# Stability Analysis of a Fractional-Order SEIR-KS Computer Virus-Spreading Model with Two Delays

Authors: Zhufeng Wang; Xiaoqian Nie; Maoxin Liao
Journal: Journal of Mathematics
 (2021)

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## 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 orderq 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.


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## Body

## 1. Introduction

The spread of computer virus is the most common problem in network security. It spreads fast, spreads widely, and does serious harm. If it is not controlled in time, it will cause huge economic losses and seriously interfere with the normal production of the society. Therefore, it is very necessary to study the propagation mechanism of computer virus, and the research on virus propagation model has important significance both in theory and practice. In 1991, Kephart and White noted that there are many similarities between biological virus propagation and computer virus propagation [1]. It was the first time of introducing an epidemiological mathematical model into the study of computer virus transmission. Since then, many scholars have also used some models and analytical methods in biology to conduct in-depth research on computer viruses [2–8].The 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]. With the development of computer technology, the complex calculation process can be completed by various software programs and the fractional model has re-entered the field of view of researchers. Since fractional calculus can describe the evolutionary process with genetic attributes, its memory characteristics can not only describe the past state of the relevant system but also affect the present and future state of the system. In recent years, researchers in the fields of mathematics, physics, biology, and finance have found that some fractional models used to describe problems better reflect the real situation of the problem[10–16].The proper “ecological memory” is an important characteristic of the biological population model; it refers to “the past state or experience affects the current or future response ability of the community” [17]. This characteristic coincides with that of fractional calculus. Many researchers have introduced fractional calculus into various models of biological mathematics and obtained a number of results with theoretical value and practical significance. For example, Djordjevic et al. used fractional-order ordinary differential equations to model a class of cell problems, and the consistency between the model and the actual data was more accurate than that of the integer-order model [18]. El-Saka studied the SIS epidemic model of fractional-order changing population and analyzed the stability of the equilibrium point and the conditions for the existence of the uniform steady-state solution [19]. Uçar et al. studied the stability of a fractional SAIDR model in the frame of the Atangana–Baleanu derivative [20]. The 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].In this paper, a fractional reciprocal SEIR-KS computer virus model, which has double delay, is proposed on the basis of reference. Based on the conclusions summarized in [22], the relationship between Hopf bifurcation and time delay appearing at the positive equilibrium point of the model is theoretically analyzed. To verify the rationality of the theoretical analysis, the appropriate system parameters were selected for numerical simulation and the variation trend of the critical value of the time delay under the condition of fractional-order change was obtained through numerical simulation.


## 2. Preliminaries and Model Description

In this part, 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].First of all, we give the definition of the Caputo fractional-order derivative:(1)Dtαft=1Γu−α∫0tt−τu−α−1fnτdτ,where u−1<α<u, μ∈N, 0≤α≤1, and Γ⋅ is the Gamma function.The definition of the Laplace transform of the Caputo fractional-order derivative is(2)LDtαft=sαFs−∑l=0m−1sα−l−1fl0.Iffl0=0, where l=0,1,…,u−1, we get LDtαft=sαFs.Fangfang Yang and Zizhen Zhang studied the following integer-order time delay computer virus-spreading model [25]:(3)dSdt=pb−β1StEt−β2StIt−ϕStKt−μSt+σRt−τ1,dEdt=rb+β1StEt+β2StIt−αEt−τ2−χEt−μEt,dIdt=αEt−τ2−δItKt−γIt−εIt−μIt,dKdt=δItKt+γIt−χEt−μKt,dRdt=ϕStKt+εIt−μRt−σRt−τ1,where S(t) represent the number of susceptible computers, E(t) represent the number of exposed computers, I(t) represent the number of infected computers, K(t) represent the number of kill signals computers, and R(t) represent the number of recovered computers; p,b,β1,β2,ϕ,μ,σ,α,χ,δ,γ,ε,r∈0,1 are the probability of interconversion between various nodes; τ1 is the temporary immunization time delay for the recovered computer to be converted into a susceptible computer; and τ2 is the latency from the exposed computer to the infected computer. They analyze the local stability and Hopf bifurcation of system (3) with time delay as the parameter and confirm the effectiveness of the conclusion. The direction of Hopf bifurcation and the stability of bifurcation periodic solution are proved.Up to now, the Hopf bifurcation of the fractional computer virus model is poorly studied. To remedy this defect, we think that it is necessary to study the Hopf branch of the fractional computer virus. Inspired by the above analysis, on the basis of previous system (3), we establish the following fractional-order SEIR-KS computer virus-spreading model with two delays:(4)DqSt=pb−β1StEt−β2StIt−ϕStKt−μSt+σRt−τ1,DqEt=rb+β1StEt+β2StIt−αEt−τ2−χEt−μEt,DqIt=αEt−τ2−δItKt−γIt−εIt−μIt,DqKt=δItKt+γIt+χEt−μKt,DqRt=ϕStKt+εIt−μRt−σRt−τ1,where q∈0,1 is the fractional order. Obviously, system (4) degenerates into the model in [25] when q = 1.


