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This paper is dedicated to Jim Cushing on the occasion of his 80th birthday. It is inspired by his work on evolutionary theory. We investigate the global dynamics of discrete-time phenotypic evolutionary models, both autonomous and periodic. We developed the theory of mixed monotone maps and applied it to show that the positive equilibrium of the autonomous evolutionary Ricker model of single and multi-species is globally asymptotically stable. Then we extend this result to the corresponding evolutionary Ricker model with periodic parameters. [ABSTRACT FROM AUTHOR]

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0 (E*) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E*) < 1 < R0. [ABSTRACT FROM AUTHOR]

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, a predator–prey model that takes into account the interactions between three species of fish, zooplankton, and phytoplankton has been developed. The logistic growth of phytoplankton has been considered in the absence of zooplankton. It is assumed that zooplankton eats phytoplankton. It is considered that the growth rate of zooplankton decreases due to the fear of fish. The release of toxins by phytoplankton and fish consumption are two further factors that are assumed to decrease the growth rate of zooplankton. The harvesting of fish has also been considered in this study. Local and global stability of the system around the equilibrium points has been investigated. The existence condition of oscillatory behaviour of the system has been studied through Hopf bifurcation. It is found that the increasing rate of minimum cost of fear η may stabilize the system. It is obtained that the rate of increase of release of toxin substance ρ by phytoplankton may be responsible for the stability of the system. It is also found that the system may become unstable for the increase of level of fear ξ of zooplankton. Some numerical simulation results are presented to test the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper studies the existence, uniqueness, asymptotic and S p -globally exponential stability of Stepanov-like pseudo almost automorphic solution to Lotka–Volterra recurrent neural networks (LVRNNs) with time-varying delays on time scales by applying D-matrix theory, Banach fixed point theorem, constructing a suitable Lyapunov functional and the theory of time scales calculations. An example and simulations are given to prove the effectiveness of main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function E f (S) is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number R 0 is calculated. When R 0 ≤ 1 the global asymptotical stability of the disease-free equilibrium is obtained, while when R 0 > 1 the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when R 0 > 1 the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions for Nicholson's blowflies equation with nonlocal dispersion subjected to Dirichlet boundary condition. We first prove the existence and uniqueness of the solution for the initial boundary value problem and its non-trivial steady state. Then we give a threshold result on global stability of equilibria: when D λ 1 + δ > p , the solution time-exponentially converges to the constant equilibrium 0 for any large initial data; when D λ 1 + δ < p , the solution time-asymptotically converges to its positive steady-state ϕ (x) for any large initial data, once 1 < p δ ≤ e , where D>0 is the diffusion coefficient, δ > 0 is the death rate, p>0 is the birth rate and λ 1 > 0 is the principal eigenvalue for the nonlocal characteristic equation. The adopted approach is the energy method and the monotonic technique. [ABSTRACT FROM AUTHOR]

We consider a neural network approach to an inverse variational inequality which is assumed to have a non-empty set of solutions. In the case of gradient mappings, we prove that every trajectory of the network converges to the solution set in the case of convex potentials and if the solution set is singleton, the network is globally asymptotically stable at the equilibrium point. We also prove that if the network has a strongly convex potential, then the network is globally exponentially stable at the equilibrium point. Another purpose of this paper is to point out certain fatal mistakes in the paper by Zou et al. [A novel method to solve inverse variational inequality problems based on neural networks. Neurocomputing. 2016;173:1163–1168]. [ABSTRACT FROM AUTHOR]

The prey–predator systems exist in almost all ecological systems and the refuge stage is highly concerned with such ecological systems in which the prey goes out of reach of the attacking region of the predator. In this paper, dynamical behaviour of predator and prey species with Holling type III response function is investigated. We add refuge in prey species and also additional predators are introduced into the system. Firstly, the crisp model is formulated with these effects and the boundedness, existence, conditions of local and global stability, local bifurcation and Hopf bifurcation are discussed. The crisp model is transformed into the fuzzy model to incorporate the unpredictable behaviour and using the Root Mean Square (RMS) defuzzification method, the fuzzy model is reduced to the defuzzified model in order to avoid the complexity of the analysis of the solution. Further, numerical simulations are performed with the help of MATLAB, MATCONT package and we examine solution curves and phase portraits in two different cases, in the presence and in the absence of additional predators along with the Hopf bifurcation scenarios. Finally, the sensitivities of the controlling parameters are plotted graphically and presented their significances in controlling the infection of the prey. [ABSTRACT FROM AUTHOR]

