Deterministic and stochastic stability of a mathematical model of smoking

Our aim in this paper, is first constructing a Lyapunov function to prove the global stability of the unique smoking-present equilibrium state of a mathematical model of smoking. Next we incorporate random noise into the deterministic model. We show that the stochastic model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. Then a stochastic Lyapunov method is performed to obtain the sufficient conditions for mean square and asymptotic stability in probability of the stochastic model. Our analysis reveals that the stochastic stability of the smoking-present equilibrium state, depends on the magnitude of the intensities of noise as well as the parameters involved within the model system.     

A compartmental model for global spread dynamics of malware under mutation

Malware mutation is pervasive among networks. Modeling and understanding its propagation characteristics have been of great importance. In this study, a new compartmental model that extends the present model by incorporating mutated malware into the modeling process as a separate dynamic variable is proposed and theoretically analyzed to deepen the understanding of the spreading mechanisms of mutated malware. The model involves two equilibria, namely, malware‐free equilibrium and malware equilibrium, wherein both have proven to be locally and globally asymptotically stable through the Routh‐Hurwitz criterion and Lyapunov functional approach, respectively. An epidemic threshold is obtained that clearly forms the boundary among the comprehensive dynamics of the model between two distinct ramifications: one with mutation infection prevalence and the other without any mutation infection. Both are incarnated via the existence and stability of the equilibria admitted by the model. Further analyses show that the mutation is related not only to the epidemic threshold, but also to the malware prevalence level. The numerical simulations based on the analytic results demonstrate that the diffusion of mutated malware can fall away or can be maintained at a suitable level.     

Threshold of Effective Degree SIR Model

The effective degree SIR model is a precise model for the SIR disease dynamics on a network. The original ODE model is only applicable for a network with finite degree distributions. The new generating function approach rewrites with model as a PDE and allows infinite degree distributions. In this paper, we first prove the existence of a global solution. Then we analyze the linear and nonlinear stability of the disease-free steady state of the PDE effective degree model, and show that the basic reproduction number still determines both the linear and the nonlinear stability. Our method also provides a new tool to study the effective degree SIS model, whose basic reproduction number has been elusive so far.     

LOCAL AND GLOBAL HOPF BIFURCATIONS IN A DELAYED HUMAN RESPIRATORY SYSTEM

This paper considers a delayed human respiratory model. Firstly, the stability of the equilibrium of the model is investigated and the occurrence of a sequence of Hopf bifurcations of the model is proved. Secondly, the explicit algorithms which determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived by applying the normal form method and the center manifold theory. Finally, the existence of the global periodic solutions is showed under some assumptions on the model.     

Virtual stability: Constructing a simulation model

After a discussion of the importance of stability and instability for complex systems theory, we define the concept of virtual stability as a state in which a system employs self‐monitoring and adaptive control to maintain itself in a configuration that would otherwise be unstable. The energy expended in this gains the system an increase in its flexibility of behavioral response to environmental changes. A model designed to illustrate virtual stability is presented, followed by a brief discussion of the evolutionary advantage this capacity provides. This leads to the suggestion that such advantage gives an argument both for the directionality of evolution and for the emergence of self‐consciousness. © 2008 Wiley Periodicals, Inc. Complexity, 2009     

具有财政赤字的OLG竞争商业周期

讨论平面上ＯＬＧ模型的均衡解的稳定性，并利用一个类似稳定流形定理的结论，把二维动力系统化为一维动力系统，从而在整体上研究带有国家财政赤字的ＯＬＧ模型．     

Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation

In this paper, a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied. The delay represents the time taken for cells to undergo mitosis. Nonnegativity of solutions is investigated. The steady-state analysis is presented with respect to the magnitude of the delay. Existence of Hopf bifurcation is proved for some parameter values. Local and global stability of the stationary solutions are proved for other ones. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non-retarded problems for some parameter values.     

一类带扩散项的HIV模型的平衡解的稳定性分析

研究了一类齐次Neumann边界条件下带扩散项的HIV模型.运用赫尔维茨判定定理得出正常数平衡解在一定条件下的局部渐近稳定性;当游离病毒达到一定量时,通过构造Lyapunov函数得出正常数平衡解全局稳定的条件.     

