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The purpose of this paper is to analyze the mechanism for the interplay of deterministic and stochastic models for contagious diseases. Deterministic models for contagious diseases are prone to predict global stability. Small natural birth and death rates in comparison to disease parameters like the contact rate and the removal rate ensures that the globally stable endemic equilibrium corresponds to a tiny average proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results. Diffusion effects force probability mass functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model predict, however, oscillatory patterns. Small and isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction. We prove that evolution maximizes the infectiousness of the disease as measured by the ability to increase the proportion of infected individuals. This holds provided the stochastic oscillations are moderate enough to keep the proportion of susceptible individuals near a deterministic equilibrium. We close our paper with a discussion of the herd-immunity concept and stress its close relation to vaccination-programs. [ABSTRACT FROM AUTHOR]

In this paper, we study the global stability of delayed genetic regulatory networks (DGRNs) with Hill-type activation (or inhibition) functions based on the mixing monotone semiflows approach, where Hill coefficients can be arbitrary positive real number. A new result on the global stability of DGRN is given in the case that all Hill coefficients are less than or equal to 1. In addition, for all Hill coefficients greater than 1, a new sufficient condition for global convergence of DGRN is given, which is less conservative than the condition of the existing correlation results. Finally, two numerical examples and simulations are given to explain the effect of obtained results. [ABSTRACT FROM AUTHOR]

Mathematical models have been considered as a robust tool to support biological and medical studies of the coronavirus disease 2019 (COVID-19). This new disease is caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). This paper develop a within-host SARS-CoV-2 dynamics model with logistic growth for healthy epithelial cells, humoral (antibody) immune response and general SARS-CoV-2-target incidence rate. The model is incorporated with four mixed (distributed/discrete) time delays, delay in the formation of latent infected epithelial cells, delay in the formation of active infected epithelial cells, delay in the activation of latent infected epithelial cells, and maturation delay of new SARS-CoV-2 particles. The model is formulated as a system of delay differential equations (DDEs). We establish that the model's solutions are non-negative and ultimately bounded. We deduce that the model has three equilibria and their existence and stability are perfectly determined by two threshold parameters. We prove the global stability of the model's equilibria by utilizing the Lyapunov method and applying the LaSalle's invariance principle. To support and illustrate our theoretical findings we present numerical simulations for the model with a special form of the general incidence rate function. The effect of time delays on the SARS-CoV-2 dynamics is addressed. We observe that increasing time delays values can have the same impact as drug therapies in suppressing viral progression. [ABSTRACT FROM AUTHOR]

Dengue is one of the vector-borne diseases spread in most parts of the world. The number of infected individuals is increased each year. This paper proposes a mathematical model describing the secondary dengue viral infection in micro-environment. This model takes into consideration that the dengue virus can infect multiple classes of target cells. Due to the secondary infection, the model incorporates two types of antibodies, heterologous and homologous. We establish the well-posedness of the model. We compute three threshold parameters which characterize the existence and stability conditions for the four steady states of the model. Global stability analysis for all steady states is carried out by formulating Lyapunov function and using Lyapunov-LaSalle asymptotic stability theorem. We demonstrate the analytical findings via numerical simulations. [ABSTRACT FROM AUTHOR]

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]

Abstract In this paper we consider the global stability of traveling curved fronts in the Belousov–Zhabotinskii (BZ for short) system in R 2. The existence and part of the stability of such curved fronts have already been proved in our recent work (Niu et al., 2018). The aim of this paper is to prove the global asymptotic stability of traveling curved fronts in space. The degeneration of the system at one of its equilibria makes its stability hard to study. We overcome this difficulty by introducing a mild subsolution which involves a perturbation that is the power of the one-dimensional traveling front. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by combining the p -order block boundary value methods with the m -th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ -order (0 < γ < 1) Caputo derivatives are obtained. It is proved under some appropriate conditions that the induced methods are convergent of order min ⁡ { p , m − γ + 1 } and globally stable. Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. Highlights • This paper deals with numerical computation and analysis for nonlinear fractional differential equations (FDEs) with γ -order Caputo derivatives. • A class of extended block boundary value methods (EBBVMs) for solving the FDEs are obtained. • The EBBVMs are proved to be convergent of order min ⁡ { p , m − γ + 1 } and globally stable under the appropriate conditions. • Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Rumor spreading is an important form of social communication, and it will cause unnecessary panic and conflict to society. This paper aims to investigate the combined impact of the internal and external factors on the efficiency of rumor spreading over the social networks. In this paper, firstly, a modified I S R rumor spreading model on heterogeneous networks by taking into account the activity and infectivity of nodes and propagation environment was introduced. Then, through theoretical analysis, we derive the mean field equations with considering nonlinear spreading rate and active network nodes. Next, we calculate the basic reproduction number ℜ 0 based on the next generation matrix method whose results reveal that if ℜ 0 < 1 , the rumor-free equilibrium is global asymptotically stable, the rumors will disappear. If ℜ 0 > 1 , the rumor-endemic equilibrium is global attractivity, then the number of spreaders will remain stable and the rumors will become endemic. Further, some numerical simulations are performed, which is consistent with the deterministic mean-field approach. Our results show that it is very important to consider the internal and external factors to control the spread of the rumors. [ABSTRACT FROM AUTHOR]

