# Dynamics and Control of Worm Epidemic Based on Mobile Networks by SEIQR-Type Model with Saturated Incidence Rate

Authors: Rui Hu; Qingwu Gao; Bairong Wang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The mobile networks have increasingly facilitated our daily life but are also breeding grounds for malicious worms, which are considered as the main threat to cyber security. The purpose of this paper is to analyze the dynamics of worm propagation and to control the worm epidemic based on mobile-phone networks. Accordingly, we establish an SEIQR-type model to explore the worm epidemic with saturated incidence rate. This paper shows that if the basic reproduction number is less than 1, the worm-free equilibrium is asymptotically stable, and the epidemic of worm will eventually disappear and remain under control; in addition, if the basic reproduction number is greater than 1, the asymptotical stability of worm-existence equilibrium is derived to imply that the epidemic will remain persistent and uncontrollable. Our results give new insights to mobile network security, namely, that is predicting the worm spreading tendency, identifying the epidemic control strategies, and estimating the worm popularity level. Numerical experiments are conducted to show the rationality of our obtained results and the effectiveness of the control strategies.


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## Body

## 1. Introduction

As everyone knows, the mobile devices, e.g., smartphones, tablets, and laptops, are increasingly pervasive in the world today but have been also both the target and victim of network worm attackers. For example, because of its convenience, the Wi-Fi is currently the most popular way for mobile devices to access to the Internet, which has considerably facilitated our daily life. Unfortunately, it exposes these devices to worm attackers from mobile Internet. A Wi-Fi worm, named Chameleon and appeared in 2014, could spread like an airborne diseases [1]. However, the majority of mobile devices do not have any effective methods to prevent worm attacks. Once a mobile device has been compromised by worms, it can cause great losses to users, including data leakage, system damage, and financial losses [2]. Hence, as considered by network experts, the outbreak of malicious worms is one of the most serious risks to network security, property, and function.Many researchers have proposed mathematical models to explore the dynamic behaviors of computer worms and biological communities [3–11]. Due to the high similarity between computer worms and biological virus, a series of epidemic models are developed to study the transmission of infectious disease in the host population [12–21]. In the literature mentioned above, we find that the bilinear incidence rate plays an important part in describing the dynamics of virus/worm transmission [4, 22]. However, in most practical situations, this bilinear rate is impractical so as to considerably restrict the availability of the results obtained. Thus, some nonlinear rates of incidence were introduced by the literature [12, 15, 19–21], among which the saturated rate can tend to its saturation level with the increase of the number of infected or susceptible individuals. Obviously, the saturated rate is more realistic as it can present the inhibition effect of susceptible individuals and the crowding effect of infectious individuals and can also guarantee the boundness of contact rate. In this study, a mathematical model with the saturated rate of incidence is taken into consideration to study the worm spreading through mobile networks.Notice that the above existing models mainly aim at the computer worms spreading through fixed networks, and because of the differences between computers and mobile devices, they could not be applied directly in mobile networks, especially in Wi-Fi ones and 5G ones. Once a mobile device connects to the networks via Wi-Fi, the Wi-Fi base station can control (continue or disrupt) the connection; then, the base station can be adopted to disrupt the connection between the infected devices by worms and other devices. If the connection is cut off, the infected devices are quarantined so that the susceptible devices can be protected from the attack of worms. Hence, in this paper, we consider an SEIQR-type (susceptible-exposed-infectious-quarantined-recovered) model, proposed by Xiao et al. [22], to describe the dynamics and characteristics of the worm propagation based on Wi-Fi networks. Compared with the previous papers, the novel ideas of our paper are that (a) the SEIQR model is along with the saturated rate of incidence; (b) the recovery process from the susceptible group to the recovered group is considered thanks to some sufficient defense mechanisms or antivirus treatments; (c) the dysfunction of devices not only relates to natural death but also depends on worm-related death, because in the real world, the crash of most devices results from the attack of malicious worms; and (d) the worm epidemic intervention strategies are established by reducing its size and speed.In this paper, we analyze the stability of worm-free and worm-existence equilibria and identify the control strategies of worm epidemic where an optimal strategy is devised to minimize the cost of quarantine intervention and the number of infected devices while maximizing the number of recovered devices. Remark that the worm-free equilibrium stability provides the rationale of worm epidemic intervention strategies; while the worm-existence equilibrium stability can predict the worm spreading tendency for a long period and estimate the worm popularity level by the final scope of infected devices. Furthermore, we carry out a series of numerical experiment in order to demonstrate the rationality of our obtained results and the effectiveness of the control strategies.The rest of our paper is constructed as follows: in Section2, we introduce the SEIQR-type model with saturated incidence rate; in Section 3, we analyze the stabilities of worm-free and worm-existence equilibria; in Section 4, we implement the numerical analysis and experiments; in Section 5, we identify the worm epidemic intervention strategies; in Section 6, we conclude and discuss the paper; and in Appendix section, we prove the main results obtained.