In this paper, a general setting is presented to study the exponential stability of discrete-time systems with bounded or unbounded delays. Based on the M-matrix theory, we establish sufficient conditions to ensure the global exponential stability of the zero equilibrium of low-order, and high-order, discrete-time Hopfield neural network models with unbounded delays and delay in the leakage terms. A comparison of the literature shows that our results generalize and improve some in recent publications. [ABSTRACT FROM AUTHOR]

This paper is concerned about a diffusive degenerate predator–prey model with Beddington–DeAngelis functional response subject to homogeneous Neumann boundary condition. First, the global bifurcation branches of positive stationary solutions are studied, which are quite different from those with different degeneracy or functional response. Second, the multiplicity and stability of positive stationary solutions are obtained as the parameter k or m in the Beddington–DeAngelis functional response is large enough, from which the effects of the functional response on the coexistence region are revealed. In particular, the global stability of the positive stationary solution is derived as it exists uniquely. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global stability of traveling waves for non-monotone infinite-dimensional lattice differential equations with time delay. By establishing the boundedness estimate of the solution of the perturbation equation, we obtain that, for any initial perturbations around the traveling wave, the noncritical traveling waves are globally stable with the exponential convergence rate t − 1 / α e − μ t (μ > 0 a n d 0 < α ≤ 2) , and the critical traveling waves are globally stable with the algebraic convergence rate t − 1 / α in a weighted Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a detailed mathematical model describing the dynamics of a prey-predator model under the influence of an SIS infectious disease by using nonlinear differential equations. We use the functional response of ratio-dependent Michaelis-Menten type to describe the predation strategy. In the presence of the disease, prey and predator population are divided into two disjointed classes, namely infected and susceptible. The first one is governed through due predation interaction, and the second one is governed through the propagation of disease in the prey and predator population via predation. Our aim is to analyze the effect of predation on the dynamic of the disease transmission. Important mathematical results resulting from the transmission of the disease under influence of predation are offered. First, results concerning boundedness, uniform persistence, existence and uniqueness of solutions have been developed. In addition, many thresholds have been computed and used to investigate local and global stability analysis by using Routh-Hurwitz criterion and Lyapunov principle. We also establish the Hopf bifurcation to highlight periodic fluctuation with persistence of the disease or without disease in the prey and predator population. Finally, numerical simulations are carried out to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global dynamics of a ratio-dependent predator–prey system with prey-taxis. We establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain with Neumann boundary conditions, and furthermore prove the global stability of homogeneous steady states under certain conditions. Finally we perform numerical simulations to show that the pattern formation may arise and prey-taxis is a factor driving the evolution of spatial inhomogeneity into homogeneity. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multi-group viral infection model with age structure and general incidence. We first give the positivity and boundedness of the solutions. Then, we study the local stability of the equilibria and uniform persistence of this model. Moreover, we establish the global results by constructing Lyapunov functionals. Finally, two special cases are presented to verify the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of R 0 by applying Lyapunov method. The results showed that if R 0 is less than 1, then the infection-free equilibrium E 0 is globally asymptotically stable. If R 0 is greater than 1, then the infection equilibrium E ∗ is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number R 0 and the basic immunity reproduction number R 1 . The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R 01 and immune reproduction number R 02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R 01 < 1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually; when R 02 > 1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional discrete-time predator–prey model that was recently developed in Ackleh et al. [Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603]. Utilizing a novel approach that is based on nullcline analysis, we derive conditions for the global stability of the interior equilibrium. This result significantly expands the parameter ranges under which global stability was shown to hold in Ackleh et al. [Long-term dynamics of discrete-time predator–prey models: Stability of equilibria, cycles and chaos, J. Differ. Equ. Appl. 26 (2020), pp. 693–726] using Lyapunov functions. We then extend these global stability results to a predator–prey model with evolution in the prey to obtain sharper conditions on the persistence of the system and to establish global-stability results for the interior equilibrium of this three-dimensional model. Numerical results corroborating these theoretical findings and demonstrating a relationship between the conditions for local asymptotic stability and global asymptotic stability are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the bearing-only formation control problem of multi-agent systems in local reference frame is studied. By considering that the inter-agent communication topology satisfies the maximal clique graph condition, the formation control can achieve global stability due to the existence of common agents among the maximal cliques. A cost function is proposed based on the target formation. By minimising the cost function under the optimal rotate matrix, a distributed globally stable controller is proposed, which only depends on the local bearing measurement. Then the rotation bias of the final formation and the target formation are analysed. The designed control law can realise the global formation control under the local reference frame. Finally the simulations demonstrate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of R n with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level R 0 ≤ 1 , the disease-free equilibrium E 0 is globally asymptotically stable. On the other hand, if R 0 > 1 then there exists a unique stable disease equilibrium E ∗ . The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R 0 . [ABSTRACT FROM AUTHOR]

