsive effects. By using the semi-tensor product, the algebraic form of the switched Boolean network with impulsive effects is obtained. Then, sufficient and necessary condition is presented for the global stability of the switched Boolean network with impulsive effects. Furthermore, sufficient and necessary conditions are provided for the stabilization of the switched Boolean network with impulsive effects under two types of controls. Finally, examples are presented to illustrate the main results. [Copyright &y& Elsevier]

Abstract: In this paper, we study a predator–prey model with a transmissible disease spreading in the predator population and a time delay representing the gestation period of the predator. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the predator-extinction equilibrium and the disease-free equilibrium and the global attractiveness of the coexistence equilibrium of the system, respectively. Numerical simulations are carried out to support the theoretical analysis. [Copyright &y& Elsevier]

In this paper, we give some sufficient conditions to guarantee global asymptotic stability of the zero solution of the third-order nonlinear differential equation: x ′ ′ ′ + g( x, x ′ , x ′ ′ ) + f( x, x ′ ) x ′ + h( x) = 0. Two examples are also given to illustrate our results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the stability of the solutions to the stochastic logistic model with distributed delay, which is represented by the equation where is a standard Brownian motion. This study shows that the above stochastic system has a global positive solution with probability 1 and establishes the sufficient conditions for stability of the zero solution and the positive equilibrium. Several numerical examples are introduced to illustrate the results. Some recent results are improved and generalized. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the dynamical behavior of in-host viral models with humoral immunity and intracellular delays. For both models, using the method of Lyapunov functional, we establish that the global dynamics are determined by two threshold parameters R 0 and R 1. If R 0 ⩽1, the uninfected equilibrium E 0 is globally asymptotically stable, and the viruses are cleared. If R 1 ⩽1< R 0, the infected equilibrium without B cells response is globally asymptotically stable, and the infection becomes chronic but with no persistent B cells response. If R 1 >1, the infected equilibrium with B cells response is globally asymptotically stable, and the infection is chronic with persistent B cells response. Alone with some numerical simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number . It is shown that, if , then the disease free equilibrium is globally asymptotically stable and the disease dies out; if , there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: For an age-structured SIR epidemic model, which is described by a system of partial differential equations, the global asymptotic stability of an endemic equilibrium in the situation where the basic reproduction number is greater than unity has been an open problem for decades. In the present paper, we construct a multigroup epidemic model regarded as a generalization of the model, and study the global asymptotic stability of each of its equilibria. By discretizing the multigroup model with respect to the age variable under some parameter assumptions, we first rewrite the PDE system into an ODE system, and then, applying the classical method of Lyapunov functions and a recently developed graph-theoretic approach with an original idea of maximum value functions, we prove that the global asymptotic stability of each equilibrium of the discretized system is completely determined by , namely, the disease-free equilibrium is globally asymptotically stable if , while an endemic equilibrium exists uniquely and is globally asymptotically stable if . A numerical example illustrates that a numerical solution of for the discretized ODE system becomes closer to that for the original PDE system as the step size of the discretization decreases. [Copyright &y& Elsevier]