An Epidemic Patch-Enabled Delayed Model for Virus Propagation: Towards Evaluating Bifurcation and White Noise

School of Control Technology, Wuxi Institute of Technology, Wuxi 214121, China Department of Mathematics, S.A. Engineering College, Chennai 600077, Tamil Nadu, India Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India Open Studies Unit, Nigeria Correctional Service, Awka, Nigeria Department of Computer Science, Ambo University, Ambo, Ethiopia


Introduction
e Internet and computer networks have greatly facilitated human e ort, education, and living since the rising notoriety of computers [1] and the fast evolution of information communication technology. In light of this, new cyber threats are arising as their actors' techniques are always advancing to retain or increase their prominence in the threat ecosystem through exploiting vulnerabilities [2]. Be it for individuals, organizations or critical national infrastructure, black hat hackers target zero-day issues in certain situations, but more likely, they target freshly xed aws and "unpatched systems" [3].
Due to the increasing growth of communication networks and their applications, virus transmission has become one of the topics of interest in computing research [4]. Besides worms and trojans [5], the virus is one way through which malicious attacks can arise in a computer network. In a networked system, whenever the computers are contaminated with viruses, the regular resident applications may lose the ability to function properly, corrupt saved les, or cause the loss of essential data on those machines. Subsequently, through the infected computers, the virus infection is transmitted to other computers through several means, which include able storage media (CD, USB, and ash drives) and e-mail attachments [6].
To curb viral spread incidences and safeguard computer networks against viruses, antimalicious programmes, rewalls, and patches are used to lter out all infections that remain in personal devices including personal laptops and other detachable storage media [7]. Cybersecurity solutions largely aim to prevent harmful malware from entering and operating on computer systems [8]. Other ways include integrating audits, performing updates to security infrastructure [9], and modelling threats. Mathematical models have been developed to grasp the spread of malicious programs fully. ese models are based on intriguing parallels between virtual viruses and their biological counterparts; wherein numerous phenomena are represented [2,6,7]. e basic inspiration for constructing the model is motivational research on mathematical modelling for Wireless Sensor Networks. Disease modelling is trending in the latest research works, particularly e-epidemic models are attractive and interesting, leading to our current proposed model with delay and stochastic dynamics. Some qualitative literature on delay models and stochastic models. In any Disease model, whether epidemic or e-epidemic, the delay is a key attribute to changing the system dynamics in terms of stability; environmental noise (stochastic) is also one of the key attributes which play a major role in the system dynamics. Motivational research works on delay and stochastic dynamics in various systems. erefore, in this paper, we propose the deterministic susceptible (S), latent (L), breaking out (B), and patched (P) model and a modification to include stochasticity in the form of noise, alongside delay and bifurcation. is paper is organized as follows: Section 2 contains the related literature; Section 3 presents the mathematical model; Section 4 contains the delay analysis, and Section 5 contains the directions of Hopf bifurcation and stability of the periodic solutions. Section 6 contains the SLBP model with noise; Section 7 contains the numerical simulations.