## 3. The Stability and the Hopf Bifurcation Analysis

In this section, some sufficient conditions are established, for the existence of Hopf bifurcation of system (4).Make the five equations in system (4) be equal to zero. Next, we can obtain that(5)pb−β1StEt−β2StIt−ϕStKt−μSt+σRt=0,rb+β1StEt+β2StIt−αEt−χEt−μEt=0,αEt−δItKt−γIt−εIt−μIt=0,δItKt+γIt+χEt−μKt=0,ϕStKt+εIt−μRt−σRt=0.Hence,(6)S∗=σR∗+pbβ1E∗+β2I∗+ϕK∗+M,E∗=rbα+χ+M−β1E∗−β2I∗,I∗=αE∗δK∗+γ+ε+M,K∗=γI∗+χE∗μ−δI∗,R∗=εI∗+ϕS∗K∗μ+σ.From [25], if R0=β1b/μα+χ+μ>1, S∗,E∗,I∗,K∗,R∗ is a unique endemic equilibrium of system (4).The linearized part of system (4) is(7)DqSt=a11St+a12Et+a13It+a14Kt+m15Rt−τ1,DqEt=a21St+a22Et+n22Et−τ2+a23It,DqIt=n32Et−τ2+a33It+a34Kt,DqKt=a42Et+a43It+a44Kt,DqRt=a51St+a53It+a54Kt+a55Rt+m55Rt−τ1,where(8)a11=−β1E∗+β2I∗+ϕK∗+μ,a12=−β1S∗,a13=−β2S∗,a14=ϕS∗,m15=σ,a21=β1E∗+β2I∗,a22=β1S∗−χ−μ,n22=−α,a23=β2S∗,n32=α,a33=−δK∗+γ+ε+μ,a34=−δI∗,a42=χ,a43=δK∗+γ,a51=ϕK∗,a53=ε,a54=ϕS∗,a55=−μ,m55=−σ,a44=δI∗−μ.We can get the characteristic equation of system (7):(9)λq−a11−a2100−a51−a12λq−a22−n22e−λτ2−n32e−λτ2−a420−a13−a23λq−a33−a43−a53−a140−a34λ−a44−a54−m15e−λτ1000λq−a55−m55e−λτ1=0.Then, the following exponential equation can be obtained:(10)λ5q+a4λ4q+a3λ3q+a2λ2q+a1λq++a0+b4λ4q+b3λ3q+b2λ2q+b1λq+b0e−λτ1+c4λ4q+c3λ3q+c2λ2q+c1λq+c0e−λτ1+d3λ3q+d2λ2q+d1λq+d0e−λτ1+τ2=0,where(11)a4=−a11+a22+a33+a44+a55,a3=a11a22−a12a21+a11a33+a11a33+a11a44+a22a33+a11a55+a22a44+a22a55+a33a44−a34a43+a33a55+a44a55,a2=−a11a22a33−a12a21a33+a11a22a44−a12a21a44+a14a21a42+a11a22a55+a11a33a44−a11a34a43−a12a21a55+a11a33a55+a22a33a44−a22a34a43+a22a34a42+a11a44a55+a22a33a55+a22a44a55+a33a44a55−a34a43a55,a1=a11a22a33a44−a11a22a34a43+a11a23a34a42−a12a21a33a44+a12a21a34a43−a13a21a34a42+a14a21a33a42+a11a22a33a55−a11a21a33a55+a11a22a44a55−a12a21a44a55+a14a21a42a55+a11a33a44a55−a11a34a43a55+a22a33a44a55−a22a34a43a55+a23a34a42a55,a0=−a11a22a33a44a55+a11a22a34a43a55−a11a23a34a42a55+a12a21a33a44a55−a12a21a34a43a55+a13a21a34a42a55−a14a21a33a42a55,(12)b4=−m55,b3=a11m55+a51m15+a22m55+a33m55+a44m55,b2=m55−a11a22+a12a21−a11a33−a11a44−a22a33−a22a44−a33a44+a34a43+m15a22a51+a33a51+a44a51,b1=m55a11a22a33−a12a21a33+a11a22a44−a12a21a44+a14a21a42+a11a33a44−a11a34a43+a22a33a44−a22a34a43+a23a34a42,b0=m55−a11a22a33a44+a11a22a34a43−a11a23a34a42+a12a21a33a44−a12a21a34a43+a13a21a34a42−a14a22a33a42+m15a21a33a42a54−a21a34a42a53−a22a33a44a51−a22a34a43a51+a23a34a42a51,(13)c4=−n22,c3=a11n22−a23n32+a33n22+a44n22+a55n22,c2=−−a11a33n22+a13a21n32+a11a44n22+a11a55n22−a23a44n32+a33a44n22−a34a43n22−a23a55n32+a33a55n22+a44a55n22,c1=n22a11a33a44−a11a34a43+a11a33a55+a11a44a55+a33a44a55−a34a43a55+n32−a11a23a44+a13a21a44−a14a21a43−a11a23a55+a13a21a55−a23a44a55,c0=a11a23a44a55a32−a11a33a44a55a22+a11a34a43a55a22−a13a21a44a55a32+a14a21a43a55a32.


### 3.1. Case 1:τ1=τ2=0

Whenτ1=τ2=0, equation (10) becomes(14)λ5+B4λ4+B3λ3+B2λ2+B1λ+B0=0,where(15)B0=a0+b0+c0+d0,B1=a1+b1+c1+d1,B2=a2+b2+c2+d2,B3=a3+b3+c3+d3,B4=a4+b4+c4+d4.By the Routh–Hurwitz criterion, a set of necessary and sufficient conditions for all roots of equation (14) to have a negative real part is given in the following form:(16)D1=B4>0,(17)D2=B4B21B3>0,(18)D3=B4B2B010B3B4B1B2>0,(19)D4=B4B21B3B10B200B401B2B0B3B1>0,(20)D5=B410B2B100B3B200B4B2B00001B3B100B4B2B0>0.Lemma 1.
Ifτ1=τ2=0,R0>1, and (16)-(20) are satisfied, the endemic equilibrium of system (4) is asymptotically stable.


