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Abstract This paper is mainly concerned with an investigation of antigen-driven T-cell infection process through a mathematical model. Several mathematical models have been introduced in the literature in order to gain insights into the dynamics of the disease progression, however, the results considering the effect of multiple factors which include antigen-driven CD4 T-cell progress, latent infection stage, activation of CTLs response, role of antiretroviral therapies (ARTs) and possibilities of multiple time-delays during the infection process are not involved. Hence, the paper introduces a six-dimensional virus infection model by involving the above factors. Particularly, (i) the effect of activation of antigen-specific T-cells; (ii) the effect of the maturation of infected cells; (iii) effect of multiple time delays, that is, during the interaction between susceptible and infectious and during the activation of immune responses; which play a significant role in preventing and modulating the Human Immunodeficiency Virus (HIV). The main aim of the paper to analyze the local and global stability of the class of mathematical models regarding the effect of time delays which provides a better pathway to the infection progress. Finally, the overall contribution of the present work is listed as follows: (1) by constructing the suitable Lyapunov functional, the global stability of the intracellular delayed model is derived; (2) detailed Hopf-bifurcation analysis is discussed with respect to immune response delay. The numerical simulation is performed to validate the effectiveness and applicability of the theoretical predictions. [ABSTRACT FROM AUTHOR]

In terms of fishery management, if it is not managed carefully, it will lead to the extinction of the species and other unfavorable conditions. In order to strengthen the protection, development and rational utilization of fishery resources, this paper proposes a population capture model with state feedback control based on previous studies, in which the small fish is influenced by the Allee effect, and the catch of the big fish is linearly related with the release of the small fish. Firstly, we give the definition of the Poincaré map of the catch system based on impulsive point series, and study the complex dynamic properties of the map, including single peak, multi-peak functions and multiple jumping points. Moreover, the conditions of existence and the number of jumping points are discussed. At the same time, we analyze its properties including monotonicity, differentiability and fixed point. Secondly, the conditions under which the existence and uniqueness of the order-1 periodic solution are provided, and we address the necessary and sufficient conditions for stability of the order-1 periodic solution. Furthermore, existence of the order- k (k ≥ 2) periodic solution is obtained. Finally, correctness of our theoretical results is illustrated by numerical simulations. The results show that under human control, the density of big fish and small fish will maintain periodic benign changes, and fishery resources are sufficient for sustainable development. • Building Poincaré map of system to discuss complex dynamics. • Discussed the properties of Poincaré map in the case of jumping points, such as the condition and number of jumping points. • The study of the big fish and small fish model has practical significance. • Numerical simulation shows the existence of periodic solutions of order 2 and order 3. [ABSTRACT FROM AUTHOR]

In this paper, to analyze the effect of the cross patch infection between different groups to the spread of gonorrhea in a community, we establish the complete global dynamics of a multi-group SIS epidemic model with varying total population size by a threshold parameter. In the proof, we use special Lyapunov functional techniques, not only one proposed by the paper [Prüss et al., 2006], but also the other one for a varying total population size with some ideas specified to our model and no longer need a grouping technique derived from the graph theory which is commonly used for the global stability analysis of multi-group epidemic models. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we have proposed and mathematically modeled an epidemic problem with vector-borne disease. We have taken three different classes for the trees, namely susceptible, exposed and infected, and two different classes for the vector population, namely susceptible and infected. In the first part of our paper, we rigorously analyze our model using the dynamical systems approach. Global stability of equilibria is resolved by using Lyapunov functional. In the second part, the model is reformulated as an optimal control problem in order to determine the significance of certain control measures on the model. We apply four control parameters, namely the tree injection control to the trees, deforestation of infected trees, eradication effort of aerial insecticide spraying and the effort of restrain of mating. Both numerical and analytical methods are employed to ascertain the existence of cost effective control measures. [Copyright &y& Elsevier]

