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In recent years, with the rapid development of the Internet and the Internet of Things, network security is urgently needed. Malware becomes a major threat to network security. Thus, the study on malware propagation model plays an important role in network security. In the past few decades, numerous researchers put up various kinds of malware propagation models to analyze the dynamic interaction. However, many works are only concerned with the integer-order malware propagation models, while the investigation on fractional-order ones is very few. In this paper, based on the earlier works, we will put up a new fractional-order delayed malware propagation model. Letting the delay be bifurcation parameter and analyzing the corresponding characteristic equations of considered system, we will establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model. The study shows that the delay and the fractional order have important effect on the stability and Hopf bifurcation of considered system. To check the correctness of theoretical analyses, we carry out some computer simulations. At last, a simple conclusion is drawn. The derived results of this paper are completely innovative and play an important guiding role in network security.

Nowadays, social networks are important platforms for disseminating information and building relationship. Different from the classical approaches of communication, social networks have fast speed of information propagation and diffusion. Furthermore, social networks have important effect on commercial negotiations, social connections, and information-sharing activities. Owing to the potential applications of social networks in many areas, many scholars pay much attention to dynamics of wireless sensor networks. For example, Deng et al.[

Malware software (malware) is an important tool to attack cybersecurity. Malware, which widely appears in the Internet, will cause some serious security risks of networks such as network paralysis, instability of society, loss of secret key, and personal information leakage. In order to understand and grasp the damage of malware, it is necessary for us to investigate the cause of malware occurrence, the harmful level to human beings, and the internal mechanism of malware propagation. During the past few decades, some researchers have obtained excellent achievements. For example, Liu et al. [

The fractional calculus is a generalization of ordinary differentiation and integration to random order (noninteger)[

Here we must point out that all the above works on Hopf bifurcation of fractional-order (integer-order) differential models are only concerned with neural networks and predator-prey models. Up to now, there are few articles that focus on the effect of delays on the Hopf bifurcation of fractional-order delayed malware propagation model.

In 2018, Du et al. [

The main objective of this article is to handle two problems: (i) the sufficient conditions that guarantee the stability and existence of Hopf bifurcation of system (

The highlights of this paper consist of four points:

(i) The integer-order delayed malware propagation model in social networks has been extended to fractional-order delayed malware propagation model in social networks, which can better describe the memory and hereditary properties of the model.

(ii) A sufficient criterion of the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model in social networks are derived. The effect of delay and fractional order on the stability and Hopf bifurcation of (

(iii) To the best of our knowledge, there are no articles that focus on the Hopf bifurcation of fractional-order malware propagation model. The obtained results of this article will enrich and develop the Hopf bifurcation theory of fractional-order delayed differential equations and supplement the previous publications.

(iv) The idea of this manuscript will provide a good reference to investigate many other fractional-order systems with delays.

The rest of this paper is organized as follows. In Section

In this section, we introduce three definitions and two lemmas.

The fractional integral of order

The Caputo fractional-order derivative of order

For the given fractional-order system

Consider the following autonomous system

For the given fractional-order delayed differential equation with Caputo derivative,

In this section, we will investigate the impact of the delay on Hopf bifurcation for system (

Obviously, system (

Let

For (

(1) If (Q2) holds and

(2) If (Q3) holds, then (

The proof of Lemma

In order to obtain the transversality condition of Hopf bifurcation, the following hypothesis is given:

Let

Differentiating (

According to the analysis above and Lemmas

For system (

Xu et al. [

Consider the following fractional-order system:

The impact of fractional-order

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In real world, malware plays an important role in economy and society. Based on the previous works, in this paper, we propose a new fractional-order delayed malware propagation model in social networks. By choosing the time delay as bifurcation parameter, we establish a sufficient condition to ensure the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model. The research shows that the positive equilibrium point of the involved model is locally asymptotically stable when the time delay is less than the critical value

The figure data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work is supported by National Natural Science Foundation of China (No. 61673008), Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004), and Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020).