Multi-phase epidemic model by a Markov chain

In this paper we propose a continuous-time Markov chain to describe the spread of an infective and non-mortal disease into a community numerically limited and subjected to an external infection. We make a numerical simulation that shows tendencies for recurring epidemic outbreaks and for fade-out or extinction of the infection.     

Mobility matrix evolution for an SIS epidemic patch model

Intercommunity disease spread can be modeled using a collection of discrete community “patches” with continuous population flow between them. In a susceptible–infected–susceptible (SIS) model residents of a community may either be classified as susceptible or infected. Infected individuals may heal and become susceptible again but are not permitted to die or become immune. The spread of disease can be controlled by modifying the rate and direction of resident movement across patch boundaries. In this work we use genetic algorithms to evolve optimal connections between patch boundaries such that the total number of infected individuals is minimized.     

A second-order phase transition in the complete graph stochastic epidemic model

A stochastic model for epidemic spread in a set of individuals placed upon the sites of a complete graph of relations is investigated. The model is defined by three parameters: the number of individuals or sites, N, the probability that an infected site transmits the disease to a susceptible site, α, and the probability of recovery of infected sites, β, both referred to the unit of time.We show that this system evolves towards a, approximately Gaussian, stationary distribution of infected sites whose mean and variance can be analytically estimated. Also, we find that the average fraction of infected sites, x, is zero for transmission probabilities below the critical value α_c=1-e^-β/N and grows linearly with α for 0<α-α_c1. A sharp peak observed in Monte Carlo simulations of the variance of the number of infected sites as a function of α allows us to classify this dynamical phase transition as second order with x playing the role of an order parameter. Some consequences of this model to the dynamics of highly connected complex systems, such as the brain cortex, are also discussed.     

一种基于元胞自动机的自适应网络病毒传播模型

自适应网络是节点动力学和网络动力学相互作用和反馈的演化网络. 基于元胞自动机建立自适应网络中易感-感染-易感(susceptible-infected-susceptible)的病毒传播模型,研究节点为了规避病毒传播所采取的多种网络重连规则对病毒传播及网络统计特征的影响. 结果表明:自适应网络中的重连规则可以有效减缓病毒传播速度,降低病毒传播规模;随机重连规则使得网络统计特征趋于随机网络;基于元胞自动机建立的传播模型清晰地表达了病毒在传播过程中的双稳态现象. 关键词： 自适应网络 传播动力学 网络动力学 元胞自动机     

Stochastic epidemics and rumours on finite random networks

In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n^−1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks).     

Optimal vaccination and treatment of an epidemic network model

In this Letter, we firstly propose an epidemic network model incorporating two controls which are vaccination and treatment. For the constant controls, by using Lyapunov function, global stability of the disease-free equilibrium and the endemic equilibrium of the model is investigated. For the non-constant controls, by using the optimal control strategy, we discuss an optimal strategy to minimize the total number of the infected and the cost associated with vaccination and treatment. Table 1 and Figs. 1–5 are presented to show the global stability and the efficiency of this optimal control.     

External Stimuli on Neural Networks: Analytical and Numerical Approaches

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested—matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.     

A Monte Carlo model for networks between professionals and society

We propose a network model with a fixed number of nodes and links and with a dynamic which favors links between nodes differing in connectivity. We observe a phase transition and parameter regimes with degree distributions following power laws, P(k)∼k^-γ$P(k)\sim k^{-\gamma }$, with γ$\gamma$ ranging from 0.2$0.2$ to 0.5$0.5$, small-world properties, with a network diameter following D(N)∼logN$D(N)\sim \log N$ and relative high clustering, following C(N)∼1/N$C(N)\sim 1/N$ and C(k)∼k^-α$C(k)\sim k^{-\alpha }$, with α$\alpha$ close to 3. We compare our results with data from real-world protein interaction networks.     

异质自适应网络中的核心-边缘结构及其对疾病传播的抑制作用

节点属性异质自适应网络中疾病传播的研究表明节点属性异质性可以很大程度上增大传播阈值, 并且自组织形成一个更鲁棒的度异质网络结构. 本文从数值模拟方面研究鲁棒的度分布异质结构的自组织形成过程, 分析发现核心-边缘结构的形成才是导致传播阈值增大的根本原因. 鉴于此, 提出一种重连策略, 能够促进核心-边缘结构的形成, 从而达到增大传播阈值的目的. 这不仅有助于深入认识节点属性异质自适应网络中的流行病传播过程, 而且为疾病传播控制策略的提出提供了新思路.     

Renormalized quasiparticles in antiferromagnetic states of the Hubbard model

We analyze the properties of the quasiparticle excitations of metallic antiferromagnetic states in a strongly correlated electron system. The study is based on dynamical mean field theory (DMFT) for the infinite dimensional Hubbard model with antiferromagnetic symmetry breaking. Self-consistent solutions of the DMFT equations are calculated using the numerical renormalization group (NRG). The low energy behavior in these results is then analyzed in terms of renormalized quasiparticles. The parameters for these quasiparticles are calculated directly from the NRG derived self-energy, and also from the low energy fixed point of the effective impurity model. From these the quasiparticle weight and the effective mass are deduced. We show that the main low energy features of the k-resolved spectral density can be understood in terms of the quasiparticle picture. We also find that Luttinger's theorem is satisfied for the total electron number in the doped antiferromagnetic state.     

Parameter estimation based synchronization for an epidemic model with application to tuberculosis in Cameroon

We propose a method based on synchronization to identify the parameters and to estimate the underlying variables for an epidemic model from real data. We suggest an adaptive synchronization method based on observer approach with an effective guidance parameter to update rule design only from real data. In order, to validate the identifiability and estimation results, numerical simulations of a tuberculosis (TB) model using real data of the region of Center in Cameroon are performed to estimate the parameters and variables. This study shows that some tools of synchronization of nonlinear systems can help to deal with the parameter and state estimation problem in the field of epidemiology. We exploit the close link between mathematical modelling, structural identifiability analysis, synchronization, and parameter estimation to obtain biological insights into the system modelled.     

