Effect of Delay in SMS Worm Propagation in Mobile Network with Saturated Incidence Rate

The mobile network has become the social standing pulverized for heaps of novel worms, including the SMS, Bluetooth-built worms and Wi-Fi based worms. Particularly, SMS worms have drawn significant attention owing to their enormous threats to the mobile network. In this article, we proposed and analysed the SMS worm propagation, a delayed SAIDR model (Susceptible-Affected-Infected-Suspended-Recovered) with a saturated incidence rate. A novel express in the SMS model, the suspended state, which gets some margin to affected node to infected node in mobile network. Furthermore, the influence of time delay during the charging behaviour of the affected state nodes needs to be considered. Hopf bifurcation is studied by discussing the time delay that is chosen as the bifurcation parameter. Direction and stability of the Hopf bifurcation are derived using normal form theory and centre manifold theorem. At last, a few numerical simulations are presented to check the analytical outcomes.


Introduction
The broadly utilization of versatile figuring and specialized gadgets phones, PCs and individual computerized collaborators is driving a progressive change in our data society. Presently elective ways of conveying the administrations are engaged around remote organization [1,2], which can cause terminals to associate with one another in the transmission 1 3 range however programmed design. The remote field has been encountering development in the current. Clients can utilize their cell phone to check email and peruse web; specialists can trade data by convenient PCs through remote organization while going to their works. As of late, cell phones have become progressively inescapable, this present circumstance draws in aggressors to engender worm agendas between cell phones. In 2004, the principal Bluetooth-based worm, named cabir, was sent off into the ecosphere, as of now, the accommodation of Wi-Fi has made it one of the generally utilized means to interface with the web among cell phones. Another Wi-Fi worm called Chameleon, which could spread in a way like that of in-flight illnesses, showed up in 2014 [3][4][5].However, most of cell phones have no powerful techniques to forestall worm assaults, which shrubberies them powerless against such assaults. Cell phones compromised by worms can make incredible misfortunes clients, including information spillage, framework harm and monetary misfortune.
In the previous many years, analysts have proposed various numerical models to investigate the powerful ways of behaving and attributes of PC worms in the web. Motivated by the scientists on organic infections, the SI [6], SIR [7,8], SIRS [9][10][11], SEIR [12], SVIRS [13] traditional model has been introduced. In these models some real effect factors are disregarded. Hence, a two-factor system was proposed to examine the engendering of worm propagation in wireless sensor networks with media access control mechanisms in [14]. One factor is the unique counter measures taken by Internet specialist organizations and clients. The other is the clog of certain courses brought about by the uncontrolled spread of worm propagation in mobile network. Lurong etal [14] assumed that outer PCs are associated with the Internet. In any case, since there are a few distinctions between PC worms and SMS centred worms, particularly their scattering ways, these models can't be straightforwardly used to portray propogation of SMS centred worms.
The vast majority of the early malware for PDAs like mosquito and skulls were either Trojans which tainted a PDA whenever they were downloaded from the web or like cabir and lasco which utilized Bluetooth to contaminate phones in its area. Another model is the commwarrior worm which utilizes the contaminated hosts mixed media informing administrations to spread itself to the telephone numbers put away in the location book of the tainted telephone. In versatile correspondence, when another communication has come into the communication box, except if the client taps the pernicious connection remembered for the message, the worm is set off and he will be contaminated on the double. In any case, the vindictive connection isn't clicked and the client can't be infected. Thus the client can't promptly convert into the tainted state regardless of whether he has gotten the malevolent message. We add another express, the impacted express (A), to address the condition of clients who have gotten the malignant messages however have not tapped the malevolent connections. Additionally, we find that the contaminated telephone might be separated for this situation it can't send malignant messages to different telephones. Thus, the suspended state (D) is acquainted with portrayed the circumstance. To definitively depict the powerful ways of behaving of SMS centred worms and portray their spread in the versatile climate. In this article we foster a dynamical model SAIDR in light of the over two states.
Dynamical examination of the framework with time delay is a fundamental subject in many fields, particularly for the versatile correspondence. Opportunity defer comes from the time sharing of the correspondence medium and the calculation time involved for versatile organization handling and actual sign coding. Wang [15] showed the way that the computational postponement can cause framework shakiness in a computerized regulator. Delay is a significant viewpoint since it straightforwardly influences the speed of the computerized gadget on a working portable. As of late PC infection model is with time delay have been examined by certain researchers [16][17][18][19][20]. Zhang [21] proposed a delayed SEIRS-V model on the spread of worms in a remote device organization. Upadhyay [22] concentrated on the bifurcation examination of an e-scourge system in remote sensor organization. In any case, as expressed in [23], one of the average elements for the vindictive codes in networks is their impacted. There is generally a deferral from the time the impacted hubs are tainted to the time they become irresistible because of the inherent impacted time of worms.
The rest of the paper is organised as follows. In Sect. 2, we formulate a delayed SAIDR SMS worm propagation model in mobile network. In Sect. 3, we find equilibrium points and investigated the stability of the SMS worm propagation with delay and without delay. In Sect. 4, we derive sufficient conditions for the existence of Hopf bifurcation. In Sect. 5, we determine the direction and stability of the Hopf bifurcation by using the normal form theory and the center manifold theorem. In Sect. 6, the analytical results verify with numerical simulation. In Sect. 7, conclusions and future research work is mentioned.