Abstract: In this paper we proposed and analyzed a prey predator system with stage-structured for the predator population. A time delay is incorporated due to the gestation for the matured predator. All the possible non-negative equilibria are obtained and their local as well as global behavior are studied. Choosing delay as a bifurcation parameter, the existence of the Hopf bifurcation of the system has been investigated. Moreover, we use the normal form method and the center manifold theorem to examine the direction of the Hopf bifurcation and the nature of the bifurcating periodic solution. Some numerical simulations are given to support the analytical results. Some interesting conclusions are obtained from our analysis and it is given at the end of the paper. [Copyright &y& Elsevier]

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]

• A cholera infection model with vaccination strategy is studied. • The basic reproduction number is obtained. • A threshold dynamics is established by using Lyapunov function technique. • Several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. In this paper, a cholera infection model with vaccination strategy is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria is established. By means of Lyapunov functions and LaSalle's invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable. In addition, by using Pontryagin's maximum principle, several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an I2S2R rumor spreading model in complex networks is established and analyzed. Firstly, the threshold value is acquired to identify the existence and stability of rumor-free equilibria state and rumor equilibria state by using the next generation matrix method. Secondly, based on Routh–Hurwitz judgment, Lyapunov stability theory and LaSalle's invariance principle, the local and global stability of rumor-free/rumor equilibrium points are discussed. Thirdly, the sensitivity analysis about the threshold value is given to analyze the impact of model parameters on rumor propagation. Finally, some numerical simulations are presented to illustrate the validity and feasibility of corresponding results. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the global stability of a delayed diffusive influenza model with two-strain and two vaccinations. First, we consider the well-posedness of solutions of the model and show that the model admits four constant equilibria: a disease-free equilibrium, two single-strain-infection equilibria, and a double-strain-infection equilibrium, which are determined by three threshold parameters R 1 , R 2 and R 3 (or R 4). Second, constructing four suitable Lyapunov functionals and using the LaSalle's invariant principle, we establish the global stability of these equilibria. Finally, we give several numerical simulations to illustrates the analytic results. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ 0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ 0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method. [ABSTRACT FROM AUTHOR]

Abstract In this paper a nonlinear mathematical model for a within-host co-infection of influenza A virus and pneumococcus is investigated. Conditions that explain the relations amid R IP and its relationship to the global asymptotic stability of the infection-free steady state are discussed. A graph-theoretic method shows that, the unique endemic steady state is globally asymptotically stable. The sensitivity analysis show that, the pathogen fitness for pneumococcus and influenza A virus are most sensitive to maximum number of bacteria an alveolar macrophage can catch, phagocytosis rate, number of infectious IAV and pneumococcus particles liberated from lysis of infected cells and infection rates of influenza A virus and pneumococcal. Numerical results of the model show that, there exists a biologically important steady state where the two infectious pathogens of unequal strength co-exist and replace each other in the epithelial cell population (with pneumococcus leading) when the pathogen fitness for each infection exceeds unity, and we find that this endemic steady state is globally asymptotically stable. Further, the impact of influenza A virus on pneumococcus and vice-visa leads to a bifurcation state. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by ideas from papers: C. Cinar, I. Yalçinkaya, S. Stević, A note on global asymptotic stability of a family of rational equations, Rostock. Math. Kolloq. 59 (2004), 41–49, and S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), 60–68, here we confirm a conjecture on a rational symmetric difference equation from the paper: K. Berenhaut, J. Foley, S. Stević, The global attractivity of the rational difference equation , Appl. Math. Lett. 20 (2007), 54–58. [Copyright &y& Elsevier]

Abstract: In this paper the global asymptotic stability of the nonlinear difference equationis investigated, where α, β, A, B, C >0 are real numbers, and the initial conditions x −1 is nonnegative real numbers and x 0 is a positive real number. We show that the unique positive equilibrium of the equation is globally asymptotically stable. In particular, our results solve two conjectures proposed by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002] and by Kulenovic et al. in their paper [M.R.S. Kulenovic, G. Ladas, L.F. Martins, I.W. Rodrigues, The dynamics of facts and conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]. [Copyright &y& Elsevier]

