The Dynamical Behaviors of a Fractional-Order Malware Propagation Model in Information Networks

Abstract

With the swift progress in communication and IT, information networks are increasingly integrated into our work and everyday life. This paper is dedicated to the study of the information network dynamics for a newly proposed fractional-order malware propagation model. Guided by the matrix theory of eigenvalues, the local stability criteria for the model described above are investigated. In addition, Hopf bifurcation is under examination with time delay serving as the bifurcation parameter. Numerical simulations are used to validate the accuracy of theoretical outcomes.

1. Introduction

It is well known that information networks have become more and more important and universal in our work and daily lives with the rapid progress of communication technology. Recently, information networks have been frequently utilised in communication protocols, intelligent transportation, medical and health care and so on [1]. However, due to the limitation of the application fields of wireless sensor, the large number of sensor nodes and other objective factors, it is impossible to pay attention to each sensor node manually, which leads to great obstacles or even inability to maintain the networks. Accompanying with it, information networks are frequently invaded by various kinds of malware. These have caused serious crisis for the communication confidentiality and security of information networks. How to prevent the spread of malware in information networks is an exigent problem to sustain the reliable performance of network systems. Therefore, finding out the spread law of malware and finding a reasonable control method of malware spread have gradually become a key topic to many scholars.

Mathematical model is a key tool to explain the law of malware propagation in information networks. As we know, there are many similarities between the spread of disease and the spread of malware. In fact, both of them can be distinguished by the differences in the attributes of the research objects. That is, the research objects generally can be divided into susceptible one, infected one, recovered one and so on. Therefore, the mathematical models of infectious diseases to study the malware propagation have been widely utilized by the domestic and foreign scholars. In recent years, Piqueira and Araujo [2] proposed a classical Susceptible-Infected-Removed (SIR) malware spread model. Tang [3] derived a modified SI model for analysis and design of information propagation mechanisms. Inspired by the compartmental biological epidemic model, the authors investigated an e-SEIRS-V model in [4]. Ren et al. [5] developed an SIR malware spread model with segmented recovery rate. Feng et al. [6] proposed an SIRS model for worm propagation that distinguishes sensor nodes as susceptible, infected, and restored. The work presented in [7] focused on a state feedback control model to study malware propagation in Mobile Wireless Sensor Networks (MWSNs). In [8,9,10], different types of SEIRS malware propagation models were developed. Liu et al. [11] developed an SEIRS model incorporating time delay to describe e-epidemic malware propagation.

In general, the mathematical models can be modeled and analyzed using traditional calculus [12,13,14] or fractional calculus [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Nevertheless, Petras [32] highlighted that fractional calculus offers greater flexibility in modeling dynamic systems than the traditional approach used for the aforementioned system. In recent decades, fractional calculus has gained significant attention with its broad applications in physics and engineering fields, encompassing biological systems, electromagnetic waves, thermal conduction, robotics, and so forth, see [15,16,17,18,19] for details. Lately, there’s a growing trend in the study of the dynamics of fractional-order systems, which involves stability analysis, bifurcations, chaos, and related areas [20,21,22,23,24,25,26,27,28,29,30,31,33]. Fractional derivatives can bring outstanding memory and describe the genetic characteristics of distinct materials and processes. In fact, lots of real-world processes show fractional-order dynamics and are depicted by the non-integer order differential models.

As is commonly understood, time delay is an essential element in the dynamics of malware propagation model, as it not only affects the dynamics of malware propagation and system stability, but also provides guidance for developing effective defense strategies and control measures. Prompted by the above dialogue, we have transitioned a delayed integer-order malware propagation model into a fractional-order model. The paper’s contributions are described hereafter:

(i) The Caputo fractional derivative is introduced to convert the SIRS model with time delay into a fractional-order model for the first occasion. To our knowledge, there have been few achievements for the latter model. Our new model more accurately reflects the actual situation. Therefore, for some practical applications, it will be more important.

($ii$) The local stability of the positive equilibrium in the fractional-order SIRS model is examined, and the precise bifurcation conditions of the model are explored with time delay as the bifurcation parameter.

