dynamical analysis and optimal control for a malware propagation model in an information network

Dynamical analysis and optimal control for a malware propagationmodel in an information network

Linhe Zhu, Hongyong Zhao n

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

a r t i c l e i n f o

Article history:Received 30 June 2014Accepted 29 August 2014Communicated by H. Jiang

Keywords:Information networksStabilityHopf bifurcationMalware propagationOptimal control strategy

a b s t r a c t

With the rapid development of network information technology, information networks security hasbecome a very critical issue in our work and daily life. This paper investigates a nonlinear malwarepropagation model in wireless sensor networks (WSNs) based on SIR epidemic model. Sufficientconditions for the local stability of the positive equilibrium point and the existence of Hopf bifurcationare obtained by analyzing the associated characteristic equation. Moreover, formulas for determining theproperties of the bifurcating periodic oscillations are derived by applying the normal form method andcenter manifold theorem. Furthermore, with the help of the Maximum Principle of Pontryagin, wedesign an optimal control strategy for the previous model to extend the region of stability and reducethe density of infected nodes in WSNs. Finally, we conduct extensive simulations to evaluate theproposed model. Numerical evidence shows that the dynamic characteristics of malware propagation inWSNs are closely related to the immune period of a recovered node and the rate constant for nodesbecoming susceptible again after recovered. Besides, we obtain that the optimal control strategyeffectively improves the performance of the networks.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

With the rapid development of the information and communica-tion technology, information networks have become more prevalentin our work and daily life. Over the years, these information networkshave been successfully used in hardware design, communicationprotocols, resource efficiency, home security, battlefield surveillanceand other aspects [1–3]. Wireless sensor network, as a novel informa-tion and communication network, has gained worldwide attention inrecent years [4–6]. In general, a WSN is composed of hundreds or eventhousands of small, low cost, low power sensor nodes, which hasfacilitated the development of smart sensors. However, as WSNs areunfolding their vast potential in a plethora of application environment,information security has become one of the most critical challengesyet to be fully addressed. Because sensor nodes are resource con-strained, they generally have weak defense capabilities and areattractive targets for software attacks (like malware attacks on theinformation networks). As malware being injected into some nodes inWSNs, the networks will fall out of stability and at the same time theoscillation by large amplitude occurring through Hopf bifurcation mayappear, which possibly leads to that the utilization of the network

decreases and the network performance declines. The oscillatoryphenomena have been observed in many other similar networks,with rhythms originating from isolated components or emerging as aproperty of a network as a whole [7–9].

To defend against the malware propagation, we need to accu-rately understand the dynamic characteristics of malware propaga-tion. In recent years, some analytical models have been brought intomalware propagation system so that large strides have been takenin the research in WSNs [10–16]. The simulations and matchingwith practical data show that most of these models can not onlydescribe the process of information and disease diffusion in humansociety, but also capture the process of malware propagation incomputer networks such as the Internet and WSNs. In [13], theauthor proposed a percolation theory based evaluation of thespread of an epidemic on graphs with given degree distributions.However, [13] had payed little attention to the temporal dynamicsof epidemic spread and only studied the final outcome of aninfection spread. The authors in [14] proposed a spatial-temporalmodel for characterizing malware propagation in networks basedon probabilistic graphs and spatial-temporal random processes. Thebasic idea was to abstract malware propagation into a probabilisticgraph, and described the statistical dependence of malware propa-gation in arbitrary topologies using a spatial-temporal randomprocess. Based on the ordinary differential equation and the SIRmodel, Wang in [15,16] derived the threshold for a piece of malware

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http://dx.doi.org/10.1016/j.neucom.2014.08.0600925-2312/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: [email protected] (H. Zhao).

Please cite this article as: L. Zhu, H. Zhao, Dynamical analysis and optimal control for a malware propagation model in aninformation network, Neurocomputing (2014), http://dx.doi.org/10.1016/j.neucom.2014.08.060i

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to propagate in WSNs, where all the nodes were supposed to bestationary. Furthermore, the author gave the sufficient conditions ofstability and Hopf bifurcation of the malware propagation model.

As is well known, the periodic oscillation occurring throughHopf bifurcation in WSNs may destroy, block regular communica-tions, or even damage the integrity of regular data packets. Thus, itis necessary to propose a control strategy to ensure that the systemis stable and reliable operation. On the other hand, in order toreduce infected nodes in WSNs at minimal cost, in this paper weconsider optimal control strategies associated with eliminationpolicy and defense policy including execution costs based on theprevious model. In fact, the optimal control theory [17], which wasdeveloped by Pontryagin and his co-workers in the late 1950s, hasbeen applied to many areas including economics, management,engineering and biology [18–21]. In [22], Gumel and Moghadasproposed a model for the dynamics of an infectious disease in thepresence of a preventive vaccine considering non-linear incidencerate and found the optimal vaccine coverage threshold needed fordisease control and eradication. Swan [23] applied control theoryto obtain maximal benefits interims of social benefits from theparsimonious use of insufficient public funds in the control ofepidemics. With the help of the Maximum Principle of Pontryagin[24,25] and an iterative method we shall develop some new modelwith optimal control strategy. The goal of this work is not toconsider a process of malware propagation but to present a methodof how to treat this class of optimization problems. Our maincontributions are summarized as follows.

(1) Through the analysis of the mechanism of malware propaga-tion in WSNs, we generally quantify the process of malwarepropagation in WSNs based on the SIR model in the epidemictheory, and then we develop a delayed malware propagationmodel with logistic growth process. At the same time, weconduct extensive simulations on large-scale WSNs to evaluatethe proposed model. Numerical evidence shows that thedynamic characteristics of malware propagation in WSNs areclosely related to the immune period of a recovered node andthe rate constant for nodes becoming susceptible again afterrecovered.

(2) Through the stability and Hopf bifurcation analysis of thepositive equilibrium point for the proposed model, we obtainthe sufficient conditions whether a series of oscillation phe-nomenon occurs through a Hopf bifurcation in WSNs. Whenthe system occurs periodic oscillation, based on the normalform method and center manifold theorem we analysis theproperty of periodic oscillation.

(3) Based on our proposed model, with time increasing thedistribution of infected nodes can be effectively predicted inadvance. On this basis, the optimal control strategy weproposed can delay Hopf bifurcation and extend the stabilityregion. Furthermore, we derive optimal state solutions asso-ciated with the optimal control variable unðtÞ for the optimalcontrol problem by means of the Pontryagin's MaximumPrinciple. Numerical results exhibit that the optimal controlstrategy ensures the security of WSNs and the regular com-munications between nodes. This will provide new insights onwhen and where countermeasures should be employed forpreventing, controlling and removing malware propagationin WSNs.

The structure of this paper is arranged as follows. In Section 2,we consider the mechanism of malware propagation and the modelformulation problem. In Section 3, we study the local stability andthe existence of Hopf bifurcation. In Section 4, we give formuladetermining the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions. In Section 5, we propose an

optimal control strategy for malware propagation in WSNs. Finally,to support our theoretical predictions, some numerical simulationsare given which support the analysis in Sections 3–5.

2. Model formulation

A WSN consists of many static and identical wireless sensors.Each wireless sensor is called a node. At any time, a node isclassified as either internal or external accordingly as it is con-nected to the networks or not at that time. Generally, the nodescan be divided into three classes depending on their states:susceptible (healthy), infected and recovered (immunized). In thispaper, we use S(t), I(t), R(t) to denote the densities of susceptiblenodes, infected nodes and recovered nodes at time t, respectively.

Based on the classical SIR epidemic model [26–28], we considerthe following four facts:

(i) The susceptible nodes are assumed to have the logistic growthwith carrying capacity K ðK40Þ as well as intrinsic increaserate constant r ðr40Þ, and the incidence term is of bilinearmass action.

(ii) Users may immunize their nodes with countermeasures instate I.

(iii) Some recovered nodes go through a temporary immunitywith probability δ.

(iv) As the energy of nodes is exhausted, more and more nodesbecome dead nodes. Any malware residing in other nodescannot infect these dead nodes. Moreover, when a node dies,it becomes a dead node. All of the malware which everresided in the dead nodes immediately disappear from thedead nodes. This means that the dead nodes no longerparticipate in the process of malware propagation in WSNs.

Our assumption on the dynamical transfer of the nodes isdepicted in Fig. 1. As a result, the SIRS model can be formulated bythe following delayed differential equations:

dSdt

¼ rS 1� SK

� ��βSI�ηSþδRðt�τÞ;dIdt

¼ βSI�εI�ηI;

dRdt

¼ εI�ηR�δRðt�τÞ;

8>>>>>>><>>>>>>>:

ð2:1Þ

with initial conditions

SðtÞ ¼ S0Z0; tA ½�τ;0�;IðtÞ ¼ I0Z0; tA ½�τ;0�;RðtÞ ¼ R0Z0; tA ½�τ;0�;

8><>: ð2:2Þ

Fig. 1. Node state transition relationship, where N and D represent new nodes anddeath nodes, respectively.

L. Zhu, H. Zhao / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





where K, r, β, η, δ, ε are positive constants and τ is non-negativeconstant. The constant K is the carrying capacity the networks, r isthe intrinsic increase rate, β is the constant contact rate between Sand I, η is the death rate of nodes, δ is the rate constant for nodesbecoming susceptible again after recovered, ε is the rate constantfor nodes leaving the infective class I for recovered class R, τ is theimmune period of a recovered node [29]. S0, I0 and R0 are theinitial density of susceptible nodes, infected nodes and recoverednodes, respectively.

3. Local stability and Hopf bifurcation

In this section, considering the delay τ as the bifurcationparameter, we will discuss the local stability and Hopf bifurcationof system (2.1) by analyzing the corresponding characteristicequation.