This paper deals with the global boundedness and stability of classical solutions to an important alarm-taxis ecosystem that is significant in understanding the behaviors of prey and predators. Specifically, it studies the case where prey attracts the secondary predators when threatened by the primary predators. The secondary consumers pursue the signal generated by the interaction between the prey and direct consumers. However, obtaining the necessary gradient estimates for global existence seems difficult in the critical case due to the strong coupled structure. Therefore, a new approach is developed to estimate the gradient of prey and primary predators, which takes advantage of slightly higher damping power. Subsequently, the boundedness of classical solutions in two-dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states are asymptotically stable, and the convergence rate is exponential under certain assumptions on the system coefficients. [ABSTRACT FROM AUTHOR]

This paper presents a non-modal bi-global linear stability analysis of an incompressible flat-plate boundary layer under the effects of oblique and non-uniform wall suction and injection. The base flow velocity profile is two-dimensional and fully non-parallel. The flow is laminar at the inflow boundary, and no reverse flow occurs in the flow domain. The Chebyshev spectral collocation method has been used for the discretization of the governing stability equations. The discretized stability equations along with appropriate boundary conditions form an initial value problem (IVP). The transient energy growth (G (t)) and associated optimal perturbations are computed for different oblique angles (θ = 30 ° , 60 ° , 90 ° , 120 ° , and 150 °), suction and injection intensities (I = 0. 5 % , 1.5%, and 2.5% of U ∞), Reynolds numbers (R e = 195 , 284 , and 411), and different spanwise wavenumbers (β = 0 − 2) for uniform and non-uniform profiles of wall suction and injection. The behaviour of the boundary layer is found to be approximately similar but opposite in nature for θ = 90 ° to 180 ° and θ = 0 ° to 90 ° due to lower transpiration intensity. The G (t) decreases with the increase in θ from 0 ° to 90 ° for wall suction due to the damping of T–S modes. However, an opposite trend has been observed for injection due to the amplification of T–S modes. The uniform profile is found to be more effective than the non-uniform profiles of suction and injection in terms of their impact on the stability of the boundary layer. The 2D spatial structures of the optimal perturbations are damped and flow becomes modally stable as the spanwise wavenumber (β) is increased. • Flat-plate boundary layer solved under the effect oblique and non-uniform of wall transpiration. • Parametric stability analysis of boundary layer performed for wall suction and injection. • The transient energy growth computed for stable and unstable global modes. • The effect of oblique angle, Reynolds number, and uniform, linear and quadratic profile of wall suction and injection with different intensities are studied. [ABSTRACT FROM AUTHOR]

This paper proposes a novel inner–outer loop control methodology to tackle the trajectory tracking problem for input-saturated quadrotors. The outer-loop, encapsulating the position tracking dynamics, is rendered input-to-state stable by designing a nested saturated control law with integrative action and applying small-gain arguments. The control strategy comprises an inner-loop that relies on a hybrid controller with integral action, designed based on the modified Rodrigues parameters attitude description, to solve the attitude tracking problem. The hybrid formulation of the controller benefits from the unique properties of the mentioned attitude description. Furthermore, it also provides a favorable framework to overcome the global stabilizing continuous feedback topological obstruction. For any given initial state, the resulting double-loop control architecture asymptotically tracks a position trajectory that satisfies some assumptions, while minimizing the distance to the desired attitude and satisfying the saturation limits of the thrust and torque inputs. The simulation results demonstrate the potential of the strategy. The experimental test carried out with a commercially available quadcopter and a motion capture system highlighted the proposed solution tracking capabilities, validating thereby the global saturated trajectory tracking controller. The proposed strategy is the first saturated MRP-based hybrid solution for quadrotor trajectory tracking. [ABSTRACT FROM AUTHOR]

