# Stochastic Stabilization of Malware Propagation in Wireless Sensor Network via Aperiodically Intermittent White Noise

Authors: Xiaojing Zhong; Baihao Peng; Feiqi Deng; Guiyun Liu
Journal: Complexity
 (2020)

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## Abstract

In this paper, we propose a novel heterogeneous model to describe the propagation dynamics of malware (viruses, worms, Trojan horses, etc.) in wireless sensor networks. Our model takes into consideration different battery-level sensor nodes contrary to existing models. In order to control the spread of malware, we design an aperiodically intermittent controller driven by white noise, which has striking advantages of lower cost and more flexible control strategy. We give a distinct condition on stability in probability one using graph-theoretical Lyapunov function and stochastic analysis method. Our results show that the nonlinear malware propagation system can be stabilized by intermittent stochastic perturbation under the intermittent time related to stochastic perturbation intensity. Our theoretical results can be applied to understand the observed mechanisms of malware and design interventions to control the spread of malware. Numerical simulations illustrate our analytical results clearly.


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## Body

## 1. Introduction

With the improvement of wireless sensor technology, wireless sensor networks (WSNs) are used in many new scenarios: intelligent transportation network, water quality monitoring, military target detection, etc. Due to wireless feature and special structure of wireless sensor network, it is vulnerable to malware attack which can eavesdrop and paralyze the network [1–3]. Consequently, mathematical modeling and control strategies of malware propagation are of vital importance in order to predict its effects and defend it. In recent years, many scientific literatures have appeared, specifically, global model and the related global behaviour of WSNs by the concept of epidemiology. In the epidemic system, all people separate into different compartments such as S (Susceptible), E (Exposed), I (Infectious), R (Recovered), and so on. Toutonji et al. [4] proposed a VEIS-V worm attack propagation model which considers security countermeasures during worm attack. Bimal and Keshri [5] established a SEIRSV model by introducing a maintenance mechanism to sleep nodes. Zhu and Zhao [6] considered a SIR model which has logistic growth. Wang et al. [7] proposed a SCI model to describe the propagation dynamics of mobile sensor worm.However, the above global models cannot cover individual feature and capability. In this sense, individual-based model will be more accurate to model malware propagation since it values individual diversity and derives both malware relationship and individual feature from malware propagation. Under the framework of heterogeneous complex network theory, the connection topology of nodes can be defined by node degree and the related degree distribution. Several complex network models have been studied for the epidemic model [8–11], alcoholism model [12], information spreading model [13–17], etc. But for wireless sensor network, there are few achievements: del Rey et al. [18] established heterogeneous SIS and SIR models for malware propagation in WSNs; all nodes are separated into three compartments based on the classic SIR epidemic model. Li et al. [19] considered a SIR network malware propagation model; Li’s SIR model is different from del Rey’s heterogeneous model which considers tree-based networks. Inspired by stifler state in the rumor spreading model, Hosseini and Azgomi [20] proposed a SEIRS model. They assumed that nodes could be vaccinated and immunized to the malware infection. All of the above models did not consider sensor’s energy consumption, which affects seriously the information transmission quality. Nodes may not have enough energy to propagate malware by sending information to their neighbors. Proceeding from this angle, infected state should be separated into infected state with high capacity level and infected state with low capacity level. It is worth to mention that most of the results concentrate on dynamic analysis of the deterministic model (global or individual-based model) by the basic reproduction number R0, which is calculated by the next generation matrix method. But there are rare results on malware control strategies if malware becomes endemic (R0>1).As is well known, stochastic noises can stabilize an unstable system [21–27]. Its application extends to many practical areas, such as sleep improvement, tone discrimination [28], and financial stability [21, 23]. From the consideration of reducing control cost and control time, discontinuous controllers have been designed to stabilize a given system such as feedback control [29–32], pinning control [33], impulsive control [34], adaptive control [35–40], and intermittent control [41–44]. As for intermittent control, control time is divided into periodic and aperiodic type. Periodically intermittent control has been studied by many authors especially in synchronization problems. Zhang et al. [41] designed a periodically intermittent linear controller driven by white noises to stabilize an unstable memristor-based system. As Liu and Chen [45, 46] have mentioned before, the requirement of periodicity is unusual in real world application, while aperiodically intermittent control strategy needs a complex management to study the dynamic behaviour of the controlled system. We highlight Liu and Chen [45, 46] who investigated aperiodically intermittent deterministic controller for complex coupled deterministic system, and sufficient conditions have been given to guarantee global synchronization.Stochastic control has been proved to be sufficiently useful in many fields. But for the best of our knowledge, there is no work on aperiodically intermittent stochastic stabilization for malware propagation; the application on malware propagation needs to be explored. This motivates us to investigate this kind of scheme oriented to the security issue for WSNs. In this paper, we will design an aperiodically intermittent control strategy. In this strategy, we will focus on quantitative conditions for network to stabilize. The main contributions can be summarized as follows:(i)
Compared with the previous works on malware spreading models on WSNs, ourSILIHA model is a new model which considers both individual behaviours and sensor capacitance. It can be used to solar heterogeneous network system.(ii)
Compared with the previous works on spreading control, the control itself is random perturbation and the control time is aperiodically intermittent rather than continuous, and starting time and control length are arbitrary. We can control steady rate autonomously by adjusting the work width. In this angle, our control strategy is more flexible.The paper is organized as follows. In Section2, we establish a new malware propagation model on WSNs at first, which considers different battery-level sensor nodes. Then, we analyze the dynamic behaviour of the model. In Section 3, we design an aperiodically intermittent stochastic noise controller to control malware spreading. The control intensity expression reflects the relationship between system parameters and network topological structure. In Section 4, numerical simulations of the proposed model are given to show the power of aperiodically intermittent noise controller. Finally, we conclude the paper in Section 5.