This paper presents a class model for neutral-type bidirectional associative memory (BAM) neural networks (NNS) with infinite distributed delay. The model extends and improves some recent works for BAM-NNS. Based on the Lyapunov–Krasovskii functional (LKF) theory, the linear matrix inequality (LMI) method and some analysis techniques, the novel stability criteria for such model are obtained. Four numerical examples prove the validity and less conservativeness of the proposed criteria. The developed results are helpful to design of an actual system. [ABSTRACT FROM AUTHOR]

In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of the Lyapunov stability theorem. As a result of this we have proved that the NSFD schemes preserve essential properties of the metapopulation models (positivity, boundedness and monotone convergence of the solutions, equilibria and their stability properties). Especially, the basic reproduction number of the continuous models is also preserved. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper. [ABSTRACT FROM PUBLISHER]

Scarlet fever is an acute respiratory infectious disease and the incidence rate is increasing from 2011 throughout the world. In this paper, the mathematical models are proposed, which incorporate both direct transmissions and indirect transmissions of scarlet fever. The threshold conditions for disease invasion are obtained in terms of the basic reproduction number. The peak value, final size and epidemic time in a seasonal prevalence are investigated numerically. Furthermore, the effects of seasonal fluctuations on disease outbreak are also studied on the basis of real data in China. [ABSTRACT FROM AUTHOR]

This paper proposes a malaria transmission model to describe the dynamics of malaria transmission in the human and mosquito populations. This model emphasizes the impact of limited resource on malaria transmission. We derive a formula for the basic reproductive number of infection and investigate the existence of endemic equilibria. It is shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium co-exists with an endemic equilibrium. Furthermore, we determine conditions under which the disease-free equilibrium of the model is globally asymptotically stable. The global stability of the endemic equilibrium is also studied when the basic reproductive number is greater than one. Finally, numerical simulations to illustrate our findings and brief discussions are provided. [ABSTRACT FROM AUTHOR]

Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show the existence, uniqueness, nonnegativity, and boundedness of solutions for the model. We study the existence and local stability of steady states, which is shown to be determined by the basic reproduction number. By showing the existence of a global compact attractor and uniform persistence of the system, we establish the threshold dynamics using the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable and otherwise the disease will be established when there is initial infection force for the hosts. We also study a model with the standard incidence rate and discuss the effect of different incidence rates on the disease dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a stage-structured predator–prey model with mutual interference, in which includes two discrete delays. By theoretical analysis, we establish the stability of the unique positive equilibrium and the existence of Hopf bifurcation when the maturation delay for predators is used as the bifurcation parameter. Our results exhibit that the maturation delay for preys does not affect the stability of the positive equilibrium. However, the maturation delay for predator is able to destabilize the positive equilibrium and causes periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and display the differential impacts of two type delays and mutual interference. [ABSTRACT FROM AUTHOR]

This paper studies a class of stochastic SIRS epidemic models with general nonlinear incidence f (S , I , R) and varying population sizes. A new threshold value is obtained. The sufficient conditions for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution in probability meaning are established. As applications of the main results, the stochastic SIRS epidemic models with standard incidence, Beddington–DeAngelis incidence and nonlinear incidence rate h (S) g (I) are discussed, and a series of new criteria for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution are established. [ABSTRACT FROM AUTHOR]

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

This paper studies the global stability of discrete-time viral infection models with humoural immunity. We consider both latently and actively infected cells. We study also a model with general production and clearance rates of all compartments as well as general incidence rate of infection. We use nonstandard finite difference method to discretize the continuous-time models. The positivity and boundedness of solutions of the discrete models are established. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoural immune response activation number R 1 . The results signify that the infection dies out if R 0 ≤ 1. Moreover, the infection persists with inactive immune response if R 1 ≤ 1 < R 0 and with active immune response if R 1 > 1. We illustrate our theoretical results by using numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R 0 is obtained such that R 0 ≤ 1 indicates the possibility of dysentery eradication in the community while R 0 > 1 represents uniform persistence of the disease. The Lyapunov–LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R 0 > 1. Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation. [ABSTRACT FROM AUTHOR]