一类具有时滞传染病模型的稳定性

讨论了具有双时滞的SIS传染病模型.研究了一个边界平衡点的全局稳定性和正平衡点的局部稳定性,得到了传染病最终消失和成为地方病的阈值.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Feedback control variables to restrain the Babesiosis disease

In this paper, we complete the study of the dynamics of a recognized continuous‐time model for the Babesiosis disease. The local and global asymptotic stability of the endemic state are established theoretically and experimentally. In addition, to restrain the disease in the original model when the endemic state exists, we propose and study the continuous model with feedback controls. The global stability of the boundary‐equilibrium point of this model is analyzed by means of rigorous mathematical methods. As an important consequence of this result, we propose a strategy to select feedback control variables in order to restrain the disease in the original model. This strategy allows us to make the disease vanish completely. In other words, the feedback controls are specially effective for restraining disease in the model. The validity of the established theoretical result is supported by a set of numerical simulations.     

Global Stability for a Dynamic model of Hepatitis B with Antivirus Treatment

An epidemic model on the basis of therapy of chronic Hepatitis B with antivirus treatment was introduced in this paper. By applying a comparison theorem and analyzing the corresponding characteristic equations, we obtain sufficient conditions on the parameters for the global stability of the disease-free state. It's proved that if the basic reproduction number \(R_0 < 1\) , the disease-free equilibrium is globally asymptotically stable. If \(R_0 > 1\), the disease-free equilibrium is unstable and the disease is uniformly permanent. Moreover, if \(R_0 > 1\), sufficient conditions are obtained for the global stability of the endemic equilibrium.     

Persistence and stability for a two-species ratio-dependent predator-prey system with distributed time delay

A two-species ratio-dependent predator-prey model with distributed time delay is investigated. It is shown that the system is persistent under some appropriate conditions, and sufficient conditions are obtained for both the local and global stability of the positive equilibrium of the system.     

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 ? i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley & Sons, Ltd.     

A host-vector model for malaria with infective immigrants

This paper considers a host-vector mathematical model for the spread of malaria that incorporates recruitment of human population through a constant immigration, with a fraction of infective immigrants. The model analysis is carried out to find the steady states and their stability. It is found that in the presence of infective immigrant humans, there is no disease-free equilibrium point. However, the model exhibits a unique endemic equilibrium state if the fraction of the infective immigrants ? is positive. When the fraction of infective immigrants approaches a small value, there is sharp threshold for which the disease can be reduced in the community. The unique endemic equilibrium for which there is a fraction of infective immigrants is globally asymptotically stable.     

Existence and stability of almost periodic solutions in impulsive neural network models

The existence and global exponential stability of an almost periodic solution of an impulsive neural network model with distributed delays is considered in a matrix setting. The approach transforms the original network into a matrix analysis problem, where a set of sufficient conditions based on spectral radius is presented. A concrete Hopfield model shows the advantages in comparison with a classical norm approach.     

Research on lateral stability and rollover mechanism of articulated wheel loader

A seven-degrees-of-freedom (DOF) dynamic model was established based on the Lagrange equation to analyse the lateral stability and instability mechanism of an articulated wheel loader. A scale wheel loader was designed and manufactured to validate the dynamic model in two conditions, namely turning on slopes and passing over obstacles. Experimental data and simulated data fitted well on the whole, so the developed dynamic model was proved to be useful and could serve as an important tool to analyse the stability of wheel loaders. At last, the lateral stability of one ZL50 wheel loader was analysed by using this dynamic model. The results showed that there is a phenomenon of sudden stability losing during the wheel loader cornering due to the relative rotation between the subframe and the rear axle. The subframe can enhance the stability when the wheel loader is passing over obstacles but reduces the stability when it is turning.     

Study of a fractional-order model of chronic wasting disease

This paper studies a fractional-order modelling chronic wasting disease (CWD). The basic results on existence, uniqueness, non-negativity, and boundedness of the solutions are investigated for the considered model. The criterion for local as well as global stability of the equilibrium points is derived. A numerical analysis for Hopf-type bifurcation is presented. Finally, numerical simulations are provided to justify the results obtained.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

房地产销售商投机行为的稳定性分析

分析了房地产销售的纳什博弈模型、房地产开发商作为领导者寡头博弈模型、房地产销售商作为领导者寡头博弈模型、房地产市场销售讨价还价博弈模型四种房地产市场销售博弈模型的利益分配机制和收益策略.结果表明,在房地产开发商作为领导者寡头博弈策略中,房地产销售商具有实施投机行为的动机,利益分配机制不稳定;在房地产销售商作为领导者寡头博弈策略中,房地产销售商没有实施投机行为的动机,利益分配机制的稳定;在房地产市场销售讨价还价博弈策略中,房地产销售商利益分配机制的稳定性与房地产产品的差异化替代率和无限期重复博弈策略的贴现率均有关.