Hepatitis C virus (HCV) can establish infection via two distinct modes: virus-to-cell infection and cell-to-cell transmission. These infections prompt the activation of two types of adaptive immune responses: the cytotoxic T lymphocyte response and the antibody response. In this paper, we study HCV dynamics by developing a multiscale model that incorporates both modes of infection as well as the two types of immune responses. We derive both the basic reproduction number of virus and four immune reproduction numbers for the model. We identify five equilibria, the existence of which depends on the values of basic reproduction number of virus and the immune reproduction numbers. We also establish the global asymptotic stability of the equilibria by employing Lyapunov functions, which further underscores the profound impact of the aforementioned reproduction numbers on the model's overall stability. [ABSTRACT FROM AUTHOR]

A k -winners-take-all (k -WTA) neural network is designed and applied to a task assignment problem in a multi-robot competitive target tracking scenario in this paper. The proposed neural network features a single neuron and a non-hard-limiting activation function, which greatly simplifies the model structure and reduces the computation cost. This neural network has finite-time convergence property and can be applied to real-time situations. The stability and convergence property of the neural network is theoretically analyzed. Simulations of handling a situation that a target moves at a higher speed than tracking robots are conducted to demonstrate the effectiveness of the designed scheme. [ABSTRACT FROM AUTHOR]

This paper studies the global stability analysis of a modified framework proposed by Mayengo et al. on alcohol-related health risks model. We first present the global stability analysis of risk-free equilibrium (RFE). Later, the global stability of the risk endemic equilibrium is studied. This goal was achieved by appropriate utilization of the symmetrical properties in the structure of Volterra–Lyapunov matrices. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in alcohol-related health risks and similar models. The numerical examples are simulated to validate the theoretical model results presented. • Alcohol-related health risk model is developed. • The equilibrium points (E 0 and E 1) are computed. • The basic risk reproduction number (R 0) of the model is computed. • The global stability analysis of E 0 is performed. • Volterra–Lyapunov matrix theory is applied in the global stability analysis of E 1. • The numerical examples are simulated to validate the model results. [ABSTRACT FROM AUTHOR]

A mathematical model for the transmission of buruli ulcer disease is proposed in this paper, incorporating three control strategies. Using the stability theory of differential equations, optimal control, and computer simulation, a nonlinear mathematical model for the problem is developed and qualitatively analyzed. Buruli ulcer free equilibrium and buruli ulcer present equilibrium were established. Using the next-generation operator method, the basic reproduction number of the model is obtained. It is found that the buruli ulcer-free equilibrium point is asymptotically stable locally and globally when R 0 < 1 and otherwise when R 0 > 1. The bifurcation analysis performed in this study indicates that the buruli ulcer model exhibits backward bifurcation. To investigate the parameters that influence the basic reproduction number R 0 , sensitivity analysis is performed. To minimize the transmission dynamics of buruli ulcer in humans and water bugs populations respectively, the buruli ulcer model is updated to include control measures attached to the parameters that affect the R 0 . In particular, the prevention measures covered are health education on the use of protective clothing, treatment of infected humans, and spraying of insecticides. The Pontryagin's maximum principle is used to characterize the controls' optimal levels. Using forward and backward sweep techniques in MATLAB, the optimality system was solved numerically. The results of the simulations show that, relative to the case of no control, there is a substantial decrease in the population of infected humans and the total population of water bugs when control measures are implemented. A study of cost-effectiveness shows that the most cost-effective approach is strategy G with the smallest ICER of 33442.44. To mitigate or eliminate buruli ulcer in the population, we conclude that the government should invest in health education on the use of protective clothing, treatment of infected humans, and the use of insecticide spray. [ABSTRACT FROM AUTHOR]