### 3.2. Case 2:τ1>0,τ2=0

Whenτ1>0andτ2=0, then we can obtain the following equation:(21)λ5q+C4λ4q+C3λ3q+C2λ2q+C1λq+C0+E4λ4q+E3λ3q+E2λ2q+E1λq+E0e−λτ1=0,where C4=a4+c4, C3=a3+c3, C2=a2+c2, C1=a1+c1, C0=a0+c0, E4=b4, E3=b3+d3, E2=b2+d2, E1=b1+d1, and E0=b0+d0.Letλ=i˙ω; we substitute this into (21), and separate the real part from imaginary part:(22)M11cosωτ1+M12sinωτ1=M13,M12cosωτ1−M11sinωτ1=M14,where(23)M11=E4ω4qcos2qπ+E3ω3qcos3qπ2+E2ω2qcosqπ+E1ωqcosqπ2+E0,M12=E4ω4qsin2qπ+E3ω3qsin3qπ2+E2ω2qsinqπ+E1ωqsinqπ2,M13=ω5qcos5qπ2+C4ω4qcos2qπ+C3ω3qcos3qπ2+C2ω2qcosqπ+C1ωqcosqπ2+C0,M14=ω5qsin5qπ2+C4ω4qsin2qπ+C3ω3qsin3qπ2+C2ω2qsinqπ+C1ωqsinqπ2.From (22), it is easy to obtain(24)cosωτ1=M13M11+M14M12M112+M122,(25)M112+M122=M132+M142.By (25), we have(26)ω10q+f9ω9q+f8ω8q+f7ω7q+f6ω6q+f5ω5q+f4ω4q+f3ω3q+f2ω2q+f1ωq+f0=0,where(27)f0=C02−E02,f1=2C0C1cosqπ2−2E0E1cosqπ2,f2=C12−E12+2C0C1cosqπ−2E0E2cosqπ,f3=2C0C3cos3qπ2−2E0E3cos3qπ2+2C1C2sinqπsinqπ2−2E1E2sinqπsinqπ2+2C1C2cosqπcosqπ2−2E1E2cosqπcosqπ2,f4=C22−E22+2C0C4cos2qπ−2E0E4cos2qπ+2C1C3sinqπ2sin3qπ2−2E1E3sinqπ2sin3qπ2+2C1C3cosqπ2cos3qπ2−2E1E3cosqπ2cos3qπ2,(28)f5=2C0cos5qπ2+2C1C4sin2qπsinqπ2+2C2C3sinqπsin3qπ2−2E1E4sin2qπsinqπ2−2E2E3sinqπsin3qπ2+2C1C4cos2qπcosqπ2+2C2C3cosqπcos3qπ2−2E1E4cos2qπcosqπ2−2E2E3cosqπsin3qπ2,f6=C32−E32+2C1cosqπ2cos5qπ2+2C1sinqπ2sin5qπ2+2C2C4sinqπsin2qπ−2E2E4sinqπsin2qπ+2C2C4cosqπcos2qπ−2E2E4cosqπcos2qπ,f7=2C2cosqπcos5qπ2+2C2sinqπsin5qπ2+2C3C4sin2qπsin3qπ2−2E3E4sin2qπsin3qπ2+2C3C4cos2qπcos3qπ2−2E3E4cos2qπcos3qπ2,f8=C42−E42−2C3cos3qπ2cos5qπ2+2C3sin3qπ2sin5qπ2,f9=2C4cos2qπcos5qπ2+2C4sin2qπsin5qπ2.Leth1ω=ω10q+f9ω9q+f8ω8q+f7ω7q+f6ω6q+f5ω5q+f4ω4q+f3ω3q+f2ω2q+f1ωq+f0; obviously, limω⟶+∞h1ω=+∞. When f0=C02−E02<0, namely, a0−C02−b0−d02<0, then (26) has at least one positive real root.Lemma 2.
Whena0−C02−b0−d02<0, (26) has at least one positive real root.
By (24), we have(29)τ10=1ω10arccosM13M11+M14M12M112+M122,where ±iω10 is a pair of purely imaginary roots of (21) with τ10. Also,(30)H1Redλdτ1−1|τ1=τ10≠0.
Taking the derivative ofλ with respect to τ1 in (21), we can obtain(31)dλdτ1=λV13V11+V12−τ1V13,where(32)V11=5qλ5q−1+4qC4λ4q−1+3qC3λ3q−1+2qC2λ2q−1+qC1λq−1,V12=4qE4λ4q−1+3qE3λ3q−1+2qE2λ2q−1+qE1λq−1e−λτ1,V13=E4λ4q+E3λ3q+E2λ2q+E1λq+E0−λτ1,dλdτ1−1=V11+V12λV13=−τ1λ,dλdτ1−1=W11+W12cosωτ1+W13sinωτ1+i−W13cosωτ1−W11sinωτ1+W14W15cosωτ1+W16sinωτ1+i−W16cosωτ1−W15sinωτ1−τiW,where(33)W11=5qω5q−1sin5qπ2+4qC4ω5q−1sin2qπ+3qC3ω3q−1sin3qπ2+2qC2ω2q−1sinqπ+qC1ωq−1sinqπ2,W12=4qE4ω4q−1sin2qπ+3qE3ω3q−1sin3qπ2+2qE2ω2q−1sinqπ+qE1ωq−1sinqπ2,W13=4qE4ω4q−1cos2qπ+2qE3ω3q−1cos3qπ2+2qE2ω2q−1cosqπ+qE1ωq−1cosqπ2,W14=−5qω5q−1cos5qπ2+4qC4ω4q−1cos2qπ+3qC3ω3q−1cos3qπ2+2qC2ω2q−1cosqπ+qC1ωqcosqπ2,W15=−ωE4ω4qsin2qπ+E3ω3qsin3qπ2+E2ω2qsinqπ+E1ωqsinqπ2,W16=ωE4ω4qcos2qπ+E3ω3qcos3qπ2+E2ω2qcosqπ+E1ωqcosqπ2+E0,Redλdτ1−1=X11X13+X12X14X132+X142,where(34)X11=W11+W12cosωτ1+W13sinωτ1,X12=−W13cosωτ1−W11sinωτ1+W14,X13=W15cosωτ1+W16sinωτ1,X14=−W16cosωτ1−W15sinωτ1.
Noticing thatsignRedλ/dτ1|τ1=τ10=signRedλ/dτ1−1|τ1=τ10, we can obtain Redλ/dτ1|τ1=τ10≠0.Theorem 1.
Suppose thatR0>1, (16)–(20), (H1), and a0−C02−b0−d02<0 hold; then, the following results hold:(i)
Whenτ1∈0,τ10, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable(ii)
Whenτ1=τ10, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ1=τ10.