## 2. SEIQR-Type Worm Propagation Model

In the paper, we consider the general SEIQR-type worm propagation model, in which the mobile devices are divided into five groups: susceptible group (S) includes the devices uninfected by malicious worms but vulnerable to worm attacks; exposed group (E) includes the devices exposed to and infected by the worms but not infectious; infectious group (I) includes the exposed and infectious devices; quarantine group (Q) includes the infectious and quarantined devices; recovered group (R) includes the recovered devices with immunity against the worms. Figure 1 shows the state transition diagram with the involved modeling parameters, Table 1 lists the baseline parameters and their initial values, and Table 2 lists the state transitions and their rates.Figure 1
State transition diagram of the SEIQR-type model.Table 1
Explanations and initial values of modeling parameter.
ParametersExplanationsInitial valuesN (t)Device number at timetN (0) =75,000S (t)Number of susceptible devices at timetNot fixedE (t)Number of exposed devices at timetNot fixedI (t)Number of infectious devices at timetNot fixedQ (t)Number of quarantined devices at timetQ (0) = 0R (t)Number of recovered devices at timetS (0) = 0ΛRate of recruitment0.75μRate of natural death0.00001θRate of worm-related death0.001αParameter of saturated incidence rate0.8βInfection rate in groupS0.053ψRate of transition fromS to RNot fixedηRate of transition fromE to I0.008εRate of transition fromE to R0.0008γRate of transition fromI to R0.05ξRate of transition fromI to Q0.05φRate of transition fromQ to R0.005Table 2
State transition events and transition rates.
Transition eventsRate of transitionS⟶EβStIt/1+αItS⟶RψStE⟶IηEtE⟶RεEtI⟶QξItQ⟶RφQtS⟶DysfunctionμStE⟶DysfunctionμEtI⟶Dysfunctionθ+μItQ⟶Dysfunctionθ+μQtR⟶DysfunctionμRtRecruitment⟶SΛAs shown in Figure1 and Tables 1 and 2, we present the state transitions of the SEIQR-type model in detail as follows:(1)
Since the susceptible devices are of no immunity against malicious worms, they can transit to the exposed group at the saturated incidence rateβSI/1+αI. Note that βI/1+αI goes to its saturation level as I increases, βI denotes the force of worm infection, and 1/1+αI denotes the crowding effect of infectious ones and guarantees the boundness of contact rate by an appropriate value for α.(2)
The exposed devices become infectious at rateη, where 1/η denotes the mean value of latent period. If .., the latent period tends to 0, which follows that the SEIQR model can reduce to a SIQR one.(3)
Due to the quarantine intervention implemented on the infectious group, the infectious ones can transit to the quarantined group at rateξ. As ξ=0, there is no quarantine intervention, and then the SEIQR model reduces to a SEIR one [13, 19].(4)
Thanks to some sufficient defense mechanisms or antivirus treatments, the susceptible, exposed, infectious, and quarantined devices transit to the recovered group at ratesψ, ε, γ, and φ, respectively. Note that 1/φ denotes the mean value of quarantine period. If φ⟶∞, then the quarantine period tends to 0, which follows that the SEIQR model reduces to an SEIR one [13, 19].(5)
The dysfunction of mobile device can take place in all groups due to not only the natural death but also the worm-related death. The dysfunction replacement is taken to the susceptible group at rateΛ.Therefore, the general SEIQR-type model considered in the paper is formulated by the following system of nonlinear ordinary differential equations:(1)S′t=Λ−βStIt1+αIt−ψ+μSt,E′t=βStIt1+αIt−η+ε+μEt,I′t=ηEt−ξ+γ+μ+θIt,Q′t=ξIt−φ+μ+θQt,R′t=ψSt+εEt+γIt+φQt−μRt.The total number of mobile devices at timet, Nt, satisfies that(2)Nt=St+Et+It+Qt+Rt.By a summation of all the equations of system (1), we have(3)N′t≤Λ−μNt,which implies that(4)Nt≤Λμ+N0−Λμe−μt,limt⟶∞Nt=Λμ.Hence, we define system (1)’s feasible region as(5)Ω=S,E,I,Q,R∈ℝ5|0≤S+E+I+Q+R≤Λμ.Now, we make a nonnegativity analysis to ensure the plausibility of SEIQR-type model (1). By the first equation in (1), we have(6)S′t≤Λ−ψ+μSt,which implies that(7)St≤Λψ+μ+S0−Λψ+μe−ψ+μt,and then in a long run,(8)St≤Λψ+μ≕Δ1.Further by the second equation in (1), we have(9)E′t≤βΔ1α−η+ε+μEt,which implies that(10)Et≤βΔ1αη+ε+μ+E0−βΔ1αη+ε+μe−η+ε+μt,and then in a long run,(11)Et≤βΔ1αη+ε+μ≕Δ2.Arguing as above, it follows that(12)It≤ηΔ2ξ+γ+μ+θ=Δ3,Qt≤ξΔ3φ+μ+θ=Δ4,Rt≤ψΔ1+εΔ2+γΔ3+φΔ4μ.Thus, by the above results, we get a proposition below.Proposition 1.
System (1) has solutions satisfying St>0, Et>0, It>0, Qt>0, and Rt>0 for all t>0, and its feasible region Ω is an attracting and positively invariant set.


## 3. Dynamics Analysis

In this section, we analyze the stabilities of worm-free and worm-existence equilibria. Firstly, we calculate the equilibrium points and the basic reproduction number. Secondly, we study the stability of worm-free equilibrium which can establish worm epidemic intervention strategies. Thirdly, we present the system uniform persistence. Finally, we study the stability of worm-existence equilibrium which can predict the worm spreading tendency and estimate the epidemic popularity level.


### 3.1. Basic Reproduction Number and Equilibrium Points

To analyze the dynamics of the SEIQR-type model described by (1), we determine the equilibria satisfying(13)S′t=0,E′t=0,I′t=0,Q′t=0,R′t=0.ByI′t=0, it follows that(14)E=ξ+γ+μ+θIη,which is substituted into E′t=0 to get that(15)βS1+αI−η+ε+μξ+γ+μ+θηI=0.Hence, we have(16)I=0,orI>0,S=η+ε+μξ+γ+μ+θ1+αIβη.IfI=0, we obtain the unique worm-free equilibrium point of system (1) as(17)P0=S0,E0,I0,Q0,R0=Λψ+μ,0,0,0,ψμS0.IfI>0, we derive by substituting S in (16) into S′t=0 that(18)η+ε+μξ+γ+μ+θβ+αψ+μI−βηΛ−ψ+μη+ε+μξ+γ+μ+θ=0.Define the basic reproduction numberR0 as(19)R0=βηΛψ+μη+ε+μξ+γ+μ+θ.Clearly, ifR0≤1, the worm-free equilibrium is the unique equilibrium; if R0>1, the worm-existence equilibrium is the unique positive equilibrium P∗=S∗,E∗,I∗,Q∗,R∗, where(20)I∗=βηΛ−ψ+μη+ε+μξ+γ+μ+θη+ε+μξ+γ+μ+θβ+αψ+μ,and then S∗, E∗, Q∗, and R∗ satisfy(21)S∗=η+ε+μξ+γ+μ+θ1+αI∗βη,E∗=ξ+γ+μ+θI∗η,Q∗=ξI∗φ+μ+θ,R∗=ψS∗+εE∗+γI∗+φQ∗μ.Hence, we give the following result regarding the equilibrium points.Theorem 1.
IfR0≤1, the worm-free equilibrium P0 is the unique equilibrium point satisfying (17); if R0>1, the worm-existence equilibrium P∗ is the unique positive equilibrium point satisfying (20) and (21).Remark 1.
The worm-existence equilibrium pointS∗,E∗,I∗,Q∗,andR∗ can access the final scales of mobile devices in all the groups for a long period. Especially, the final scale of infected devices, namely, E∗+I∗+Q∗, can estimate the popularity level of worm epidemic.
In fact, the basic reproduction numberR0 is an important concept in virus/worm epidemic dynamics, which is “the average number of secondary cases generated by one primary case at the start of the epidemic in a completely susceptible population” [23, 24] and can also be obtained by the next generation matrix method [23] or the next generation operator theory [24]. From Theorem 1, the sharp threshold value, R0, can completely determine the existence of two equilibrium points.


### 3.2. Stability of Worm-Free Equilibrium

In this section, we study the local and global stabilities for worm-free equilibriumP0, and their proofs can be found in Appendixes A.1 and A.2.Theorem 2.
IfR0≤1, the worm-free equilibrium P0 is locally asymptotically stable. Otherwise, P0 is an unstable point.Theorem 3.
IfR0≤1, the worm-free equilibrium P0 is globally asymptotically stable. Otherwise, P0 is an unstable point.Combining Theorems2 and 3, we know that the local and global stabilities of worm-free equilibrium are equivalent.Remark 2.
From Theorem2 and 3, we conclude that if R0≤1, the exposed, infectious, and quarantined devices all tend to 0, namely, that the infected devices tend to 0. Accordingly, R0≤1 indicates that the epidemic of worm will eventually disappear and remain under control, and then to prevent and control the epidemic, some related intervention strategies should be adopted to make R0 smaller than 1, and see those in Section 5. Clearly, Theorems 2 and 3 provide the theoretical basis of those worm epidemic intervention strategies.
We find out from Theorems2 and 3 and Remark 2 that the basic reproduction number R0, as a sharp threshold value, will play a critical role to control the worm epidemic based on mobile networks, because the threshold value R0 can completely determine the local and global dynamics for worm-free equilibrium point P0 and also can govern if the mobile devices infected by worms become extinct for time locally and globally.