The structure of this entire paper is highlighted as follows. Section 2 provides the definition of the Caputo derivative and the primary model. Section 3 delves into the local linear stability of the new model at the positive equilibrium point and presents the conditions for Hopf bifurcation. Section 4 displays numerical examples to validate our theoretical findings. Finally, Section 5 offers conclusions and future perspectives.

2. Model Description

As we know, a wireless sensor network represents an innovative type of information network, composed of numerous stationary and identical wireless sensors. We call the above wireless sensors as nodes. In general, according to whether the sensor nodes have been invaded by malware, the nodes can be categorized into three groups: susceptible (S), infected (I), and recovered (R) nodes. Each node has a correspondent variable at time t. For simplicity, $S\left(t\right)$, $I\left(t\right)$ and $R\left(t\right)$ represent the density of S, I and R at time t, respectively. In view of the SIR epidemic model [34,35,36], Zhu and Zhao [1] constructed an SIRS model to analyze malware propagation in information networks. The aforementioned specific model is detailed below:

$$\left\{\begin{array}{c}{\displaystyle \frac{dS\left(t\right)}{dt}}=a_{1}S\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)}{a_2}}\right)-a_{3}S\left(t\right)I\left(t\right)-a_{4}S\left(t\right)+a_{5}R\left(t-\iota \right)\hfill \\ {\displaystyle \frac{dI\left(t\right)}{dt}}=a_{3}S\left(t\right)I\left(t\right)-a_{6}I\left(t\right)-a_{4}I\left(t\right)\hfill \\ {\displaystyle \frac{dR\left(t\right)}{dt}}=a_{6}I\left(t\right)-a_{4}R\left(t\right)-a_{5}R\left(t-\iota \right)\hfill \end{array}\right.,$$

(1)

where

$$\left\{\begin{array}{c}S\left(t\right)=S_0\ge 0,t\in \left[-\iota ,0\right]\hfill \\ I\left(t\right)=I_0\ge 0,t\in \left[-\iota ,0\right]\hfill \\ R\left(t\right)=R_0\ge 0,t\in \left[-\iota ,0\right]\hfill \end{array}\right.,$$

$a_i$ with $i=1,...,6$ are positive constants, and $\iota$ is a nonnegative constant. $a_1$ is the intrinsic growth rate of susceptible nodes. The constant $a_2$ indicates the maximum environmental carrying capacity of the network. $a_3$ is the infection rate of malware. $a_4$ is the node mortality rate. $a_5$ is the conversion rate of recovered nodes. $a_6$ is the self-immunity repair coefficient of infected nodes. $\iota$ describes the latency of the transition of recovered nodes. $S_0$, $I_0$ and $R_0$ are the initial density of $S\left(t\right)$, $I\left(t\right)$ and $R\left(t\right)$, respectively.

It is common knowledge that fractional-order calculus has more degrees of freedom when modeling dynamic systems. Next, the Model (1) is extended to the fractional-order case. There exist several shared definitions concerning fractional derivatives. We know that Caputo derivative is more widely chosen and used due to its convenience in practical applications and compatibility with integer-order derivative. Therefore, we choose Caputo derivative to generalize the Model (1) in this paper.

Definition 1. 

([32]). The Caputo derivative of $g\left(t\right)$ with order ξ is delineated below:

$$\begin{array}{c}\hfill {}_C{D}_{t_0,t}^{\xi }g\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\xi )}}{\int }_{t_0}^t{(t-x)}^{n-\xi -1}{g}^{\left(n\right)}\left(x\right)dx,\end{array}$$

where $n-1<\xi \le n\in Z^{+}$, $\Gamma (·)$ is the Gamma function and $\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dx$.

Then we have ${}_C{D}_{t_0,t}^{\xi }g\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\xi )}}{\int }_0^t{(t-\iota )}^{-\xi }{g}^{\prime }\left(\iota \right)d\iota$ if $0<\xi \le 1$.

For the sake of brevity, we employ the notation $D^{\xi }g\left(t\right)$ to signify the Caputo derivative operator ${}_C{D}_{t_0,t}^{\xi }g\left(t\right)$.

