Julia Sets and Their Control of Discrete Fractional SIRS Models

Abstract

It is of crucial significance to study infectious disease phenomenon by using the discrete SIRS model with the Caputo deltas sense and fractal viewpoint. In this paper, Julia set of the discrete fractional SIRS model is established to analyze the fractal dynamics of this model. Then three different controllers, which are, respectively, added to different parts of the model as a whole, a part, and a product factor, are designed to change the Julia set, and the graphs illustrate the complexity of the model. Simulation results show the efficacy of these methods.

Fractal Control and Synchronization of the Discrete Fractional SIRS Model

“…Consider system (2) of the SIRS model in [33], in which the effective contact rate coefficient is affected by seasonal factors:…”

“…where the initial values of τ 3 and τ 4 are τ 3 (0) � 0 and τ 4 (0) � 1. According to the method and discretization process of the literature [33], discretize the equation by using…”

“…where s 0 � 0, r 0 � 1, and η, b, c, q, e, z, and δ are system parameters. Now, refer to the method of converting the integer order equation into fractional order equation in [33], subtract x n , y n , s n , and r n from both sides of the 4 equations of system (4), rewrite the equation as a fractional one from the perspective of discrete fractional calculus, and take the momentums x(n), y(n), s(n), and r(n) into account. erefore, the numerical equations can be accurately proposed to the following one, the fractional difference about 4 equations is introduced:…”

“…In [32], the authors utilize active control technique to synchronize different fractional order chaotic dynamical systems. e main content of this paper is the follow-up work of the literature [33], which applies the ideas of Julia set in fractal theory and basic theoretical knowledge of fractional calculus to the SIRS model. e remainder of this article is arranged as follows.…”

“…Moreover, in [33], the authors researched the Julia set of the discrete fractional SIRS model and designed three different controllers, which are added to different parts of the model as a whole, a part, and a product factor, respectively, to change the Julia set. Nevertheless, in this paper, two kinds of control items including fixed points are designed for the fractional order model, which are added into the system as a whole and a part, respectively.…”