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In this paper, a fractional derivative with short-term memory properties is defined, which can be viewed as an extension of Caputo fractional derivative. Then, some properties of the short memory fractional derivative are discussed. Also, a comparison theorem for a class of short memory fractional systems is shown, via which some relationship between short memory fractional systems and Caputo fractional systems can be established. By applying the comparison theorem and Lyapunov direct method, some sufficient criteria are obtained, which can ensure the asymptotic stability of some short memory fractional equations. Moreover, a special result is presented, by which the stability of some special systems can be judged directly. Finally, three examples are provided to demonstrate the effectiveness of the main results.

This note considers the constrained H-infinity consensus of multi-agent networks with nonidentical constraint sets. An improved distributed algorithm is adopted and a nonlinear controlled output function is defined to evaluate the effect of disturbances. Then, it is shown that the constrained H-infinity consensus can be achieved if some linear matrix inequality has positive solution.

This paper is mainly devoted to the distributed second-order multi-agent optimization problem with unbalanced and directed networks. To deal with this problem, a new distributed algorithm is proposed based on the local neighbor information and the private objective functions. By a coordination transformation, the closed-loop system is divided into two first-order subsystems, which is easier to be dealt with. Under the assumption of the strong connectivity of networks, it is proved that all agent can collaboratively converge to some optimal solution of the team objective function, where the gradient of the private objective functions is not assumed to be bounded.

In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like COVID-19 and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number $R_{0}$ is derived. When $R_{0}<1$, the disease-free equilibrium point is unique and locally asymptotically stable. When $R_{0}>1$, the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asymptotic stability of disease-free and endemic equilibrium points. The trend of COVID-19 spread in the United States is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model. According to the real data of the United States, it is found that our improved model has a better prediction ability for the epidemic trend in the next two weeks. Hence, the epidemic trend of the United States in the next two weeks is investigated, and the peak of isolated cases are predicted. The modified SEIQRP model successfully capture the development process of COVID-19, which provides an important reference for understanding the trend of the outbreak.

In this paper, a mathematical model is proposed to analyze the dynamic behavior of COVID-19. Based on inter-city networked coupling effects, a fractional-order SEIHDR system with the real-data from 23 January to 18 March, 2020 of COVID-19 is discussed. Meanwhile, hospitalized individuals and the mortality rates of three types of individuals (exposed, infected and hospitalized) are firstly taken into account in the proposed model. And infectivity of individuals during incubation is also considered in this paper. By applying least squares method and predictor-correctors scheme, the numerical solutions of the proposed system in the absence of the inter-city network and with the inter-city network are stimulated by using the real-data from 23 January to $18-m$ March, 2020 where $m$ is equal to the number of prediction days. Compared with integer-order system ($\alpha=0$), the fractional-order model without network is validated to have a better fitting of the data on Beijing, Shanghai, Wuhan, Huanggang and other cities. In contrast to the case without network, the results indicate that the inter-city network system may be not a significant case to virus spreading for China because of the lock down and quarantine measures, however, it may have an impact on cities that have not adopted city closure. Meanwhile, the proposed model better fits the data from 24 February to 31, March in Italy, and the peak number of confirmed people is also predicted by this fraction-order model. Furthermore, the existence and uniqueness of a bounded solution under the initial condition are considered in the proposed system. Afterwards, the basic reproduction number $R_0$ is analyzed and it is found to hold a threshold: the disease-free equilibrium point is locally asymptotically stable when $R_0\le 1$, which provides a theoretical basis for whether COVID-19 will become a pandemic in the future., Comment: 11 pages, 10 figures, ND Special Issue paper submitted

L\'{e}vy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that L\'{e}vy flights can maximize the efficiency of resource searching in uncertain environments, and also movements of many foragers and wandering animals have been shown to follow a L\'{e}vy distribution. The reason mainly comes from that the L\'{e}vy distribution, has an infinite second moment, and hence is more likely to generate an offspring that is farther away from its parent. However, the investigation into the efficiency of other different heavy-tailed probability distributions in swarm-based searches is still insufficient up to now. For swarm-based search algorithms, randomness plays a significant role in both exploration and exploitation, or diversification and intensification. Therefore, it's necessary to discuss the optimal randomness in swarm-based search algorithms. In this study, CS is taken as a representative method of swarm-based optimization algorithms, and the results can be generalized to other swarm-based search algorithms. In this paper, four different types of commonly used heavy-tailed distributions, including Mittag-Leffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, are considered to enhance the searching ability of CS. Then four novel CS algorithms are proposed and experiments are carried out on 20 benchmark functions to compare their searching performances. Finally, the proposed methods are used to system identification to demonstrate the effectiveness.