### 3.3. Uniform Worm Persistence

With the use of the acyclicity theorem [25], we present the uniform persistence of system (1) if R0>1, whose proof is given in Appendix A.3.Definition 1 (see [26]).
System (1) is uniformly persistent in Ω, if there is c∈0,1 such that any solution St,Et,It,Qt,andRt of (1) with initial value satisfies(22)minliminft⟶∞St,liminft⟶∞Et,liminft⟶∞It,liminft⟶∞Qt,liminft⟶∞Rt≥c.Theorem 4.
IfR0>1, the system (1) is uniform persistent in the bounded set Ω.Remark 3.
The uniform persistence of (1) in Ω means that there exists a compact set K⊂Ω which is absorbent for system (1) (see [27]).


### 3.4. Stability of Worm-Existence Equilibrium

We now perform the stability analysis for the worm-existence equilibriumP∗ and prove its related main results in Appendixes A.4 and A.5. From an epidemiological point of view; P∗ is more necessary to study than its counterpart, i.e., the worm-free equilibrium P0.Theorem 5.
IfR0>1, the worm-existence equilibrium P∗ is locally asymptotically stable.It is well-known that the local stability forP∗ may be insignificant in practice, because it only assures the stability of microvariations relative to its initial state of P∗. Hence, we turn to the global stability for P∗ by the Li–Muldowney geometric approach [28].Theorem 6.
IfR0>1, the worm-existence equilibrium P∗ is globally asymptotically stable.Remark 4.
The global stability forP∗∈intΩ is equivalent to that the set Ω−S,E,I,Q,R|I=0 is a globally stable set for P∗.Remark 5.
From Theorem5 and 6, the stability for P∗ shows that if R0>1, all the groups of mobile devices will tend to S∗, E∗, I∗, Q∗, and R∗, respectively, which are those in (20) and (21). Accordingly, we conclude that R0>1 assures that the epidemic of mobile network-based worm remains persistent and uncontrollable, where we can predict the worm spreading tendency in a long run and access the epidemic popularity level by the final scope of infected devices (i.e., E∗+I∗+Q∗).
From the arguments of Theorem5 and 6 and Remark 5, we also see that the basic reproduction number R0 is an exact threshold value which guarantees the local and global stabilities for P∗ and governs whether the network-based worm keeps always transmitting locally and globally.


## 3.1. Basic Reproduction Number and Equilibrium Points

To analyze the dynamics of the SEIQR-type model described by (1), we determine the equilibria satisfying(13)S′t=0,E′t=0,I′t=0,Q′t=0,R′t=0.ByI′t=0, it follows that(14)E=ξ+γ+μ+θIη,which is substituted into E′t=0 to get that(15)βS1+αI−η+ε+μξ+γ+μ+θηI=0.Hence, we have(16)I=0,orI>0,S=η+ε+μξ+γ+μ+θ1+αIβη.IfI=0, we obtain the unique worm-free equilibrium point of system (1) as(17)P0=S0,E0,I0,Q0,R0=Λψ+μ,0,0,0,ψμS0.IfI>0, we derive by substituting S in (16) into S′t=0 that(18)η+ε+μξ+γ+μ+θβ+αψ+μI−βηΛ−ψ+μη+ε+μξ+γ+μ+θ=0.Define the basic reproduction numberR0 as(19)R0=βηΛψ+μη+ε+μξ+γ+μ+θ.Clearly, ifR0≤1, the worm-free equilibrium is the unique equilibrium; if R0>1, the worm-existence equilibrium is the unique positive equilibrium P∗=S∗,E∗,I∗,Q∗,R∗, where(20)I∗=βηΛ−ψ+μη+ε+μξ+γ+μ+θη+ε+μξ+γ+μ+θβ+αψ+μ,and then S∗, E∗, Q∗, and R∗ satisfy(21)S∗=η+ε+μξ+γ+μ+θ1+αI∗βη,E∗=ξ+γ+μ+θI∗η,Q∗=ξI∗φ+μ+θ,R∗=ψS∗+εE∗+γI∗+φQ∗μ.Hence, we give the following result regarding the equilibrium points.Theorem 1.
IfR0≤1, the worm-free equilibrium P0 is the unique equilibrium point satisfying (17); if R0>1, the worm-existence equilibrium P∗ is the unique positive equilibrium point satisfying (20) and (21).Remark 1.
The worm-existence equilibrium pointS∗,E∗,I∗,Q∗,andR∗ can access the final scales of mobile devices in all the groups for a long period. Especially, the final scale of infected devices, namely, E∗+I∗+Q∗, can estimate the popularity level of worm epidemic.
In fact, the basic reproduction numberR0 is an important concept in virus/worm epidemic dynamics, which is “the average number of secondary cases generated by one primary case at the start of the epidemic in a completely susceptible population” [23, 24] and can also be obtained by the next generation matrix method [23] or the next generation operator theory [24]. From Theorem 1, the sharp threshold value, R0, can completely determine the existence of two equilibrium points.


## 3.2. Stability of Worm-Free Equilibrium

In this section, we study the local and global stabilities for worm-free equilibriumP0, and their proofs can be found in Appendixes A.1 and A.2.Theorem 2.
IfR0≤1, the worm-free equilibrium P0 is locally asymptotically stable. Otherwise, P0 is an unstable point.Theorem 3.
IfR0≤1, the worm-free equilibrium P0 is globally asymptotically stable. Otherwise, P0 is an unstable point.Combining Theorems2 and 3, we know that the local and global stabilities of worm-free equilibrium are equivalent.Remark 2.
From Theorem2 and 3, we conclude that if R0≤1, the exposed, infectious, and quarantined devices all tend to 0, namely, that the infected devices tend to 0. Accordingly, R0≤1 indicates that the epidemic of worm will eventually disappear and remain under control, and then to prevent and control the epidemic, some related intervention strategies should be adopted to make R0 smaller than 1, and see those in Section 5. Clearly, Theorems 2 and 3 provide the theoretical basis of those worm epidemic intervention strategies.
We find out from Theorems2 and 3 and Remark 2 that the basic reproduction number R0, as a sharp threshold value, will play a critical role to control the worm epidemic based on mobile networks, because the threshold value R0 can completely determine the local and global dynamics for worm-free equilibrium point P0 and also can govern if the mobile devices infected by worms become extinct for time locally and globally.


## 3.3. Uniform Worm Persistence

With the use of the acyclicity theorem [25], we present the uniform persistence of system (1) if R0>1, whose proof is given in Appendix A.3.Definition 1 (see [26]).
System (1) is uniformly persistent in Ω, if there is c∈0,1 such that any solution St,Et,It,Qt,andRt of (1) with initial value satisfies(22)minliminft⟶∞St,liminft⟶∞Et,liminft⟶∞It,liminft⟶∞Qt,liminft⟶∞Rt≥c.Theorem 4.
IfR0>1, the system (1) is uniform persistent in the bounded set Ω.Remark 3.
The uniform persistence of (1) in Ω means that there exists a compact set K⊂Ω which is absorbent for system (1) (see [27]).


