Global stability of a delayed SEIRS epidemic model with saturation incidence rate

In this paper, an SEIRS epidemic model with a saturation incidence rate and a time delay describing a latent period is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is established. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Numerical simulations are carried out to illustrate the main theoretical results.     

Global stability of a delayed HIV infection model with nonlinear incidence rate

Under the assumption that the incidence rate of the infection and the removal rate of the infective by cytotoxic T lymphocytes are nonlinear, we study the global dynamics of a HIV infection model with the response of the immune system using characteristic equation, the Fluctuation lemma, and the direct Lyapunov method. The existence of a threshold parameter, i.e., the basic reproduction number or basic reproductive ratio is established and the global stability of the equilibria is discussed.     

QUALITATIVE ANALYSIS OF AN SEIS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE

By means of limit theory and Fonda's theorem, an SEIS epidemic model with constant recruitment and the disease-related rate is considered. The incidence term is of the nonlinear form, and the basic reproduction number is found. If the basic reproduction number is less than one, there exists only the disease-free equilibrium, which is globally asymptotically stable, and the disease dies out eventually. If the basic reproduction number is greater than one, besides the unstable disease-free equilibrium, there exists also a unique endemic equilibrium, which is locally asymptotically stable, and the disease is uniformly persistent.     

Impacts of the Cell-Free and Cell-to-Cell Infection Modes on Viral Dynamics

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

Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments

The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.      

The pulse treatment of computer viruses: a modeling study

Unlike new medical procedures, new antivirus software can be disseminated rapidly through the Internet and takes effect immediately after it is run. As a result, a considerable number of infected computers can be cured almost simultaneously. Consequently, it is of practical importance to understand how pulse treatment affects the spread of computer viruses. For this purpose, an impulsive malware propagation model is proposed. To the best of our knowledge, this is the first computer virus model that takes into account the effect of pulse treatment. The dynamic properties of this model are investigated comprehensively. Specifically, it is found that (a) the virus-free periodic solution is globally asymptotically stable when the basic reproduction ratio (BRR) is less than unity, (b) infections are permanent when the BRR exceeds unity, and (c) a locally asymptotically stable viral periodic solution bifurcates from the virus-free periodic solution when the BRR goes through unity. A close inspection of the influence of different model parameters on the BRR allows us to suggest some feasible measures of eradicating electronic infections.     

Dynamics and control of COVID-19 pandemic with nonlinear incidence rates

World Health Organization (WHO) has declared COVID-19 a pandemic on March 11, 2020. As of May 23, 2020, according to WHO, there are 213 countries, areas or territories with COVID-19 positive cases. To effectively address this situation, it is imperative to have a clear understanding of the COVID-19 transmission dynamics and to concoct efficient control measures to mitigate/contain the spread. In this work, the COVID-19 dynamics is modelled using susceptible–exposed–infectious–removed model with a nonlinear incidence rate. In order to control the transmission, the coefficient of nonlinear incidence function is adopted as the Governmental control input. To adequately understand the COVID-19 dynamics, bifurcation analysis is performed and the effect of varying reproduction number on the COVID-19 transmission is studied. The inadequacy of an open-loop approach in controlling the disease spread is validated via numerical simulations and a robust closed-loop control methodology using sliding mode control is also presented. The proposed SMC strategy could bring the basic reproduction number closer to 1 from an initial value of 2.5, thus limiting the exposed and infected individuals to a controllable threshold value. The model and the proposed control strategy are then compared with real-time data in order to verify its efficacy.

     

Feedback stabilization of a class of nonlinear second-order systems

The goal of this paper is to study stabilization techniques for a system described by nonlinear second-order differential equations. The problem is to determine the feedback control as a function of the state variables. It is shown that the following controllers can asymptotically stabilize the system: linear position feedback, linear velocity feedback and a group of nonlinear feedbacks. The stability of the corresponding closed-loop system is proved by imposing a suitable Lyapunov function and then using LaSalle’s invariance principle. The results of numerical computations are included to verify theoretical analysis and mathematical formulation. Some application examples from robotics, mechanics and electronics are presented.     

Effect of the rate dependence of nonlinear kinematic hardening rule on relaxation behavior

Relaxation experiments for metallic materials and solid polymers have exhibited nonlinear dependence of stress relaxation on prior loading rate; the relaxed stress associated with the fastest prior strain rate has the smallest magnitude at the end of the same relaxation periods. Modeling capability for the basic feature of relaxation behavior is qualitatively investigated in the context of unified state variable theory. Unified constitutive models are categorized into three general classes according to the rate dependence of kinematic hardening rule, which defines the evolution of the back (equilibrium) stress and is the major difference among constitutive models. The first class of models adopts the nonlinear kinematic hardening rule proposed by Armstrong and Frederick. In this class, the back stress appears to be rate-independent under loading and subsequent relaxation conditions. In the second class of models, a stress rate term is incorporated into the Armstrong–Frederick rule and the back stress then becomes rate-dependent during relaxation condition even though it remains rate-independent under loading condition. The final class proposed here includes a new nonlinear kinematic hardening rule that causes the back stress to be rate-dependent all the time. It is shown that the apparent rate dependence of the back stress during relaxation enables constitutive models to predict the influence of prior loading rate on relaxation behavior.     

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

A two scaled numerical method for global analysis of high dimensional nonlinear systems

This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.     

