Robust synchronization for stochastic delayed complex networks with switching topology and unmodeled dynamics via adaptive control approach

In this paper, the robust synchronization problem for a class of stochastic delayed complex networks with switching topology and unmodeled dynamics is investigated by using the adaptive control scheme. Different from some existing uncertain complex network models, the uncertain terms in the complex networks are only needed to be bounded. Based on the stability theory of stochastic delayed differential equation, a novel adaptive controller which ensures that all the nodes in the complex networks robustly synchronize with the target node is derived. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection

We investigate the behaviour of a neural network model consisting of two coupled oscillators with delays and inhibitory-to-inhibitory connections. We consider the absolute synchronization and show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including fold bifurcation and Hopf bifurcation) and codimension two bifurcation (including fold-Hopf bifurcations and Hopf–Hopf bifurcations). Based on the normal form theory and center manifold reduction, we obtain detailed information about the bifurcation direction and stability of various bifurcated equilibria as well as periodic solutions with some kinds of spatio-temporal patterns. Numerical simulation is also given to support the obtained results.     

Collective behavior of interacting locally synchronized oscillations in neuronal networks

Local circuits in the cortex and hippocampus are endowed with resonant, oscillatory firing properties which underlie oscillations in various frequency ranges (e.g. gamma range) frequently observed in the local field potentials, and in electroencephalography. Synchronized oscillations are thought to play important roles in information binding in the brain. This paper addresses the collective behavior of interacting locally synchronized oscillations in realistic neural networks. A network of five neurons is proposed in order to produce locally synchronized oscillations. The neuron models are Hindmarsh–Rose type with electrical and/or chemical couplings. We construct large-scale models using networks of such units which capture the essential features of the dynamics of cells and their connectivity patterns. The profile of the spike synchronization is then investigated considering different model parameters such as strength and ratio of excitatory/inhibitory connections. We also show that transmission time-delay might enhance the spike synchrony. The influence of spike-timing-dependence-plasticity is also studies on the spike synchronization.     

Dynamics of an SIR epidemic model with limited medical resources revisited

The dynamics of an SIR epidemic model is explored in this paper in order to understand how the limited medical resources and their supply efficiency affect the transmission of infectious diseases. The study reveals that, with varying amount of medical resources and their supply efficiency, the target model admits both backward bifurcation and Hopf bifurcation. Sufficient criteria are established for the existence of backward bifurcation, the existence, the stability and the direction of Hopf bifurcation. The mechanism of backward bifurcation and its implication for the control of the infectious disease are also explored. Numerical simulations are presented to support and complement the theoretical findings.     

Mean-field modeling approach for understanding epidemic dynamics in interconnected networks

Modern systems (e.g., social, communicant, biological networks) are increasingly interconnected each other formed as ‘networks of networks’. Such complex systems usually possess inconsistent topologies and permit agents distributed in different subnetworks to interact directly/indirectly. Corresponding dynamics phenomena, such as the transmission of information, power, computer virus and disease, would exhibit complicated and heterogeneous tempo-spatial patterns. In this paper, we focus on the scenario of epidemic spreading in interconnected networks. We intend to provide a typical mean-field modeling framework to describe the time-evolution dynamics, and offer some mathematical skills to study the spreading threshold and the global stability of the model. Integrating the research with numerical analysis, we are able to quantify the effects of networks structure and epidemiology parameters on the transmission dynamics. Interestingly, we find that the diffusion transition in the whole network is governed by a unique threshold, which mainly depends on the most heterogenous connection patterns of network substructures. Further, the dynamics is highly sensitive to the critical values of cross infectivity with switchable phases.     

Analysis of SE τ IR ω S epidemic disease models with vertical transmission in complex networks

When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SEτ IRωS epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R0 ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R0 > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R0 > 1. In a scale-free (SF) network we obtain the condition R1 > 1 under which the system will be of non-zero stationary prevalence.     

Equation-Free multiscale computational analysis of individual-based epidemic dynamics on networks

The surveillance, analysis and ultimately the efficient long-term prediction and control of epidemic dynamics appear to be some of the major challenges nowadays. Detailed individual-based mathematical models on complex networks play an important role towards this aim. In this work, it is shown how one can exploit the Equation-Free approach and optimization methods such as Simulated Annealing to bridge detailed individual-based epidemic models with coarse-grained, system-level analysis within a pair-wise representation perspective. The proposed computational methodology provides a systematic approach for analyzing the parametric behavior of complex/multiscale epidemic simulators much more efficiently than simply simulating forward in time. It is shown how steady state and (if required) time-dependent computations, stability computations, as well as continuation and numerical bifurcation analysis can be performed in a straightforward manner. The approach is illustrated through a simple individual-based SIRS epidemic model deploying on a random regular connected graph. Using the individual-based simulator as a black box coarse-grained timestepper and with the aid of Simulated Annealing I compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level.     

Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate

In this paper we study an SEIR epidemic model with saturated recovery rate. A backward bifurcation leading to bistability possibly occurs, and global dynamics are shown by compound matrices and geometric approaches. Numerical simulations are presented to illustrate the results.     

On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE

Normal form method is first employed to study the Hopf-pitchfork bifurcation in neutral functional differential equation (NFDE), and is proved to be an efficient approach to show the rich dynamics (periodic and quasi-periodic oscillations) around the bifurcation point. We give an algorithm for calculating the third-order normal form in NFDE models, which naturally arise in the method of extended time delay autosynchronization (ETDAS). The existence of Hopf-pitchfork bifurcation in a van der Pol’s equation with extended delay feedback is given and the unfoldings near this critical point is obtained by applying our algorithm. Some interesting phenomena, such as the coexistence of several stable periodic oscillations (or quasi-periodic oscillations) and the existence of saddle connection bifurcation on a torus, are found by analyzing the bifurcation diagram and are illustrated by numerical method.     

Coarse-grained bifurcation analysis and detection of criticalities of an individual-based epidemiological network model with infection control

We present and discuss how the so called Equation-free approach for multi-scale computations can be used to systematically study certain aspects of the dynamics of detailed individual-based epidemiological simulators. As our illustrative example, we choose a simple individual-based stochastic epidemic model evolving on a fixed random regular network (RRN). We show how control policies based on the isolation of the infected population can dramatically influence the dynamics of the disease resulting to big-amplitude oscillations. We also address the development of a computational framework that enables detailed epidemiological simulators to converge to their coarse-grained critical points, which mark the onset of the emergent time-dependent solutions as well as to trace branches of coarse-grained unstable equilibria. Using the individual-based simulator we construct the coarse-grained bifurcation diagrams illustrating the dependence of the solutions on the disease characteristics.