## 3.4. Stability of Worm-Existence Equilibrium

We now perform the stability analysis for the worm-existence equilibriumP∗ and prove its related main results in Appendixes A.4 and A.5. From an epidemiological point of view; P∗ is more necessary to study than its counterpart, i.e., the worm-free equilibrium P0.Theorem 5.
IfR0>1, the worm-existence equilibrium P∗ is locally asymptotically stable.It is well-known that the local stability forP∗ may be insignificant in practice, because it only assures the stability of microvariations relative to its initial state of P∗. Hence, we turn to the global stability for P∗ by the Li–Muldowney geometric approach [28].Theorem 6.
IfR0>1, the worm-existence equilibrium P∗ is globally asymptotically stable.Remark 4.
The global stability forP∗∈intΩ is equivalent to that the set Ω−S,E,I,Q,R|I=0 is a globally stable set for P∗.Remark 5.
From Theorem5 and 6, the stability for P∗ shows that if R0>1, all the groups of mobile devices will tend to S∗, E∗, I∗, Q∗, and R∗, respectively, which are those in (20) and (21). Accordingly, we conclude that R0>1 assures that the epidemic of mobile network-based worm remains persistent and uncontrollable, where we can predict the worm spreading tendency in a long run and access the epidemic popularity level by the final scope of infected devices (i.e., E∗+I∗+Q∗).
From the arguments of Theorem5 and 6 and Remark 5, we also see that the basic reproduction number R0 is an exact threshold value which guarantees the local and global stabilities for P∗ and governs whether the network-based worm keeps always transmitting locally and globally.


## 4. Performance Evaluation

In this section, we carry out numerical simulations to verify the feasibility of the dynamics of worm propagation via the SEIQR-type model.


### 4.1. Simulation Setting

In the realistic scenario, it is quite hard for researchers to get the corresponding data of worm traffic traces. Even for some available data from MIT or CAIDA (http://www.caida.org), the legitimate flow of traffic traces is not contained since these traces have been intentionally filtered out before they are available. Hence, we could not easily determine the necessary and appropriate parameters of the constructed SEIQR-type model. However, in the paper, we plan to choose the same parameter values as those in [29] to fit a real worm epidemic based on mobile networks.As a typical example, we take Slammer worm in this experiment. As one of the fastest spreading worms, Slammer had attacked75,000 devices at the early stage after its outbreak in January 2003 [30]. Thus, the total number of devices is set to be 75,000. The infection rate β is a key parameter that determines the spreading speed of worms among mobile devices. Consider that Slammer is a bandwidth-limited worm with an average scan rate s=4,000 scans/second [31]; then, by the same method in [32], we get the worm-infection rate as(23)β=sN≈0.053.The rate of natural death isμ=0.00001, and the rate of worm-related death is θ=0.001. The rates of state transitions from E to I, from I to R, and from Q to R are η=0.008, γ=0.05, and φ=0.005, respectively. These parameters come from the average value of more than 10 simulation operations [31].The rest parameters involved of the SEIQR-type model are determined as follows. According to the definition of regionΩ, the recruitment rate is Λ=0.75. Given the quarantine intervention on the group of infectious devices, the quarantine rate is ξ=0.05 [22]. Based on the fact that the average value of recovery period from the exposed group is approximately 1,200 seconds [7], we get the rate of recovery from group E to group R as(24)ε=11200≈0.0008.Due to one or more security countermeasures which have been taken initially for the susceptible devices to prevent the worm attack, the recovery rate from states S toR is set to ψ=0.5. Finally, we set the parameter α of the saturated incidence rate as 0.8 [29].


### 4.2. Sensitivity Analysis for Parameterα of Saturated Incidence Rate

In this section, we make a sensitivity analysis for the constant parameterα of the saturated incidence rate by a numerical experiment, where we choose α=0, 0.8, and 2, and the rest parameters involved as the same as the ones in Section 4.1. Please see Figure 2, which gives the impact of α on the numbers of infectious and recovered devices. More specifically, we observe from Figure 2(a) that as the parameter α increases, the number of infectious devices decreases and the duration of worm epidemic becomes shorter (see Figure 2(a)), and the number of recovered devices increases at a higher speed (see Figure 2(b)), which shows that increasing the parameter α can be taken as an effectively approach to control the worm epidemic based on mobile networks.Figure 2
Sensitivity analysis of parameter in saturated incidence rate.
(a)(b)Especially, if the parameterα=0, in other words, the SEIQR-type model considered is accompanied by a bilinear incidence rate rather than a saturated incidence rate, we know that by Figure 2 that the mobile network-based worm can infect much more devices and its epidemic keeps ongoing for a longer period, and the devices becoming recovered from all groups are less and reach their maximum size in an even longer period. Obviously, the saturated rate is more effective and reasonable than the bilinear rate to control the worm epidemic based on mobile networks.


### 4.3. Dynamics of Worm-Free and Worm-Existence Equilibria

In this section, we present simulation results to describe the tendency of worm propagation and then to validate the plausibility of the obtained results.Firstly, we set the initial numbers in all device groups asS0=45000, E0=20000, I0=10000, Q0=0, and R0=0, while the modeling parameters as the same as those in Section 4.1. Then, using equation (17), P0=1.5,0,0,0,74998.5, and using equation (19), R0=0.715<1. Based on Theorems 2 and 3, P0 is locally and globally asymptotically stable, respectively, implying that as t⟶∞, St, and Rt will tend to steady states 1.5 and 74998.5, while Et, It, and Et will all tend to 0. Hence, the final size of infected devices, E0+I0+Q0, is 0, which tell us that the worm will gradually become extinct and its epidemic remain under control in the end. As seen in Figure 3, the worm propagation tendency is fairly depressive for a long period, and the final scope of infected devices will vanish eventually, which is in accordance with the results of Theorems 2 and 3.Figure 3
Dynamics of the worm-free equilibrium.
(a)(b)Secondly, all the parameters and initial numbers are set as those in the first part of this section except forψ=0.05. By equations (20) and (21), P∗=7.64,41.67,3.3,27.45,71758.6, and by equation (19), R0=7.146>1. Accordingly, we conclude from Theorems 5 and 6 that P∗ is locally and globally stable, namely, that as t⟶∞, St, Et, It, Qt, and Rt will tend to 7.64, 41.67, 3.3, 27.45, and 71758.6, respectively. Please see Figure 4, which gives the worm propagation trend for a long period. Particularly, we can get the final size of the infected devices, i.e., E∗+I∗+Q∗=72.42, to evaluate the worm epidemic popularity level. In other words, the worm epidemic remains always propagating within a population of about 72.42. From Figure 4(a), we get the rough curves of St,Et,It,Qt, and Rt, and further from Figure 4(b), we get the tendency of St,Et,It, and Qt in more detail. The numerical results of Figure 4 are in accordance with Theorems 5 and 6.Figure 4
Dynamics of worm-existence equilibrium.
(a)(b)Thirdly, as shown in Figures3 and 4, the infected devices will eventually vanish or approach to a negligible size, and nearly all groups of devices will get recovered in the end, which suggests that the worm epidemic does exist in mobile networks but be of negligible scale owing to the quarantine strategy and other measures taken to the susceptible and infected devices. Consequently, these simulations validate the proposed SEIQR-type model in defending against malicious worms.