With the analysis of noise-induced synchronization of opinion dynamics with bounded confidence (BC), a natural and fundamental question is what opinion structures can be synchronized by noise. In the traditional Hegselmann-Krause (HK) model, each agent examines the opinion values of all the other ones and then choose neighbors to update its own opinion according to the BC scheme. In reality, people are more likely to interchange opinions with only some individuals, resulting in a predetermined local discourse relationship as introduced by the DeGroot model. In this paper, we consider an opinion dynamics that combines the schemes of BC and local discourse topology and investigate its synchronization induced by noise. The new model endows the heterogeneous HK model with a time-varying discourse topology. With the proposed definition of noise-synchronizability, it is shown that the compound noisy model is almost surely noise-synchronizable if and only if the time-varying discourse graph is uniformly jointly connected, taking the noise-induced synchronization of the classical heterogeneous HK model as a special case. As a natural implication, the result for the first time builds the equivalence between the connectivity of discourse graph and the beneficial effect of noise for opinion consensus., Comment: We need to make a substantial revision about this paper. Will submit it again two monthes later

Designing feasible control strategies for opinion dynamics in complex social systems has never been an easy task. It requires a control protocol which 1) is not enforced on all individuals in the society, and 2) does not exclusively rely on specific opinion values shared by the social system. Thanks to the recent studies on noise-induced consensus in opinion dynamics, the noise-based intervention strategy has emerged as the only one meeting both of the above requirements, yet its underlying general theory is still lacking. In this paper, we perform rigorous theoretical analysis and simulations of a noise-based control strategy for opinion formation in which only a fraction of individuals is affected by randomly generated noise. We found that irrespective of the number of noise-driven individuals, including the case of only one single noise-affected individual, the system can attain a quasi-consensus in finite time, and the critical noise strength can be obtained. Our results highlight the efficiency of noise-driven mechanisms for the control of complex social dynamics., Comment: arXiv admin note: text overlap with arXiv:1711.01432

How can we approach the truth in a society? It may depend on various factors. In this paper, using a well-established truth seeking model, we show that the persistent free information flow will bring us to the truth. Here the free information flow is modeled as the environmental random noise that could alter one's cognition. Without the random noise, the model predicts that the truth can only be captured by the truth seekers who own active perceptive ability of the truth and their believers, while the other individuals may stick to falsehood. But under the influence of the random noise, we strictly prove that even there is only one truth seeker in the group, all individuals will finally approach the truth.

Eliminating disagreement in a group is usually beneficial to the social stability. In this paper, using the well-known Hegselmann-Krause (HK) model, we design a quite simple strategy that could resolve the opinion difference of the system in finite time and induce the opinions synchronized to a targeted value. To be specific, we intentionally introduce weak random noise to only one agent to affect its opinion in the evolution and also a leader agent with fixed opinion in a divisive group, then strictly prove that the disagreement is finally eliminated and the opinions get synchronized to the leader's opinion in finite time. Other than that, we calculate the finite stopping time when the opinions synchronously reach the objective opinion value, which could provide further guide for designing more efficient noise intervention strategy.

In opinion dynamics, the convergence of the heterogeneous Hegselmann-Krause (HK) dynamics has always been an open problem for years which looks forward to any essential progress. In this short note, we prove a partial convergence conclusion of the general heterogeneous HK dynamis. That is, there must be some agents who will reach static states in finite time, while the other opinions have to evolve between them with a minimum distance if all the opinions does not reach consensus. And this result leads to the convergence of two special case of heterogeneous HK dynamics: the minimum confidence threshold is large enough, or the initial opinion difference is small enough., Comment: 6pages,no figure