We propose a novel epidemic model based on two-layered multiplex networks to explore the influence of positive and negative preventive information on epidemic propagation. In the model, one layer represents a social network with positive and negative preventive information spreading competitively, while the other one denotes the physical contact network with epidemic propagation. The individuals who are aware of positive prevention will take more effective measures to avoid being infected than those who are aware of negative prevention. Taking the microscopic Markov chain (MMC) approach, we analytically derive the expression of the epidemic threshold for the proposed epidemic model, which indicates that the diffusion of positive and negative prevention information, as well as the topology of the physical contact network have a significant impact on the epidemic threshold. By comparing the results obtained with MMC and those with the Monte Carlo (MC) simulations, it is found that they are in good agreement, but MMC can well describe the dynamics of the proposed model. Meanwhile, through extensive simulations, we demonstrate the impact of positive and negative preventive information on the epidemic threshold, as well as the prevalence of infectious diseases. We also find that the epidemic prevalence and the epidemic outbreaks can be suppressed by the diffusion of positive preventive information and be promoted by the diffusion of negative preventive information.

A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance \(k\) to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity.

Shannon entropies for networks have been widely introduced. However, entropies for weighted graphs have been little investigated. Inspired by the work due to Eagle et al., we introduce the concept of graph entropy for special weighted graphs. Furthermore, we prove extremal properties by using elementary methods of classes of weighted graphs, and in particular, the one due to Bollobás and Erdös, which is also called the Randi´c weight. As a result, we derived statements on dendrimers that have been proven useful for applications. Finally, some open problems are presented.

The extended state observer (ESO) used to estimate the unknown model for nonlinear systems has been widely applied in model-free control. In the ESO, three factors: 1) nonderivative structure; 2) insufficient usage of innovation; and 3) no compensation of the unknown character of a nonlinear function, causing low type and low accuracy of observation are identified. The convergence problem of the ESO is also raised. A simple linear compensation function observer (CFO) is proposed. The CFO adopts the derivative form overcoming the structure problem of the ESO, fully uses the available innovation, and then introduces a compensator to replace the unknown function. The filtering compensator counteracts the effect of the time-varying nonlinear unknown model to make the error equation behave with desired poles. The convergence of the CFO is theoretically proved. The CFO is the Type-III system, whereas the ESO is a Type-I system; therefore, the CFO has a qualitative leap over the ESO in accuracy improvement of observation. The CFO can observe nonlinear functions, including lumped disturbances, uncertainties, and unmodeled parts for nonlinear systems. The CFO decouples the multi-input and multioutput (MIMO) systems. Five unknown typical nonlinear functions are tested and demonstrates that the CFO qualitatively prevails over the ESO. The MIMO chaotic attitude system of a small-scale unmanned aerial vehicle (UAV) helicopter is conducted with results that CFO performance of function observation is highly improved over that of ESO about 29 times.