### 3.3. Case 3:τ1=0,τ1<0

Whenτ1=0,τ1<0, equation (10) becomes(35)λ5q+F4λ4q+F3λ3q+F2λ2q+F1λq+F0+G4λ4q+G3λ3q+G2λ2q+G1λq+G0e−λτ1=0.where F0=a0+b0, F1=a1+b1, F2=a2+b2, F3=a3+b3, F4=a4+b4, G0=c0+d0, G1=c1+d1, G2=c2+d2, G3=c3+d3, and G4=c4.Since (19) and (25) are the same equation, we can draw the conclusion directly:(36)H2Redλdτ2−1|τ2=τ20≠0.Theorem 2.
Suppose thatR0>1, (16)–(20), (H2), and a0+b02−c0+d02<0 hold; then, the following results hold:(i)
Whenτ2∈0,τ20, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable.(ii)
Whenτ2=τ20, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ2=τ20.


### 3.4. Case 4:τ1=τ2=τ∗>0

Whenτ1=τ2=τ∗>0, we can easily obtain the following equation:(37)λ5q+a4λ4q+a3λ3q+a2λ2q+a1λq+a0eλτ∗+J4λ4q+J3λ3q+J2λ2q+J1λq+J0+d3λ3q+d2λ2q+d1λq+d0e−λτ∗=0,where J0=b0+c0, J1=b1+c1, J2=b2+c2, J3=b3+c3, and J4=b4+c4.Supposingλ=i˙ω, next we can obtain equation (38) by separating the real part from imaginary part:(38)M31cosωτ∗+M32sinωτ∗=M33,M34cosωτ∗−M35sinωτ∗=M36,where(39)M31=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,M32=−ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2,M33=J4ω4qcos2qπ+J3ω3qcos3qπ2+J2ω2qcosqπ+J1ωqcosqπ2+J0,M34=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2+d3ω3qsin3qπ2+da2ω2qsinqπ+d1ωqsinqπ2,M35=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0−d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,M36=J4ω4qsin2qπ+J3ω3qsin3qπ2+J2ω2qsinqπ+J1ωqsinqπ2.Suppose that (38) has roots, the expression of sinωτ and cosωτ can be obtained as follows:(40)cosωτ∗=M33M35−M32M36M31M35−M32M34,(41)sinωτ∗=M31M36−M33M34M31M35−M32M34.Sincesin2ωτ∗+cos2ωτ∗=1, we have(42)ω20q+k19ω19q+⋯+k1ωq+k0=0,where(43)k0=a0+d02a0−d02−J02a0−d02,k1=2a0+d0a0−d0a0+d0a1−d1cosqπ2+a1+d1a0−d0cosqπ2−2J0a0−d0J1a0+d0cosqπ2+J0a1−d1cosqπ2,k19=2a4cos2qπcos5qπ2−a4−d4sin2qπsin5qπ2.Leth3ω=ω20q+k19ω19q+⋯+k1ωq+k0; obviously, limω⟶+∞h3ω=+∞. When k0=a0+d02a0−d02−J02a0−d02<0, then (42) has at least one positive real root.Lemma 3.
Whena0+d02a0−d02−J02a0−d02<0, (42) has at least one positive real root.
By (40), we have(44)τ∗=1ω30arccosM33M35−M32M36M31M35−M32M34,where ±iω30 is a pair of purely imaginary roots of (37) with τ∗0.(45)H3Redλdτ∗−1|τ∗=τ∗0≠0.
The derivative ofλ with respect to τ∗ in (37) is easily obtained:(46)dλdτ∗=λV32+V35V31+V33+V34−τ∗V13,where(47)V31=5qλ5q−1+4qa4λ4q−1+3qa3λ3q−1+2qa2λ2q−1+qa1λq−1eλτ∗,V32=λ5q+a4λ4q+a3λ3q+a2λ2q+a1λq+a0−eλτ∗,V33=4qJ4λ4q−1+3qJ3λ3q−1+2qJ2λ2q−1+qJ1λq−1,V34=3qd3λ3q−1+2qd2λ2q−1+qd1λq−1e−λτ∗,V35=d3λ3q+d2λ2q+d1λq+d0e−λτ∗,dλdτ∗−1=V31+V33+V34λV32+V35=−τ∗λ,dλdτ∗−1=W30+W31cosωτ∗+W32sinωτ∗+iW33cosωτ∗+W34sinωτ∗+W35W36cosωτ∗+W37sinωτ∗+i−W38cosωτ∗−W39sinωτ∗−τ∗iW,where(48)W30=4qJ4ω4q−1sin2qπ+3qJ3ω3q−1sin3qπ2+2qJ2ω2q−1sinqπ+qJ1ωq−1sinqπ2,W31=5qω5q−1sin5qπ2+4qa4ω4q−1sin2qπ+3qa3ω3q−1sin3qπ2+2qa2ω2q−1sinqπ+qa1ωq−1sinqπ2+3qd3ω3q−1sin3qπ2+2qd2ω2q−1sinqπ+qd1ωq−1sinqπ2,W32=5qω5q−1cos5qπ2+4qa4ω4q−1cos2qπ+3qa3ω3q−1cos3qπ2+2qa2ω2q−1cosqπ+qa1ωq−1cosqπ2−3qd3ω3q−1cos3qπ2+2qd2ω2q−1cosqπ+qd1ωq−1cosqπ2,W33=−4qJ4ω4q−1cos2qπ+3qJ3ω3q−1cos3qπ2+2qJ2ω2q−1cosqπ+qJ1ωq−1cosqπ2,W34=−5qω5q−1cos5qπ2+4qa4ω4q−1cos2qπ+3qa3ω3q−1cos3qπ2+2qa2ω2q−1cosqπ+qa1ωq−1cosqπ2+3qd3ω3q−1cos3qπ2+2qd2ω2q−1cosqπ+qd1ωq−1cosqπ2,W35=5qω5q−1sin5qπ2+4qa4ω4q−1sin2qπ+3qa3ω3q−1sin3qπ2+2qa2ω2q−1sinqπ+qa1ωq−1sinqπ2−3qd3ω3q−1sin3qπ2+2qd2ω2q−1sinqπ+qd1ωq−1sinqπ2,(49)W36=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2−d3ω3qsin3qπ2+d2ω2qsinqπ+d1ωqsinqπ2,W37=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,W38=−ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,W39=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2+d3ω3qsin3qπ2+d2ω2qsinqπ+d1ωqsinqπ2,Redλdτ∗−1=X31X33+X32X34X332+X342,where(50)X31=W30+W31cosωτ∗+W32sinωτ∗,X32=W33+W34cosωτ∗+W35sinωτ∗,X33=W36cosωτ∗+W37sinωτ∗,X34=W38cosωτ∗−W39sinωτ∗.
Noticing thatsignRedλ/dτ∗|τ∗=τ∗0=signRdλ/dτ∗−1|τ∗=τ∗0, we can obtain Redλ/dτ∗|τ∗=τ∗0≠0.Theorem 3.
Suppose thatR0>1, (16)–(20), (H3), and a0+d02a0−d02−J02a0−d02<0 hold; then, the following results hold:(i)
Whenτ∗∈0,τ∗0, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable.(ii)
Whenτ∗=τ∗0, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ∗=τ∗0.