This paper studies the global stability of a discrete-time pathogen infection model with humoral immunity. The incidence rate of infection as well as the production and clearance rates of the cells, pathogens and antibodies are modelled by general functions. We use nonstandard finite difference method to discretize the continuous-time model. Two threshold parameters are derived, the basic reproduction number R 0 and the humoral immune response activation number R 1 . The basic and global properties of the model are established. A global stability analysis of the equilibria is performed using the Lyapunov method. Theoretical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper gives global stability criteria for a controlled stochastic system with randomly distributed time-delay. The delay function is governed by a homogeneous Poisson process and is independent of the stochastic perturbation on the system. A delay-dependent gain controller is constructed and used to develop a stabilization criterion for the system. Based on the Lyapunov-Krasovskii functional method and stochastic analysis theory, notions of stochastic stability are examined in terms of linear matrix inequality and global stability criteria derived. [ABSTRACT FROM AUTHOR]

Existing adaptive control algorithms at best guarantee that the tracking error is a l2 sequence. This paper presents globally stable adaptive control algorithms for linear discrete-time systems providing l1 tracking performance. Two algorithms with different degree of complexity are proposed, one for the case of known control directions, and a separate algorithm for the case of unknown control directions. It is demonstrated that in both cases the tracking error is l1 sequence, while the input and output signals are uniformly bounded. [ABSTRACT FROM AUTHOR]

In the recent paper [E. C. Balreira, S. Elaydi, and R. Luís, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luís established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luís' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of traveling wave fronts for a coupled system of non-local delayed lattice differential equations with a quiescent stage. It shows that under certain conditions all non-critical traveling wave fronts are globally exponentially stable, and critical ftraveling wave fronts are globally algebraically stable by applying the weighted energy method and the semi-discrete Fourier transform. [ABSTRACT FROM AUTHOR]

In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability analysis of latent Chikungunya virus (CHIKV) dynamics models. The incidence rate between the CHIKV and the uninfected monocytes is modelled by a general nonlinear function which satisfies a set of conditions. The model is incorporated by intracellular discrete or distributed time delays. Using the method of Lyapunov function, we established the global stability of the steady states of the models. The theoretical results are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes two modified susceptible-infected-recovered-susceptible models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, it is proved that if the basic reproduction numberof the model is less than one, then the disease-free equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the condition that the threshold valueis less than one. Otherwise, ifis more than one, the endemic equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the assumption that the total populationwill tend to a specific plane. In the study of the heterogeneous network model, this paper discusses the existences of the disease-free equilibrium and endemic equilibrium of the model. It is proved that if the threshold valueis less than one, then the disease-free equilibrium is globally asymptotically stable. Otherwise, ifis more than one, the system is permanent. A series of numerical experiments are given to illustrate the theoretical results. We also numerically predict the effect of vaccination ratio on the size of HBV-infected mainland Chinese population. [ABSTRACT FROM AUTHOR]

In this paper,we formulated an age-dependent model for the transmission dynamics of HBV with vaccination. The class of acutely infectious individuals,asymptomatic carrier of host population is stratified by age. Mathematically, we established that basic reproduction number can govern the global stability of equilibria. Biologically, we verify the impacts of the asymptomatic carriers and the effectiveness of vaccination on the disease transmission through numerical simulation. Our results indicated that the more number of infectious individuals specific to frequently progressed to asymptomatic carriers, the more likely the disease can be eradicated by continuous vaccination strategies. [ABSTRACT FROM AUTHOR]

Several nonlinear proportional-integral-derivative (PID) controllers for robot manipulators that ensure global asymptotic stability have been proposed in the literature. However, the tuning criteria obtained are expressed in terms of conditions so restrictive that they have avoided, until now, carrying out experimental tests with such controllers. Tuning criteria of some PID controllers for robot manipulators with conditions more relaxed than those presented previously in the literature have been proposed in two recent works by the authors. This was achieved by setting the tuning conditions individually for each joint instead of general conditions for the whole robot. In this paper we extend these results to a class of nonlinear PID global regulators for robot manipulators. The obtained tuning criteria are given in terms of conditions so relaxed that they have allowed to carry out, for the first time, experimental essays with these controllers. Such experiments are presented in this paper using a two-degrees-of-freedom robot manipulator. [ABSTRACT FROM PUBLISHER]