Related Literature
A sophisticated virus's primary purpose is to damage more computer systems; to achieve that purpose, the malware would attempt to infiltrate as many computers without being detected. For purposes of infection modelling Yang and Yang [10], two typical stages of a virus are the latent (L) and breaking out (B) stages. While the former signifies the entry period, when the virus inhabits the host, the latter represents the time the infection starts causing harm to the host computer. Here, models representing virus propagation in computers and their networks using this SLB format are reviewed.
Yang et al. [11] employed the SLB model internal computers and derived the reproduction number, equilibria, and stability of points. Yang et al. [12] modelled the spread of computer viruses in the complex World-Wide-Web using the SLB model. ey discovered that a greater heterogeneity and a scale-free graph with smaller power-law exponents promote virus growth. Representing recovered computers in separate compartments and reinfection, Yang et al. [13] derived the global stability of the susceptible-latent-breaking-recovered-susceptible (SLBRS) model. Zhang et al. [14] employed the SLBRS model to represent virus dissemination but used the time delay of antivirus cleaning as a bifurcating parameter. Due to the ubiquity of the mass action infection rate, Yang and Yang [15] adopted a nonlinear type of incidence for virus growth using the SLB model.
Contrary to homogenous mixing, prevalent in most models, Yang et al. [16] used the SLB model to represent a scenario where distinct nodes have varied infectious, exploding, and remedial rates. e SLBR model was used by Zhang [17] to model both storage media and internal/ external computers. Zhang and Bi [18] investigated the existence and properties of Hopf bifurcation using the SLB model. Zhang and Wang [19] considered isolation of contaminated hosts and thus developed the susceptiblelatent-breaking out-quarantined-recovered (SLBQR) model, which was later used to investigate Hopf bifurcation. Zhang [20] applied the SLB model for multilayer computer subnetworks. Similarly, Zhao and Bi [21] used the SLBQR to model virus spread with two-time delays and the existence of Hopf bifurcation. Another delayed version of the SLBS model was developed by Zhang et al. [22] and after studying its stability analyses. Because most models only represent horizontal transmission (HT), Zhu et al. [23] utilized the SLBR model to represent both HT and vertical transmission of viruses in a computer network.
More SLB models were observed in the following studies; [24,25]. Other models which have been used to represent virus propagation in computers alongside detachable storage, external computers, and age structure using the susceptible-infected-countermeasure (SIC) model [7], strongly protected susceptible-weakly protected susceptible-infective-external (SWIE) model [6] and susceptible-infected-recovered (SIR) model [2]. A critical evaluation of the above-given models shows that besides considering the deterministic SLB model with their delay analysis, this work presents its SLB's stochastic version, wherein the impact of white noise is studied. Our model also includes the patch (P) compartment that prevents the inundation [26] of the computer network as a result of virus infestation. e following sections contain the proposed models and further analysis.

Mathematical Model
In this part, a patch-enabled SLBP model is developed to capture the dynamics of virus and patch transmission. Computers are divided into two groups for virus propagation offline and online: internal computers linked to the Internet (World Wide Web) and external computers not connected to the world wide web. e host computers are called nodes for the sake of simplicity. All internal nodes are connected to the world wide web and divided into four categories, namely, susceptible nodes without virus (S), latently infected (L), breaking out infected (B), and nodes that have received patches (P) at the time t. In the latent state, the virus infection is inactive. erefore, S(t) + L(t) + B(t) + P(t) ≡ 1. We suggested the following conditions alongside the mathematical model based on the inspiration: (A1) When an external node connects to the world wide web, it becomes vulnerable. At rate δ, nodes are added to or withdrawn from the network. (A2) Both infected nodes (L, B) can potentially infect susceptible nodes. Infection rates from L and B to S nodes are β1 and β2, respectively. It is assumed that the incidence function is of the mass-action type. (A3) With a rate of α, the L node becomes B. (A4) Remediation of L nodes occurs at rate c 2 and then becomes vulnerable. (A5) e patched node will become invalid due to the development of new viruses. As a result, it loses immunity at a rate of c 1 . (A6) Patches are acquired at a rate of βP by nodes S, L, or B.
With the initial conditions By direct computation, the system equation (1) has a unique positive equilibrium point E * (S * , L * , B * , P * ) in which

Delay Analysis
e linear system of equation (1) about endemic equilibrium point E * (S * , L * , B * , P * ) is given by the following equations: where en, the associated characteristic equation is

Mathematical Problems in Engineering
Put τ � 0 in equation (8), we get the following equation: From equation (9), By using Routh-Hurwitz criteria, sufficient conditions for all roots of equation (11) to be negative real parts are given in the following form: is, if conditions equations (11), (13)- (15) hold, E * is locally asymptotically stable in the absence of delay.
For τ > 0, Put λ � iω in equation (8) we have the following equation: Equating real and imaginary parts we have the following equation: Squaring and Adding equations (17) and (18) we get the following equation: where Now, by assuming ω 2 � u then the equation (19) becomes e function is defined as follows: clearly lim u⟶∞ f(u) � ∞. us, if C 4 < 0, then equation (22) has at least one positive root.