In this paper, we consider a transmission model of hepatitis B virus by taking into account media coverage. First, we formulate the model and find the basic reproduction number R 0 by using the next generation matrix method. We show that the disease free equilibrium is locally asymptotically stable for R 0 < 1 and unstable if R 0 > 1 . We also prove that the system is globally asymptotically stable for R 0 < 1 . In order to control the spread of this disease in community, we devise an optimal control problem by introducing three control functions, that is, the educational campaign, vaccination and the media coverage. To do this, we solve analytically the control problem with characterization of control variables using the Portraying’s Maximum Principle. Finally, we present some numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n ( n  ≥ 2) patches. We derive the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable if R 0  ≤ 1. In the case of R 0  > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global dynamics of a general SIRS epidemic model with a ratio-dependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the basic reproduction number R 0 determines whether there is an endemic outbreak or not: if R 0 < 1 , the disease dies out; while if R 0 > 1 , the disease persists. For the stochastic model, we show that its related reproduction number R 0 S can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness and permanence/extinction. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the threshold dynamics of an HIV-1 viral infection model with cell-mediated immune responses and direct cell-to-cell transmission mechanism. The model demonstrates global threshold dynamics with respect to the reproductive numbers for viral infection ℜ 0 and for CTL immune response ℜ 1 . The proofs of main results come from suitable uses of analyzing the characteristic equation and constructing Lyapunov functionals. Specifically, if ℜ 0 < 1, the infection-free equilibrium E 0 is locally and globally asymptotically stable, and the viruses are cleared. If ℜ 1 < 1 < ℜ 0 , the CTL-inactivated equilibrium E 1 is locally and globally asymptotically stable, and the infection becomes chronic but without persistent CTLs response. If ℜ 1 > 1, the CTL-activated equilibrium E 2 is locally and globally asymptotically stable, and the infection is chronic with persistent CTLs response. Numerical simulations are performed to support our results. [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 study a model of HCV with saturation and delay, we stablish the local and global stability of system also we stablish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. We present a sensitivity analysis for the basic reproductive number. [ABSTRACT FROM AUTHOR]

As a tobacco victimized country in the world, China is being burdened by smoking-related illnesses. This paper concentrates on the problem of controlling smoking in China. We propose a mathematical model described by ordinary differential equations with a saturated incidence rate to explore the effect of controlling smoking by setting up designed smoking areas and raising the price of cigarettes. The sufficient criteria for the local and global stabilities of the equilibria are obtained. Furthermore, a sensitivity analysis is performed to gain insight into which factors are the most important in controlling smoking. Lastly, we carry out numerical simulations to illustrate and verify the analytic conclusions, and provide a control measure. [ABSTRACT FROM AUTHOR]

Based on some important biological meanings, we propose a class of HIV infection models incorporating both classical cell-free virus diffusion and direct cell-to-cell transmission. According to recent studies, the direct cell-to-cell transfer of HIV is a significantly more efficient mode of retroviral dissemination. In the first part of our analysis, we show that our model possesses non-negative solutions. Then, we derive sufficient conditions for the asymptotic stability of equilibriums. The analytical solutions are verified by simulation results. At the end of the paper, some important conclusions are given. [ABSTRACT FROM AUTHOR]

• An SVEITR model with vaccination, fast and slow progression, incomplete treatment and relapse for TB is proposed. • Based on the data from 1988 to 2019, the basic reproduction number (R 0)of TB in the United States is 1.3763. • If R 0 > 1, the unique endemic equilibrium is globally asymptotically stable under certain conditions. • Optimal control strategies are prevention and control education, timely treatment and enhanced efficacy. • The most cost-effective strategies are timely treatment and enhanced efficacy. Tuberculosis is a chronic infectious disease with a high death rate, which has attracted worldwide attention. In this paper, we focus on the annual data of tuberculosis in the United States from 1988 to 2019. Firstly, we propose an S V E I T R dynamical model with vaccination, fast and slow progression, incomplete treatment, and relapse. Mathematical analyses show that the disease-free equilibrium is globally asymptotically stable (unstable) if R 0 ≤ 1 (if R 0 > 1). There exists a unique endemic equilibrium that is globally asymptotically stable when R 0 > 1. Next, we use the Markov-chain Monte-Carlo (MCMC) method to acquire the model's optimal parameters. From R 0 = 1.3763 , tuberculosis will not be eliminated without control measures taken in the United States. Through partial rank correlation analysis, the most sensitive parameters to the basic reproduction number are found. Based on this, we take the actual epidemic situation of tuberculosis in the United States into considerations to get the required conditions for optimal control using Pontryagin's maximum principle and cost-effectiveness analysis. Implementation of three public health measures: tuberculosis prevention and control education, timely treatment, and enhanced efficacy can reduce the prevalence of tuberculosis in the United States. But at present, it is challenging for us to achieve the goal of eliminating tuberculosis by 2030 as called for by WHO. [ABSTRACT FROM AUTHOR]