It can be seen that the zero equilibrium point always being themalware-free equilibrium point for any feasible parameters,E1 ¼ ððKðr�ηÞ=rÞ;0;0ÞT always being the boundary equilibriumpoint under the condition ðH1ÞðH1Þ r�η40;

and system (2.1) having a unique positive equilibrium pointEn ¼ ðSn; In;RnÞT provided that the condition

ðH2Þ rðKβ�ε�ηÞ�Kβη40

holds, where

Sn ¼ εþηβ

; In ¼ ðηþδÞðηþεÞ½rðKβ�ε�ηÞ�Kβη�Kηβ2ðεþηþδÞ

;

Rn ¼ εðηþεÞ½rðKβ�ε�ηÞ�Kβη�Kηβ2ðεþηþδÞ

:

Under the condition ðH2Þ, let ~S ¼ S�Sn, ~I ¼ I� In, ~R ¼ R�Rn, anddrop bars for the simplicity of notations. Then system (2.1) can betransformed into the following form:

dSdt

¼ r�2rSn

K�βIn�η

� �S�βSnIþδRðt�τÞ�βSI� r

KS2;

dIdt

¼ βInSþðβSn�η�εÞIþβSI;

dRdt

¼ εI�ηR�δRðt�τÞ:

8>>>>>>><>>>>>>>:

ð3:1Þ

Thus, the positive equilibrium point En ¼ ðSn; In;RnÞT of system (2.1)is transformed into the zero equilibrium point E0 ¼ ð0;0;0ÞT ofsystem (3.1). In the following, we will analysis stability andbifurcation of the zero equilibrium point E0 of system (3.1).

The linearizing system of Eq. (3.1) at E0 is

dSdt

¼ r�2rSn

K�βIn�η

� �S�βSnIþδRðt�τÞ;

dIdt

¼ βInSþðβSn�η�εÞI;dRdt

¼ εI�ηR�δRðt�τÞ:

8>>>>>>><>>>>>>>:

ð3:2Þ

Then the corresponding characteristic equation of system (3.2) is��λþ2rSn

K þβInþη�r βSn �δe�λτ

�βIn λþεþη�βSn 00 �ε λþηþδe�λτ

��¼ 0:

That is

λ3þ 2rSn

KþβInþ2η�r

� �λ2þ η

2rSn

KþβInþη�r

� �þβ2SnIn

� �λ

þβ2ηSnInþ λ2þ 2rSn

KþβInþη�r

� �λþðβ2SnIn�βInεÞ

� �δe�λτ ¼ 0:

ð3:3Þ

It is well know that the stability of the zero equilibrium point E0depends on the distribution of the characteristic root of (3.3). It isstable if all roots of (3.3) have negative real parts and unstable ifone root has positive real part. In the following, we will analysisthe distribution of characteristic roots of system (3.2).

Firstly, we make the following assumption:

ðH3Þ ðεþηÞ 2rSn

KþβInþ2η�r�ε

� �þεδ40:

Lemma 1. Assume that ðH2Þ and ðH3Þ hold, then the zero equili-brium point E0 of system (3.1) with τ¼ 0 is locally asymptoticallystable.

Proof. When τ¼ 0, Eq. (3.3) is equivalent to the following cubicequation:

λ3þ 2rSn

KþβInþ2ηþδ�r

� �λ2

þ ðηþδÞ 2rSn

KþβInþη�r

� �þβ2SnIn

� �λ

þβInðβSnεþβSnδ�εδÞ ¼ 0: ð3:4Þ

Clearly, according to ðH2Þ, we have

2rSn

KþβInþ2ηþδ�r4

2rSn

KþβInþη�r

¼ ðηþδÞðηþεÞ½rðKβ�η�εÞ�Kβη��ηðηþεþδÞ½rðKβ�2η�2ε�Kβη�Kηβ2ðηþεþδÞ

40

and

ðηþδÞ 2rSn

KþβInþη�r

� �þβ2SnIn40:

In addition, under the condition ðH3Þ, it is easy to show that

2rSn

KþβInþ2ηþδ�r

� �ðηþδÞ 2rSn

KþβInþη�r

� �þβ2SnIn

� �

�βInðβSnεþβSnδ�εδÞ40:

Furthermore, a simple calculation shows that

βInðβSnεþβSnδ�εδÞ ¼ βInðηεþηδþε2Þ40:

Therefore, by the Routh–Hurwitz criteria, all the roots ofEq. (3.4) have negative real parts. This completes the proof. □

Now we discuss the effect of the delay τ on the stability of E0.Assume that iω is a root of Eq. (3.3). Then ω should satisfy thefollowing equation:

� iω3� 2rSn

KþβInþ2η�r

� �ω2

þ η2rSn

KþβInþη�r

� �þβ2SnIn

� �iωþβ2ηSnIn

þ �ω2þ 2rSn

KþβInþη�r

� �iωþβ2SnIn�βInε

� �

�δð cos ωτ� i sin ωτÞ ¼ 0; ð3:5Þ

L. Zhu, H. Zhao / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





which implies that

δ ð�ω2�βInεþβ2SnInÞ cos ωτ þ 2rSn

KþβInþη�r

� �ω sin ωτ

� �

¼ 2rSn

KþβInþ2η�r

� �ω2�β2ηSnIn;

δ2rSn

KþβInþη�r

� �ω cos ωτ þðω2�β2SnInþβInεÞ sin ωτ

� �

¼ω3� η 2rSnK þβInþη�r

þβ2SnIn

h iω;

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð3:6Þtaking square on both sides of the equations of (3.6) and summingthen up, we obtain

ω6þða21þ2a3þη2�δ2Þω4þð2a3η2þa21η2

þa23�a21δ2�2a2δ

2Þω2þa23η2�a22δ

2 ¼ 0; ð3:7Þwhere

a1 ¼2rSn

KþβInþη�r; a2 ¼ βInε�β2SnIn; a3 ¼ �β2SnIn:

Set z¼ω2, Eq. (3.7) is transformed into the following equation:

z3þða21þ2a3þη2�δ2Þz2þð2a3η2þa21η2þa23�a21δ

2�2a2δ2Þz

þa23η2�a22δ

2 ¼ 0: ð3:8ÞDenode

hðzÞ ¼ z3þA1z2þA2zþA3; ð3:9Þwhere

A1 ¼ a21þ2a3þη2�δ2; A2 ¼ 2a3η2þa21η2þa23�a21δ

2�2a2δ2;

A3 ¼ a23η2�a22δ

2:

Furthermore, a simple calculation shows that

A3 ¼ βηInðδa2þηa3Þðδ�η�εÞ:Clearly, the sign of A3 is consistent with that of δ�η�ε.

Case 1. δ�η�ε40.

Lemma 2. If ðH4Þ δ�η�ε40 holds, then Eq. (3.8) has at least onepositive root.

Proof. According to ðH4Þ, it is easy to show that

hð0Þ ¼ A3 ¼ βηInðδa2þηa3Þðδ�η�εÞo0; limz-1

hðzÞ ¼1:

Thus, it is obvious that Eq. (3.8) has at least one positive root. Thiscompletes the proof. □

Without loss of generality, we assume that it has three positiveroots, defined by z1, z2, z3. Then we have

ω1 ¼ffiffiffiffiffiz1

p; ω2 ¼

ffiffiffiffiffiz2

p; ω3 ¼

ffiffiffiffiffiz3

p:

By (3.6), we obtain

cos ωkτk ¼a1ω2

k ðω2kþa3�a1ηÞ�ðω2

kþa2Þ½ða1þηÞω2kþa3η�

δ½a21ω2kþðω2

kþa2Þ2�;

thus, if we denote

τjk ¼1ωk

arccosa1ω2

k ðω2kþa3�a1ηÞ�ðω2

kþa2Þ½ða1þηÞω2kþa3η�

δ½a21ω2kþðω2

kþa2Þ2�þ2jπ

!;

ð3:10Þwhere k¼ 1;2;3; j¼ 0;1;2;…; then 7 iωk is a pair of purelyimaginary roots of (3.3) with τkj . Define

τ0 ¼ τ0k0 ¼minfτ0kg; ω0 ¼ωk0 : ð3:11Þ

Lemma 3. [37] Let λðτÞ ¼ αðτÞ7 iωðτÞ be the root of (3.3) near τ¼ τ0satisfying αðτ0Þ ¼ 0, ωðτ0Þ ¼ω0. Suppose that h0ðω2

0Þa0, where h(z)

is defined by (3.9). Then 7 iω0 is a pair of simple purely imaginaryroots of Eq. (3.3). Moreover, the following transversality conditionholds:

dðRe λðτÞÞdτ

��τ ¼ τ0 ; λ ¼ iω0

a0; ð3:12Þ

and the sign of

dðReλðτÞÞdτ

��τ ¼ τ0 ;λ ¼ iω0

is consistent with that of h0ðω20Þ.