## 3.1. Case 1:τ1=τ2=0

Whenτ1=τ2=0, equation (10) becomes(14)λ5+B4λ4+B3λ3+B2λ2+B1λ+B0=0,where(15)B0=a0+b0+c0+d0,B1=a1+b1+c1+d1,B2=a2+b2+c2+d2,B3=a3+b3+c3+d3,B4=a4+b4+c4+d4.By the Routh–Hurwitz criterion, a set of necessary and sufficient conditions for all roots of equation (14) to have a negative real part is given in the following form:(16)D1=B4>0,(17)D2=B4B21B3>0,(18)D3=B4B2B010B3B4B1B2>0,(19)D4=B4B21B3B10B200B401B2B0B3B1>0,(20)D5=B410B2B100B3B200B4B2B00001B3B100B4B2B0>0.Lemma 1.
Ifτ1=τ2=0,R0>1, and (16)-(20) are satisfied, the endemic equilibrium of system (4) is asymptotically stable.


## 3.2. Case 2:τ1>0,τ2=0

Whenτ1>0andτ2=0, then we can obtain the following equation:(21)λ5q+C4λ4q+C3λ3q+C2λ2q+C1λq+C0+E4λ4q+E3λ3q+E2λ2q+E1λq+E0e−λτ1=0,where C4=a4+c4, C3=a3+c3, C2=a2+c2, C1=a1+c1, C0=a0+c0, E4=b4, E3=b3+d3, E2=b2+d2, E1=b1+d1, and E0=b0+d0.Letλ=i˙ω; we substitute this into (21), and separate the real part from imaginary part:(22)M11cosωτ1+M12sinωτ1=M13,M12cosωτ1−M11sinωτ1=M14,where(23)M11=E4ω4qcos2qπ+E3ω3qcos3qπ2+E2ω2qcosqπ+E1ωqcosqπ2+E0,M12=E4ω4qsin2qπ+E3ω3qsin3qπ2+E2ω2qsinqπ+E1ωqsinqπ2,M13=ω5qcos5qπ2+C4ω4qcos2qπ+C3ω3qcos3qπ2+C2ω2qcosqπ+C1ωqcosqπ2+C0,M14=ω5qsin5qπ2+C4ω4qsin2qπ+C3ω3qsin3qπ2+C2ω2qsinqπ+C1ωqsinqπ2.From (22), it is easy to obtain(24)cosωτ1=M13M11+M14M12M112+M122,(25)M112+M122=M132+M142.By (25), we have(26)ω10q+f9ω9q+f8ω8q+f7ω7q+f6ω6q+f5ω5q+f4ω4q+f3ω3q+f2ω2q+f1ωq+f0=0,where(27)f0=C02−E02,f1=2C0C1cosqπ2−2E0E1cosqπ2,f2=C12−E12+2C0C1cosqπ−2E0E2cosqπ,f3=2C0C3cos3qπ2−2E0E3cos3qπ2+2C1C2sinqπsinqπ2−2E1E2sinqπsinqπ2+2C1C2cosqπcosqπ2−2E1E2cosqπcosqπ2,f4=C22−E22+2C0C4cos2qπ−2E0E4cos2qπ+2C1C3sinqπ2sin3qπ2−2E1E3sinqπ2sin3qπ2+2C1C3cosqπ2cos3qπ2−2E1E3cosqπ2cos3qπ2,(28)f5=2C0cos5qπ2+2C1C4sin2qπsinqπ2+2C2C3sinqπsin3qπ2−2E1E4sin2qπsinqπ2−2E2E3sinqπsin3qπ2+2C1C4cos2qπcosqπ2+2C2C3cosqπcos3qπ2−2E1E4cos2qπcosqπ2−2E2E3cosqπsin3qπ2,f6=C32−E32+2C1cosqπ2cos5qπ2+2C1sinqπ2sin5qπ2+2C2C4sinqπsin2qπ−2E2E4sinqπsin2qπ+2C2C4cosqπcos2qπ−2E2E4cosqπcos2qπ,f7=2C2cosqπcos5qπ2+2C2sinqπsin5qπ2+2C3C4sin2qπsin3qπ2−2E3E4sin2qπsin3qπ2+2C3C4cos2qπcos3qπ2−2E3E4cos2qπcos3qπ2,f8=C42−E42−2C3cos3qπ2cos5qπ2+2C3sin3qπ2sin5qπ2,f9=2C4cos2qπcos5qπ2+2C4sin2qπsin5qπ2.Leth1ω=ω10q+f9ω9q+f8ω8q+f7ω7q+f6ω6q+f5ω5q+f4ω4q+f3ω3q+f2ω2q+f1ωq+f0; obviously, limω⟶+∞h1ω=+∞. When f0=C02−E02<0, namely, a0−C02−b0−d02<0, then (26) has at least one positive real root.Lemma 2.