In this paper, the global stability of a schistosomiasis infection model that involves human and intermediate snail hosts as well as an additional mammalian host and a competitor snail species is studied by constructing Lyapunov functions and using properties of K monotone systems. We derive the basic reproduction number R 0 for the deterministic model, and establish that the global dynamics are completely determined by the values of R 0 . We show that the disease can be eradicated when R 0 ⩽ 1 . In the case where R 0 > 1 , we prove the existence, uniqueness and global asymptotic stability of an endemic steady state. This mathematical analysis of the model gives insight about the epidemiological consequences of the introduction of a competitor resistant snail species. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of the asymptotic behavior for a non-autonomous randomized food-limited logistic population model. Several sufficient conditions are formulated and proved for p -moment persistence and extinction of the population, as well as in sense of almost sure. Results show that food-limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model. Some previously known results are improved. Numerical simulations are provided to support the results. [ABSTRACT FROM AUTHOR]

This paper deals with a robust stage structured predator–prey model with Beddington–DeAngelis-type functional response. The proposed mathematical model consists of a system of three nonlinear ordinary differential equations to stimulate the interactions between prey population, juvenile predators and adult predator population. The positivity, boundedness and the conditions for uniform persistence have been derived. The dynamical behavior of the system both analytically and numerically investigated from the point of view of local stability, persistence and global stability. The global stability of the system has been derived by using the theory of competitive systems, stability of periodic orbits and compound matrices for the interior equilibrium point. Depending on the conversion rate of the prey population to juvenile predator, the model exhibits Hopf-bifurcation. The model admit periodic solutions which is produced from the stage structure of the predator populations. Numerical simulations have been accomplished to validate our analytical findings. [ABSTRACT FROM AUTHOR]

It has been recently conjectured in Yang et al. (2013) [27] that the unique virose equilibrium of a network-based model is globally asymptotically stable in the feasible region. The aim of this paper is to investigate the global stability of the virose equilibrium by using the direct Lyapunov method and appropriate grouping techniques for Lyapunov function derivatives. Some sufficient conditions are established, which improves the relevant results. Moreover, we provide a new brief proof for global stability of the virus-free equilibrium by using comparison method, and discuss quantitatively the effect of various immunization schemes, including uniform immunization, targeted immunization and high-risk immunization, on the viruses spread, which suggests promising strategy to halt viruses spread in computer networks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze two structured models for the transmission of a vector-borne infectious disease. The first of these models assumes that the level of contagiousness and the rate of removal (recovery) of infected hosts depends on the infection age. In the second model the hosts population is structured with respect to the physical age of the hosts, and the susceptibility of the hosts is assumed to be age-dependent. For these models, the threshold parameter for the existence of a positive (endemic) equilibrium state is determined, and the global asymptotic stability of the equilibrium states are established by the Lyapunov's direct method. [ABSTRACT FROM AUTHOR]