Hopf Bifurcation and the Periodic Solution's Stability
Using the theories of normal form and centre manifold [27] of the system, we explore stability and Hopf bifurcation's direction.

Theorem 2.
If μ H > 0 then the Hopf bifurcation is supercritical otherwise it is subcritical. Here, sign determines the direction of the Hopf bifurcation.
If Ω p > 0 then the bifurcated periodic solutions are stable otherwise it is unstable. Here, the sign of Ω p determines the stability of the bifurcated periodic solutions.
If Θ I > 0 then the period of the bifurcated solutions increases otherwise it decreases. Here, the sign of Θ I determines the period of the bifurcated periodic solutions.
Proof. Let 4 � P(t) − P * and normalize the delay with t � t/τ. Let τ � τ 0 + ξ, ξ ∈ R, then ξ � 0 is the Hopf-bifurcation value of system equation (2) and system equation (2) can be transformed into a functional differential equation in C � C([−1,0], R 4 ) as follows: where 4 ) and L ξ : C ⟶ R 4 and G: RXC ⟶ R 4 are given, respectively, by the following equation: where By the Riesz representation theorem, there exists a function η(θ, ξ) of bounded variation for θ ∈ [−1, 0] such that In fact, we choose e system equation (1) is equivalent to For Next, we define the bilinear inner form for A and A * .
According to the algorithms in [27] and a similar computation process to that in [2], we can obtain the following expressions: 6 Mathematical Problems in Engineering with where E 1 and E 2 can be obtained by the following two equations: en, we can obtain the following equation: us, we can obtain the results described in theorem (2). e proof is completed.

The SLBP Model with Noise
In this section, we considered a SLBP stochastic epidemic model, which includes four compartments i.e., susceptible, latent, breaking out, and patched with the mass action incidence rate. Specifically, we are investigating the effect of Gaussian white noise on the model (without delay) proposed by Zhang and Upadhyay [27] for various low, medium, and high intensities. e schematic representation of the proposed model is as follows.
e following system of nonlinear differential equations with noise describes the dynamics of the proposed model: where S(t), L(t), B(t) , and P(t) represent nodes of susceptible, latent, breaking out, and patch at time t, respectively. In this analysis, we emphasis on the dynamics of the model about the interior equilibrium point D * (S * , L * , B * , P * ) only according to the method introduced by Nisbet and Gurney [28] and Carletti [29].

Mathematical Problems in Engineering
And, by focusing solely on the effects of linear stochastic perturbations As a result, the model (47)-(50) is reduced to the linear system shown as follows: Taking the Fourier transform of equations (47)-(50) we get the following equation: Equations (51) and (54) have a matrix form as follows: where, M(ω) � Alternatively, equation (55) can be written as follows:

If the function's Y(t) mean value is zero, the fluctuation intensity (variance) of its components in frequency intervals
is the spectral density Y and is defined as follows: e auto covariance function is the inverse transform of S Y (ω) if Y has a zero mean value.
And, the variance of the corresponding fluctuations in Y(t) is given by the following equation: e normalised auto covariance function is the auto correlation function.
For a Gaussian white noise process, it is e components of equation (57)'s solutions are as follows: e range of u i , i � 1, 2, 3, 4 is provided by the following equation: Hence the intensities of fluctuations in the variable u i , i � 1, 2, 3, 4 are given. 8 Mathematical As a result, the intensities of the variable's u i , i � 1, 2, 3, 4 fluctuations are given by the following equation: In other words, the variances of u i , i � 1, 2, 3, 4 are calculated as follows:

Here, M(ω) � R(ω) + iI(ω), where R(ω) is the real part of M(ω) and I(ω) is the an imaginary part of
We can derive the following from these numbers and equation (68): (70)