Proof. Denote

HðλÞ ¼ λ3þ 2rSn

KþβInþ2η�r

� �λ2þ η

2rSn

KþβInþη�r

� �þβ2SnIn

� �λ

þβ2ηSnIn;

GðλÞ ¼ λ2þ 2rSn

KþβInþη�r

� �λþβ2SnIn�βInε

� �δ:

Then Eq. (3.3) can be written as

HðλÞþGðλÞe�λτ ¼ 0; ð3:13Þand Eq. (3.7) is equivalent to:

HðiωÞHðiωÞ�GðiωÞGðiωÞ ¼ 0: ð3:14Þ

Thus, together with (3.8) and (3.9), it is easy to show that

hðω2Þ ¼HðiωÞHðiωÞ�GðiωÞGðiωÞ: ð3:15ÞDifferentiating both sides of Eq. (3.15) with respect to ω, we obtain

2ωh0ðω2Þ ¼ i½H0ðiωÞHðiωÞ�H0ðiωÞHðiωÞ�G0ðiωÞGðiωÞþG

0ðiωÞGðiωÞ�:ð3:16Þ

If iω0 is not simple, then ω0 must satisfy

ddλ

½HðλÞþGðλÞe�λτ0 �∣λ ¼ iω0¼ 0:

That is

H0ðiω0ÞþG0ðIω0Þe� iω0τ0 �τ0Gðiω0Þe� iω0τ0 ¼ 0:

With Eq. (3.13), a simple calculation shows

τ0 ¼G0ðiω0ÞGðiω0Þ

�H0ðiω0ÞHðiω0Þ

:

Thus, by (3.14) and (3.16) we obtain

Im τ0 ¼ ImG0ðiω0ÞGðiω0Þ

�H0ðiω0ÞHðiω0Þ

�

¼ ImG0ðiω0ÞGðiω0ÞGðiω0ÞGðiω0Þ

�H0ðiω0ÞHðiω0ÞHðiω0ÞHðiω0Þ

( )

¼ ImG0ðiω0ÞGðiω0Þ�H0ðiω0ÞHðiω0Þ

Hðiω0ÞHðiω0Þ

( )

¼ i½G0ðiω0ÞGðiω0Þ�H0ðiω0ÞHðiω0Þ�G0ðiω0ÞGðiω0ÞþH

0ðiω0ÞHðiω0Þ�2Hðiω0ÞHðiω0Þ

¼ω0h0ðω2

0ÞjHðiω0Þj2

:

Since τ0 is real, i.e. Im τ0 ¼ 0, we have

h0ðω20Þ ¼ 0:

We get a contradiction to the condition h0ðω20Þa0. This proves the

first conclusion. Differentiating both sides of Eq. (3.13) withrespect to τ, we obtain

½H0ðλÞþG0ðλÞe�λτ�τGðλÞe�λτ �dλdτ

�λGðλÞe�λτ ¼ 0;






which implies

Re

24dðλðτÞÞ

dτ

35

�1��τ ¼ τ0 ; λ ¼ iω0

¼ Re

24H0ðλÞeλτþG0ðλÞ

λGðλÞ �τλ

��τ ¼ τ0 ; λ ¼ iω0

35

¼ Re

"½H0ðλÞeλτþG0ðλÞ�λGðλÞ

jλGðλÞj2 �τλ

��τ ¼ τ0 ; λ ¼ iω0

#

¼ Reiω0½H0ðiω0ÞF ðiω0Þ�G0ðiω0ÞGðiω0Þ�

jiω0Gðiω0Þj2� τiω0

" #

¼ iω0½H0ðiω0ÞHðiω0�H0ðiω0ÞHðiω0Þ�G0ðiω0ÞGðiω0ÞþG

0ðiω0ÞGðiω0Þ�2jiω0Gðiω0Þj2

¼ ω20h

0ðω20Þ

jiω0Gðiω0Þj2a0:

Clearly, the sign of

dðReλðτÞÞdτ

��τ ¼ τ0 ;λ ¼ iω0

is determined by that of h0ðω20Þ. This completes the proof. □

Applying Lemmas 1–3, it is easy to obtain the followingconclusion.

Theorem 1. Let τkj and τ0 be defined by (3.10) and (3.11), respec-tively. If ðH2Þ–ðH4Þ hold. Assume furthermore that the conditions ofLemma 3 are satisfied. Then for system (3.1), the following statementsare true:

(i) When τA ½0; τ0Þ, the zero equilibrium point E0 of system (3.1) islocally asymptotically stable.

(ii) The Hopf bifurcation occurs when τ¼ τjk. That is, system (3.1) hasa branch of periodic solutions bifurcating from the zero equili-brium point E0 near τ¼ τjk.

Case 2. δ�η�εr0.In this case, the discussion about the roots of Eq. (3.8) is similar

to that of Wei [30], we have the following lemma.

Lemma 4. If ðH5Þ δ�η�εr0, Δ¼ A21�3A240, zn ¼ ð�A1þffiffiffiffi

Δp

Þ=340, hðznÞr0 holds, then Eq. (3.8) has positive roots.

Furthermore, according to Descartes’ rule of signs [31], it is easyto show that Eq. (3.8) has two positive roots. Without loss ofgenerality, we assume that Z1 and Z2 are the two positive roots,and Z14Z2. Then we have

ω1 ¼ffiffiffiffiffiZ1

p4ω2 ¼

ffiffiffiffiffiZ2

p:

By (3.6), we obtain

cos ωlτl ¼a1ω2

l ðω2l þa3�a1ηÞ�ðω2

l þa2Þ½ða1þηÞω2l þa3η�

δ½a21ω2l þðω2

l þa2Þ2�ðl¼ 1;2Þ;

thus

τn;1 ¼1ω1

arccosa1ω2

1ðω21þa3�a1ηÞ�ðω2

1þa2Þ½ða1þηÞω21þa3η�

δ½a21ω21þðω2

1þa2Þ2�þ2nπ

!;

ð3:17Þ

τn;2 ¼1ω2

arccosa1ω2

1ðω22þa3�a1ηÞ�ðω2

2þa2Þ½ða1þηÞω22þa3η�

δ½a21ω22þðω2

2þa2Þ2�þ2nπ

!;

ð3:18Þwhere n¼ 0;1;2;…; then 7 iωl is a pair of purely imaginary rootsof (3.3) with τ¼ τn;l. Furthermore, similar to Lemma 3 we caneasily prove that 7 iωl is a pair of simple purely imaginary roots ofEq. (3.3).

Theorem 2. If ðH2Þ–ðH3Þ and ðH5Þ hold. Assume furthermore thath0ðZ1Þ40 and h0ðZ2Þo0. Then for system (3.1), there is a positiveinteger m, such that the zero equilibrium point E0 switches m timesfrom stability to instability to stability; that is, E0 is locally asympto-tically stable when

τA ½0; τ0;1Þ [ ðτ0;2; τ1;1Þ [ ⋯ [ ðτm�1;2; τm;1Þ;and unstable when

τA ðτ0;1; τ0;2Þ [ ðτ1;1; τ1;2Þ [ ⋯ [ ðτm�1;1; τm�1;2Þ and τ4τm;1:

System (3.1) undergoes a Hopf bifurcation at E0 when τ¼ τn;lðl¼ 1;2; n¼ 0;1;2…Þ, where τ¼ τn;l is defined by (3.17) and (3.18).

Proof. Similar to Lemma 3, it is easy to show that 7 iωl is a pair ofsimple purely imaginary roots of Eq. (3.3). Moreover, the followingtransversality conditions hold:

dðRe λðτÞÞdτ

��τ ¼ τn;1 ; λ ¼ iω1

40 anddðRe λðτÞÞ

dτ

��τ ¼ τn;2 ; λ ¼ iω2

o0:

Thus, for system (3.1), at each value τ¼ τn;1, a pair of roots crossesthe imaginary axis at iω1 into the right half-plane, and at eachτ¼ τn;2, a pair of roots crosses at iω2 back into the left-plane.According to the discussion above, we can see that the phenom-enon of stability switches occurs with τ increasing. Next we willprove that the phenomenon of stability switches can be found in afinite number.

In fact, we note immediately that since

τnþ1;1�τn;1 ¼2πω1

; τnþ1;2�τn;2 ¼2πω2

and ω14ω2, this alternation cannot persist for the whole of thesequences. Eventually, therefore, there is an integer m such that

τm�1;1oτm�1;2oτm;1oτmþ1;1oτm;2:

Hence for τ4τm;1, the multiplicity of roots in the right half-plane is at least two and the system is unstable. This provesTheorem 2. □

4. Properties of bifurcating periodic oscillations

In the previous section, we have shown that system (3.1)admits a series of periodic solutions bifurcating from the zeroequilibrium point E0 at the critical value τkj ðjAN0Þ. As we all know,the periodic oscillation in the wireless sensor networks maydestroy, block regular communications, or even damage theintegrity of regular data packets. Thus, understanding the proper-ties of the periodic oscillation, such as stability, direction andmonotonicity of the periodicity, is necessary. In this section, wederive explicit formulae to determine the properties of the Hopfbifurcation at critical value τkj ðjAN0Þ by using the normal formtheory and center manifold reduction for PFDEs developed by[32–34].