Whena0−C02−b0−d02<0, (26) has at least one positive real root.
By (24), we have(29)τ10=1ω10arccosM13M11+M14M12M112+M122,where ±iω10 is a pair of purely imaginary roots of (21) with τ10. Also,(30)H1Redλdτ1−1|τ1=τ10≠0.
Taking the derivative ofλ with respect to τ1 in (21), we can obtain(31)dλdτ1=λV13V11+V12−τ1V13,where(32)V11=5qλ5q−1+4qC4λ4q−1+3qC3λ3q−1+2qC2λ2q−1+qC1λq−1,V12=4qE4λ4q−1+3qE3λ3q−1+2qE2λ2q−1+qE1λq−1e−λτ1,V13=E4λ4q+E3λ3q+E2λ2q+E1λq+E0−λτ1,dλdτ1−1=V11+V12λV13=−τ1λ,dλdτ1−1=W11+W12cosωτ1+W13sinωτ1+i−W13cosωτ1−W11sinωτ1+W14W15cosωτ1+W16sinωτ1+i−W16cosωτ1−W15sinωτ1−τiW,where(33)W11=5qω5q−1sin5qπ2+4qC4ω5q−1sin2qπ+3qC3ω3q−1sin3qπ2+2qC2ω2q−1sinqπ+qC1ωq−1sinqπ2,W12=4qE4ω4q−1sin2qπ+3qE3ω3q−1sin3qπ2+2qE2ω2q−1sinqπ+qE1ωq−1sinqπ2,W13=4qE4ω4q−1cos2qπ+2qE3ω3q−1cos3qπ2+2qE2ω2q−1cosqπ+qE1ωq−1cosqπ2,W14=−5qω5q−1cos5qπ2+4qC4ω4q−1cos2qπ+3qC3ω3q−1cos3qπ2+2qC2ω2q−1cosqπ+qC1ωqcosqπ2,W15=−ωE4ω4qsin2qπ+E3ω3qsin3qπ2+E2ω2qsinqπ+E1ωqsinqπ2,W16=ωE4ω4qcos2qπ+E3ω3qcos3qπ2+E2ω2qcosqπ+E1ωqcosqπ2+E0,Redλdτ1−1=X11X13+X12X14X132+X142,where(34)X11=W11+W12cosωτ1+W13sinωτ1,X12=−W13cosωτ1−W11sinωτ1+W14,X13=W15cosωτ1+W16sinωτ1,X14=−W16cosωτ1−W15sinωτ1.
Noticing thatsignRedλ/dτ1|τ1=τ10=signRedλ/dτ1−1|τ1=τ10, we can obtain Redλ/dτ1|τ1=τ10≠0.Theorem 1.
Suppose thatR0>1, (16)–(20), (H1), and a0−C02−b0−d02<0 hold; then, the following results hold:(i)
Whenτ1∈0,τ10, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable(ii)
Whenτ1=τ10, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ1=τ10.


## 3.3. Case 3:τ1=0,τ1<0

Whenτ1=0,τ1<0, equation (10) becomes(35)λ5q+F4λ4q+F3λ3q+F2λ2q+F1λq+F0+G4λ4q+G3λ3q+G2λ2q+G1λq+G0e−λτ1=0.where F0=a0+b0, F1=a1+b1, F2=a2+b2, F3=a3+b3, F4=a4+b4, G0=c0+d0, G1=c1+d1, G2=c2+d2, G3=c3+d3, and G4=c4.Since (19) and (25) are the same equation, we can draw the conclusion directly:(36)H2Redλdτ2−1|τ2=τ20≠0.Theorem 2.
Suppose thatR0>1, (16)–(20), (H2), and a0+b02−c0+d02<0 hold; then, the following results hold:(i)
Whenτ2∈0,τ20, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable.(ii)
Whenτ2=τ20, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ2=τ20.