## 2. Model Formulation

According to topological structures of WSNs, all nodes partition inton groups based on different nodal degrees. Each group i is further compartmentalized into four states:(1)
Susceptible (Si): the nodes in this state are prone to being infected by the implantation of malware and will become an infected one if it connects to an infected node.(2)
Infectious with high energy level (IiH): the nodes in this state have already been infected by malware successfully and have stored up enough energy to continue propagating the malware.(3)
Infectious with low energy level (IiL): the nodes in this state have already been infected by malware successfully but do not have sufficient energy to propagate malware by sending information to their neighbors.(4)
Antimalware program activated (Ai): when nodes in state IH and state IL are detected to be infected, the nodes will activate antimalware program to clear up the infection. The node activated by the antimalicious program will no longer be invaded by the same malware.Based on the sensor features and transition characters, the malware propagation process is described as follows: when a susceptible node connects to an infected node, the susceptible node with high energy level becomes an infected node with probabilityβ1 and the susceptible with low energy level node becomes an infected node with probability β2. Both the contact rates β1 and β2 are positive constants. The malware itself will not cause additional cost to infected node. The infected node with low energy level can become infected node with high level if it is recharged, and the charge rate is a positive constant γ. The influence of the malware on the natural mortality rate of infected nodes should be ignored and nodes in four states are assigned equal natural mortality rate η which satisfies η≥0. Figure 1 shows state transition diagram of the model.Figure 1
The flow diagram of the propagation model.Thus, our model can be formulated by the following ordinary differential equations:(1)Sit′=Λ−β1iΘtSit−β2iΘtSit−ηSit,IiLt′=β1iΘtSit−γIiLt−μ1IiLt−ηIiLt,IiHt′=β2iΘtSit+γIiLt−μ2IiHt−ηIiHt,Ait′=μ1IiLt+μ2IiHt−ηAit,with initial values(2)Si0≥0,IiL0≥0,IiH0≥0,Ai0≥0,Si0+IiL0+IiH0+Ai0=Λη,where i=1,…,n, represents the nodal degree. Θt represents the probability that an edge connects to an infectious sensor node at time t. Its expression is(3)Θt=1j∑j=1njptIiHtNit,where j=∑j=1njpj represents the average nodal degree. pj represents the probability that a randomly chosen sensor node has degree j, ∑j=1npj=1. Obviously, for all i and t≥0,(4)Nit=Sit+IiLt+IiHt+Ait=Λη.Then, we can deduce system (1) into(5)Sit′=Λ−β1iΘtSit−β2iΘtSit−ηSit,IiLt′=β1iΘtSit−γIiLt−μ1IiLt−ηIiLt,IiHt′=β2iΘtSit+γIiLt−μ2IiHt−ηIiHt.About the dynamic behaviour of the above deterministic model (5), we give the following corollary.Corollary 1.
Define(6)R0=β1γ+β2γ+μ1+ηj2η+μ2γ+η+μ1j.(1)
IfR0<1, the malware-free equilibrium is globally asymptotically stable.(2)
IfR0>1, there is an endemic equilibrium, and it is persistent.Remark 1.
The threshold is obtained byR0=β1γ+β2γ+μ1+ηj2/η+μ2γ+η+μ1j, which depends on the fluctuations of the degree distribution and system parameters. Since malware will die out when R0<1, malware prevention measures can be designed to decrease R0, such as decreasing node connectivity to decrease the contact rates β1,β2 and increasing monitoring efforts on infective nodes to increase the detective rate μ2.


