Global properties of a class of virus infection models with multitarget cells

In this paper, we propose a class of virus infection models with multitarget cells and study their global properties. We first study three models with specific forms of incidence rate function, then study a model with a more general nonlinear incidence rate. The basic model is a (2n+1)-dimensional nonlinear ODEs that describes the population dynamics of the virus, n classes of uninfected target cells, and n classes of infected target cells. Model with exposed state and model with saturated infection rate are also studied. For these models, Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models. We have proven that if the basic reproduction number is less than unity then the uninfected steady state is globally asymptotically stable, and if the basic reproduction number is greater than unity then the infected steady state is globally asymptotically stable. For the model with general nonlinear incidence rate, we construct suitable Lyapunov functions and establish the sufficient conditions for the global stability of the uninfected and infected steady states of this model.     

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.      

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.     

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.     

GLOBAL BIFURCATION AND CHAOS IN A VAN DER POL-DUFFING-MATHIUE SYSTEM WITH A SINGLE-WELL POTENTIAL OSCILLATOR

The global bifurcation and chaos are investigated in this paper for a van der Pol-Duff-ing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The au-tonomous system corresponding to the system under discussion is analytically studied to draw all globalbifureation diagrams in every parameter space, These diagrams are called basic bifurcation ones. Thenfixing parameter in every space and taking the parametrically excited amplitude as a bifurcation param-eter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of nu-merical methods. The results are sufficient to show that the system has distinct dynamic behavior, Fi-nally, the properties of the basins of attraction are observed and the appearance of fractal basin bound-aries heralding the onset of a loss of structural integrity is noted in order to consider how to control theextent and the rate of the erosion in the next paper.     

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 bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods. The project supported by the National Natural Science Foundation of China     

球壳轴对称弯曲问题共轭二阶挠度微分方程

提出球壳轴对称弯曲问题共轭二阶挠度微分方程并给出了初等函数解. 球壳微分方程是薄壳理论三大壳之一旋转壳的典型方程. 共轭二阶挠度微分方程是球 壳中微分方程形式最简单的, 是人们最喜爱的挠度微分方程. 挠度微分方程满足边 界条件非常简单, 使球壳的计算得到很大的简化.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

????????Ч?????????????????·???

在研究矩独立基本变量对响应分布影响的重要性测度的基础上,定义了基本变量对失效概率 的重要性测度. 基本变量对失效概率的重要性测度可以直接全面地给出基本变量对结构安全 影响的重要程度,因而该重要性测度将更具有工程指导作用. 文中分析了所定义基本变量对 失效概率影响重要性测度的基本性质,并基于鞍点线抽样在求解可靠度时不受随机变量分布 形式的限制及其效率和精度较高的优点,提出了求解该重要性测度的鞍点线抽样方法.     

Non-linear and linear evolution of perturbation in stochastic basic flows

In this paper, we examine the non-linear and linear evolutions of perturbation in stochastic basic flows with two-dimensional quasi-geostrophic equations on a sphere. As the analytic solutions for the considered quasi-geostrophic equations are not available, the Fourier finite volume element method is used to perform numerical simulation. It is found that, the non-linear and linear evolutions of perturbation in stochastic basic flow will be consistent for a short period of time and small stochastic fluctuations when they are consistent in the deterministic basic flow. However, the tangent linear model will fail to approximate the original non-linear model when the time period is considerably long and stochastic fluctuation becomes large. Moreover, the global energy decays faster for stochastic basic flow with stronger fluctuations.     

A material force method for inelastic fracture mechanics

A material force method is proposed for evaluating the energy release rate and work rate of dissipation for fracture in inelastic materials. The inelastic material response is characterized by an internal variable model with an explicitly defined free energy density and dissipation potential. Expressions for the global material and dissipation forces are obtained from a global balance of energy-momentum that incorporates dissipation from inelastic material behavior. It is shown that in the special case of steady-state growth, the global dissipation force equals the work rate of dissipation, and the global material force and J-integral methods are equivalent. For implementation in finite element computations, an equivalent domain expression of the global material force is developed from the weak form of the energy-momentum balance. The method is applied to model problems of cohesive fracture in a remote K-field for viscoelasticity and elastoplasticity. The viscoelastic problem is used to compare various element discretizations in combination with different schemes for computing strain gradients. For the elastoplastic problem, the effects of cohesive and bulk properties on the plastic dissipation are examined using calculations of the global dissipation force.     