Therefore, Model (1) is transformed into the following fractional-order SIRS model.

$$\left\{\begin{array}{c}{D}^{{\xi }_1}S\left(t\right)=a_{1}S\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)}{a_2}}\right)-a_{3}S\left(t\right)I\left(t\right)-a_{4}S\left(t\right)+a_{5}R\left(t-\iota \right)\hfill \\ D^{{\xi }_2}I\left(t\right)=a_{3}S\left(t\right)I\left(t\right)-a_{6}I\left(t\right)-a_{4}I\left(t\right)\hfill \\ D^{{\xi }_3}R\left(t\right)=a_{6}I\left(t\right)-a_{4}R\left(t\right)-a_{5}R\left(t-\iota \right)\hfill \end{array}\right.,$$

(2)

where ${\xi }_i\in (0,1]$ for $i=1$, 2 and 3.

As we know, the issue of stability and bifurcation in the fractional-order SIRS model remains a challenging and unresolved problem. In the following, the dynamical properties of the fractional-order SIRS model, such as local stability and Hopf bifurcation, will be analyzed with time delay $\iota$ serving as the bifurcation parameter.

3. Theoretical Analysis

Firstly, in order to facilitate the discussion of the existence of the positive equilibrium point of system (2), the subsequent assumption is presented.

Assume that

$\left(\mathbf{H}\mathbf{1}\right)$$a_1(a_2{a}_3-a_6-a_4)-a_2{a}_3{a}_4>0$

holds.

Then there is one unique positive equilibrium point $E^{\diamond }=(S^{\diamond },I^{\diamond },R^{\diamond }{)}^T$ in system (2), where

$$S^{\diamond }={\displaystyle \frac{a_6+a_4}{a_3}},$$

$$I^{\diamond }={\displaystyle \frac{(a_4+a_5)(a_4+a_6)[a_1(a_2{a}_3-a_6-a_4)-a_2{a}_3{a}_4]}{a_2{a}_4{a}_3^2(a_6+a_4+a_5)}},$$

$$R^{\diamond }={\displaystyle \frac{a_6(a_4+a_6)[a_1(a_2{a}_3-a_6-a_4)-a_2{a}_3{a}_4]}{a_2{a}_4{a}_3^2(a_6+a_4+a_5)}}.$$

In what follows, we assign time delay $\iota$ the role of bifurcation parameter, and then we analyze the dynamic behavior of the fractional delayed malware propagation model, give the delayed dependent stability criterion and the bifurcation condition. Since $\left(\mathbf{H}\mathbf{1}\right)$ holds, we let $\overline{S}=S-S^{\diamond }$, $\overline{I}=I-I^{\diamond }$ and $\overline{R}=R-R^{\diamond }$. For convenience, S is still substituted for $\overline{S}$, I is still substituted for $\overline{I}$ and R is still substituted for $\overline{R}$. Then one finds that

$$\left\{\begin{array}{c}{D}^{{\xi }_1}S\left(t\right)=\left(a_1-{\displaystyle \frac{2a_1{S}^{\diamond }}{a_2}}-a_3{I}^{\diamond }-a_4\right)S\left(t\right)-a_3{S}^{\diamond }I\left(t\right)+a_{5}R\left(t-\iota \right)-a_{3}S\left(t\right)I\left(t\right)-{\displaystyle \frac{a_1}{a_2}}{\left(S\left(t\right)\right)}^2\hfill \\ D^{{\xi }_2}I\left(t\right)=a_3{I}^{\diamond }S\left(t\right)+\left(a_3{S}^{\diamond }-a_4-a_6\right)I\left(t\right)+a_{3}S\left(t\right)I\left(t\right)\hfill \\ D^{{\xi }_3}R\left(t\right)=a_{6}I\left(t\right)-a_{4}R\left(t\right)-a_{5}R\left(t-\iota \right)\hfill \end{array}\right..$$

(3)

Thus, the positive equilibrium point of system (2) $E^{\diamond }=(S^{\diamond },I^{\diamond },R^{\diamond }{)}^T$ is converted to the system’s zero equilibrium point $E_0={(0,0,0)}^T$. Next, we will subsequently analyze the dynamics of system (3) at the zero equilibrium point $E_0$.