## 3. Control Strategy

In this section, we mainly concentrate on the control strategy of malware propagation. In the context of aperiodically intermittent stochastic noise stabilization of the complex heterogeneous network system, we propose a design procedure for the noise control input to stabilize system (5). At first, we consider a stochastic noise driven by Brownian motions and a g-type structure of the controller. The controller’s mathematical expression is gt,xtB˙t, where Bt is a 3n-dimensional Brownian motion corresponding to system variables and B˙t is the related white noise. To be flexible and to save cost, we design the control time as aperiodically intermittent type, which admits uncertain rest times. We give a sketch of aperiodically intermittent control strategy in Figure 2, under the perturbation of g-type; the time span tk,tk+1 contains the work time tk,tk+ck and the rest time tk+ck,tk+1, and ck denotes the k-th noise width. Naturally, the noise widths satisfy 0≤ck≤tk+1−tk. The stochastic control input system is(7)dxt=ft,xtdt+gt,xtdBt,where(8)xt=I1Lt,I1Ht,A1t,I2Lt,I2Ht,A2t,…,InLt,InHt,AntT,fx=β1iΘtSit−γIiLt−μ1IiLt−ηIiLtβ2iΘtSit+γIiLt−μ2IiHt−ηIiHtμ1IiLt+μ2IiHt−ηAiti=1n,gt,xt=σxt,t∈tk,tk+ck,0,t∈tk+ck,tk+1,with i,k∈N, g1:t0,∞×Rn⟶Rn×m. For the aperiodically intermittent perturbation strategy, the start time and the noise width might be different, but the total perturbation time ratio should be fixed in the long term. Mathematically, there exists a positive scalar τ, such that the above time nodes satisfy the following assumption:(9)limn⟶∞∑k=0ncktn+1−t0=τ.Figure 2
The sketch of aperiodically intermittent control strategy.We callτ as the control time ratio. Moreover, we assume gt,0≡0 for stochastic stability analysis, which guarantees the existence of a trivial solution xt,t0,0≡0.Theorem 1.
If the aperiodically perturbed intensityσ satisfies σ2>2β1γ+β2γ+μ1+ηj2/τγ+η+μ1j−2η+μ2/τ, then the infected nodes will die out with probability one.Proof.
Define a matrixM:(10)M=β1γ+β2γ+μ1+ηγ+η+μ1j1p12p2⋯npn2p122p2⋯2npn⋮⋮⋱⋮np12np2⋯n2pn.
We can calculate thatρM=β1γ+β2γ+μ1+ηj2/γ+η+μ1j. Let ω=ω1,ω2,…,ωn be the eigenvector of M corresponding to ρM; then, we have(11)ω1,ω2,…,ωnρM=ω1,ω2,…,ωnM,where ωi>0,i=1,2,…,n. Define a Lyapunov function:(12)VI1L,I1H,I2L,I2L,…,InL,InH=ϖx¯a,where(13)ϖ=γω1γ+μ1+η,ω1,γω2γ+μ1+η,ω2,…,γωnγ+μ1+η,ωn,x¯=I1L,I1H,I2L,I2H,…,InL,InHT,and 0<a<1/2 is a constant. Using Itô formula, we obtain(14)ℒV=aϖx¯a−1ϖf¯x¯+aa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯,Vx¯=Vt0,x¯0+∫t0tℒVx¯sds+∫t0taϖx¯a−1ϖg¯x¯dBs,where(15)f¯x¯=β1ΘtS1−γI1L−μ1I1Lt−ηI1Ltβ2ΘtS1+γI1Lt−μ2I1Ht−ηI1Ht⋮nβ1ΘtSn−γInL−μ1InLt−ηInLtnβ2ΘtSn+γInLt−μ2InHt−ηInHtT,g¯x¯=σx¯,t∈tk,tk+ck,0,t∈tk+ck,tk+1.