A delayed HIV-1 infection model with Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV-1 infection with intracellular delay and Beddington–DeAngelis functional response is investigated. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give some sufficient conditions for the local stability of the infected equilibrium.     

Dynamic and wear characteristics of self-lubricating bearing cage: effects of cage pocket shape

34,354,966 active cases and 460,787 deaths because of COVID-19 pandemic were recorded on November 06, 2021, in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible (S), asymptomatic infected (A), clinically ill or symptomatic infected (I), quarantine (Q), isolation (J) and recovered (R), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin’s maximum principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore, the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario.

     

The energy release rate for linear viscoelastic materials

For viscoelastic materials, the energy release rate is a fairly important parameter for determining whether the crack extends, but so far its meaning for viscoelastic materials is not yet clear enough. In this paper, the thermodynamic mechanical theory is used to derive the local and global energy release rate of viscoelastic materials when constitutive equations are given. Moreover, the method of deriving the viscoelastic energy release rate is discussed. The relation between the energy release rate and internal energy, Helmholz free energy is described. The equivalence between the local and global energy release rates is also proved after some doubts in previous papers are dispelled, although the upshot is not original. Furthermore, the concrete forms of the energy release rate for anisotropic, orthotropic and isotropic viscoelastic media are presented. The forms of the local and global energy release rates in the failure zone are discussed and the difference between the equation presented here and the one given in previous papers is pointed out.     

Global exponential stabilization for chaotic brushless DC motors with a single input

In this paper, the global exponential stabilization for the chaotic brushless DC motor (BLDCM) is considered. Based on Lyapunov-like Theorem with differential and integral inequalities, a single and linear feedback control is proposed to realize the global stabilization of BLDCM with exponential convergence rate. The guaranteed exponential convergence rate can be also correctly estimated. Computer simulation results show that the proposed method is effective.     

Closed set of the uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity for small deformations

This paper reports the results of a study into global and local conditions of uniqueness and the criteria excluding the possibility of bifurcation of the equilibrium state for small strains. The conditions and criteria are derived on the basis of an analysis of the problem of uniqueness of a solution involving the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This work forms a follow-up of previous research (?loderbach in Bifurcations criteria for equilibrium states in generalized thermoplasticity, IFTR Reports, 1980, Arch Mech 3(35):337–349, 351–367, 1983), but contains a new derivation of global and local criteria excluding a possibility of bifurcation of an equilibrium state regarding a comparison body dependent on the admissible fields of stress rate. The thermal elasto-plastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem.     

On suppression of chaotic motion of a nonlinear MEMS oscillator

Nonlinear Dynamics - The present novel coronavirus (SARS-CoV-2) infection has created a global emergency situation by spreading all over the world in a large scale within very short time period....     

Drop size spectra in sprays from pressure-swirl atomizers

A study on the characterization of sprays from Newtonian liquids produced by pressure-swirl atomizers is presented. The global drop size spectra of the sprays are measured with phase-Doppler anemometry, and global mean drop sizes are derived as moments of the spectra for varying atomizer geometry, liquid flow rate, and physical properties of the liquids. Dimensional analysis provides a correlation for the non-dimensional global Sauter mean diameter. A relationship between the global Sauter mean drop size and the global drop size RMS is established. A method is developed for predicting the global drop size spectra in the sprays, using easily accessible experimental input parameters. The basis for the function defining the spectrum is a gamma distribution, which is known from the literature as physically relevant for ligament-mediated sprays.     

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

In this paper, a delay-differential mathematical model that described HIV infection of CD4⁺ T cells is analyzed. The effect of time delay on stability of the equilibria of the infection model has been studied. And the sufficient criteria for stability switch of the infected equilibrium and the local and global asymptotic stability of the uninfected equilibrium are given. By using the geometric stability switch criterion in the delay-differential system with delay-dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.