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models offacultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable. [ABSTRACT FROM AUTHOR]

This paper addresses the global finite-time regulation problem of robotic manipulators. A simple nonlinear proportional-integral-derivative (PID) control is proposed by adding a nonlinear proportional and derivative term to the commonly used PID controller. Lyapunov's stability theory and geometric homogeneity technique are employed to prove global finite-time stability. Advantages of the proposed control include the absence of modelling information in the control law formulation and the global finite-time stability featuring fast transient and high-precision positioning. Explicit conditions on the controller parameters to ensure global finite-time regulation stability are obtained. Simulations are presented to demonstrate the effectiveness and the improved performances of the proposed approach. [ABSTRACT FROM PUBLISHER]

In this paper we construct nonstandard difference schemes, which are dynamically consistent with a metapopulation model formulated by Keymer et al. in 2000, i.e. preserve all dynamical properties of the differential equations of the model. These properties are: monotone convergence, boundedness, local asymptotic stability and especially, global stability of equilibria and non-periodicity of solutions. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. [ABSTRACT FROM PUBLISHER]

This paper proposes and analyzes a discrete-time deterministic SIR model with information dependent immunization behaviour, where vaccination coverage at birth during any period of time is a general phenomenological function of the risk of infection that is perceived at the beginning of the period. Results on existence of equilibria, their local stability, and system persistence are proved. Then, by considering the noteworthy subcase of a piecewise linear ‘prevalence-dependent’ coverage function, the local stability of the endemic state is proved and conditions for its global asymptotic stability are given. Some insight on both Neimarck-Sacher and period-doubling bifurcations are provided. Overall we show that prevalence-dependent coverage is an essentially stabilising force. However period-doubling bifurcations are possible though under stressed parameter constellations. [ABSTRACT FROM PUBLISHER]

This paper deals with the global property of a drinking model with public health educational campaigns. With the help of Lyapunov function, global stability of equilibria of the model is derived. The alcohol-free equilibrium is globally asymptotically stable and the alcohol problems are eliminated from population if R0 ≤ 1. A unique alcohol present equilibrium is globally asymptotically stable if R0 > 1. Furthermore, the basic reproductive R0 for the model is compared with the basic reproductive number R1 for the absence of public health educational campaigns. We conclude that the public health educational campaigns of drinking individuals can slow down the drinking dynamics. Some numerical simulations are also given to explain our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a virus dynamics model with humoral immune response including latently infected cells. The incidence rate is given by Beddington–DeAngelis functional response. We have derived two threshold parameters, the basic infection reproduction numberand the humoral immune response activation numberwhich completely determined the basic and global properties of the virus dynamics model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have proven that if, then the infection-free equilibrium is globally asymptotically stable (GAS), if, then the chronic-infection equilibrium without humoral immune response is GAS, and if, then the chronic-infection equilibrium with humoral immune response is globally asymptotically stable. These results are further illustrated by numerical simulations. [ABSTRACT FROM PUBLISHER]

In this paper, an Susceptible-Vaccines-Exposed-Infectious-Recovered model with continuous age-structure in the exposed and infectious classes is investigated. These two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the exposed and the infectious classes. We investigate the global dynamics of this model in the sense of basic reproduction number via constructing Lyapunov functions. The asymptotic smoothness of solutions and uniform persistence of the system is shown from reformulating the system as a system of Volterra integral equations. [ABSTRACT FROM PUBLISHER]

This paper aims to study the combined impact of external computers and network topology on the spread of computer viruses over the Internet. By assuming that the network underlying a recently proposed model capturing virus spreading behaviour under the influence of external computers follows a power-law degree distribution, a new virus epidemic model is proposed. A comprehensive study of the model shows the global stability of the virus-free equilibrium or the global attractivity of the viral equilibrium, depending on the basic reproduction numberR0. Next, the impacts of different model parameters onR0are analysed. In particular, it is found that (a) higher network heterogeneity benefits virus spreading, (b) higher-degree nodes are more susceptible to infections than lower-degree nodes, and (c) a lower rate at which external computers enter the Internet could restrain virus spreading. On this basis, some practical measures of inhibiting virus diffusion are suggested. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc. [ABSTRACT FROM AUTHOR]