Note thatf¯x¯ can be regarded as a linear growth function of Si; as Si≤Ni=Λ/η for t≥t0, we can derive that(16)∫t0tℒVx¯sds=∫t0tecs−t0aϖx¯a−1ϖf¯x¯dt+∫t0taa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯dt≤∫t0taϖx¯a−1M−μ2+ηEI1,I2,…,InTdt+∫t0taa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯dt,where E represents an elementary matrix or an identity matrix. By condition (11), the above inequation becomes(17)∫t0tℒVx¯sds≤∫t0taρM−μ2−ηϖx¯adt+∫t0taa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯dt.
Then, we discuss the timet in different time intervals. Obviously, there exists one positive integer m such that t∈tm,tm+1.(1)
Ift∈tm,tm+cm, then(18)∫t0taa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯dt=∫t0t0+c0aa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt+0+∫t1t1+c1aa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt+0+⋯+∫tmtaa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt=∫t0t0+c0aa−1σ2ϖx¯a2dt+0+∫t1t1+c1aa−1σ2ϖx¯a2dt+0+⋯+∫tmtaa−1σ2ϖx¯a2dt,which means(19)∫t0tℒVx¯sds=∫t0t0+c0ρM−η−μ2+12a−1σ2aω¯x¯adt+∫t0+c0t1ρM−η−μ2aω¯x¯adt+∫t1t1+c1ρM−η−μ2+12a−1σ2aω¯x¯adt+∫t1+c1t2ρM−η−μ2aω¯x¯adt+⋯+∫tmtρM−η−μ2+12a−1σ2aω¯x¯adt≤supt0≤s≤taω¯x¯aρM−η−μ2t−t0+12a−1σ2∑i=0m−1ci+t−tm.(2)
Ift∈tm+cm,tm+1, then(20)∫t0taa−1ϖx¯a−22Trg¯Tx¯ϖTϖg¯x¯dt=∫t0t0+c0aa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt+0+∫t1t1+c1aa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt+0+⋯+∫tmtm+cmaa−1ϖx¯a−22Trg¯1Tx¯ϖTϖg¯1x¯dt+0=∫t0t0+c0aa−1σ2ϖx¯a2dt+0+∫t1t1+c1aa−1σ2ϖx¯a2dt+0+⋯+∫tmtm+cmaa−1σ2ϖx¯a2dt+0,which means(21)∫t0tLVx¯sds=∫t0t0+c0ρM−η−μ2+12a−1σ2aω¯x¯adt+∫t0+c0t1ρM−η−μ2aω¯x¯adt+∫t1t1+c1ρM−η−μ2+12a−1σ2aω¯x¯adt+∫t1+c1t2ρM−η−μ2aω¯x¯adt+⋯+∫tmtm+cmρM−η−μ2+12a−1σ2aω¯x¯adt+∫tm+cmtρM−η−μ2aω¯x¯adt≤supt0≤s≤taω¯x¯aρM−η−μ2t−t0+12a−1σ2∑i=0mci.
Comparing Case (1) and Case (2), we can conclude fortm≤t<tm+1 that(22)ω¯x¯a≤ω¯x¯0a+supt0≤s≤taω¯x¯aρM−η−μ2tm+1−t0+12a−1σ2∑i=0mci+∫t0taϖx¯a−1ϖg¯x¯dBs.
Taking the expectation on both sides of the above inequation gives(23)Eω¯x¯a≤ω¯x¯0a+Esupt0≤s≤taω¯x¯aρM−η−μ2tm+1−t0+12a−1σ2∑i=0mci=ω¯x¯0a+aEsupt0≤s≤tω¯x¯aρM−η−μ2tm+1−t0+12a−1σ2∑i=0mci=ω¯x¯0a+atm+1−t0ρM−η−μ2+12a−1σ2τmEsupt0≤s≤tω¯x¯a=ω¯x¯0a+ζm,aEsupt0≤s≤tω¯x¯a,where ζm,a=atm+1−t0ρM−η−μ2+1/2a−1σ2τm. Since limm⟶∞τm=τ, there exists a positive integer m0 such that when m>m0,(24)1−ζm,aEω¯x¯a≤Eω¯x¯a−ζm,aEsupt0≤s≤tω¯x¯a≤ω¯x¯0a,which implies(25)Ex¯a≤Eω¯x¯aξa≤ω¯x¯0aξa1−ζm,a,where ξ=minγ/γ+η+μ1ω1,γ/γ+η+μ1ω2,…,γ/γ+η+μ1ωn. By Chebyshev inequality, for any ε>0, we have(26)Px¯≥ε≤Ex¯aεa≤ω¯x¯0aεξa1−ζm,a.
IfρM−η−μ2+1/2a−1σ2τm<0, from the definition of ζm,a, we can obtain ζm,a⟶−∞ when m⟶∞. Consequently, we have(27)limt⟶∞Px¯≥ε≤limt⟶∞ω¯x¯0aεξa1−ζm,a=limt⟶∞ω¯x¯0aεξa1−ζm,a=0.
Then, we deduce that(28)Plimt⟶∞x¯≥ε=0.
Sinceε is arbitrary, we obtain(29)Plimt⟶∞x¯=0=1.