Making a local linearization to system (3) at $E_0$, we find that

$$\left\{\begin{array}{c}{D}^{{\xi }_1}S\left(t\right)=\left(a_1-{\displaystyle \frac{2a_1{S}^{\diamond }}{a_2}}-a_3{I}^{\diamond }-a_4\right)S\left(t\right)-a_3{S}^{\diamond }I\left(t\right)+a_{5}R\left(t-\iota \right)\hfill \\ D^{{\xi }_2}I\left(t\right)=a_3{I}^{\diamond }S\left(t\right)+\left(a_3{S}^{\diamond }-a_4-a_6\right)I\left(t\right)\hfill \\ D^{{\xi }_3}R\left(t\right)=a_{6}I\left(t\right)-a_{4}R\left(t\right)-a_{5}R\left(t-\iota \right)\hfill \end{array}\right..$$

(4)

Then, we obtain

$$\left|\begin{array}{ccc}{s}^{{\xi }_1}-m_{11}& -m_{12}& -m_{13}{e}^{-s\iota }\\ -m_{21}& s^{{\xi }_2}-m_{22}& -m_{23}\\ -m_{31}& -m_{32}& s^{{\xi }_3}-m_{33}-{m^{\prime }}_{33}{e}^{-s\iota }\end{array}\right|=0,$$

(5)

where

$$\begin{array}{c}{m}_{11}=a_1-{\displaystyle \frac{2a_1{S}^{\diamond }}{a_2}}-a_3{I}^{\diamond }-a_4,m_{12}=-a_3{S}^{\diamond },m_{13}=a_5,\hfill \\ m_{21}=a_3{I}^{\diamond },m_{22}=a_3{S}^{\diamond }-a_4-a_6,m_{23}=0,\hfill \\ m_{31}=0,m_{32}=a_6,m_{33}=-a_4,{m_{33}}^{\prime }=-a_5.\hfill \end{array}$$

Then, we have

$${\mathbb{P}}_1\left(s\right)+{\mathbb{P}}_2\left(s\right)e^{-s\iota }=0,$$

(6)

$$\begin{array}{c}{\mathbb{P}}_1\left(s\right)=s^{{\xi }_1+{\xi }_2+{\xi }_3}-m_{33}{s}^{{\xi }_1+{\xi }_2}-m_{22}{s}^{{\xi }_1+{\xi }_3}-m_{11}{s}^{{\xi }_2+{\xi }_3}+m_{22}{m}_{33}{s}^{{\xi }_1}\hfill \\ +m_{11}{m}_{33}{s}^{{\xi }_2}+\left(m_{11}{m}_{22}-m_{12}{m}_{21}\right)s^{{\xi }_3}-m_{11}{m}_{22}{m}_{33}+m_{12}{m}_{21}{m}_{33},\hfill \\ {\mathbb{P}}_2\left(s\right)=-{m_{33}}^{\prime }{s}^{{\xi }_1+{\xi }_2}+m_{22}{m_{33}}^{\prime }{s}^{{\xi }_1}+m_{11}{m_{33}}^{\prime }{s}^{{\xi }_2}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}.\hfill \end{array}$$

Assume that Equation (6) has one root $s=\varpi (cos{\displaystyle \frac{\pi }{2}}+isin{\displaystyle \frac{\pi }{2}})$. Hence, we have

$$\left\{\begin{array}{c}{\mathbb{A}}_{2}cos\varpi \iota +{\mathbb{B}}_{2}sin\varpi \iota =-{\mathbb{A}}_1\hfill \\ {\mathbb{B}}_{2}cos\varpi \iota -{\mathbb{A}}_{2}sin\varpi \iota =-{\mathbb{B}}_1\hfill \end{array}\right.,$$

(7)

where ${\mathbb{A}}_j$ represents the real part of $P_j\left(s\right)$ and ${\mathbb{B}}_j$ represents the imaginary part of $P_j\left(s\right)$ with $j=1$ and 2. See Appendix A for details of ${\mathbb{A}}_j$ and ${\mathbb{B}}_j$ with $j=1$ and 2.

