A dynamic systems model of dyadic interaction

We develop a differential equation model of dyadic interaction that embodies the basic assumption that members of intimate couples form an interactive system in which the behavior of each member of a couple is influenced by the other's behavior and by goals that each person has for herself or himself. The dynamic solutions of this system suggest that when each person in the dyad is “cooperative”, then an equilibrium can be approached. The equilibrium represents a compromise position between the individuals’ own ideals and those of the partner. On the other hand, if one individual, or both, is uncooperative, then this system often, but not always, becomes unstable. One paradoxical deduction from the model is that, through mutual cooperation, couples can experience periods of stability, but such stable situations are not necessarily satisfying.     

Asymptotic behavior of an SEI epidemic model with diffusion

An SEI epidemic model with constant recruitment and infectious force in the latent period is investigated. This model describes the transmission of diseases such as SARS. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by the method of upper and lower solutions and its associated monotone iterations. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.     

Avian–human influenza epidemic model with diffusion,nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.     

Global attractivity of Cohen-Grossberg model with finite and infinite delays

In this paper, we have studied the global attractivity of the equilibrium of Cohen-Grossberg model with both finite and infinite delays. Criteria for global attractivity are also derived by means of Lyapunov functionals. As a corollary, we show that if the delayed system is dissipative and the coefficient matrix is VL-stable, then the global attractivity of the unique equilibrium is maintained provided the delays are small. Estimates on the allowable sizes of delays are also given. Applications to the Hopfield neural networks with discrete delays are included.     

Dynamics of a Mutualistic Model with Diffusion

This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when ε_1+ ε_2 ≠ 0. Moreover sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When ε_1+ε_2= 0, grow up property is derived if the geometric mean of the interaction coefficients is greater than 1(α_1α_2 1),while if the geometric mean of the interaction coefficients is less than 1(α_1α_2 1), there exists a global solution. Finally, numerical simulations are given.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

考虑消费者社会影响下的企业返利策略研究

研究了存在社会影响下的企业返利策略。以返利的持续性为依据, 建立了前期返利、后期返利、始终返利以及始终不返利四种返利模型。分别分析了垄断和竞争两种情形, 通过研究发现:在垄断情况下, 较小的返利促使企业选择在前期返利而后期不返利, 然而较大的返利迫使企业放弃返利策略;在存在竞争情形下, 会出现四种可能的均衡返利策略, 如果企业间竞争较强且返利较大, 均衡结果是两企业会做出差异化的返利策略选择;如果竞争较弱或返利较小, 则均衡结果是两企业会选择相同的返利策略。虽然两个企业均选择后期返利总不是均衡策略, 但是在某些情形下与其它返利策略相比, 两个企业会陷入囚徒困境, 致使两企业利润的减少。     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

On stabilizing volatile product returns

As input flows of secondary raw materials show high volatility and tend to behave in a chaotic way, the identification of the main drivers of the dynamic behavior of returns plays a crucial role. Based on a stylized production-recycling system consisting of a set of nonlinear difference equations, we explicitly derive parameter constellations where the system will or will not converge to its equilibrium. Using a constant elasticity of substitution production function, the model is then extended to enable coverage of real world situations. Using waste paper as a reference raw material, we empirically estimate the parameters of the system. By using these regression results, we are able to show that the equilibrium solution is a Lyapunov unstable saddle point. This implies that the system is sensitive on initial conditions that will hence impede the predictability of product returns. Small variations of production input proportions could however stabilize the whole system.     