Then, sendinga⟶0, we obtain the minimum control intensity expression:(30)σ¯2=2β1γ+β2γ+μ1+ηj2τγ+η+μ1j−2η+μ2τ.
Therefore, system (7) is almost surely stable if σ2 is bigger than the minimum control intensity. The proof is complete.
In Theorem1, we give the global stability of the malware-free equilibrium. It is worth to mention that we separate interaction terms in the stochastic model (7) by using the group-theoretic approaching method, and then the infected groups IiLt and IiHt can be considered separately. Calculating the Lyapunov exponents, we obtain the global stability of the malware-free equilibrium.Proof of Corollary 1.
Setσ=0; then, stochastic model (7) becomes deterministic model (5). The control intensity mathematical expression becomes(31)0>2β1γ+β2γ+μ1+ηj2τγ+η+μ1j−2η+μ2τ.
It implies(32)2Tτβ1γ+β2γ+μ1+ηj2δγ+η+μ1j>2η+μ2τ,which means R0<1. Applying Theorem 1, the malware-free equilibrium is globally asymptotically stable.
To get the endemic equilibrium solutionSi∗,IiL∗,IiH∗,Ai∗, the right side of system (1) should equal to zero. The solution definitely satisfies the restrictive condition on the total quality of nodes:(33)Si∗+IIL∗+IiH∗+Ai∗=Λη,for alli.
Combining and solving all the above equations, we can obtain an equation containingΘt and Si∗,IiL∗,IiH∗,Ai∗. To guarantee a nontrivial solution of it, R0 need to be bigger than one, and thus we obtain that there is an endemic equilibrium when R0>1.
Then, we prove the malware will be persistent whenR0>1. If R0>1, the derivative of function V becomes positive for Siti=1,2,…,n sufficiently close to Λ/η except when IiL=IiH=0i=1,2,…,n. Otherwise, system (1) reduces to dSi=Λ−ηSi, which implies Si⟶Λ/η as t⟶∞. This establishes the malware-free equilibrium’s unstable property. Since the necessary and sufficient condition for uniform persistence is equivalent to the malware-free equilibrium being unstable, system (1) is persistent.Remark 2.
Iftk+1−tk=T and ck=δ for all k∈N, system (7) becomes a periodic intermittent system. The minimum control intensity becomes(34)σ2>2Tβ1γ+β2γ+μ1+ηj2δγ+η+μ1j−2Tη+μ2δ.
This agrees with Theorem1 in Zhang et al. [41]. Our results can be regarded as a generalization of Zhang et al. [41].Remark 3.
According to the expression ofσ, the control intensity depends on the network topological structure and system parameters. Apparently, if the degree distribution of WSNs follows a power law (like a scale-free network) and the number of nodes is huge enough, then j2/j⟶∞, so the absence of control intensity, i.e., σ⟶∞, is observed. Moreover, the disappearance of malware is related to perturbation time ratios τ. Thus, we can enhance or weaken noise intensity to decrease or increase the corresponding control time ratios. Control strategy would be designed based on our theoretical results.