## 4.1. Simulation Setting

In the realistic scenario, it is quite hard for researchers to get the corresponding data of worm traffic traces. Even for some available data from MIT or CAIDA (http://www.caida.org), the legitimate flow of traffic traces is not contained since these traces have been intentionally filtered out before they are available. Hence, we could not easily determine the necessary and appropriate parameters of the constructed SEIQR-type model. However, in the paper, we plan to choose the same parameter values as those in [29] to fit a real worm epidemic based on mobile networks.As a typical example, we take Slammer worm in this experiment. As one of the fastest spreading worms, Slammer had attacked75,000 devices at the early stage after its outbreak in January 2003 [30]. Thus, the total number of devices is set to be 75,000. The infection rate β is a key parameter that determines the spreading speed of worms among mobile devices. Consider that Slammer is a bandwidth-limited worm with an average scan rate s=4,000 scans/second [31]; then, by the same method in [32], we get the worm-infection rate as(23)β=sN≈0.053.The rate of natural death isμ=0.00001, and the rate of worm-related death is θ=0.001. The rates of state transitions from E to I, from I to R, and from Q to R are η=0.008, γ=0.05, and φ=0.005, respectively. These parameters come from the average value of more than 10 simulation operations [31].The rest parameters involved of the SEIQR-type model are determined as follows. According to the definition of regionΩ, the recruitment rate is Λ=0.75. Given the quarantine intervention on the group of infectious devices, the quarantine rate is ξ=0.05 [22]. Based on the fact that the average value of recovery period from the exposed group is approximately 1,200 seconds [7], we get the rate of recovery from group E to group R as(24)ε=11200≈0.0008.Due to one or more security countermeasures which have been taken initially for the susceptible devices to prevent the worm attack, the recovery rate from states S toR is set to ψ=0.5. Finally, we set the parameter α of the saturated incidence rate as 0.8 [29].


## 4.2. Sensitivity Analysis for Parameterα of Saturated Incidence Rate

In this section, we make a sensitivity analysis for the constant parameterα of the saturated incidence rate by a numerical experiment, where we choose α=0, 0.8, and 2, and the rest parameters involved as the same as the ones in Section 4.1. Please see Figure 2, which gives the impact of α on the numbers of infectious and recovered devices. More specifically, we observe from Figure 2(a) that as the parameter α increases, the number of infectious devices decreases and the duration of worm epidemic becomes shorter (see Figure 2(a)), and the number of recovered devices increases at a higher speed (see Figure 2(b)), which shows that increasing the parameter α can be taken as an effectively approach to control the worm epidemic based on mobile networks.Figure 2
Sensitivity analysis of parameter in saturated incidence rate.
(a)(b)Especially, if the parameterα=0, in other words, the SEIQR-type model considered is accompanied by a bilinear incidence rate rather than a saturated incidence rate, we know that by Figure 2 that the mobile network-based worm can infect much more devices and its epidemic keeps ongoing for a longer period, and the devices becoming recovered from all groups are less and reach their maximum size in an even longer period. Obviously, the saturated rate is more effective and reasonable than the bilinear rate to control the worm epidemic based on mobile networks.


## 4.3. Dynamics of Worm-Free and Worm-Existence Equilibria

In this section, we present simulation results to describe the tendency of worm propagation and then to validate the plausibility of the obtained results.Firstly, we set the initial numbers in all device groups asS0=45000, E0=20000, I0=10000, Q0=0, and R0=0, while the modeling parameters as the same as those in Section 4.1. Then, using equation (17), P0=1.5,0,0,0,74998.5, and using equation (19), R0=0.715<1. Based on Theorems 2 and 3, P0 is locally and globally asymptotically stable, respectively, implying that as t⟶∞, St, and Rt will tend to steady states 1.5 and 74998.5, while Et, It, and Et will all tend to 0. Hence, the final size of infected devices, E0+I0+Q0, is 0, which tell us that the worm will gradually become extinct and its epidemic remain under control in the end. As seen in Figure 3, the worm propagation tendency is fairly depressive for a long period, and the final scope of infected devices will vanish eventually, which is in accordance with the results of Theorems 2 and 3.Figure 3
Dynamics of the worm-free equilibrium.
(a)(b)Secondly, all the parameters and initial numbers are set as those in the first part of this section except forψ=0.05. By equations (20) and (21), P∗=7.64,41.67,3.3,27.45,71758.6, and by equation (19), R0=7.146>1. Accordingly, we conclude from Theorems 5 and 6 that P∗ is locally and globally stable, namely, that as t⟶∞, St, Et, It, Qt, and Rt will tend to 7.64, 41.67, 3.3, 27.45, and 71758.6, respectively. Please see Figure 4, which gives the worm propagation trend for a long period. Particularly, we can get the final size of the infected devices, i.e., E∗+I∗+Q∗=72.42, to evaluate the worm epidemic popularity level. In other words, the worm epidemic remains always propagating within a population of about 72.42. From Figure 4(a), we get the rough curves of St,Et,It,Qt, and Rt, and further from Figure 4(b), we get the tendency of St,Et,It, and Qt in more detail. The numerical results of Figure 4 are in accordance with Theorems 5 and 6.Figure 4
Dynamics of worm-existence equilibrium.
(a)(b)Thirdly, as shown in Figures3 and 4, the infected devices will eventually vanish or approach to a negligible size, and nearly all groups of devices will get recovered in the end, which suggests that the worm epidemic does exist in mobile networks but be of negligible scale owing to the quarantine strategy and other measures taken to the susceptible and infected devices. Consequently, these simulations validate the proposed SEIQR-type model in defending against malicious worms.


## 5. Control Strategies on Worm Epidemic

To control biological viruses and network worms, the optimal control strategy is a frequently-used mathematical approach [33], and its related numerical simulations and region plots have also been considered in tails [33, 34]. So in what follows, we firstly construct the optimal control strategy based on Pontryagin’s Minimum Principle [35] to minimize the cost of quarantine intervention and the number of infected devices as well as maximize the number of recovered devices, and secondly we establish other control strategies with the use of Theorems 2 and 3 to control the worm epidemic based on mobile networks. Besides, a series of corresponding numerical simulations are implemented to validate the effectiveness of control strategies mentioned above.