It follows from (7) that

$$\left\{\begin{array}{c}cos\varpi \iota =-{\displaystyle \frac{{\mathbb{A}}_1{\mathbb{A}}_2+{\mathbb{B}}_1{\mathbb{B}}_2}{{\mathbb{A}}_2^2+{\mathbb{B}}_2^2}}=G_1\left(\varpi \right)\hfill \\ sin\varpi \iota ={\displaystyle \frac{{\mathbb{A}}_2{\mathbb{B}}_1-{\mathbb{A}}_1{\mathbb{B}}_2}{{\mathbb{A}}_2^2+{\mathbb{B}}_2^2}}=G_2\left(\varpi \right)\hfill \end{array}\right..$$

(8)

It follows that

$$G_1^2\left(\varpi \right)+G_2^2\left(\varpi \right)=1.$$

(9)

With the assumption that (8) has at least one positive real root, the bifurcation point can be characterized as

$${\iota }_0=min\left\{{\iota }^{\left\{j\right\}}\right\},j=0,1,2,\dots$$

(10)

We assume that there is at least one positive real root of (9), and then the bifurcation point can be defined as

$${\tau }_0=min\left\{{\tau }^{\left\{j\right\}}\right\},j=0,1,2,\dots$$

(11)

In order to explore the crossing conditions of Hopf bifurcation, we make the following hypothesis.

$\left(\mathbf{H}\mathbf{2}\right)$${\displaystyle \frac{{\mathbb{M}}_1{\mathbb{N}}_1+{\mathbb{M}}_2{\mathbb{N}}_2}{{\mathbb{N}}_1^2+{\mathbb{N}}_2^2}}\ne 0,$

where Appendix B provides the descriptions of ${\mathbb{M}}_j$ and ${\mathbb{N}}_j$ for $j=1$ and $j=2$.

Lemma 1. 

Assume that Equation (6) has one root $s\left(\iota \right)=\gamma \left(\iota \right)+i\varpi \left(\iota \right)$ near ${\iota }_0$ where $\gamma \left({\iota }_0\right)=0$ and $\varpi \left({\iota }_0\right)={\varpi }_0$. Then $\mathrm{Re}\left[{\displaystyle \frac{ds}{d\iota }}\right]{|}_{(\iota ={\iota }_0,\varpi ={\varpi }_0)}\ne 0$.

It follows Equation (6) that

$$\begin{array}{c}\hfill {\mathbb{P}}_1^{\prime }\left(s\right){\displaystyle \frac{ds}{d\iota }}+{\mathbb{P}}_2^{\prime }\left(s\right)e^{-s\iota }{\displaystyle \frac{ds}{d\iota }}+{\mathbb{P}}_2\left(s\right)e^{-s\iota }(-\iota {\displaystyle \frac{ds}{d\iota }}-s)=0.\end{array}$$

Therefore, we have

$${\displaystyle \frac{ds}{d\iota }}={\displaystyle \frac{\mathbb{M}\left(s\right)}{\mathbb{N}\left(s\right)}},$$

(12)