In this section, let UðtÞ ¼ ðu1ðtÞ;u2ðtÞ;u3ðtÞÞT ¼ ðSðtÞ; IðtÞ;RðtÞÞT . Forfixed jAN0, denote τkj by τn and introduce the new parameterμ¼ τ�τn. Normalizing the delay τ by the time-scaling t-t=τ. Then(3.1) can be written as a functional differential equation (FDE) inC¼ Cð½�1;0�;R3ÞdUðtÞdt

¼ LðτnÞðUtÞþFðUt ;μÞ; ð4:1Þ

where LðμÞðφÞ : C-R3 and Fð�;μÞ : C � R-R3 are given by

LðμÞðφÞ ¼ μ

ð�βIn�ηÞφ1ð0Þ�βSnφ2ð0Þþδφ3ð�1ÞβInφ1ð0ÞþðβSn�ε�k1Þφ2ð0Þεφ2ð0Þ�ηφ3ð0Þ�δφ3ð�1Þ

0B@

1CA; ð4:2Þ

Fðφ;μÞ ¼ LðμÞφþ f ðφ;μÞ ð4:3Þ






and

f ðφ;μÞ ¼ ðτnþμÞ�βφ1ð0Þφ2ð0Þ� r

Kφ21ð0Þ

βφ1ð0Þφ2ð0Þ0

0B@

1CA; ð4:4Þ

for φ¼ ðφ1;φ2;φ3ÞT AC:Then linearized system of (4.1) at E0 is

dUðtÞdt

¼ LðτnÞðUtÞ: ð4:5Þ

Obviously, LðτnÞ is a continuous linear function mappingCð½�1;0�;R3Þ into R3. By the Riesz representation theorem, thereexists a 3�3 matrix function ηðθ; τÞ (�1rθr0), whose elementsare of bounded variation such that

LðτnÞðφÞ ¼Z 0

�1½dηðθ; τnÞ�φðθÞ for φAC: ð4:6Þ

In fact, we can choose

ηðθ; τnÞ ¼ τnr�2rSn

K �βIn�η �βSn 0

βIn βSn�ε�η 00 ε �η

0BB@

1CCAνðθÞ

�τn0 0 δ0 0 00 0 �δ

0B@

1CAνðθþ1Þ; ð4:7Þ

where ν is the Dirac delta function.Let AðτnÞ denote the infinitesimal generator of the semigroup

induced by the solutions of (4.5) and An be the formal adjoint ofAðτnÞ under the bilinear pairing

ðψ ;ϕÞ ¼ ðψ ð0Þ;ϕð0ÞÞ�Z 0

�1

Z θ

ζ ¼ 0ψ ðζ�θÞ dηðθÞϕðζÞ dζ

¼ ðψ ð0Þ;ϕð0ÞÞþτnZ 0

�1ψ ðθþ1Þ

0 0 δ0 0 00 0 �δ

0B@

1CAϕðθÞ dθ; ð4:8Þ

for ϕ;ψAC. Then AðτnÞ and An are a pair of adjoint operators [32].From the discussion in Section 3, we know that AðτnÞ has a pair ofsimple purely imaginary eigenvalues 7 iω0τn and they are alsoeigenvalues of An since AðτnÞ and An are a pair of adjoint operators.Let P and Pn be the center spaces, that is, the generalizedeigenspaces of AðτnÞ and An associated with Λ0, respectively. ThenPn is the adjoint space of P and dim P ¼ dim Pn ¼ 2. Direct compu-tations give the following results.

Lemma 5. Let

ξ¼ βIn

iω0; ς¼ βεIn

iω0ðηþδe� iω0τn þ iω0Þ;

ξn ¼ δε�βSnðiω0þδþηÞiω0ðiω0þδþηÞ ; ςn ¼ δ

iω0þδþη:

8>>><>>>:

ð4:9Þ

Then

p1ðθÞ ¼ eiω0τnθð1;ξ; ςÞT ; p2ðθÞ ¼ p1ðθÞ; �1rθr0 ð4:10Þis a basis of P associated with Λ0 and

q1ðsÞ ¼ ð1; ξn; ςnÞe� iω0τns; q2ðsÞ ¼ q1ðsÞ; 0rsr1 ð4:11Þis a basis of Q associated with Λ0.

Let Φ¼ ðΦ1;Φ2Þ and Ψ n ¼ ðΨ n

1;Ψn

2ÞT with

Φ1ðθÞ ¼p1ðθÞþp2ðθÞ

2¼

Refeiω0τnθgRefξeiω0τnθgRefςeiω0τnθg

0B@

1CA

¼

cos ω0τnθβIn

ω0sin ω0τnθ

βInε cos ω0τnθðδ sin ω0τn�ω0ÞþβInε sin ω0τnθðηþδ cos ω0τnÞω0½ðηþδ cos ω0τnÞ2þðδ sin ω0τn�ω0Þ2�

0BBBBBB@

1CCCCCCA;

Φ2ðθÞ ¼p1ðθÞ�p2ðθÞ

2i

¼

Imfeiω0τnθgImfξeiω0τnθgImfςeiω0τnθg

0BBBB@

1CCCCA

¼

sin ω0τnθ

�βIn

ω0cos ω0τnθ

βInε sin ω0τnθðδ sin ω0τn �ω0Þ�βInε cos ω0τnθðηþδ cos ω0τnÞω0 ½ðηþδ cos ω0τnÞ2 þðδ sin ω0τn �ω0Þ2 �

0BBB@

1CCCA

and

Ψ n

1ðsÞ ¼q1ðsÞþq2ðsÞ

2

¼Refe� iω0τnsgRefξne� iω0τnsgRefςne� iω0τnsg

0B@

1CA

T

¼

cos ω0τns�δεω0 cos ω0τnsþ sin ω0τns½βSnðηþδÞ2 þβSnω2

0 �δεðηþδÞ�ω0 ½ðηþδÞ2 þω2

0 �δðηþδÞ cos ω0τns�δω0 sin ω0τns

ðηþδÞ2 þω20

0BBBB@

1CCCCA

T

;

Ψ n

2ðsÞ ¼q1ðsÞ�q2ðsÞ

2i

¼Imfe� iω0τnsgImfξne� iω0τnsgImfςne� iω0τnsg

0B@

1CA

T

¼

� sin ω0τnsδεω0 sin ω0τnsþ cos ω0τns½βSnðηþδÞ2 þβSnω2

0 �δεðηþδÞ�ω0 ½ðηþδÞ2 þω2

0 ��δðηþδÞ sin ω0τns�δω0 cos ω0τns

ðηþδÞ2 þω20

0BBBB@

1CCCCA

T

;

for sA ½0;1�. From (4.8), we can obtain ðΨ n

1;Φ1Þ and ðΨ n

1;Φ2Þ.Noting that

ðq1; p1Þ ¼ ðΨ n

1;Φ1Þ�ðΨ n

2;Φ2Þþ i½ðΨ n

1;Φ2ÞþðΨ n

2;Φ1Þ�and

ðq1; p1Þ ¼ 1þξξnþςςnþτnδςe� iω0τn ð1�ς�Þ≔Dn:

Therefore, we have

ðΨ n

1;Φ1Þ�ðΨ n

2;Φ2Þ ¼ RefDng;ðΨ n

1;Φ2ÞþðΨ n

2;Φ1Þ ¼ ImfDng:Now, we define ðΨ n

;ΦÞ ¼ ðΨ n

j ;ΦkÞ ðj; k¼ 1;2Þ and construct anew basis Ψ for Q by Ψ ¼ ðΨ 1;Ψ 2ÞT ¼ ðΨ n

;ΦÞ�1Ψ n.Obviously, ðΨ ;ΦÞ is a second order identity matrix. In addition,

define f 0 ¼ ðβ10;β

20;β

30ÞT , where

β10 ¼

100

0B@

1CA; β2

0 ¼010

0B@

1CA; β3

0 ¼001

0B@

1CA:

Let c � f 0 be defined by

c � f 0 ¼ c1β10þc2β

20þc3β

30;

for c¼ ðc1; c2; c3ÞT , cjAR ðj¼ 1;2;3Þ.






Then the center space of linear equation (4.5) is given by PCNC,where

PCNφ¼ΦðΨ ; ⟨φ; f 0⟩Þ � f 0; φAC ð4:12Þ

and

C¼ PCNC � PSC;

here PSC denotes the complementary subspace of PCNC.

Let Aτn be defined by

AτnφðθÞ ¼ _φðθÞþX0ðθÞ½LðτnÞðφðθÞÞ� _φð0Þ�; φAC; ð4:13Þ

where X0 is given by

X0ðθÞ ¼0; �1rθo0;I; θ¼ 0:

(ð4:14Þ

Then Aτn is the infinitesimal generator induced by the solution of(4.5) and (4.1) can be rewritten as the following operator differ-ential equation:

_Ut ¼ AτnUtþX0FðUt ;μÞ: ð4:15Þ

Using the decomposition C¼ PCNC � PSC and (4.12), the solu-tion of (4.1) can be rewritten as

Ut ¼Φx1ðtÞx2ðtÞ

!� f 0þhðx1; x2;μÞ; ð4:16Þ

where

x1ðtÞx2ðtÞ

!¼ ðΨ ; ⟨Ut ; f 0⟩Þ; ð4:17Þ

and hðx1; x2;μÞAPsC with hð0;0;0Þ ¼Dhð0;0;0Þ ¼ 0. In particular,the solution of (4.1) on the center manifold is given by

Un

t ¼Φx1ðtÞx2ðtÞ

!� f 0þhðx1; x2;0Þ: ð4:18Þ

Setting z¼ x1� ix2 and noticing that p1 ¼Φ1þ iΦ2, then (4.18)can be rewritten as

Un

t ¼12Φ

zþziðz�zÞ

!� f 0þwðz; zÞ ¼ 1

2ðp1zþp1zÞ � f 0þWðz; zÞ; ð4:19Þ

where Wðz; zÞ ¼ hððzþzÞ=2; �ðz�zÞ=2i;0Þ. Moreover, by [32], zsatisfies

_z ¼ iω0τnzþgðz; zÞ; ð4:20Þ

where

gðz; zÞ ¼ ðΨ 1ð0Þ� iΨ 2ð0ÞÞ⟨FðUn

t ;0Þ; f 0⟩: ð4:21Þ

Let

Wðz; zÞ ¼W20z2

2þW11zzþW02

z2

2þ⋯ ð4:22Þ

and

gðz; zÞ ¼ g20z2

2þg11zzþg02

z2

2þ⋯: ð4:23Þ

From (4.19), we have

where

⟨WnijðθÞ;1⟩¼

1π

Z π

0Wn

ijðθÞðxÞ dx; iþ j¼ 2; nAN:

Let ðψ1;ψ2Þ ¼Ψ 1ð0Þ� iΨ 2ð0Þ. Then by (4.20)–(4.22), we canobtain the following quantities:

g20 ¼τn

2� βξþ r

K

ψ1þβξψ2

h i;

g11 ¼τn

4� βξþβξþ2r

K

� �ψ1þðβξþβξÞψ2

� �;

g02 ¼τn

2� βξþ r

K

ψ1þβξψ2

h i;

g21 ¼τn

2�β 2W ð2Þ

11 ð0ÞþW ð2Þ20 ð0ÞþξW ð1Þ

20 ð0Þþ2ξW ð1Þ11 ð0Þ

Dh�2r

K2W ð1Þ

11 ð0ÞþW ð1Þ20 ð0Þ

;1�ψ1

þ ⟨βð2W ð2Þ11 ð0ÞþW ð2Þ

20 ð0ÞþξW ð1Þ20 ð0Þþ2ξW ð1Þ

11 ð0ÞÞ;1⟩ψ2

i:

Since W20ðθÞ;W11ðθÞ for θA ½�1;0� appear in g21, we still needto compute them. It easily follows from (4.21) that

_W ðz; zÞ ¼W20z _zþW11ð_zzþz _z ÞþW02z _zþ⋯ ð4:24Þand

AτnW ¼ AτnW20z2

2þAτnW11zzþAτnW02

z2

2þ⋯: ð4:25Þ

In addition, by [32], WðzðtÞ, zðtÞÞ satisfies_W ¼ AτnWþHðz; zÞ; ð4:26Þwhere

Hðz; zÞ ¼H20z2

2þH11zzþH02

z2

2þ⋯

¼ X0FðUn

t ;0Þ�ΦðΨ ; oX0FðUn

t ;0Þ; f 04Þ � f 0;with HijAPSC; iþ j¼ 2.

Thus, from (4.18) and (4.23)–(4.25), we can obtain that

ð2iω0τn�Aτn ÞW20 ¼H20;

�AτnW11 ¼H11:

(ð4:27Þ

⟨FðUn

t ;0Þ; f 0⟩¼τn

4

�ðβξþ rKÞz2�½βðξþξÞþ2r

K �zz�ðβξþ rKÞz2

βξz2þðβξþβξÞzzþβξz2

0

0BB@

1CCA

þτn

4

⟨�βð2W ð2Þ11 ð0ÞþW ð2Þ


11 ð0ÞÞ�2rK ð2W ð1Þ

11 ð0ÞþW ð1Þ20 ð0ÞÞ;1⟩

⟨βð2W ð2Þ11 ð0ÞþW ð2Þ


11 ð0ÞÞ;1⟩0

0BB@

1CCAz2zþ⋯;






Noticing that Aτn has only two eigenvalues 7 iω0τn with zeroreal parts, therefore, (4.25) has unique solution Wijðiþ j¼ 2Þ in PSCgiven by

W20 ¼ ð2iω0τn�Aτn Þ�1H20;

W11 ¼ �A�1τn H11:

(ð4:28Þ

From (4.26), we know that for �1rθo0,

Hðz; zÞ ¼ �ΦðθÞΨ ð0ÞoFðUn

t ;0Þ; f 04 � f 0¼ � p1ðθÞþp2ðθÞ

2;p1ðθÞ�p2ðθÞ

2i

� �ðΨ 1ð0Þ;Ψ 2ð0ÞÞT � ⟨FðUn

t ;0Þ; f 0⟩ � f 0

¼ �12½p1ðθÞðΨ 1ð0Þ� iΨ 2ð0ÞÞþp2ðθÞðΨ 1ð0Þþ iΨ 2ð0ÞÞ� � ⟨FðUn

t ;0Þ; f 0⟩ � f 0

¼ �14½g20p1ðθÞþg02p2ðθÞ�z2 � f 0�

12½g11p1ðθÞþg11p2ðθÞ�zz � f 0þ⋯

Therefore, for �1rθo0;

H20ðθÞ ¼ �12 ½g20p1ðθÞþg02p2ðθÞ� � f 0; ð4:29Þ

H11ðθÞ ¼ �12 ½g11p1ðθÞþg11p2ðθÞ� � f 0 ð4:30Þ

and

Hðz; zÞð0Þ ¼ FðUn

t ;0Þ�ΦðΨ ; ⟨FðUn

t ;0Þ; f 0⟩Þ � f 0;

H20ð0Þ ¼τn

2

�βξ� rK

βξ0

0B@

1CA�1

2½g20p1ð0Þþg02p2ð0Þ� � f 0; ð4:31Þ

H11ð0Þ ¼τn

4

�βξ�βξ�2rK

βξþβξ0

0BB@

1CCA�1

2½g11p1ð0Þþg11p2ð0Þ� � f 0: ð4:32Þ

By the definition of Aτn , we get from (4.27) that

_W 20ðθÞ ¼ 2iω0τnW20ðθÞþ12 ½g20p1ðθÞþg02p2ðθÞ� � f 0; �1rθo0:

Noting that p1ðθÞ ¼ p1ð0Þeiω0τn , �1rθr0: Hence

W20ðθÞ ¼i2

g20ω0τn

p1ðθÞþg02

3ω0τnp2ðθÞ

� �� f 0þEe2iω0τnθ ð4:33Þ

and

E¼W20ð0Þ�i2

g20ω0τn

p1ð0Þþg02

3ω0τnp2ð0Þ

� �� f 0: ð4:34Þ

Using the definition of Aτn , and combining (4.27) and (4.33), weget

2iω0τnig20

2ω0τnp1ð0Þ � f 0þ

ig02

6ω0τnp2ð0Þ � f 0þE

� �

�LðτnÞ ig202ω0τn

p1ðθÞ � f 0þig02

6ω0τnp2ðθÞ � f 0þEe2iω0τnθ

� �

¼ τn

2

�βξ� rK

βξ0

0B@

1CA�1

2½g20p1ð0Þþg02p2ð0Þ� � f 0:

Notice that

LðτnÞ½p1ðθÞ � f 0� ¼ iω0τnp1ð0Þ � f 0;LðτnÞ½p2ðθÞ � f 0� ¼ � iω0τnp2ð0Þ � f 0:

(

Then

2iω0τnE�LðτnÞðEe2iω0τnθÞ ¼ τn

2

�βξ� rK

βξ0

0B@

1CA:

From the above expression, we can see easily that

E¼ 12

2iω0þ2rSnK þηþβIn�r βSn �δ

�βIn 2iω0þεþη�βSn 00 �ε 2iω0þηþδ

0BB@

1CCA

�1

��βξ� r

K

βξ0

0B@

1CA:

In a similar way, we have

_W 11ðθÞ ¼ 12 ½g11p1ðθÞþg11p2ðθÞ� � f 0; �1rθo0

and

W11ðθÞ ¼i

2ω0τn½�g11p1ðθÞþg11p2ðθÞ� � f 0þF:

Similar to the above, we can obtain that

F ¼ 14

2rSnK þηþβIn�r βSn �δ

�βIn ηþε�βSn 00 �ε ηþδ

0BB@

1CCA

�1

�m�βξ�βξ�2r

K

βξþβξ0

0BB@

1CCA:

So far, W20ðθÞ and W11ðθÞ have been expressed by the para-meters of system (3.1), Therefore, g21 can be expressed explicitly.

Theorem 3. System (3.1) have the following Poincaré normal form:

_ϱ ¼ iω0τnϱþc1ð0Þϱ∣ϱ∣2þoð∣ϱ∣5Þ;where

c1ð0Þ ¼i

2ω0τng20g11�2∣g11∣

2� ∣g02∣2

3

� �þg21

2;

so we can compute the following result:

σ2 ¼ � Reðc1ð0ÞÞReðλ0ðτnÞÞ;

β2 ¼ 2 Reðc1ð0ÞÞ;

T2 ¼ � Imðc1ð0ÞÞþσ2 Imðλ0ðτnÞÞω0τn

;

which determine the properties of bifurcating periodic solutions atthe critical values τn, i.e., σ2 determines the directions of the Hopfbifurcation: if σ240 ðσ2o0Þ, then the Hopf bifurcation is super-critical (subcritical) and the bifurcating periodic solutions exist forτ4τn; β2 determines the stability of the bifurcating periodic solu-tions: the bifurcating periodic solutions on the center manifold arestable (unstable), if β2o0 ðβ240Þ; and T2 determines the period ofthe bifurcating periodic solutions: the periodic increase(decrease), ifT240 ðT2o0Þ.

5. Optimal control strategy

Optimal control techniques are of great use in developingoptimal strategies to control various kinds of malwares. In thiswork, to minimize the total density of infected nodes and the costassociated with the intervention measure, we use the optimalcontrol theory [17].

For our purpose, let ½0; T � denote the time period during acontrol strategy that is imposed on system (2.1), and introduce aLebesgue square integrable control function uðtÞ ð0rtrTÞ, thebudget for buying antivirus software at time t, which is






normalized to fall between 0 and 1 [29]. Then the admissible set ofcontrol function is given by

U ¼ fuðtÞAL2½0:T � : 0ruðtÞr1;0rtrTg:Therefore, system (2.1) can be transformed into the followingcontrol system:

dSdt

¼ rSð1� SKÞ�βSI�ηSþδRðt�τÞþγuðtÞI;

dIdt

¼ βSI�εI�ηI�uðtÞI;dRdt

¼ εI�ηR�δRðt�τÞþð1�γÞuðtÞI;

8>>>>>>><>>>>>>>:

ð5:1Þ

with initial conditions

SðtÞ ¼ S0Z0; tA ½�τ;0�;IðtÞ ¼ I0Z0; tA ½�τ;0�;RðtÞ ¼ R0Z0; tA ½�τ;0�;

8><>: ð5:2Þ

where γ denotes the control coefficient and γA ½0;1�. Next, we shalltry to find a control function u(t) so that the functional

JðuÞ ¼Z T

0IðtÞþαu2ðtÞ

2

� �dt ð5:3Þ

is minimized, where α is s tradeoff factor. That is to minimize thetotal density of infected nodes and the cost associated with theintervention measure.