## 3.4. Case 4:τ1=τ2=τ∗>0

Whenτ1=τ2=τ∗>0, we can easily obtain the following equation:(37)λ5q+a4λ4q+a3λ3q+a2λ2q+a1λq+a0eλτ∗+J4λ4q+J3λ3q+J2λ2q+J1λq+J0+d3λ3q+d2λ2q+d1λq+d0e−λτ∗=0,where J0=b0+c0, J1=b1+c1, J2=b2+c2, J3=b3+c3, and J4=b4+c4.Supposingλ=i˙ω, next we can obtain equation (38) by separating the real part from imaginary part:(38)M31cosωτ∗+M32sinωτ∗=M33,M34cosωτ∗−M35sinωτ∗=M36,where(39)M31=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,M32=−ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2,M33=J4ω4qcos2qπ+J3ω3qcos3qπ2+J2ω2qcosqπ+J1ωqcosqπ2+J0,M34=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2+d3ω3qsin3qπ2+da2ω2qsinqπ+d1ωqsinqπ2,M35=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0−d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,M36=J4ω4qsin2qπ+J3ω3qsin3qπ2+J2ω2qsinqπ+J1ωqsinqπ2.Suppose that (38) has roots, the expression of sinωτ and cosωτ can be obtained as follows:(40)cosωτ∗=M33M35−M32M36M31M35−M32M34,(41)sinωτ∗=M31M36−M33M34M31M35−M32M34.Sincesin2ωτ∗+cos2ωτ∗=1, we have(42)ω20q+k19ω19q+⋯+k1ωq+k0=0,where(43)k0=a0+d02a0−d02−J02a0−d02,k1=2a0+d0a0−d0a0+d0a1−d1cosqπ2+a1+d1a0−d0cosqπ2−2J0a0−d0J1a0+d0cosqπ2+J0a1−d1cosqπ2,k19=2a4cos2qπcos5qπ2−a4−d4sin2qπsin5qπ2.Leth3ω=ω20q+k19ω19q+⋯+k1ωq+k0; obviously, limω⟶+∞h3ω=+∞. When k0=a0+d02a0−d02−J02a0−d02<0, then (42) has at least one positive real root.Lemma 3.
Whena0+d02a0−d02−J02a0−d02<0, (42) has at least one positive real root.
By (40), we have(44)τ∗=1ω30arccosM33M35−M32M36M31M35−M32M34,where ±iω30 is a pair of purely imaginary roots of (37) with τ∗0.(45)H3Redλdτ∗−1|τ∗=τ∗0≠0.
The derivative ofλ with respect to τ∗ in (37) is easily obtained:(46)dλdτ∗=λV32+V35V31+V33+V34−τ∗V13,where(47)V31=5qλ5q−1+4qa4λ4q−1+3qa3λ3q−1+2qa2λ2q−1+qa1λq−1eλτ∗,V32=λ5q+a4λ4q+a3λ3q+a2λ2q+a1λq+a0−eλτ∗,V33=4qJ4λ4q−1+3qJ3λ3q−1+2qJ2λ2q−1+qJ1λq−1,V34=3qd3λ3q−1+2qd2λ2q−1+qd1λq−1e−λτ∗,V35=d3λ3q+d2λ2q+d1λq+d0e−λτ∗,dλdτ∗−1=V31+V33+V34λV32+V35=−τ∗λ,dλdτ∗−1=W30+W31cosωτ∗+W32sinωτ∗+iW33cosωτ∗+W34sinωτ∗+W35W36cosωτ∗+W37sinωτ∗+i−W38cosωτ∗−W39sinωτ∗−τ∗iW,where(48)W30=4qJ4ω4q−1sin2qπ+3qJ3ω3q−1sin3qπ2+2qJ2ω2q−1sinqπ+qJ1ωq−1sinqπ2,W31=5qω5q−1sin5qπ2+4qa4ω4q−1sin2qπ+3qa3ω3q−1sin3qπ2+2qa2ω2q−1sinqπ+qa1ωq−1sinqπ2+3qd3ω3q−1sin3qπ2+2qd2ω2q−1sinqπ+qd1ωq−1sinqπ2,W32=5qω5q−1cos5qπ2+4qa4ω4q−1cos2qπ+3qa3ω3q−1cos3qπ2+2qa2ω2q−1cosqπ+qa1ωq−1cosqπ2−3qd3ω3q−1cos3qπ2+2qd2ω2q−1cosqπ+qd1ωq−1cosqπ2,W33=−4qJ4ω4q−1cos2qπ+3qJ3ω3q−1cos3qπ2+2qJ2ω2q−1cosqπ+qJ1ωq−1cosqπ2,W34=−5qω5q−1cos5qπ2+4qa4ω4q−1cos2qπ+3qa3ω3q−1cos3qπ2+2qa2ω2q−1cosqπ+qa1ωq−1cosqπ2+3qd3ω3q−1cos3qπ2+2qd2ω2q−1cosqπ+qd1ωq−1cosqπ2,W35=5qω5q−1sin5qπ2+4qa4ω4q−1sin2qπ+3qa3ω3q−1sin3qπ2+2qa2ω2q−1sinqπ+qa1ωq−1sinqπ2−3qd3ω3q−1sin3qπ2+2qd2ω2q−1sinqπ+qd1ωq−1sinqπ2,(49)W36=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2−d3ω3qsin3qπ2+d2ω2qsinqπ+d1ωqsinqπ2,W37=ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,W38=−ω5qcos5qπ2+a4ω4qcos2qπ+a3ω3qcos3qπ2+a2ω2qcosqπ+a1ωqcosqπ2+a0+d3ω3qcos3qπ2+d2ω2qcosqπ+d1ωqcosqπ2+d0,W39=ω5qsin5qπ2+a4ω4qsin2qπ+a3ω3qsin3qπ2+a2ω2qsinqπ+a1ωqsinqπ2+d3ω3qsin3qπ2+d2ω2qsinqπ+d1ωqsinqπ2,Redλdτ∗−1=X31X33+X32X34X332+X342,where(50)X31=W30+W31cosωτ∗+W32sinωτ∗,X32=W33+W34cosωτ∗+W35sinωτ∗,X33=W36cosωτ∗+W37sinωτ∗,X34=W38cosωτ∗−W39sinωτ∗.
Noticing thatsignRedλ/dτ∗|τ∗=τ∗0=signRdλ/dτ∗−1|τ∗=τ∗0, we can obtain Redλ/dτ∗|τ∗=τ∗0≠0.Theorem 3.
Suppose thatR0>1, (16)–(20), (H3), and a0+d02a0−d02−J02a0−d02<0 hold; then, the following results hold:(i)
Whenτ∗∈0,τ∗0, the endemic equilibrium S∗,E∗,I∗,K∗,R∗ is asymptotically stable.(ii)
Whenτ∗=τ∗0, system (4) experiences a Hopf bifurcation at the endemic equilibrium. In other words, system (4) has a bifurcation of periodic solution bifurcating near τ∗=τ∗0.