### 5.1. The Optimal Control Strategy

In this section, the optimal control strategy is constructed for the SEIQR-type model described by system (1), where a control variable ut, 0≤ut≤1, is chosen to denote the inhibiting effect of quarantine strategy on the worm epidemic. Especially, when ut=1, the quarantine strategy is perfectly effective; when ut=0, the quarantine strategy has no effect at all. Hence, the optimal control problem is formulated as(25)Ju=∫t0tfKu2+Et+It−Rtdt,where the parameter K≥0 denotes the weight of the benefit of quarantine cost, and this problem is subject to the following state system:(26)S′t=Λ−βStIt1+αIt−ψ+μSt,E′t=βStIt1+αIt−η+ε+μEt,I′t=ηEt−ut+γ+μ+θIt,Q′t=utIt−φ+μ+θQt,R′t=ψSt+εEt+γIt+φQt−μRt.The main goal of optimal control is to determine the control variableu∗ satisfying(27)Ju∗=minJu:u∈U,where U is the control set defined by U=ut:utis measurable,0≤ut≤1,t∈t0,tf.According to Fleming and Rishel (1975, [36]), we now consider the existence of the optimal control variable u∗.Theorem 7.
The optimal control variableu∗∈U does exist such that (27) holds under its state system (26) with initial conditions at t0.
The Hamiltonian for state system (26) is characterized as(28)H=Ku2+E+I−R+w1Λ−βSI1+αI−ψ+μS+w2βSI1+αI−η+ε+μE+w3ηE−u+γ+μ+θI+w4uI−φ+μ+θQ+w5ψS+εE+γI+φQ−μR,which, along with the Pontryagin’s Minimum Principle [35], leads to a theorem below.Theorem 8.
Given an optimal control variableu and corresponding solution of state system (26), there are adjoint variables wi, i=1,…,5, such that the following adjoint system of equations holds:(29)w1′t=−∂H∂S=w1−w2βI1+αI+w1μ+w1−w5ψ,w2′t=−∂H∂E=−1+w2−w3η+w2−w5ε+w2μ,w3′t=−∂H∂I=−1+w1−w2βS1+αI2+w3−w4u+w3−w5γ+w3μ+θ,w4′t=−∂H∂Q=w4−w5φ+w4μ+θ,w5′t=−∂H∂R=1+w5μ,with boundary conditions witf=0, i=1,…,5. Also, the optimal control variable u∗, subject to (26) and satisfying (27), is given by(30)u∗=max0,min1,w3−w4I2K.Remark 6.
Note here that the optimality system contains state system (26), adjoint system (11) with its initial and boundary conditions, and the optimal control variable u∗ given by (30). Hence, the optimality system with quarantine intervention is expressed by(31)S′t=Λ−βSI1+αI−μS,E′t=βSI1+αI−η+ε+μE,I′t=ηE−u∗+γ+μ+θI,Q′t=u∗I−φ+μ+θQ,R′t=εE+γI+φQ−μR,w1′t=−∂H∂S=w1−w2βI1+αI+w1μ+w1−w5ψ,w2′t=−∂H∂E=−1+w2−w3η+w2−w5ε+w2μ,w3′t=−∂H∂I=−1+w1−w2βS1+αI2+w3−w4u∗+w3−w5γ+w3μ+θ,w4′t=−∂H∂Q=w4−w5φ+w4μ+θ,w5′t=−∂H∂R=1+w5μ,with initial conditions at time t0 and boundary conditions, witf=0, i=1,…,5.
Please see Appendixes A.6 and A.7 for the proofs of Theorems7 and 8. According to Lemma 4.1 in [37] and the proof of Theorem 2 in [38], the uniqueness of solution for the optimality system follows easily.Theorem 9.
The solution for the optimality system is unique for a small enoughtf.We finally carry out a set of numerical experiments to validate the obtained-above control strategy. The modeling parameters and initial numbers are taken as those ofSection 4.1. From Figure 5, the dynamics of the system without/with quarantine are presented as follows:(1)
Figure5(a) shows the trend of the susceptible number without/with quarantine, where we see that the susceptible number will decay sharply because most of them become recovered with quarantine intervention.(2)
Figure5(b) shows the trend of the exposed number without/with quarantine, which indicates that the exposed number with quarantine gets much smaller, and the device exposed to worms persists for a shorter time.(3)
Figure5(c) shows the trend of the infectious number without/with quarantine. We observed that the infectious maximum number is nearly 8000 without quarantine, but this number is less than 500 with quarantine.(4)
Figure5(d) shows the trend of the recovered number without/with quarantine, which gives that with quarantine, a larger number of mobile devices become recovered more quickly.Figure 5
Dynamics of system without/without quarantine.
(a)(b)(c)(d)


### 5.2. Control Strategies Based onR0

The basic reproduction numberR0, defined as (19), is a sharp threshold value that determines completely the dynamics of worm propagation and performs an important role in controlling the worm epidemic. As described in Theorems 2 and 3, the worm-free equilibrium point P0 is stable locally and globally under condition that R0≤1. Hence, we adopt the corresponding parameters such that R0≤1 to prevent the worm epidemic. From equation (19), the number R0 relates to the parameters as follows: the recruitment rate Λ, the quarantine rate ξ, the recovery rates ψ, ε, and γ, the infection rates β and η, and the death rates μ and θ.According to Theorems2 and 3, the following corollary regarding the parameters of R0 is given to prevent the worm epidemic, and its proof is delayed to Appendix A.8.Corollary 1.
To control the epidemic of mobile network-based worms, the following assertions for all the parameters in (19) should be true:(32)Λ<ψ+μξ+γ+θ+μβ;β<ψ+μξ+γ+θ+μΛ;ψ>βηΛη+ε+μξ+γ+θ+μ−μ;ξ>βηΛψ+μη+ε+μ−γ+μ+θ;γ>βηΛψ+μη+ε+μ−ξ+μ+θ;θ>βηΛψ+μη+ε+μ−ξ+γ+μ;ε>βηΛψ+μξ+γ+θ+μ−η+μ;η<ψ+με+μξ+γ+θ+μβΛ−ψ+μξ+γ+θ+μ,ifβΛ>ψ+μξ+γ+θ+μ;orη>ψ+με+μξ+γ+θ+μβΛ−ψ+μξ+γ+θ+μ,ifβΛ<ψ+μξ+γ+θ+μ.Considering the parameter values appearing inSection 4.1 except for ψ, we find out that the transition from the worm-free regime to the worm-existence one happens when ψ=0.3573, which is in line with the lower-bound of ψ, obtained by Corollary 1. And the similar conclusions still hold for the other parameters.By using the partial derivatives ofR0 in equation (19), it follows that(33)∂R0∂β>0,∂R0∂η>0,∂R0∂Λ>0,∂R0∂μ<0,∂R0∂ε<0,∂R0∂ψ<0,∂R0∂ξ<0,∂R0∂γ<0,∂R0∂θ<0.Hence, if all the parameters ofR0, except for only one, are set to be constant, the function R0 decreases as β, η, and Λ decrease, or as μ, ε, ψ, ξ, γ, and θ increase. This result, along with Theorems 2 and 3, shows that to reduce the value of R0 to be less than 1, it suffices to decrease the values of β, η, and Λ, or increase the values of μ, ε, ξ, γ, and θ.By the above analysis onR0, a series of control strategies are established to control the epidemic of worms based on mobile networks, and a set of numerical experiments are conducted to demonstrate the effectiveness of obtained strategies, especially to illustrate the influences of modeling parameters on the infectious device number and the worm spreading speed, which, in fact, are two critical factors to judge the severity of worm epidemic:(1)
Decreasing the rateβ infected in group S via using immunization procedures and/or antivirus software: conduct a numerical experiment to study the impact on the infectious number by different values β=0.1, 0.0531, and 0.01. As shown in Figure 6(a), the infectious device number and the worm spreading time both get smaller with the reduction of the infection rate β.(2)
Decreasing the infectious rateη from group E: set different values for η as 0.008, 0.003, and 0.001 to consider its impact on the infectious number. The simulation result is shown from Figure 6(b), namely, that as the rate η becomes smaller, the malicious worms will infect fewer devices in a longer period.(3)
Increasing the rateξ quarantined to group Q via strengthening the capabilities of quarantine strategy: consider the effect on the worm propagation by ξ=0.03, 0.05, and 0.09. From Figure 6(c), we see that a larger rate of quarantine can lead to a small number of infectious cases, a lower speed of worm spreading, and a shorter time it takes to control the worm epidemic.(4)
Increasing the rateψ recovered from group S via improving the quality and the user scale of antivirus software: Set ψ=0.5, 0.7, and 0.9, and from Figure 6(d), we observe that as the rate ψ increases, the infectious number will be smaller, the speed of worm propagation will become lower, and the time to control the epidemic of worm will get shorter.(5)
Increasing the rateε recovered from group E via reminding to install antivirus software: consider the impact of rate ε on the infectious number by ε=0, 0.0008, and 0.0018. From Figure 6(e), it follows that a larger rate ε can result in reducing the size of mobile devices infected by the network-based worms.(6)
Increasing the rateγ recovered from group I via enhancing the efficacy of antiviral treatment: to analyze the impact of rate γ on the number of infectious devices with γ=0.05, 0.10, and 0.15. See Figure 6(f), which indicates that with the increase of γ, the malicious worm will infect fewer mobile devices and spread in mobile networks for a shorter period.(7)
Increasing the rateθ of worm-related death via stopping and even eliminating infectious devices: set θ=0.001,0.01, and 0.09, and from Figure 6(g), we show that as the rate θ becomes larger, the worm spreads more slowly, the epidemic lasts in a shorter term, and the mobile devices infected by this worm becomes fewer.Figure 6
Impacts of modeling parameters on the infectious devices number.
(a)(b)(c)(d)(e)(f)(g)