Lemma 6. The system (5.1) with any initial conditions has a uniquesolution.

Proof. We can rewrite (5.1) in the following form:

dVðtÞdt

¼ AVðtÞþBVτðtÞþFðVðtÞ;uðtÞÞ; ð5:4Þ

where

VðtÞ ¼ ðSðtÞ; IðtÞ;RðtÞÞT ; VτðtÞ ¼ ðSðt�τÞ; Iðt�τÞ;Rðt�τÞÞT ;

A¼r�η 0 00 �η�ε 00 ε �η

0B@

1CA; B¼

0 0 δ0 0 00 0 �δ

0B@

1CA;

F ¼� r

KS2ðtÞ�βSðtÞIðtÞþγuðtÞIðtÞβSðtÞIðtÞ�uðtÞIðtÞð1�γÞuðtÞIðtÞ

0B@

1CA:

A simple calculation shows that

jFðV1ðtÞ;u1ðtÞÞ�FðV2ðtÞ;u2ðtÞÞjrMjV1ðtÞ�V2ðtÞj;where M is some positive constant, independent of state variablesS(t), I(t) and R(t). Furthermore, we have

jV1ðtÞ�V2ðtÞj ¼ jS1ðtÞ�S2ðtÞjþjI1ðtÞ� I2ðtÞjþjR1ðtÞ�R2ðtÞjand

jðV1ÞτðtÞ�ðV2ÞτðtÞj ¼ jðS1ÞτðtÞ�ðS2ÞτðtÞjþjðI1ÞτðtÞ�ðI2ÞτðtÞjþjðR1ÞτðtÞ�ðR2ÞτðtÞj:

Therefore, it is easy to show that

AVðtÞþBVτðtÞþFðVðtÞ;uðtÞÞrLðjV1ðtÞ�V2ðtÞjþjðV1ÞτðtÞ�ðV2ÞτðtÞjÞ;ð5:5Þ

where L¼maxfM; JAJ ; JBJgo1. Thus the terms on the righthand side of Eq. (5.4) are uniformly Lipschitz continuous. Thesolution of system (5.1) exists from (5.5) and taking into accountthe constraints on the controls u(t) and the restrictions on thenon-negativeness of the state variables.

In order to find an optimal solution, first we find the Lagrangianand Hamiltonian for the optimal control problem (5.1)–(5.3). Herethe Lagrangian of the problem is

LðS; I;R;uÞ ¼ IðtÞþαu2ðtÞ2

:

We seek the minimal value of the Lagrangian. To accomplish this,we define the Hamiltonian H for the control problem as

HðS; I;R;u; λ1; λ2; λ3; tÞ ¼ LðS; I;R;uÞþλ1ðtÞ rS 1� SK

� ��

�βSI�ηSþδRðt�τÞþγuðtÞI�þλ2ðtÞfβSI�εI�ηI�uðtÞIgþλ3ðtÞfεI�ηR�δRðt�τÞþð1�γÞuðtÞIg; ð5:6Þ

where λ1ðtÞ, λ2ðtÞ and λ3ðtÞ are the adjoint functions to bedetermined suitable. □

Lemma 7. There exists an optimal control unðtÞ such thatJðunðtÞÞ ¼minu JðuððtÞÞ; subject to the control system (5.1) with initialconditions.

Proof. In fact, the following conditions are satisfied:

1. The set of control and corresponding state variables isnot empty.

2. The control space U is convex and closed by definition.3. Each right hand side of the state system is continuous and is

bounded by a sum of the bounded control and the state.Furthermore, it can be written as a linear function of thecontrol variate u(t) with coefficients depending on time andthe state.

4. The integrand in the functional (5.3) is convex on the controlset U and is bounded below.

Thus, according to Lukes [35], there exists an optimal controlunðtÞ. This completes the proof. □

Next, let us derive a necessary condition for the optimal controlstrategy by means of the Pontryagin's Maximum Principle [24,25].

Theorem 4. Let S⋆ðtÞ, I⋆ðtÞ, R⋆ðtÞ be optimal state solutions asso-ciated with the optimal control variable unðtÞ for the optimal controlproblem (5.1)–(5.3). Then there exist adjoint variables λ1ðtÞ, λ2ðtÞ andλ3ðtÞ satisfyingdλ1ðtÞdt

¼ λ1ðtÞ2rKS⋆ðtÞþβI⋆ðtÞþη�r

� ��λ2ðtÞβI⋆ðtÞ; ð5:7Þ

dλ2ðtÞdt

¼ λ1ðtÞ½βS⋆ðtÞ�γunðtÞ�þλ2ðtÞ½εþηþunðtÞ�βS⋆ðtÞ��λ3ðtÞ½εþð1�γÞunðtÞ��1; ð5:8Þ

dλ3ðtÞdt

¼ λ3ðtÞþχ ½0;T�τ�δ½λ3ðtþτÞ�λ1ðtþτÞ�; ð5:9Þ

with transversality conditions

λiðTÞ ¼ 0; i¼ 1;2;3:

Furthermore, the optimal control is given as follows:

unðtÞ ¼max min ½�γλ1ðtÞþλ2ðtÞ�ð1�γÞλ3ðtÞ�I⋆ðtÞα

;1�

;0�

:

ð5:10Þ

Proof. To determine the adjoint equations and the transversalityconditions we use the Hamiltonian (5.6). By using the necessarycondition for optimal control problems which is found in [34], and






differentiating the Hamiltonian (5.6) with S, I, and R, we obtain

dλ1ðtÞdt

¼ �HSðtÞ�χ ½0;T�τ�HSτ ðtþτÞ; ð5:11Þ

dλ2ðtÞdt

¼ �HIðtÞ�χ ½0;T� τ�HIτ ðtþτÞ; ð5:12Þ

dλ3ðtÞdt

¼ �HRðtÞ�χ ½0;T� τ�HRτ ðtþτÞ; ð5:13Þ

where Hx and Hxτ denote the derivative with respect tox ðx¼ S; I;RÞ and xτ ðxτ ¼ Sðt�τÞ; Iðt�τÞ;Rðt�τÞÞ, respectively. Thus,if we set SðtÞ ¼ S⋆ðtÞ, IðtÞ ¼ I⋆ðtÞ and RðtÞ ¼ R⋆ðtÞ, then by substitut-ing the corresponding derivatives in the above inequations andrearranging we obtain the adjoint equations (5.7)–(5.9). By usingthe optimality condition, we get

∂H∂u

��uðtÞ ¼ unðtÞ

¼ αunðtÞþγλ1ðtÞI⋆ðtÞ�λ2ðtÞI⋆ðtÞþð1�γÞλ3ðtÞI⋆ðtÞ:

Considering the feature of the admissible set U, we have

unðtÞ ¼max min ½�γλ1ðtÞþλ2ðtÞ�ð1�γÞλ3ðtÞ�I⋆ðtÞα

;1�

;0�

:

This completes the proof. □

Therefore, according to the analysis above, we can easily obtainthe following optimality system:

dS⋆

dt¼ rS⋆ 1�S⋆

K

� ��βS⋆I⋆�ηS⋆þδR⋆ðt�τÞ

þγmax min ½�γλ1ðtÞþλ2ðtÞ�ð1�γÞλ3ðtÞ�I⋆

α;1

� ;0

� I⋆;

dI⋆

dt¼ βS⋆I⋆�εI⋆�ηI⋆

�max min ½�γλ1ðtÞþλ2ðtÞ�ð1�γÞλ3ðtÞ�I⋆

α;1

� ;0

� I⋆;

dR⋆

dt¼ εI⋆�ηR⋆�δR⋆ðt�τÞþð1�γÞ

max min ½�γλ1ðtÞþλ2ðtÞ�ð1�γÞλ3ðtÞ�I⋆

α;1

� ;0

� I⋆;

SðtÞ ¼ S0Z0; IðtÞ ¼ I0Z0; RðtÞ ¼ R0Z0; tA ½�τ;0�;

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

ð5:14Þ

with the Hamiltonian Hn at ðS⋆; I⋆;R⋆;un; λ1; λ2; λ3; tÞ:In order to ensure the correctness of the conclusion above, in

next section, we will numerically solve the optimal control system(5.14) by using the Runge–Kutta fourth order scheme.

6. Numerical simulation and discussion

In this section, first we simulate and analyze the dynamiccharacteristics of the proposed SIRS malware propagation modelwithout the optimal control strategy through Matlab, includingthe trend in the quantity of infected nodes. Furthermore, by usinga Matlab code dde23 which is based on a standard Runge–Kuttascheme, we consider the optimal control problem (5.1) and verifythe validity of our controller.