Mathematical Formulation
A delayed SMS-based worm propagation model in mobile network is presented and investigate the impact of delay in worm propagation through SMS. In this mobile network SMS based worm propagation states is divided into five compartments (S-A-I-D-R). The susceptible state(S) shows the comparing nodes are healthy worms, affected state (A) is the transitional state before the infected state, infected state (I) is the affected node snaps the malevolent linkage contained in the message and thus activates the worm, suspended state (D) is the infected node is broken down by the worm, it turns to the suspended state, recovered state (R) is the people can repair it and install security tools afterwards. The physical description of the parameters in the system (1) is depicted in Table 1.

3
The system (1) have the primary nodes Also, the parameters , , , , , m, a, , are all assumed to be non-negative.
Then from Eqs. (1) and (2), the equivalent four dimensional system SAID (Susceptible-Affected-Infected-Suspended) as follows The affected nodes remains a definite time delay ( ) before they change into infectious nodes with rate ( ) . We track down that the system (3) can tune itself to show different ways of behaving by changing the time delay. The considering the impact of time delay in affected node because of the period that the worm spread in infected node. The system (3) immediately can show a Hopf bifurcation with increment of the time defer then the oscillations behaviour occurs.

Equilibrium Analysis of SMS Worm Propagation Model
In a system (3), there are two positive equilibrium points. The following are the existence and stability conditions for them.

Stability Analysis of SMS Worm Propagation Model
The linearized system of (3) is where The characteristics equation of system (3) is

Stability Analysis of SMS Worm Propagation Model Without Delay
Obviously if condition (7-10) holds, if E * is locally asymptotically stable in the absence of delay according to Routh-Hurwitz criterion

Stability Analysis of SMS Worm Propagation Model with Delay
Put > 0, Let = i ( > 0) be the root of (5), we have, equating real, imaginary parts Eliminating sin from (11) and (12), Also by solving (11) and (12), we have

Hopf Bifurcation Analysis of SMS Worm Propagation Model
Differentiate (5) with respect to we get Put = i in above equation Therefore the transvers ability conditions hold and hence Hopf bifurcation occurs at = 0 .

Direction and Stability of Periodic Solutions of Hopf Bifurcation
Let where where In fact, we choose The terms of Ψ, Υ and Φ are described in the following The bilinear inner form for O and O * is expressed as Following the same algorithms introduced in [24] and the similar computation used in [25][26][27], we can obtain the expressions of ℵ 20 , ℵ 11 , ℵ 02 and ℵ 22 as follows Consequently, we can acquire the outcomes described in hypothesis (2). The verification is finished.
Remark It should be emphasized that the normal form theory and the center manifold theorem dictate the direction of stability of the Hopf bifurcation at 0 of system (3). Although the number of calculations required by this method is excessive, it can assist us in obtaining explicit coefficients specifying the direction of stability of the Hopf bifurcation. This method has been used to examine Hopf bifurcation features in a variety of other domains, including epidemic models, neural networks etc.