Dynamics at the interface dividing collective chaotic and synchronized periodic states in a CML

A study is developed focusing the loss of stability of the interface dividing two regions of different spatial patterns on a coupled map lattice using coupling as the parameter guiding the transition. These patterns are constructed over local periodic/chaotic attractors generating regions of synchronized/collective behavior. The discrete feature of the underlying lattice, the anisotropy that stems from such discreteness and its possible change to an isotropic system through coupling with large number of neighbors are also investigated.     

Entanglement dynamics of a moving multi-photon Jaynes-Cummings model in mixed states

Using the algebraic dynamical method,the entanglement dynamics of an atom-field bipartite system in a mixed state is investigated.The atomic center-of-mass motion and the field-mode structure are also included in this system.We find that the values of the detuning and the average photon number are larger,the amplitude of the entanglement is smaller,but its period does not increase accordingly.Moreover,with the increase of the field-mode structure parameter and the transition photon number,the amplitude of the entanglement varies slightly while the oscillation becomes more and more fast.Interestingly,a damping evolution of the entanglement appears when both the detuning and the atomic motion are considered simultaneously.     

Thermodynamic limit of theq-state Potts-Hopfield model with infinitely many patterns

We prove the almost sure convergence of the free energy and of the overlap order parameters in aq-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the systemN like (/lnq) lnN. We study the limiting behavior of the extremal states of the model that are the measures induced on the Gibbs measures by the overlap parameters.     

Adopting epidemic model to optimize medication and surgical intervention of excess weight

We combined an epidemic model with an objective function to minimize the weighted sum of people with excess weight and the cost of a medication and surgical intervention in the population. The epidemic model is consisted of ordinary differential equations to describe three subpopulation groups based on weight. We introduced an intervention using medication and surgery to deal with excess weight. An objective function is constructed taking into consideration the cost of the intervention as well as the weight distribution of the population. Using empirical data, we show that fixed participation rate reduces the size of obese population but increases the size for overweight. An optimal participation rate exists and decreases with respect to time. Both theoretical analysis and empirical example confirm the existence of an optimal participation rate, $u^{?}$. Under $u^{?}$, the weighted sum of overweight (S) and obese (O) population as well as the cost of the program is minimized. This article highlights the existence of an optimal participation rate that minimizes the number of people with excess weight and the cost of the intervention. The time-varying optimal participation rate could contribute to designing future public health interventions of excess weight.     

Hybrid control of bifurcation and chaos in stroboscopic model of Internet congestion control system

Interaction between transmission control protocol （TCP） and random early detection （RED） gateway in the Internet congestion control system has been modelled as a discrete-time dynamic system which exhibits complex bifurcating and chaotic behaviours. In this paper, a hybrid control strategy using both state feedback and parameter perturbation is employed to control the bifurcation and stabilize the chaotic orbits embedded in this discrete-time dynamic system of TCP/RED. Theoretical analysis and numerical simulations show that the bifurcation is delayed and the chaotic orbits are stabilized to a fixed point, which reliably achieves a stable average queue size in an extended range of parameters and even completely eliminates the chaotic behaviour in a particular range of parameters. Therefore it is possible to decrease the sensitivity of RED to parameters. By using the hybrid strategy, we may improve the stability and performance of TCP/RED congestion control system significantly.     

Hybrid control of bifurcation and chaos in stroboscopic model of Internet congestion control system

Interaction between transmission control protocol (TCP) and random early detection (RED) gateway in the Internet congestion control system has been modelled as a discrete-time dynamic system which exhibits complex bifurcating and chaotic behaviours. In this paper, a hybrid control strategy using both state feedback and parameter perturbation is employed to control the bifurcation and stabilize the chaotic orbits embedded in this discrete-time dynamic system of TCP/RED. Theoretical analysis and numerical simulations show that the bifurcation is delayed and the chaotic orbits are stabilized to a fixed point, which reliably achieves a stable average queue size in an extended range of parameters and even completely eliminates the chaotic behaviour in a particular range of parameters. Therefore it is possible to decrease the sensitivity of RED to parameters. By using the hybrid strategy, we may improve the stability and performance of TCP/RED congestion control system significantly.     

Phase behavior of flowerlike micelles in a SCF cell model

We study the interactions between flowerlike micelles, self-assembled from telechelic associative polymers, using a molecular self-consistent field (SCF) theory and discuss the corresponding phase behavior. In these calculations we do not impose properties such as aggregation number, micellar structure and number of bridging chains. Adopting a SCF cell model, we calculate the free energy of interaction between a central micelle surrounded by others. Based on these results, we predict the binodal for coexistence of dilute and dense liquid phases, as a function of the length of the hydrophobic and hydrophilic blocks. In the same cell model we compute the number of bridges between micelles, allowing us to predict the network transition. Several quantitative trends obtained from the numerical results can be rationalized in terms of transparent scaling arguments.     

Ground state and vortex states of bosons in an anisotropic trap: A variational approach

We propose a simple variational form of the wave function to describe the ground state and vortex states of a weakly interacting Bose gas in an anisotropic trap. The proposed wave function is valid for a wide range of the particle numbers in the trap. It also works well in the case of attractive interaction between the atoms. Further, it provides an easy and fast method to calculate the physical quantities of interest. The results compare very well with those obtained by purely numerical techniques. Using our wave function we have been able to verify, for the first time, the predicted behaviour of the aspect ratio. Received 7 December 1998 and Received in final form 4 February 1999