6.1. Impact of delays on the malware propagation model

To observe the impact of delays on the density of the infectednodes, we consider system (2.1) with r¼0.3, K¼2, β¼ 0:5, ε¼ 0:4,η¼ 0:1, δ¼ 0:2. It is obvious that system (2.1) has a unique positiveequilibrium point En ¼ ð1;0:2143;0:2857ÞT . By calculating, theparameters satisfy the conditions of Theorem 2. According toTheorem 2, substituting these parameters into (3.8) yields twopositive and simple roots, i.e., ω1 ¼ 0:2638, ω2 ¼ 0:1860. From(3.17) and (3.18), we can obtain the critical values of time delays τas following:

τn;1 ¼ 9:3646;33:1809;56:9973;80:8136;…;

τn;2 ¼ 16:5713;50:3499;84:1285;117:9071;…:

These critical time delays can be ranked as

0oτ0;1 ¼ 9:3646oτ0;2 ¼ 16:5713oτ1;1 ¼ 33:1809oτ1;2 ¼ 50:3499

oτ2;1 ¼ 56:9973oτ3;1 ¼ 80:8136oτ2;2 ¼ 84:1285oτ3;2 ¼ 117:9071o⋯:

Furthermore,

dðRe λðτÞÞdτ

��τ ¼ τn;1 ;λ ¼ iω1

¼ 0:002940 and

dðRe λðτÞÞdτ

��τ ¼ τn;2 ;λ ¼ iω2

¼ �0:0011o0:

Thus, a pair of eigenvalues are crossing the imaginary axis fromthe left to the right when τ¼ τn;1 and from the right to the leftwhen τ¼ τn;2. The time histories for the delay chosen fromdifferent regions are shown in Figs. 2–7. The positive equilibriumpoint En of system (2.1) is locally asymptotically stable whenτA ½0; τ0;1Þ [ðτ0;2; τ1;1Þ [ ðτ2;2; τ2;1Þ, and it loses its stability forτA ðτ0;1; τ0;2Þ [ðτ1;1; τ1;2Þ [ ðτ2;1;1Þ. In a word, time delay leadssystem (2.1) to exhibit the multiple switches phenomenon of reststate from stable to unstable, then back to stable. The positiveequilibrium point En of system (2.1) is unstable in the end.

On the other hand, when τ¼ τ0;1 ¼ 9:3646 we can computec1ð0Þ ¼ 1:7083�2:7922i, σ2 ¼ �ð1:7083=Reðλ0ðτ0;1ÞÞÞ ¼ �594:4211o0, β2 ¼ 2 Reðc1ð0ÞÞ ¼ 3:416740. Therefore, from the discussionsin Section 4, we know that the bifurcated periodic solutions

0 200 400 600 800 10000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

t

The

dens

ity o

f sus

cept

ible

nod

es τ=7

0 200 400 600 800 10000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des

τ=7

0 200 400 600 800 10000.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes τ=7

Fig. 2. The positive equilibrium point En is locally asymptotically stable when τ¼ 7A ½0; τ0;1Þ.






are orbitally unstable on the center manifold. In addition, from Theorem3, we know that system (2.1) has an unstable center manifold near thepositive equilibrium point En for τ near τ0;1. Therefore, the bifurcating

periodic solution of system (2.1) is unstable on the center manifold andthe Hopf bifurcation is subcritical. We cannot get the bifurcating periodicsolutions from simulations [36]. Similarly, for the other critical values of

0 500 1000 1500 20000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

t

The

dens

ity o

f sus

cept

ible

nod

es τ=10

0 500 1000 1500 20000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

t

The

dens

ity o

f inf

ecte

d no

des τ=10

0 500 1000 1500 20000.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

t

The

dens

ity o

f rec

over

ed n

odes τ=10

Fig. 3. The periodic oscillation derived from Hopf bifurcation around the positive equilibrium point En occurs when τ¼ 10Aðτ0;1 ; τ0;2Þ.

0 500 1000 15000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

t

The

dens

ity o

f sus

cept

ible

nod

es τ=25

0 500 1000 15000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des

τ=25

0 500 1000 15000.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes

τ=25

Fig. 4. The positive equilibrium point En is locally asymptotically stable when τ¼ 25Aðτ0;2; τ1;1Þ.

0 500 1000 1500 20000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

t

The

dens

ity o

f sus

cept

ible

nod

es

0 500 1000 1500 20000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des

0 500 1000 1500 20000.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes

Fig. 5. The periodic oscillation derived from Hopf bifurcation around the positive equilibrium point En occurs when τ¼ 33:2Aðτ1;1 ; τ1;2Þ.

0

2000

4000

6000

8000

1000

012

000

1400

016

000

1800

00.9

0.92

0.94

0.96

0.98

1

1.02

1.04

t

The

dens

ity o

f sus

cept

ible

nod

es τ=55

0

2000

4000

6000

8000

1000

012

000

1400

016

000

1800

00.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des τ=55

0

2000

4000

6000

8000

1000

012

000

1400

016

000

1800

00.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes τ=55

Fig. 6. The positive equilibrium point En is locally asymptotically stable when τ¼ 55Aðτ1;2; τ2;1Þ.






time delay τ we can also analyze their direction and stability ofbifurcating periodic oscillations.

Furthermore, we choose τ¼ 82. According to Theorem 2, aperiodic oscillation derived from Hopf bifurcation around the positiveequilibrium point En occurs, as shown in Fig. 8. From Fig. 8, we cansee that system (2.1) has an attractive quasi-periodic solution, whichis different from the discussion above (such as Fig. 7). This meansthat malware propagation is sensitive to the immune period τ of arecovered node. The above observations provide new insights intothe malware propagation in WSNs that the impact of delays on themalware propagation model cannot be ignored, namely, with thedelay τ increasing the wireless sensor networks lose stability through

a Hopf bifurcation and some oscillations occur, then come back to astable state again, and in the end it becomes unstable. The phenom-enon possibly leads to that the utilization of the network decreasesand the network performance declines.

6.2. Impact of optimal control strategy on the malware propagationmodel

In this part, we will numerically illustrate that the optimalcontrol strategy we proposed can effectively extend the region ofstability of system (2.1) and also decrease the density of infectednodes in WSNs.

0 500 1000 1500 20000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

t

The

dens

ity o

f sus

cept

ible

nod

es

0 500 1000 1500 20000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des τ =57τ =57

0 500 1000 1500 20000.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes τ =57

0.99 0.995 1 1.005 1.01 1.015

0.21

0.215

0.220.275

0.28

0.285

0.29

0.295

0.3

Density of susceptible nodesDensity of infected nodes

Den

sity

of r

ecov

ered

nod

es

Fig. 7. The periodic oscillation derived from Hopf bifurcation around the positive equilibrium point En occurs when τ¼ 57A ðτ2;1 ; τ3;1Þ.

0 500 1000 1500 20000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

t

The

dens

ity o

f inf

ecte

d no

des

0 500 1000 1500 20000.2

0.22

0.24

0.26

0.28

0.3

0.32

t

The

dens

ity o

f rec

over

ed n

odes

0 500 1000 1500 20000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

t

The

dens

ity o

f sus

cept

ible

nod

es

0.99 0.995 1 1.005 1.010.21

0.2150.22

0.2250.275

0.28

0.285

0.29

0.295

0.3

0.305

Density of susceptible nodesDensity of infected nodes

Den

sity

of r

ecov

ered

nod

es

Fig. 8. The attractive quasi-periodic oscillation derived from Hopf bifurcation around the positive equilibrium point En occurs when τ¼ 82A ðτ3;1 ; τ2;2Þ.






Example 1 (Impact of optimal control strategy on the region ofstability). Consider system (5.1) with r¼0.7, K¼2, β¼ 0:3, ε¼ 0:3,η¼ 0:05, δ¼ 0:4, α¼ 6 and γ ¼ 0:3. It is easy to obtain thatthe positive equilibrium point of uncontrolled system (2.1) isEn ¼ ð1:1667;3:3833;2:2556ÞT and the critical value of delay isτ0 ¼ 6:8345. Without loss of generality, we assign τ¼ 4;5;6oτ0with the other parameters unchanged. The simulation results areshown in Fig. 9(a). From Fig. 9(a), we notice that the density ofinfected nodes converges to the positive equilibrium point En ofsystem (2.1). Thus the malware continuously propagates in WSNs.In addition, from Fig. 9(a) we notice that a larger τ implies a longerconvergence time. For optimal control system (5.1) we also chooseτ¼ 4;5;6. As shown in Fig. 9(b), we can see that system (5.1)quickly converges to the positive equilibrium point and it isindependent on delay. Thus, the optimal control strategy is moreeffective to guarantee the system to work normally.

Furthermore, we set τ¼ 6:84;7:24;7:94. According to Theorem 1,the periodic oscillation derived from Hopf bifurcation around thepositive equilibrium point En occurs, as shown in Figs. 10(a), 11(a)and 12(a). In addition, with τ increasing we find that the amplitudeof uncontrolled system (2.1) increases, which possibly leads to thatthe utilization of the network decreases and the network perfor-mance declines. For the optimal control system (5.1), it is easy toshow that system (5.1) is asymptotically stability and the density ofinfected nodes is lower than the uncontrolled system (2.1), as shownin Figs. 10(b), 11(b) and 12(b). Thus, the optimal control strategyeffectively extends the region of stability of system (2.1).

Example 2 (Impact of optimal control strategy on the density ofdifferent nodes). In this part, we have plotted susceptible nodes,infected nodes and recovered nodes with and without control byconsidering values of parameters as r¼0.7, K¼6, β¼ 0:5, ε¼ 0:3,

0 100 200 300 400 500 600 700 8003.25

3.3

3.35

3.4

t

The

dens

itty

of in

fect

ed n

odes

SIRS model without control

0 50 100 150 2001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

t

The

dens

ity o

f inf

ecte

d no

des

SIRS model with control

τ=4τ=5τ=6τ=4

τ=5τ=6

Fig. 9. Uncontrolled system (2.1) and optimal control system (5.1) are asymptotically stable when τ¼ 4;5;6.

0 100 200 300 400 5003.26

3.28

3.3

3.32

3.34

3.36

3.38

3.4

3.42

3.44

t

The

dens

ity o

f inf

ecte

d no

des


τ =6.84

0 100 200 300 400 5001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6SIRS model with control

t

The

dens

ity o

f inf

ecte

d no

des

τ =6.84

Fig. 10. (a) The periodic oscillation derived from Hopf bifurcation around the positive equilibrium point En occurs when τ¼ 6:844τ0. (b) Optimal control system (5.1) isasymptotically stable when τ¼ 6:84.

