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In this paper, an impulsive Lotka–Volterra competitive system with the effect of toxic substances is considered. By using some analytical technique, we prove that under certain conditions the species x 2 of the system will be driven to extinction while the species x 1 will be globally attractive with any positive solution of an impulsive Logistic equation. As a result, we extend the principle of competitive exclusion to a T - periodic competitive system with impulse and toxic substances. The main result in this paper is a good extension and supplement that of Li and Chen (2006). Furthermore, the obtained results supplement the results of Liu et al. (2009). [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]

• An viral infection model with intracellular delay and nonlocal diffusion is proposed. • An explicit formulation for basic reproduction number is given. • The global dynamics of the system is established. • Numerical simulations on the effects of intracellular delay and diffusion rate are performed. In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively. [ABSTRACT FROM AUTHOR]

• We propose and analyse a delayed nonautonomous predator-prey model with disease in prey. • Derived the sufficient conditions for global asymptotic stability of the positive periodic solutions. • Autonomous system develops only limit cycle oscillations through a Hopf-bifurcation for increasing the values of delay. • Corresponding nonautonomous system shows chaotic dynamics for increasing the delay parameter. • We draw Poincare map and maximum Lyapunov exponent to identify the chaotic behaviour of the system. In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov's functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system's behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system. [ABSTRACT FROM AUTHOR]

• We fixed the exponential function to describe the progression rate of TB. • The threshold value of disease persistent or not is given. • The existence and global stability of disease free equilibrium and endemic equilibrium both are proved. • The existence of global attractor, and persistence of disease are studied. This paper deals with the global dynamic analysis for a tuberculosis transmission model with age-structure and the exponential progress rate. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. The basic reproduction number R 0 of the model is defined. It is proved that R 0 = 1 is a threshold to determine the disease extinction or persistence. The disease free equilibrium is globally stable (unstable) if R 0 < 1 (if R 0 > 1). There exists an endemic equilibrium and the system is uniformly persistent if R 0 > 1. The global stability of the endemic equilibrium is given when R 0 > 1. At end, the model is applied to describe tuberculosis transmission in China. The number of total population, the number of the annual new TB cases, the annual PPD positive rate, and the prevalent rate all match the statistical data well. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new concept called α-inverse Lipschitz function is introduced. Based on the topological degree theory and Lyapunov functional method, we investigate global convergence for a novel class of neural networks with impulses where the neuron activations belong to the class of α-inverse Lipschitz functions. Some sufficient conditions are derived which ensure the existence, and global exponential stability of the equilibrium point of neural networks. Furthermore, we give two results which are used to check the stability of uncertain neural networks. Finally, two numerical examples are given to demonstrate the effectiveness of results obtained in this paper. [Copyright &y& Elsevier]

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 the present paper we formulate a mathematical model of two predators living on a single biotic prey. The two predators interfere directly with each other. The predation function for the two predators are assumed to be different - one follows the mass action kinetics while the other obeys Holling type-II functional response. We also assume that one of the predators is economically viable and undergo harvesting at a rate proportional to its density. We perform local stability analysis about the boundary states where one of the predator species is absent. It is seen that, the existence and/or extinction criteria of each predator is controlled by the predation rates of one or both the predators. The interior equilibrium point exhibits unstable oscillatory behavior in the local analysis. We also perform a global analysis using Lyapunov function and LaSalle’s principle which shows global stability around this point for limited carrying capacity of prey. Finally we model the phenomenon of random harvesting by perturbing the harvest rate using Gaussian white noise. The resulting stochastic differential equation model is analyzed for exponential mean square stability which is shown to depend on the harvesting effort. Numerical simulation study of the model equations are also carried out. [ABSTRACT FROM AUTHOR]

In this paper, using the Li–Muldowney geometric approach, we establish the global stability of the worm-epidemic equilibrium for a VEISV network worm attack model. This improves the related results presented in Toutonji et al. (2012) [1]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

The mass action infection law, the most frequently used transmission process in the theoretical studies of disease dynamics, has been challenged in various ways. Hill type infection rate is supposed to be a better alternative to mass action law. In the first phase of this paper, we study a basic HIV model with Hill type infection rate. In the second phase, we modify our basic model with intracellular delay τ that measures the time between the first effective contact between a virus and a healthy CD4 + T cell and the latter becomes productively infective. Mathematical results like well-posedness, permanence, local stability and global stability of both the delayed and non-delayed systems are studied. It is observed that the endemic equilibrium is locally and globally asymptotically stable if the virus replication factor is greater than a threshold value and unstable otherwise. In the latter case, the disease-free steady state occurs and is proved to be globally asymptotically stable. Our simulation results shed different insights on drug therapy when various perturbations are given to the system. It is shown that multi-blockers drug therapy is more appropriate in the treatment of HIV patients in comparison to any mono-blocker drug therapy. [ABSTRACT FROM AUTHOR]