where

$$\begin{array}{cc}\hfill & \mathbb{M}\left(s\right)=\left(-{m_{33}}^{\prime }{s}^{{\xi }_1+{\xi }_2}+m_{22}{m_{33}}^{\prime }{s}^{{\xi }_1}+m_{11}{m_{33}}^{\prime }{s}^{{\xi }_2}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}\right)s{e}^{-s\iota },\hfill \\ \hfill & \begin{array}{c}\mathbb{N}\left(s\right)=\left[\left({\xi }_1+{\xi }_2+{\xi }_3\right)s^{{\xi }_1+{\xi }_2+{\xi }_3-1}-m_{33}\left({\xi }_1+{\xi }_2\right)s^{{\xi }_1+{\xi }_2-1}-m_{22}\left({\xi }_1+{\xi }_3\right)s^{{\xi }_1+{\xi }_3-1}\right.\hfill \\ -m_{11}\left({\xi }_2+{\xi }_3\right)s^{{\xi }_2+{\xi }_3-1}\left.+m_{22}{m}_{33}{\xi }_1{s}^{{\xi }_1-1}+m_{11}{m}_{33}{\xi }_2{s}^{{\xi }_2-1}+\left(m_{11}{m}_{22}-m_{12}{m}_{21}\right){\xi }_3{s}^{{\xi }_3-1}\right]\hfill \\ +\left[-{m_{33}}^{\prime }\left({\xi }_1+{\xi }_2\right)s^{{\xi }_1+{\xi }_2-1}+m_{22}{m_{33}}^{\prime }{\xi }_1{s}^{{\xi }_1-1}+m_{11}{m_{33}}^{\prime }{\xi }_2{s}^{{\xi }_2-1}\right]e^{-s\iota }\hfill \\ +\left(-{m_{33}}^{\prime }{s}^{{\xi }_1+{\xi }_2}+m_{22}{m_{33}}^{\prime }{s}^{{\xi }_1}+m_{11}{m_{33}}^{\prime }{s}^{{\xi }_2}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}\right)\iota e^{-s\iota }.\hfill \end{array}\hfill \end{array}$$

Proof. 

Then, we have

$${\mathrm{Re}\left[{\displaystyle \frac{ds}{d\iota }}\right]}_{\left|\left(\iota ={\iota }_{0,}w=w_0\right)\right.}={\displaystyle \frac{{\mathbb{M}}_1{\mathbb{N}}_1+{\mathbb{M}}_2{\mathbb{N}}_2}{{{\mathbb{N}}_1}^2+{{\mathbb{N}}_2}^2}},$$

(13)

where the real and imaginary parts of $\mathbb{M}\left(s\right)$ are denoted by ${\mathbb{M}}_1$ and ${\mathbb{M}}_2$, and the real and imaginary parts of $\mathbb{N}\left(s\right)$ are denoted by ${\mathbb{N}}_1$ and ${\mathbb{N}}_2$.

According to Hypothesis $\left(\mathbf{H}\mathbf{2}\right)$, Lemma 1 holds true. □

In order to study whether the system (2) is stable with $\iota =0$, the subsequent hypotheses are considered:

$\left(\mathbf{H}\mathbf{3}\right)$$m_{11}+m_{22}+m_{33}+{m_{33}}^{\prime }<0$,

$\left(\mathbf{H}\mathbf{4}\right)$$\begin{array}{c}\left(m_{11}+m_{22}+m_{33}+{m_{33}}^{\prime }\right)\left(m_{11}{m}_{22}+m_{11}{m}_{33}+m_{11}{m_{33}}^{\prime }-m_{12}{m}_{21}+m_{22}{m}_{33}+m_{22}{m_{33}}^{\prime }\right)\hfill \\ -m_{11}{m}_{22}{m}_{33}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m}_{33}+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}<0,\hfill \end{array}$

$\left(\mathbf{H}\mathbf{5}\right)$$-m_{11}{m}_{22}{m}_{33}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m}_{33}+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}>0$.

Lemma 2. 

Assume that $\left(\mathbf{H}\mathbf{3}\right)$–$\left(\mathbf{H}\mathbf{5}\right)$ hold. Then system (2) exhibits asymptotic stability at the positive equilibrium point $E^{\diamond }=(S^{\diamond },I^{\diamond },R^{\diamond }{)}^T$ when $\iota =0$.

Proof. 

Since $\iota =0$, we have

$$\begin{array}{c}{s}^3-\left(m_{11}+m_{22}+m_{33}+{m_{33}}^{\prime }\right)s^2\hfill \\ +\left(m_{11}{m}_{22}+m_{11}{m}_{33}+m_{11}{m_{33}}^{\prime }-m_{12}{m}_{21}+m_{22}{m}_{33}+m_{22}{m_{33}}^{\prime }\right)s\hfill \\ -m_{11}{m}_{22}{m}_{33}-m_{11}{m}_{22}{m_{33}}^{\prime }+m_{12}{m}_{21}{m}_{33}+m_{12}{m}_{21}{m_{33}}^{\prime }-m_{13}{m}_{21}{m}_{32}=0\hfill \end{array}.$$

(14)

According to the Hypotheses $\left(\mathbf{H}\mathbf{3}\right)$–$\left(\mathbf{H}\mathbf{5}\right)$ and Routh-Hurwitz criteria, we get all roots of Equation (14) having negative real parts. Then system (2) is asymptotically stable at $E^{\diamond }=(S^{\diamond },I^{\diamond },R^{\diamond }{)}^T$. □

In conclusion, the following outcomes have been obtained.