The ANALYZE rulebase for supporting LP analysis

This paper describes how to design rules to support linear programming analysis in three functional categories: postoptimal sensitivity, debugging, and model management. The ANALYZE system is used to illustrate the behavior of the rules with a variety of examples. Postoptimal sensitivity analysis answers not only the paradigmWhat if …? question, but also the more frequently askedWhy …? question. The latter is static, asking why some solution value is what it is, or why it is not something else. The former is dynamic, asking how the solution changes if some element is changed. Debugging can mean a variety of things; here the focus is on diagnosing an infeasible instance. Model management includes documentation, verification, and validation. Rules are illustrated to provide support in each of these related functions, including some that require reasoning about the linear program's structure. Another model management function is to conduct a periodic review, with one of the goals being to simplify the model, if possible. The last illustration is how to test new rule files, where there is a variety of ways to communicate a result to someone who is not expert in linear programming.     

Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria

We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called "boundary equilibria", the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in L^1 toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.     

More about empirical evaluation of formal theory†

The purpose of this discussion is to transform the implicit equilibrium assumption endemic to network analysis into an explicit instrument for such analysis. I propose a formal model that brings together Coleman's restriction of Walras’ general equilibrium model and recent developments in describing the “social topology” of a multiple network system of actors such that a class of relational equilibria is defined. The specific equilibrium expected in a system is a function of the previously existing stratification of actors in the system. Corresponding to multiple observed networks, the model generates multiple equilibrium networks. The structural analysis of the observed networks can therefore be repeated on the equilibrium networks so as to assess the extent to which the analysis would differ if the observed relations were actually in an equilibrium state. Numerical illustration is provided by an analysis of alternative relational equilibria in the system of elite experts in methodological and mathematical sociology as such a system existed in 1975.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

Control by Interconnection and Energy Shaping of the Timoshenko Beam

In this paper, the dynamical control of a mixed finite and infinite dimensional mechanical system is approached within the framework of port Hamiltonian systems. In particular, a flexible beam, modeled according to the Timoshenko theory and in distributed port Hamiltonian form, with a mass under gravity field connected at a free end, is considered. The control problem is approached by generalization of the concept of structural invariant (Casimir function) to the infinite dimensional case and the so-called control by interconnection technique is extended to the infinite dimensional case. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

Global analysis of virus dynamics model with logistic mitosis,cure rate and delay in virus production

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, composite quadratic functions and Volterra—type functionals, we provide the global stability for this model. If R₀, the basic reproductive number, satisfies R₀ ≤ 1, then the infection‐free equilibrium state is globally asymptotically stable. Our system is persistent if R₀ > 1. On the other hand, if R₀ > 1, then infection‐free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R₀ > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

An optimization-based conjectured supply function equilibrium model for network constrained electricity markets

This paper proposes a model to compute nodal prices in oligopolistic markets. The model generalizes a previous model aimed at solving the single-bus problem by applying an optimization procedure. Both models can be classified as conjectured supply function models. The conjectured supply functions are assumed to be linear with constant slopes. The conjectured price responses (price sensitivity as seen for each generating unit), however, are assumed to be dependent on the system line's status (congested or not congested). The consideration of such a dependence is one of the main contributions of this paper. Market equilibrium is defined in this framework. A procedure based on solving an optimization problem is proposed. It only requires convexity of cost functions. Existence of equilibrium, however, is not guaranteed in this multi-nodal situation and an iterative search is required to find it if it exists. A two-area multi-period case study is analysed. The model reaches equilibrium for some cases, mainly depending on the number of periods considered and on the value of conjectured supply function slopes. Some oscillation patterns are observed that can be interpreted as quasi-equilibria. This methodology can be applied to the study of the future Iberian electricity market.     

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

In this paper, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied. Using the method of Liapunov–LaSalle invariance principle, we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.     

A remark on the Harsanyi-Selten theory of equilibrium selection

We consider a class of 3-person games in normal form with two pure strategies for each player and two strict equilibrium points. To select one of these two strict equilibrium points as the solution, the equilibrium selection theory of Harsanyi and Selten is applied. The games are constructed in such a way that the a priori probabilities reflect somewhat poorly the risk situation of the players. It is argued and illustrated by examples that this might yield unreasonable results. The a priori probabilities would describe the risk situation of the players more completely if their definition were not based on the expectation of correlated decision behavior.