We consider a nonlinear non-autonomous system with time-varying delays xi(t)=-ai(t)xi(hi(t)) + Σj=imFij(t,xj(gij(t))), i=1,...,m which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed using the system's parameters, is an M-matrix. Comparison with the tests obtained by Gopalsamy (2007) and Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the dynamics of a five-dimensional virus model with immune responses and an intracellular delay which describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is an improvement of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function f for virus infections, and (iii) antibody responses, which are implemented by the functioning of immunocompetent B lymphocytes, play a critical role in preventing and modulating infections. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we show that the global dynamics of the model is determined by the reproductive numbers for viral infection , for CTL immune response , for antibody immune response , for CTL immune competition and for antibody immune competition . The global stability of the model precludes the existence of Hopf bifurcation and other complex dynamical behaviors in long time. Numerical simulations are also performed in order to illustrate the dynamical behavior. [Copyright &y& Elsevier]

Abstract: In this paper, applying Lyapunov functional techniques to a nonlinear delayed Lotka–Volterra system with feedback controls and patch structure, we establish sufficient conditions of the global stability for a trivial equilibrium and a positive equilibrium, respectively. Moreover, we offer new techniques to prove the permanence and the existence of positive equilibrium of this system. The influence of the feedback controls can be essentially eliminated from the Lyapunov functional to prove the global stability, whereas feedback controls change the position of a unique positive equilibrium. These generalize the known results of recent literature. [Copyright &y& Elsevier]

Abstract: In this paper, for a class of nonlinear functional-integro-differential equations, a type of mixed Runge–Kutta methods are presented by combining the underlying Runge–Kutta methods and the compound quadrature rules. Based on the non-classical Lipschitz condition, a global stability criterion is derived. Numerical experiments illustrate applicability of the theory, efficiency of the methods, and difference of the mixed Runge–Kutta methods from the Pouzet–Runge–Kutta methods. [Copyright &y& Elsevier]

Abstract: In this paper, an HIV-1 infection model with saturation incidence and time delay due to the CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the CTL-activated infection equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, it is shown that the infection-free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the immune response reproductive ratio is less than unity and the basic reproductive ratio is greater than unity, the CTL-inactivated infection equilibrium of the system is globally asymptotically stable. [Copyright &y& Elsevier]

Abstract: As drug treatment allows more and more people with HIV/AIDS to live longer, the trade-off between benefits to drug treatment and potential threat infectivity individuals needs to be carefully evaluated. In this paper, we shall extend and investigate an HIV/AIDS treatment model (Cai et al., 2009) [14]. The infection force of the extended model is assumed to be of density dependent form. The resulting incidence term contains, the bilinear and the standard incidence. Mathematical analysis establishes that the global dynamics of the HIV infectious disease is completely determined by the basic reproduction number . If , the disease always dies out and the disease-free equilibrium is globally stable. If , the disease persists and the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region. [Copyright &y& Elsevier]

Abstract: In this paper, we have considered a dynamical model of Chlamydia disease with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a time lag (τ) to account for the fact that an individual infected with bacterium Chlamydia trachomatis is not infectious until after some time after exposure. The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for and value zero for . The probability that an individual in the latency period has survived is given by , where μ denotes the natural mortality rate in all epidemiological classes. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. We have defined two positive numbers and . It is proved that there exists an infection-free periodic solution which is globally attractive if and the disease is permanent if . Our analytical findings are illustrated through computer simulation using MATLAB, which show the reliability of our model from the epidemiological point of view. [Copyright &y& Elsevier]

Abstract: Vaccination is important for the elimination of infectious diseases. In this paper, a basic SVIR epidemic model with age of vaccination is considered. The model allows the vaccinated individuals to become susceptible again when vaccine loses its protective properties with time. The vaccination classes satisfy partial differential equations for the vaccination age. By means of LaSalle’s invariance principle and constructing suitable Lyapunov function(al)s, the dynamical property of the model is established. It is shown that the global stability results of the infection-free equilibrium and the endemic equilibrium depend only on the basic reproductive number . Finally, some numerical simulations are carried out to illustrate the main results. The combined effects of the vaccination rate and the age factor on the dynamics of the disease are displayed. [Copyright &y& Elsevier]

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]

Abstract: An autonomous stage-structured predator–prey system (stage structure for both predator and prey) with discrete delay is studied in this paper. By using an iterative method, the global stability of the interior equilibrium point of the system is investigated. Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones. [Copyright &y& Elsevier]