Numerical Simulations
In this section, we present a numerical example to demonstrate the validity of the theoretical results gained in Sects. 3 and 4. The numerical solutions are derived using the MAT-LAB software suite and the well-known RK4 numerical approach. By selecting some system parameters of ( This property can be shown as in Fig. 1b. We discovered that when the time delay is small enough, malware spread on mobile wireless sensor networks may be controlled. However, a Hopf bifurcation will occur at = 0 = 3.95 and shown in Fig. 1c and furthermore, we expanded the delay value = 5.5 > 0 , then system (3) oscillates and becomes unstable it shown in Fig. 1d. Meanwhile, numerical simulations are run to demonstrate the usefulness of the obtained measures, particularly the effects of the specified parameters on the number of infectious devices and the pace of worm spread. It's worth  . 1 a Shows the time series evaluation of mobile network nodes with attributes mentioned in Table 1 for absence of delay ( = 0) . b Shows the time series evaluation of mobile network nodes with attributes mentioned in Table 1 for presence of delay ( = 3.5) . c Shows the time series evaluation of mobile network nodes with attributes mentioned in Table 1 for presence of delay ( = 3.95) . d Shows the time series evaluation of mobile network nodes with attributes mentioned in Table 1 for presence of delay ( = 5.5) noting that the infected quantity and propagation speed are two important aspects in defining the severity of a worm attack.
Contrasting the 3D charts of Figs. 2a, b and 3a, b for S-A-I and S-A-D for delay values of = 3.5, = 5.5 respectively, it is obvious that the last option has a larger number of motions than the previous. This is so while contrasting Figs. 4a, b and 5a, b, which are plots A-I-D and S-I-D. The firmly stuffed oscillations showing instability when the delay value esteem is more than the threshold value are likewise apparent.
We have established that, under certain conditions, a critical value of the delay exists below which the system (3) is stable and above which the system (3) is unstable. Particularly, when = 0 = 3.95 system (3) experiences a Hopf bifurcation at the positive equilibrium. When Hopf bifurcation occurs, the prevalence of SMS worm propagation model, shifts from a condition of positive equilibrium to a limit cycle, which is undesirable in SMS worm propagation model.
When the time-delay value exceeds the critical point, malware spread on mobile wireless sensor networks becomes uncontrollable. In the real network environment, users of mobile wire-less sensor networks should update and run anti-virus software on a regular basis to reduce the time delay caused by the time interval that anti-virus software uses to purge malware and effectively limit the spread of malware on mobile devices.

Conclusion
In this article, a delayed SAIDR (susceptible-affected-infected-suspended-recovered) SMS worm propagation model in mobile network investigated by incorporating the time delay in affected nodes before they change into infectious nodes. The main results are given interms of local stability of the system (3) and make system (3) undergo a Hopf bifurcation under some certain conditions. We have proven that the system (3) is stable when the time delay is less than the critical value 0 and above the critical value 0 , the system (3) is unstable. Especially, system (3) undergoes a Hopf bifurcation at the positive equilibrium E * when = 0 . The occurrence of Hopf bifurcation means that characteristics of the propagation of worms in system (3) can be easily predicted and eliminated when the time delay is below the critical value 0 and the propagation of worms in system (3) may be out of control once the delay passes through the corresponding critical value 0 . Thus, we can conclude that the worm's propagation can be controlled by postponing occurrence of a Hopf bifurcation. The direction and stability of the bifurcating periodic solutions were examined in Theorem (2) by using normal form theory and centre manifold theorem. Finally we conclude that the time delay value gradually increases, the SAIDR SMS system loss the stability along with presence of periodic oscillations that leads to the origin of a Hopf bifurcation.