## 4. Numerical Simulation and Discussion

In this part, for confirming our theoretical analysis, we give some numerical results of system (4). Based on time delay fractional-order estimation method [26], we obtain the simulation results with step length h = 0.05.In this case, the following case of system (4) with the parameters were considered: p=0.5, p=0.5, b=1, β1=0.25, β2=0.3, ϕ=0.1, μ=0.05, σ=0.26, α=0.18, χ=0.15, δ=0.35, γ=0.25, ε=0.15, r=0.5, q=0.95, and initial values (0.5, 4, 0.1, 30, 5) and then we can get a system as follows:(51)D0.95St=0.5−0.25StEt−0.3StIt−0.1StKt−0.05St+0.26Rt−τ1,D0.95Et=0.5+0.25StEt+0.3StIt−0.18Et−τ2−0.2Et,D0.95It=0.18Et−τ2−0.35ItKt−0.45It,D0.95Kt=0.35ItKt+0.25It+0.15Et−0.05Kt,D0.95Rt=0.1StKt+0.15It−0.05Rt−0.26Rt−τ1,which has a unique equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823).After calculation, we can obtainω10=0.0779 and τ10=16.1132. Figure 1 manifests that system (4) is locally asymptotically stable when τ1∈0,τ10. Figure 2 shows that a Hopf bifurcation exists when τ1∈τ10,+∞. Table 1 indicates the relation of fractional order q to the critical frequency ω10 and bifurcation point τ10. From Table 1, with decrease in fractional order q, the critical frequency ω10 is decreasing and bifurcation point τ10 is increasing.Figure 1
State trajectories of fractional system (4). When τ1=15<τ10, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is asymptotically stable.Figure 2
Phase portrait of fractional system (4). When τ1=17>τ10, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is not asymptotically stable.Table 1
Relation of parameters ofq,ω10,andτ10 of model (4).
Fractional orderqCritical frequencyω10Bifurcation pointτ1010.087412.94170.950.077916.11320.900.071118.42410.850.064122.01370.800.057026.95460.750.049931.34560.700.042734.5276In the same way, we getω20=0.0322 and τ20=29.4167. Figure 3 indicates that system (4) is locally asymptotically stable when τ2∈0,τ20. From Figure 4, system (4) becomes unstable when τ2∈τ20,+∞. We can see the relation of fractional order q to the critical frequency ω20 and bifurcation point τ20 in Table 2. With decrease in fractional order q, the critical frequency ω20 is decreasing and bifurcation point τ20 is increasing, as shown in Table 2.Figure 3
State trajectories of fractional system (4). When τ2=26<τ20, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is asymptotically stable.Figure 4
Phase portrait of fractional system (4). When τ2=32>τ20, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is not asymptotically stable.Table 2
Relation of parameters ofq,ω20,andτ20 of model (4).
Fractional orderqCritical frequencyω20Bifurcation pointτ2010.036118.32480.950.032229.41670.900.028635.56680.850.025345.78150.800.022452.06680.750.020161.38410.700.019076.4499To verify the validity of Theorem3, we take ω∗0=0.0567 and τ∗0=14.4547. From Figures 5 and 6, we can find that system (4) is locally asymptotically stable when τ∗∈0,τ∗0; but once τ∗>τ∗0, system (4) becomes unstable. We can find the relation of fractional order q to the critical frequency ω∗0 and bifurcation point τ∗0 in Table 3. With decrease in fractional order q, the critical frequency ω∗0 is decreasing and bifurcation point τ∗0 is increasing, as shown in Table 3.Figure 5
State trajectories of fractional system (4). When τ∗=14<τ∗0, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is asymptotically stable.Figure 6
Phase portrait of fractional system (4). When τ∗=16>τ∗0, the equilibrium (0.5941, 2.2161, 0.0730, 14.3346, 2.7823) is not asymptotically stable.Table 3
Relation of parameters ofq,ω∗0,andτ∗0 of model (4).
Fractional orderqCritical frequencyω∗0Bifurcation pointτ∗010.072611.41610.950.056714.45470.900.042718.41840.850.030822.46850.80.021426.00110.750.014232.62670.70.009037.2772From Figures1–6, we can see the correctness of the theoretical results. All the tables show that the bifurcation point increases gradually with the decrease in fractional order q, among which the influence on τ2 is the most obvious. In other words, τ2 is more inclusive than τ1 at the same fractional order q. Therefore, when controlling the spread of computer viruses, we should find ways to reduce the temporary immunity time delay in converting recovered computers into susceptible ones.


## 5. Conclusions

We consider a fractional-order model of Susceptible-Exposed-Infected-Kill Signals Recovery (SEIR-KS) computer virus propagation with two delays, based on the model in [25]. The linear method and Laplace transform were used to transform the model. We analyze the coefficient matrix characteristic equation and the system characteristic equation of the linearized system, and then, we get the relationship between the equilibrium stability of the time delay and the fractional-order system; the calculation formula of the critical value of the time delay was derived when Hopf bifurcation appeared. To verify the rationality of the theoretical analysis, the appropriate system parameters are selected for numerical simulation. The simulation results not only verify the correctness of the conclusion but also reveal the influence trend of fractional order on the critical value of time delay. We can find that, with the decrease in fractional order q, the bifurcation point increases gradually. The bifurcation point of case 3 is larger than that of case 2 at the same fractional order q, which means that we should aim to reduce the temporary immunity time delay for converting recovered computers into susceptible ones. Therefore, the application of this paper is helpful for the computer virus emergency center to make immediate decision and response plan when computing virus outbreak. The research results of this paper extend the research scope and method of the computer virus model to some extent.



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