Theorem 1. 

Assume that $\left(\mathbf{H}\mathbf{1}\right)$–$\left(\mathbf{H}\mathbf{5}\right)$ are true. The results that can be deduced for system (2) are listed hereafter.

(1) System (2) is asymptotically stable at $E^{\diamond }$ when $\iota \in [0,{\iota }_0)$.

(2) System (2) undergoes a Hopf bifurcation at $E^{\diamond }$ when $\iota ={\iota }_0$. Namely, system (2) has a branch of periodic solutions bifurcating from the equilibrium point $E^{\diamond }$ near $\iota ={\iota }_0$.

Remark 1. 

In this study, we develop a fractional-order SIRS-based malware propagation model for information networks, distinct from the integer-order model examined in [1]. This new fractional SIRS model offers a more precise representation of the practical dynamics of malware spread in information networks.

4. Numerical Simulation

In this section, we will demonstrate the theoretical results through relevant numerical simulations and investigate the impact of time delay on the dynamics of the aforementioned Model (2). For convenience, the system parameters are set as follows. $a_1=0.3,a_2=2,a_3=0.5,a_4=0.1,a_5=0.2$ and $a_6=0.4$, which are from the literature [1]. Then we take into account the following system:

$$\left\{\begin{array}{c}{D}^{{\xi }_1}S\left(t\right)=0.3S\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)}{2}}\right)-0.5S\left(t\right)I\left(t\right)-0.1S\left(t\right)+0.2R\left(t-\iota \right)\hfill \\ D^{{\xi }_2}I\left(t\right)=0.5S\left(t\right)I\left(t\right)-0.4I\left(t\right)-0.1I\left(t\right)\hfill \\ D^{{\xi }_3}R\left(t\right)=0.4I\left(t\right)-0.1R\left(t\right)-0.2R\left(t-\iota \right)\hfill \end{array}\right..$$

(15)

Hence, it is easy to calculate the unique positive equilibrium $E^{\diamond }={(1,0.2143,0.2857)}^T$.

When the influence of time delay in the dynamic system (15), the bifurcation point needs to be determined first. We choose ${\xi }_1=0.97,{\xi }_2=0.98$ and ${\xi }_3=0.99$. Then we can figure out ${\varpi }_0=0.2860$ and ${\iota }_0=10.6719$. Applying Theorem 1, system (15) exhibits asymptotic stability at $E^{\diamond }$ when choosing $\iota =8<{\iota }_0$, which is illustrated in Figure 1, Figure 2 and Figure 3. It can be observed that system (15) is unstable at $E^{\diamond }$, and Hopf bifurcation generates from $E^{\diamond }$ when selecting $\iota =11>{\iota }_0$, as revealed in Figure 4, Figure 5 and Figure 6. On the other hand, Figure 7 and Figure 8 are phase diagrams of system (15) with different values of $\iota$. Figure 7 shows that system (15) exhibits local asymptotic stability at $E^{\diamond }$ when $\iota =8<{\iota }_0$. And we can see from Figure 8 that (15) is unstable at $E^{\diamond }$ and a periodic oscillation bifurcates from $E^{\diamond }$ when $\iota =11>{\iota }_0$.

5. Conclusions

Fractional malware propagation models in information networks bring abundant dynamical behaviors. We have utilised fractional calculus theory to study the delayed fractional-order malware propagation model, i.e., the fractional-order SIRS model. The model’s stability and Hopf bifurcation have been examined by considering time delay as the bifurcation parameter. The simulation results have shown the effectiveness of the proposed criteria. Future work will introduce different time delays into the fractional-order SIRS model and analyze the role of multiple time delays in shaping the system’s dynamic changes.