The Lyapunov direct method is a common and effective tool for discussing the global stability of dynamical systems. In this paper, we propose an analytical approach to apply this method for the global stability of the positive equilibrium of a polynomial differential system with discrete time delays, which includes how to construct an applicable Lyapunov functional and to verify the negative (semi-)definiteness of its derivative. Here the linear combination of the Volterra-type functions/functionals and their integrals plays an important role. Moreover, when the delayed system also has a boundary equilibrium, we show that the Lyapunov functional for the global stability of the positive equilibrium can be reformulated to establish the global stability of the boundary equilibrium. As an illustration, we apply this approach to a virus infection model with two delays. [ABSTRACT FROM AUTHOR]

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]

In this paper, a homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission, which incorporates anti-HIV preventive vaccines, is proposed. The dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when there is no vaccine. However, it has been shown that when the efficacy or dosage of vaccines is low, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium point (DFE) coexists with a stable endemic equilibrium point (EE) when the associated reproduction number is less than unity. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. A new critical value at the turning point should be deduced as a new threshold of disease eradication. We have generalized the integer LaSalle invariant set theorem into fractional system and given some sufficient conditions for the disease-free equilibrium point being globally asymptotical stability. Mathematical results in this paper suggest that improving the efficiency and dosage of vaccines are all valid methods for the control of disease. [ABSTRACT FROM AUTHOR]

We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data. [ABSTRACT FROM AUTHOR]

This paper investigates a new mathematical SQIR model for COVID-19 by means of four dimensions; susceptible, quarantine, infected and recovered (SQIR) via Non-linear Saturated Incidence Rate. First of all the model is formulated in the form of differential equations. Disease-free, endemic equilibriums and Basic Reproduction Number are found for the said model. Local Stability is analyzed through Jacobean Matrix while Lyapunov Function is constructed for the study of Global Stability of the Model. Using nonstandard finite difference method, numerical results are simulated. By Simulation, we mean how protection, exposure, death and cure rates affect the Susceptible, Quarantined, Infected and recovered population with the passage of time. [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]

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]

• 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]

This paper investigates the existence and uniqueness of the equilibrium point and its global µ -stability for neutral-type impulsive complex-valued bidirectional associative memory neural networks with leakage delay and unbounded time-varying delays, by considering two types of Lipschitz conditions to be satisfied by the activation functions. The homeomorphism lemma is used to obtain a sufficient condition for the existence and uniqueness of the equilibrium point. Sufficient delay-dependent conditions both in terms of complex-valued and real-valued linear matrix inequalities which ensure the global µ -stability of the equilibrium point are obtained by constructing appropriate Lyapunov–Krasovskii functionals with simple, double, and triple integral terms, and using the free weighting matrix method, simple and double complex-valued Jensen inequalities, the complex-valued reciprocally convex combination inequality, and the complex-valued Wirtiger-based integral inequality. Lastly, two numerical examples are provided to illustrate the effectiveness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion multi-group SIR epidemic model with nonlinear incidence in spatially heterogeneous and homogeneous environment is investigated. In general spatially heterogeneous environments, the well-posedness of solutions, including the nonnegativity and ultimate boundedness of solutions, firstly is established. The basic reproduction number R 0 is defined. The threshold criteria on the global dynamics of the model are established. That is, if R 0 < 1 , the disease-free steady state is globally asymptotically stable, while if R 0 > 1 , the model is uniformly persistent. Furthermore, for the homogeneous space and heterogeneous diffusion model, by using the Lyapunov functions method, the global asymptotic stability for the disease-free and endemic steady states is also established. [ABSTRACT FROM AUTHOR]

• 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]

• 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]

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]

This paper is concerned with the numerical solutions of second-order delay initial value problems (DIVPs). By adapting the generalized Störmer–Cowell methods (GSCMs) for second-order ordinary differential equations, a class of efficient numerical algorithms for solving second-order DIVPs are derived. It is proved under some appropriate conditions that the adapted GSCMs are uniquely solvable, globally stable and convergent of order p , where p is the consistence order of the methods. A numerical example is presented to illustrate the computational effectiveness of the adapted GSCMs and the obtained theoretical results. [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 propose and analyze an epidemic problem which can be controlled by vaccination as well as treatment. In the first part of our analysis we study the dynamical behavior of the system with fixed control for both vaccination and treatment. Basic reproduction number is obtained in all possible cases and it is observed that the simultaneous use of vaccination and treatment control is the most favorable case to prevent the disease from being epidemic. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both the infected populations and the associated costs. All the analytical results are verified by simulation works. Some important conclusions are given at the end of the paper. [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]