0 100 200 300 400 5003.263.28

3.33.323.343.363.38

3.43.423.443.46

t

The

dens

ity o

f inf

ecte

d no

des


0 100 200 300 400 5001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4


t

The

dens

ity o

f inf

ecte

d no

des







η¼ 0:1, δ¼ 0:3, α¼ 6, γ ¼ 0:6 and τ¼ 0:5. The control nodes aremarked by red dashed line while the nodes without control aremarked by blue solid line.

In Fig. 13(a), we have plotted susceptible nodes with and withoutcontrol. The numerical results show that the density of susceptiblenodes quickly decreases and goes to a stable state. In addition, we can

0 100 200 300 400 5003.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

3.6

t

The

dens

ity o

f inf

ecte

d no

des


τ =7.94

0 100 200 300 400 5001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4


t

The

dens

ity o

f inf

ecte

d no

des

τ =7.94


0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

t

The

dens

ity o

f sus

cept

ible

nod

es

without controlwith control

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

t

The

dens

ity o

f inf

ecte

d no

des


0 1 2 3 4 5 6 7 83

3.5

4

4.5

5

5.5

6

6.5

7

t

The

dens

ity o

f rec

over

ed n

odes


Fig. 13. The plots represent the density of nodes in three different states both with control and without control. (a) Susceptible nodes. (b) Infected nodes. (c) Recoverednodes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Table 1The dynamic behavior of system (2.1) for different constant rate δ.

δ Positive equilibrium point Stability domain σ2 β2 T2

0.35 ð0:8250;0:6277;0:3649ÞT [0, 4.4388) �67.8808 3.5794 �5.5097

0.40 ð0:8250;0:6527;0:3399ÞT [0, 3.8609) 442.4170 �33.4101 43.1185

0.45 ð0:8250;0:6744;0:3181ÞT [0, 3.4224) 15.3341 �1.5498 1.8049






see that the density of infected nodes in the optimal control system(5.1) is smaller than that in the uncontrolled system (2.1).

Fig. 13(b) represents the density of infected nodes in the twosystems (2.1) without control and (5.1) with control. Obviously, thedensity of infected nodes without control is larger than the density ofinfected nodes with optimal control strategy, so we need treatment tocontrol the infected nodes.

In Fig. 13(c), we obtain that the density of recovered nodes inthe optimal control system (5.1) is larger, which is more effectiveto guarantee the system to work normally.

7. Discussion and concluding remarks

In this paper, we have developed a SIRS model for malwarepropagation in WSNs, and have discussed the stability and Hopfbifurcation. Considering the discrete delay τ as bifurcation parameter,the existence, stability, and direction of bifurcation periodic solutionsare investigated in detail by applying the theorem of partial functiondifferential equation, the normal form method and center manifoldtheorem. In order to minimize the density of infected nodes and thecost associated with the intervention measure, we propose a optimalcontrol strategy (see the optimal control system (5.1)). Numericalsimulations reveal that the discrete delay is responsible for thestability switches of the model system, and a Hopf bifurcation occursas the delay increasing through a certain threshold. In addition, weobtain that the optimal control strategy effectively extends the region

of stability and minimizes the density of infected nodes. Thus, the SIRSmalware propagation model with a optimal control strategy providesnew insights into the malware propagation in WSNs.

In order to reduce the density of infected nodes and guaranteesystem (2.1) to work normally, we must pay more addition to thefactor of δ (the rate constant for nodes becoming susceptible againafter recovered). In fact, the dependence of the density In of infectednodes on the contact rate (the parameter δ) may be found byobserving that

dIn

dδ¼ εðηþεÞ½rðKβ�ε�ηÞ�Kβη�

Kηβ2ðεþηþδÞ240

under the condition ðH2Þ, so that In is an increasing function about δ.In this section, we will further discuss the effect of δ on the densityof infected nodes and the dynamic characteristic of system (2.1).

0 100 200 300 400 5000.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

t

The

dens

ity o

f inf

ecte

d no

des

δ=0.35δ=0.4δ=0.45

Fig. 14. The positive equilibrium point En is locally asymptotically stable with δ

increasing when τ¼ 3oτ0.

0 100 200 300 400 500

0.55

0.6

0.65

0.7

0.75

t

The

dens

ity o

f inf

ecte

d no

des

δ=0.35δ=0.4

0 100 200 300 400 5000.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

t

The

dens

ity o

f inf

ecte

d no

des

δ=0.4δ=0.45

Fig. 15. (a) The dynamic characteristic of system (2.1) with different δ and time delay τ. (a) Set τ¼ 3:87, and δ¼ 0:35;0:4. (b) Set τ¼ 3:43, and δ¼ 0:4;0:45.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

Den

sity

Recovered nodes

Infected nodes

Susceptible nodes

Fig. 17. The positive equilibrium points of system (2.1) varies with the constant rateδ increasing.

0.4 0.5 0.6 0.7 0.8 0.9 11.5

2

2.5

3

3.5

4

4.5

δ

τ

StableHopf bifurcation

Unstable

Fig. 16. The region of stability of system (2.1) varies with the constant rate δ

increasing.






Consider system (2.1) with r¼0.3, K¼2, β¼ 0:4, ε¼ 0:25,η¼ 0:08 and assign 0.35, 0.40 and 0.45 to δ, respectively.Obviously, the conditions ðH2Þ–ðH4Þ hold. According to Theorems 1and 3, by calculating, it is easy to obtain the corresponding positiveequilibrium points and the dynamic behavior of system (2.1) fordifferent δ, as shown in Table 1.

From Table 1, we can obtain that with the increasing of δ, thecritical value of τ decreases. This means that the region of stabilityof system (2.1) decreases and the malware propagation is sensitiveto δ. Moveover, if δ¼ 0:40 or δ¼ 0:45, then when τ¼ τ0, we haveσ240, β2o0. Therefore, from the discussions in Section 4, weknow that the bifurcating periodic solutions are orbitally asympto-tically stable on the center manifold. In addition, from Theorem 3,we know that system (2.1) has a stable center manifold near thepositive equilibrium point En for τ near τ0. Thus, the center manifoldtheory implies that the bifurcated periodic solutions of system (2.1)when τ¼ τ0 in the whole phase space are orbitally asymptoticallystable, and the Hopf bifurcation is supercritical for σ240. Whenδ¼ 0:35, we have σ2o0, β240. Therefore, from the discussions inSection 4, we know that the bifurcating periodic solutions areorbitally unstable on the center manifold. In addition, we have thatin the whole phase space the Hopf bifurcation is subcritical.

Without loss of generality, we assign τ¼ 3oτ0 with the otherparameters unchanged. The simulation results are shown in Fig. 14.From Fig. 14, we notice that the density of infected nodes convergesto the positive equilibrium point En of system (2.1). Thus at last themalware continuously propagates in WSNs. The simulation resultsin Fig. 14 verify Theorem 1. In addition, from Fig. 14 we notice thatwith the increasing of δ, the convergence time becomes longer andat last the density of infected nodes increases. Furthermore, formFig. 15 we can see that with the increasing of the constant rate δ theregion of system (2.1) decreases. This means that the dynamicbehave is sensitive to δ. Thus, if we want to guarantee the system towork normally, we must strengthen prevention measures to thewireless sensor networks.

Remark 1. Keep the parameters r¼0.3, K¼2, β¼ 0:4, ε¼ 0:25,η¼ 0:08. In this part, we discuss the effect of δ on the region ofstability of system (2.1) when δ varies from 0.3 to 1 continuously,as shown in Fig. 16. From Fig. 16, we can see that with δ increasing,the region of system (2.1) decreases quickly, which implies thatWSNs possibly decrease the utilization and decline the perfor-mance when δ increases. The above observations provide newinsight into the malware propagation in WSNs.

Remark 2. Keep the parameters r¼0.3, K¼2, β¼ 0:4, ε¼ 0:25,η¼ 0:08, but δ varies in ½0;1�, the corresponding situations ofpositive equilibrium points are shown in Fig. 17. Fig. 17 shows thatwith the increase of the constant rate δ, the density of thesusceptible nodes and the recovered nodes is decreasing and thedensity of the infected nodes is increasing. Obviously, the increas-ing of δ is harmful for the wireless sensor networks. Thus, we musttake the necessary measures to decrease δ, such as enhance thestrength of the antivirus software.

Acknowledgments

This research is supported by National Natural Science Founda-tion of China under Grants 61174155 and 11032009. The work isalso sponsored by Qing Lan Project to Jiangsu.

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http://refhub.elsevier.com/S0925-2312(14)01117-5/sbref1
















































































Linhe Zhu received his B.S. degree in Applied Mathe-matics from Xinzhou Teachers University, Xinzhou,China. Now he is studying for a Ph.D. degree in AppliedMathematics at the Department of Mathematics,Nanjing University of Aeronautics and Astronautics,Jiangsu province, China. His research interests includedynamical system, neural networks, informationsciences and control theory.

Hongyong Zhao received the Ph.D. degree fromSichuan University, Chengdu, China, and the Post-Doctoral Fellow in the Department of Mathematics atNanjing University, Nanjing, China.He was with the Department of Mathematics at

Nanjing University of Aeronautics and Astronautics,Nanjing, China. He is currently a Professor of NanjingUniversity of Aeronautics and Astronautics, Nanjing,China. He is also the author or coauthor of more than60 journal papers. His research interests include non-linear dynamic systems, neural networks, control the-ory, and applied mathematics.






dynamical analysis and optimal control for a malware propagation model in an information network

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