Abstract: This paper aims to explore the influence of vaccination on the diffusion of computer viruses under more reasonable assumptions. By incorporating a vaccination probability in the SIRS model with generalized nonlinear incidence rate, a novel epidemic model of computer viruses is established. A thorough analysis of this model shows that, depending on the value of the basic reproduction number, either the virus-free equilibrium or the endemic equilibrium is globally asymptotically stable. Simulation results not only justify the proposed model, but also demonstrate the effectiveness of vaccination. Based on a parameter analysis for the model, some effective strategies for inhibiting the virus prevalence are posed. [Copyright &y& Elsevier]

Abstract: This paper studies an impulsive nonautonomous two species predator–prey Lotka–Volterra type dispersal system 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. On the basis of inequality estimation techniques and the comparison theory of impulsive differential equations, sufficient conditions of integrable form for the permanence and extinction are obtained. By constructing appropriate Lyapunov function, conditions for the global attractivity of the system are established. [Copyright &y& Elsevier]

Abstract: Using parasitic wasps that deliver lethal viruses to crop pests is a new way to protect crops. In this paper, we propose and analyze an epidemic model with birth pulse and pulse infection by virus, which periodic birth pulse and pulse infection occur at different fixed time. An exact cycle of the system and the threshold conditions for its stability are obtained. Some numerical simulations are given. Moreover, the sufficient conditions of global stability of semi-trivial periodic solutions are investigated. Our proposed strategy of delivering deadly viruses is effective and adequate from the environmental point of view. [Copyright &y& Elsevier]

Abstract: This paper deals with a delayed three-species Lotka–Volterra food chain model with diffusion effects and homogeneous Neumann boundary conditions. By taking the sum of delays as the bifurcation parameter, spatially homogeneous and nonhomogeneous Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay parameter. In addition, sufficient conditions for global asymptotic stability of the positive constant steady state are derived by using an iteration technique. Furthermore, in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions, the formulas are given by using the normal form theory and the center manifold reduction for PFDEs. Finally, to verify our theoretical predictions, some numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: This paper describes a prey–predator model with stage structure for predator and selective harvesting effort on predator population. The Holling type II functional response function is taken into consideration. All the equilibria of the proposed system are determined and the behavior of the system is investigated near them. Local stability of the system is analyzed. Geometric approach is used to derive the sufficient conditions for global stability of the system. The occurrence of Hopf bifurcation of the model system in the neighborhood of the co-existing equilibrium point is shown through considering maximal relative increase of predation as bifurcation parameter. Fishing effort used to harvest predator population is considered as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a delay differential equations model of plankton allelopathy. Using an iterative method, the global stability of the interior equilibrium point of the system is investigated, which shows that toxic substances are harmless for the stability of the interior equilibrium point. The results we obtained improve and complement those of Chattopadhyay [J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecol. Model. 84 (1996) 287–289], Samanta [G.P. Samanta, A two-species competitive system under the influence of toxic substances, Appl. Math. Comput. 216 (2010) 291–299] and Mukhopadhyay et al. [A. Mukhopadhyay, J. Chattopadhyay, P.K. Tapaswi, A delay differential equations model of plankton allelopathy, Math. Biosci. 149 (1998) 167–189]. We also present some examples to verify our main result. [Copyright &y& Elsevier]

Abstract: This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one. [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, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R 0. It shows that, the basic reproduction number R 0 is a global threshold parameter in the sense 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, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result. [Copyright &y& Elsevier]

Abstract: In this paper, we have considered a nonautonomous dynamical model of diseases that spread by droplet infection and also through direct contact (with a lower risk) with varying total population size and distributed time delay to become infectious. It is assumed that there is a time lag due to incubation period of pathogens, i.e. the development of an infection from the time the pathogen enters the body until signs or symptoms first appear. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness. [Copyright &y& Elsevier]

Abstract: In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number . It is shown that if is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively. [Copyright &y& Elsevier]