Abstract
In this paper, the stability and Hopf bifurcation of a fractional-order model of the Susceptible-Exposed-Infected-Kill Signals Recovered (SEIR-KS) computer virus with two delays are studied. The sufficient conditions for solving the stability and the occurrence of Hopf bifurcation of the system are established by using Laplace transform, stability theory, and Hopf bifurcation theorem of fractional-order differential systems. The research shows that time delays and fractional order q have an important effect on the stability and the emergence of Hopf bifurcation of the fractional computer virus model. In addition, the validity of the theoretical analysis is verified by selecting appropriate system parameters for numerical simulation and the biological correlation of the equilibrium point is discussed. The results show that the bifurcation point of the model increases with the decrease in the model fractional order q. Under the same fractional order q, the effects of different types of delays on bifurcation points are obviously different.
Highlights
Preliminaries and Model DescriptionWe will introduce the relevant theories of this study, which are about the Caputo fractional derivative and the theoretical research methods for the fractional-order computer virus-spreading model [23, 24]
E fractional-order model was first proposed in the study of basic mathematical theory, but it has not been widely used for a long time because of the complexity of its calculation [9]
Uçar et al studied the stability of a fractional SAIDR model in the frame of the Atangana–Baleanu derivative [20]. e computer virus model based on epidemiology has a strong “memory” attribute because of its evolution process and control strategy and can be studied by using fractional calculus theory, so that new dynamic characteristics such as chaos can be obtained [21]
Summary
We will introduce the relevant theories of this study, which are about the Caputo fractional derivative and the theoretical research methods for the fractional-order computer virus-spreading model [23, 24]. We give the definition of the Caputo fractional-order derivative: Dαt f(t). E definition of the Laplace transform of the Caputo fractional-order derivative is m− 1. Fangfang Yang and Zizhen Zhang studied the following integer-order time delay computer virus-spreading model [25]:. Ey analyze the local stability and Hopf bifurcation of system (3) with time delay as the parameter and confirm the effectiveness of the conclusion. The Hopf bifurcation of the fractional computer virus model is poorly studied. (4) degenerates into the model in [25] when q 1
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