## 5.1. The Optimal Control Strategy

In this section, the optimal control strategy is constructed for the SEIQR-type model described by system (1), where a control variable ut, 0≤ut≤1, is chosen to denote the inhibiting effect of quarantine strategy on the worm epidemic. Especially, when ut=1, the quarantine strategy is perfectly effective; when ut=0, the quarantine strategy has no effect at all. Hence, the optimal control problem is formulated as(25)Ju=∫t0tfKu2+Et+It−Rtdt,where the parameter K≥0 denotes the weight of the benefit of quarantine cost, and this problem is subject to the following state system:(26)S′t=Λ−βStIt1+αIt−ψ+μSt,E′t=βStIt1+αIt−η+ε+μEt,I′t=ηEt−ut+γ+μ+θIt,Q′t=utIt−φ+μ+θQt,R′t=ψSt+εEt+γIt+φQt−μRt.The main goal of optimal control is to determine the control variableu∗ satisfying(27)Ju∗=minJu:u∈U,where U is the control set defined by U=ut:utis measurable,0≤ut≤1,t∈t0,tf.According to Fleming and Rishel (1975, [36]), we now consider the existence of the optimal control variable u∗.Theorem 7.
The optimal control variableu∗∈U does exist such that (27) holds under its state system (26) with initial conditions at t0.
The Hamiltonian for state system (26) is characterized as(28)H=Ku2+E+I−R+w1Λ−βSI1+αI−ψ+μS+w2βSI1+αI−η+ε+μE+w3ηE−u+γ+μ+θI+w4uI−φ+μ+θQ+w5ψS+εE+γI+φQ−μR,which, along with the Pontryagin’s Minimum Principle [35], leads to a theorem below.Theorem 8.
Given an optimal control variableu and corresponding solution of state system (26), there are adjoint variables wi, i=1,…,5, such that the following adjoint system of equations holds:(29)w1′t=−∂H∂S=w1−w2βI1+αI+w1μ+w1−w5ψ,w2′t=−∂H∂E=−1+w2−w3η+w2−w5ε+w2μ,w3′t=−∂H∂I=−1+w1−w2βS1+αI2+w3−w4u+w3−w5γ+w3μ+θ,w4′t=−∂H∂Q=w4−w5φ+w4μ+θ,w5′t=−∂H∂R=1+w5μ,with boundary conditions witf=0, i=1,…,5. Also, the optimal control variable u∗, subject to (26) and satisfying (27), is given by(30)u∗=max0,min1,w3−w4I2K.Remark 6.
Note here that the optimality system contains state system (26), adjoint system (11) with its initial and boundary conditions, and the optimal control variable u∗ given by (30). Hence, the optimality system with quarantine intervention is expressed by(31)S′t=Λ−βSI1+αI−μS,E′t=βSI1+αI−η+ε+μE,I′t=ηE−u∗+γ+μ+θI,Q′t=u∗I−φ+μ+θQ,R′t=εE+γI+φQ−μR,w1′t=−∂H∂S=w1−w2βI1+αI+w1μ+w1−w5ψ,w2′t=−∂H∂E=−1+w2−w3η+w2−w5ε+w2μ,w3′t=−∂H∂I=−1+w1−w2βS1+αI2+w3−w4u∗+w3−w5γ+w3μ+θ,w4′t=−∂H∂Q=w4−w5φ+w4μ+θ,w5′t=−∂H∂R=1+w5μ,with initial conditions at time t0 and boundary conditions, witf=0, i=1,…,5.
Please see Appendixes A.6 and A.7 for the proofs of Theorems7 and 8. According to Lemma 4.1 in [37] and the proof of Theorem 2 in [38], the uniqueness of solution for the optimality system follows easily.Theorem 9.
The solution for the optimality system is unique for a small enoughtf.We finally carry out a set of numerical experiments to validate the obtained-above control strategy. The modeling parameters and initial numbers are taken as those ofSection 4.1. From Figure 5, the dynamics of the system without/with quarantine are presented as follows:(1)
Figure5(a) shows the trend of the susceptible number without/with quarantine, where we see that the susceptible number will decay sharply because most of them become recovered with quarantine intervention.(2)
Figure5(b) shows the trend of the exposed number without/with quarantine, which indicates that the exposed number with quarantine gets much smaller, and the device exposed to worms persists for a shorter time.(3)
Figure5(c) shows the trend of the infectious number without/with quarantine. We observed that the infectious maximum number is nearly 8000 without quarantine, but this number is less than 500 with quarantine.(4)
Figure5(d) shows the trend of the recovered number without/with quarantine, which gives that with quarantine, a larger number of mobile devices become recovered more quickly.Figure 5
Dynamics of system without/without quarantine.
(a)(b)(c)(d)