Abstract: Wireless sensor networks (WSNs) have received extensive attention due to their great potential in civil and military applications. The sensor nodes have limited power and radio communication capabilities. As sensor nodes are resource constrained, they generally have weak defense capabilities and are attractive targets for software attacks. Cyber attack by worm presents one of the most dangerous threats to the security and integrity of the computer and WSN. In this paper, we study the attacking behavior of possible worms in WSN. Using compartmental epidemic model, we propose susceptible – exposed – infectious – recovered – susceptible with a vaccination compartment (SEIRS-V) to describe the dynamics of worm propagation with respect to time in WSN. The proposed model captures both the spatial and temporal dynamics of worms spread process. Reproduction number, equilibria, and their stability are also found. If reproduction number is less than one, the infected fraction of the sensor nodes disappears and if the reproduction number is greater than one, the infected fraction persists and the feasible region is asymptotically stable region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the systems of equations developed and also to validate our model. A critical analysis of vaccination class with respect to susceptible class and infectious class has been made for a positive impact of increasing security measures on worm propagation in WSN. [Copyright &y& Elsevier]

Abstract: This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results. [Copyright &y& Elsevier]

Abstract: In this paper, a generalized epidemic model on complex heterogeneous networks is proposed. To give a theoretical explanation for the simulation results established on networks, mathematical analysis of the epidemic dynamics is presented via mean-field approximation. Stabilities of the disease-free equilibrium and the endemic equilibrium are studied. The results explain why the heterogeneous connectivity patterns impact the epidemic threshold and reveal how the host parameters and the underlying network structures determine disease propagation. [Copyright &y& Elsevier]

Abstract: In this paper, a mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period and waning vaccine-induced immunity is investigated. The basic reproduction number is found by applying the method of the next generation matrix. It is shown that the global dynamics of the model is completely determined by the basic reproduction number. By means of appropriate Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable and the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. [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, an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated. The role that temporary immunity (natural, disease induced, vaccination induced) plays in the spread of disease, is incorporated in the model. The total host population is bounded and the incidence term is of the Holling-type II form. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The global dynamics are completely determined by the basic reproduction number . If , the disease-free equilibrium is globally stable which leads to the eradication of disease from population. If , a unique endemic equilibrium exists and is globally stable in the feasible region under certain conditions. Further, the transcritical bifurcation at is explored by projecting the flow onto the extended center manifold. We use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. Further, we obtain the threshold vaccination coverage required to eradicate the disease. Finally, taking biologically relevant parametric values, numerical simulations are performed to illustrate and verify the analytical 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: This paper presents an epidemic model aiming at the prevalence of HIV/AIDS in Yunnan, China. The total population in the model is restricted within high risk population. By the epidemic characteristics of HIV/AIDS in Yunnan province, the population is divided into two groups: injecting drug users (IDUs) and people engaged in commercial sex (PECS) which includes female sex workers (FSWs), and clients of female sex workers (C). For a better understanding of HIV/AIDS transmission dynamics, we do some necessary mathematical analysis. The conditions and thresholds for the existence of four equilibria are established. We compute the reproduction number for each group independently, and show that when both the reproduction numbers are less than unity, the disease-free equilibrium is globally stable. The local stabilities for other equilibria including two boundary equilibria and one positive equilibrium are figured out. When we omit the infectivity of AIDS patients, global stability of these equilibria are obtained. For the simulation, parameters are chosen to fit as much as possible prevalence data publicly available for Yunnan. Increasing strength of the control measure on high risk population is necessary to reduce the HIV/AIDS in Yunnan. [Copyright &y& Elsevier]

Abstract: In this paper, a mathematical model for HILV-I infection of CD4+ T-cells is investigated. The force of infection is assumed be of a function in general form, and the resulting incidence term contains, as special cases, the bilinear and the saturation incidences. The model can be seen as an extension of the model [Wang et al. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci. 179 (2002) 207-217; Song, Li, Global stability and periodic solution of a model for HTLV-I infection and ATL progression, Appl. Math. Comput. 180(1) (2006) 401-410]. Mathematical analysis establishes that the global dynamics of T-cells infection is completely determined by a basic reproduction number . If , the infection-free equilibrium is globally stable; if , the unique infected equilibrium is globally stable in the interior of the feasible region. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions. [Copyright &y& Elsevier]

Abstract: In this paper, a prey-dependent consumption predator–prey (natural enemy-pest) model with age structure for the predators and infectious disease in the prey, is considered. Infectious pests, immature natural enemies and mature natural enemies are released impulsively. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. The results provide a reliable theoretical tactics for pest management. [Copyright &y& Elsevier]

Abstract: In this paper, a SEIV epidemic model with a nonlinear incidence rate is investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is shown that if the basic reproduction number , the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist. Moreover, we show that if the basic reproduction number , the disease is uniformly persistent and the unique endemic equilibrium of the system with saturation incidence is globally asymptotically stable under certain conditions. [Copyright &y& Elsevier]

Abstract: This paper further develops a method, originally introduced by Mori et al., for proving local stability of steady states in linear systems of delay differential equations. A nonlinear nonautonomous system of delay differential equations with several delays is considered. Explicit delay-independent sufficient conditions for global attractivity of the solutions with an extremely simple form are provided. The above-mentioned conditions make the stability test quite practical. We illustrate application of this test to the Hopfield neural network models. The results obtained were also applied to a new marine protected areas model with delay that describes the ecological linkage between the reserve and fishing ground. [Copyright &y& Elsevier]