## 4. Numerical Simulation

The aperiodically intermittent noise controller can stabilize system (1) under the intensity σ. From the expression, control intensity is directly proportional to the contact rates β1,β2 and inversely proportional to the control time ratio. Using Milstein’s higher order method [47], the discretized equations for the system (7) are(35)Ii,k+1L=Ii,kL+β1iΘtSi−γIiLt−μ1IiLt−ηIiLtΔt+σIi,kLΔtξi+12σ2Ii,kL2ξi2−1Δt,Ii,k+1H=Ii,kH+β2iΘtSi+γIiLt−μ2IiHt−ηIiHtΔt+σIi,kHΔtξi+12σ2Ii,kH2ξi2−1Δt,Ai,k+1=Ai,k+μ1IiLt+μ2IiHt−ηAiΔt+Ai,kΔtξi+12σ2Ai,k2ξi2−1Δt,where ξi are independent random variables N (0, 1), i=1,2,…,n. Generally, we assume WSN is a scale-free network with the power law distribution Pj=aj3, where a satisfies ∑j=1naj3=1. We set the initial values to Si0=900,IiL0=50,Iih=50,Ai=0,i=1,2,…,n and n=50.Choosing parameter values in Table1, the reproduction number is(36)R0=β1γ+β2γ+μ1+ηj2η+μ2γ+η+μ1j=3.2678>1.Table 1
System parameters.
Ληβ1β2μ1μ2γ10.010.0090.010.00150.00160.01According to Corollary1, the malware is persistent. The top two pictures in Figure 3 show that the solution of system (1) converges to the endemic equilibrium. The last two pictures in Figure 3 show clearly that the number of infected nodes IkH and IkL converge to a positive constant.Figure 3
The time series and orbits of system (1) with R0>1. The malware persists in the end.To stabilize the above deterministic model, we choose different control intensitiesσ by adjusting control time to compare the stabilization effects.(i)
Let control time ratioτ=0.5 and initial value (900, 50, 50, 0, …, 900, 50, 50, 0). Generating the aperiodically intermittent time intervals randomly, we give two examples in the middle two pictures of Figure 4. We can calculate that σ2=0.04>0.0254; according to Theorem 1, IkH and IkL all tend to zero; Figure 4 shows clearly that the malware disappears.(ii)
Increase the control time ratio toτ=0.8; specific aperiodically intermittent time intervals are shown in the middle two pictures of Figure 5. We can obtain that σ2=0.04>0.0211, the condition in Theorem 1 is satisfied, and IkH and IkL all tend to zero. The top two and the last two pictures of Figure 5 show clearly that all the infected nodes go to zero.(iii)
Adding control time to full time. Figure6 shows the solution of system (7) converges to the malware-free equilibrium, IkH and IkL all tend to zero, and the malware nodes will disappear. Comparing Figures 4–6, we can conclude that the bigger the perturbation intensity is, the faster the steady speed is.Figure 4
The time series and orbits of system (7) with R0>1 and stochastic control. The top two pictures are the sample paths of S15t,I15Lt,I15Ht,A15t and S35t,I35Lt,I35Ht,A35t. The middle pictures are two different time strategies with τ=0.5 and σ=0.2. The last two pictures are sample paths of ILt and IHt with degree k=5,15,25,35,40,45. The malware will be extinct in the end.Figure 5
The time series and orbits of system (7) with R0>1 and stochastic control. The top two pictures are the sample paths of S15t,I15Lt,I15Ht,A15t and S35t,I35Lt,I35Ht,A35t. The middle pictures are two different time strategies with τ=0.8 and σ=0.2. The last two pictures are sample paths of ILt and IHt with degree k=5,15,25,35,40,45. The malware will be extinct in the end.Figure 6
The time series and orbits of system (7) with R0>1 and stochastic control. The top two pictures are the sample paths of S15t,I15Lt,I15Ht,A15t and S35t,I35Lt,I35Ht,A35t. The last two pictures are sample paths of ILt and IHt with degree k=5,15,25,35,40,45. The malware will be extinct in the end.


## 5. Conclusion

Based on network topology and malware propagation property, we established a heterogeneous wireless sensor network model which considers the influence of energy consumption to node communication. The basic reproduction numberR0 has been given which determines the extinction or persistence of malware: if R0<1, the malware will die out; if R0>1, the malware will be persistent. To control malware spreading, we design a kind of aperiodically intermittent stochastic noise controller, starting time and control length of which are arbitrary. We can control steady rate autonomously by adjusting the work width. Using the straightening operator and Markov inequality, we calculate the specific stochastic noise intensity σ, which is inversely proportional to intermittent time. When stochastic noise controller’s intensity is greater than σ, malware propagation will be controlled with probability 1. Otherwise, the malware program will destroy the wireless sensor networks. This stochastic stabilization method presents a new theoretical strategy for wireless sensor network security: stochastic noise controller. Numerical simulations are given to illustrate the stabilization effect of stochastic noise.



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