## 5.2. Control Strategies Based onR0

The basic reproduction numberR0, defined as (19), is a sharp threshold value that determines completely the dynamics of worm propagation and performs an important role in controlling the worm epidemic. As described in Theorems 2 and 3, the worm-free equilibrium point P0 is stable locally and globally under condition that R0≤1. Hence, we adopt the corresponding parameters such that R0≤1 to prevent the worm epidemic. From equation (19), the number R0 relates to the parameters as follows: the recruitment rate Λ, the quarantine rate ξ, the recovery rates ψ, ε, and γ, the infection rates β and η, and the death rates μ and θ.According to Theorems2 and 3, the following corollary regarding the parameters of R0 is given to prevent the worm epidemic, and its proof is delayed to Appendix A.8.Corollary 1.
To control the epidemic of mobile network-based worms, the following assertions for all the parameters in (19) should be true:(32)Λ<ψ+μξ+γ+θ+μβ;β<ψ+μξ+γ+θ+μΛ;ψ>βηΛη+ε+μξ+γ+θ+μ−μ;ξ>βηΛψ+μη+ε+μ−γ+μ+θ;γ>βηΛψ+μη+ε+μ−ξ+μ+θ;θ>βηΛψ+μη+ε+μ−ξ+γ+μ;ε>βηΛψ+μξ+γ+θ+μ−η+μ;η<ψ+με+μξ+γ+θ+μβΛ−ψ+μξ+γ+θ+μ,ifβΛ>ψ+μξ+γ+θ+μ;orη>ψ+με+μξ+γ+θ+μβΛ−ψ+μξ+γ+θ+μ,ifβΛ<ψ+μξ+γ+θ+μ.Considering the parameter values appearing inSection 4.1 except for ψ, we find out that the transition from the worm-free regime to the worm-existence one happens when ψ=0.3573, which is in line with the lower-bound of ψ, obtained by Corollary 1. And the similar conclusions still hold for the other parameters.By using the partial derivatives ofR0 in equation (19), it follows that(33)∂R0∂β>0,∂R0∂η>0,∂R0∂Λ>0,∂R0∂μ<0,∂R0∂ε<0,∂R0∂ψ<0,∂R0∂ξ<0,∂R0∂γ<0,∂R0∂θ<0.Hence, if all the parameters ofR0, except for only one, are set to be constant, the function R0 decreases as β, η, and Λ decrease, or as μ, ε, ψ, ξ, γ, and θ increase. This result, along with Theorems 2 and 3, shows that to reduce the value of R0 to be less than 1, it suffices to decrease the values of β, η, and Λ, or increase the values of μ, ε, ξ, γ, and θ.By the above analysis onR0, a series of control strategies are established to control the epidemic of worms based on mobile networks, and a set of numerical experiments are conducted to demonstrate the effectiveness of obtained strategies, especially to illustrate the influences of modeling parameters on the infectious device number and the worm spreading speed, which, in fact, are two critical factors to judge the severity of worm epidemic:(1)
Decreasing the rateβ infected in group S via using immunization procedures and/or antivirus software: conduct a numerical experiment to study the impact on the infectious number by different values β=0.1, 0.0531, and 0.01. As shown in Figure 6(a), the infectious device number and the worm spreading time both get smaller with the reduction of the infection rate β.(2)
Decreasing the infectious rateη from group E: set different values for η as 0.008, 0.003, and 0.001 to consider its impact on the infectious number. The simulation result is shown from Figure 6(b), namely, that as the rate η becomes smaller, the malicious worms will infect fewer devices in a longer period.(3)
Increasing the rateξ quarantined to group Q via strengthening the capabilities of quarantine strategy: consider the effect on the worm propagation by ξ=0.03, 0.05, and 0.09. From Figure 6(c), we see that a larger rate of quarantine can lead to a small number of infectious cases, a lower speed of worm spreading, and a shorter time it takes to control the worm epidemic.(4)
Increasing the rateψ recovered from group S via improving the quality and the user scale of antivirus software: Set ψ=0.5, 0.7, and 0.9, and from Figure 6(d), we observe that as the rate ψ increases, the infectious number will be smaller, the speed of worm propagation will become lower, and the time to control the epidemic of worm will get shorter.(5)
Increasing the rateε recovered from group E via reminding to install antivirus software: consider the impact of rate ε on the infectious number by ε=0, 0.0008, and 0.0018. From Figure 6(e), it follows that a larger rate ε can result in reducing the size of mobile devices infected by the network-based worms.(6)
Increasing the rateγ recovered from group I via enhancing the efficacy of antiviral treatment: to analyze the impact of rate γ on the number of infectious devices with γ=0.05, 0.10, and 0.15. See Figure 6(f), which indicates that with the increase of γ, the malicious worm will infect fewer mobile devices and spread in mobile networks for a shorter period.(7)
Increasing the rateθ of worm-related death via stopping and even eliminating infectious devices: set θ=0.001,0.01, and 0.09, and from Figure 6(g), we show that as the rate θ becomes larger, the worm spreads more slowly, the epidemic lasts in a shorter term, and the mobile devices infected by this worm becomes fewer.Figure 6
Impacts of modeling parameters on the infectious devices number.
(a)(b)(c)(d)(e)(f)(g)


## 6. Conclusions

This paper investigates dynamics and control of worm epidemic based on mobile networks via a general SEIQR-type model with a saturated incidence rate, in which this rate can tend to its saturation level with the increase of the infectious number, and obviously, it is more realistic as it can present the crowding effect of infectious individuals and also guarantee the boundness of contact rate. The innovations of this paper are (a) to consider the recovery process from the susceptible group to the recovered group owing to sufficient defense mechanisms or/and antivirus treatments; (b) to discuss the dysfunction of devices related to worm-related death because the crash of most devices results from the attack of malicious worms; and (c) to establish the control strategies in defending against the epidemic of worms.Our results suggest that the basic reproduction numberR0, as a sharp threshold value, can completely determine the dynamic behaviors of worm propagation and the control strategies for worm epidemic. More specifically, if R0≤1, the worm-free equilibrium is asymptotically stable, and the epidemic of worm will eventually disappear and remain under control; if R0>1, the asymptotical stability of worm-existence equilibrium is derived to imply that the epidemic will remain persistent and uncontrollable. This study gives new insights to mobile network security, namely, that is predicting the worm spreading tendency, identifying the epidemic control strategies, and estimating the worm popularity level. A series of numerical experiments are conducted to show the rationality of our obtained results.With the use of Pontryagin’s Minimum Principle [35], we devise the optimal control strategy to minimize the quarantine cost and the infected number as well as maximize the recovered number. Furthermore, by the analysis for R0, we present a set of control strategies to fight against the epidemic of worm based on mobile networks, including to decrease the rates of infection from groups S and E and to increase the rate of quarantine, the rate of worm-related death, and the rates of recovery from groups S, E, and I. Also, increasing the parameter α of saturated incidence rate can be used to stop the worm epidemic. Some numerical experiments are also implemented to validate the effectiveness of these obtained strategies.



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*Source: 6637263-2021-01-18.xml*