Truncated predictor stabilization control for interconnected nonlinear systems with time-varying input delay

This paper deals with control design for interconnected nonlinear systems with time-varying input delay. Based on the truncated prediction of the system state over the delay period, the state feedback control law is constructed. In the framework of the Lyapunov–Krasovskii function, the stability equations of closed-loop system under state feedback law are established, and the feasibility of the controller is transformed into the problem of establishing a set of linear matrix inequality (LMI) conditions. Based on the Lyapunov stability theorem, it is proved that the closed-loop system is asymptotically stable. Finally, a simulation example is provided to demonstrate the effectiveness of the control scheme.

     

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity,Thermal Diffusivity,and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three- dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Chaotic Vibrations of a Cylindrical Shell-Panel with an In-Plane Elastic-Support at Boundary

This paper presents numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation. The shell, with a rectangular boundary, is simply supported for deflection and the shell is constrained elastically in an in-plane direction. Using the Donnell--Mushtari--Vlasov equation, modified with an inertia force, the basic equation is reduced to a nonlinear differential equation of a multiple-degree-of-freedom system by the Galerkin procedure. To estimate regions of the chaos, first, nonlinear responses of steady state vibration are calculated by the harmonic balance method. Next, time progresses of the chaotic response are obtained numerically by the Runge--Kutta--Gill method. The chaos accompanied with a dynamic snap-through of the shell is identified both by the Lyapunov exponent and the Poincaré projection onto the phase space. The Lyapunov dimension is carefully examined by increasing the assumed modes of vibration. The effects of the in-plane elastic constraint on the chaos of the shell are discussed.     

Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control

The model and the normalized state equations of the novel version of the Colpitts oscillator designed to operate in the ultra-high frequency range are presented. The circuit is investigated numerically and simulations demonstrate chaos in the microwave frequency range. Typical phase portrait, Lyapunov exponent and Lyapunov dimension are calculated using a piece-wise linear approximation of nonlinear I–V characteristic of the bipolar junction transistor. In addition, the feedback controller is applied to achieve chaos synchronization for two identical improved chaotic Colpitts oscillators. In the frame the nonlinear function of the system is used as a nonlinear feedback term for the stability of the error dynamics. Finally, numerical simulations show that this control method is feasible for this oscillator.     

The divergence of nearby trajectories in soft-sphere DEM

The n-body instability is investigated with the soft-sphere discrete element method. The divergence of nearby trajectories is quantified by the dynamical memory time. Using the inverse proportionality between the dynamical memory time and the largest Lyapunov exponent, the soft-sphere discrete element method results are compared to previous hard-sphere molecular dynamics data for the first time. Good agreement is observed at low concentrations and the degree of instability is shown to increase asymptotically with increasing spring stiffness. At particle concentrations above 30%, the soft-sphere Lyapunov exponents increase faster than the corresponding hard-sphere data. This paper concludes with a demonstration of how this case study may be used in conjunction with regression testing and code verification activities.     

Global Analysis of Two Tuberculosis Models

Two models for tuberculosis (TB) that include treatment of latent and infective individuals are considered. The first model assumes constant recruitment with a fixed fraction entering each class, having the consequences that TB never dies out and that the usual threshold condition does not apply. The unique endemic equilibrium is locally asymptotically stable for all parameter values and is shown to be globally asymptotically stable under certain parameter restrictions. The second model has a general recruitment function, but all recruitment is into the susceptible class. Three threshold parameters determine the existence and local stability of equilibria. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than or equal to one. The endemic equilibrium, when it exists, is shown to be globally asymptotically stable under certain parameter restrictions. Global stability results for the endemic equilibria are proved using the geometric approach of Li and Muldowney.     

Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R ₀ and the immune response reproduction number R ⁰ for the virus infection model, and establish that the global dynamics are completely determined by the values of R ₀. We obtain the global stabilities of the disease-free equilibrium E ₀, immune-free equilibrium E ₁ and endemic equilibrium E ^∗ when R ₀≤1, R ₀>1, R ⁰>1, respectively.     

Nonlinear analysis of spacecraft thermal models

We study the differential equations of lumped-parameter models of spacecraft thermal control. Firstly, we consider a satellite model consisting of two isothermal parts (nodes): an outer part that absorbs heat from the environment as radiation of various types and radiates heat as a black body, and an inner part that just dissipates heat at a constant rate. The resulting system of two nonlinear ordinary differential equations for the satellite’s temperatures is analyzed with various methods, which prove that the temperatures approach a steady state if the heat input is constant, whereas they approach a limit cycle if it varies periodically. Secondly, we generalize those methods to study a many-node thermal model of a spacecraft: this model also has a stable steady state under constant heat inputs that becomes a limit cycle if the inputs vary periodically. Finally, we propose new numerical analyses of spacecraft thermal models based on our results, to complement the analyses normally carried out with commercial software packages.     

Lyapunov-Like Solutions to Stability Problems for Discrete-Time Systems on Asymptotically Contractive Sets

This paper presents new criteria for stability properties of discrete-time non-stationary systems. The criteria are based on the concept of asymptotically contractive sets. As a result, general necessary conditions are established for asymptotic stability of the zero equilibrium state, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions including the invariance principle by LaSalle cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the zero state in such a case novel criteria are established. Under the criteria the total first time difference of a system Lyapunov function may be non-positive only and still can guarantee asymptotic stability of the zero state. The results are illustrated by examples.