Numerical Simulations
In this section, we present a Numerical Simulation to validate of our analytical findings in this paper with help of Matlab software [30].
For the parameters, λ � 4, In the case of the absence of delay, the endemic equilibrium point E * E * (19.6227, 0.5138, 1.9337, 16.1464) is locally asymptotically stable, and corresponding time series is shown in Figure 1 [31].
In the presence of delay, for the value of τ � 30.50 < 40.50, the endemic equilibrium point is E * (19.6227, 0. 5138, 1.9337, 16.1464) locally asymptotically stable and the dynamical behavior of the time series as shown in Figure 2 [32].
Furthermore, we increase the delay value the system equation (1) Figure 3.
Finally, if τ � 40.50 > τ * , the system losses is stability and becomes unstable, then the corresponding time series as shown in Figure 4 [34].

Numerical Observations.
If there is no delay, the endemic equilibrium E * is really E * (19.6227, 0.5138, 1.9337, 16.1464) is locally asymptotically stable and the corresponding time series is shown in Figure 1. Figure 9 represents the time series evaluation of nodes for the values of example 1 in the presence of stochastic parameters [36]. At the values of noise intensities α 1 � 0.01;α 2 � 0.02;α 3 � 0.01;α 4 � 0.02 time series evaluation of four nodes, which are S(t), L(t), B(t), and P(t) are captured in Figure 9 [37]. At these low noise values, the proposed system (SLBP) is also less affected and clearly shown as less fluctuating [38]. Figure 10 represents the time series evaluation of nodes for the values of example 1 in the presence of stochastic parameters [39]. At the values of noise intensities α 1 � 0.04;α 2 � 0.05;α 3 � 0.04;α 4 � 0.05 time series evaluation of four nodes, which are S(t), L(t), B(t), and P(t) are captured in Figure 10 [40].
At these low noise values, the proposed system (SLBP) is also a little less affected and clearly shown as a little less fluctuating [41]. Figure 11 represents the time series evaluation of nodes for the values of example 1 in the presence of stochastic parameters. At the values of noise intensities, α 1 � 0.1; α 2 � 0.2; α 3 � 0.1; α 4 � 0.2 time series evaluation of four nodes [42], which are S(t), L(t), B(t), and P(t) are captured in Figure 11. At these values of noise, the proposed system (SLBP) is affected remarkably [43] and clearly notable fluctuations in the projections as P(t) increases and S(t) is started decreasing. Both P(t) and B(t) are affected and fluctuates more rapidly when compared with S(t) and L(t) [44]. Figures 11(a)   and P(t) [45]. Figure 11   show that noise intensity greatly affects the system. Figure 11(c) shows that noise intensity is not affecting much the system [46]. Figure 12 represents the time series evaluation of nodes for the values of example 1 in the presence of stochastic parameters. e values of noise intensities/time series evaluation of four nodes, which are S(t), L(t), B(t), and P(t), is captured in Figure 12. At these values of noise, the proposed system (SLBP) is affected remarkably and notable fluctuations in the projections as P(t) is increased and S(t) is started decreasing. Both P(t) and B(t) affected greatly and fluctuated more rapidly when compared with S(t) and L(t). Figure 13 represents the time series evaluation of nodes for the values of example 1 in the presence of stochastic parameters. At the values of noise intensities α 1 � 4;α 2 � 5;α 3 � 4;α 4 � 5 time series evaluation of four nodes, which are S(t), L(t), B(t), and P(t) are captured in Figure 13. At these values of noise, the proposed system (SLBP) is affected greatly, and oscillatory fluctuations in the projections as P(t) increases and S(t) decreases, and S(t) moves very close to B(t) and L(t), Both P(t), and B(t) affected greatly and fluctuates more rapidly when compared with S(t), L(t).        in a very short span in the absence of stochastic parameters in the proposed system. Whereas Figures 9-14 clearly show that noise affects the system SLBP greatly and the figures exhibit the disturbances in the form of oscillatory behaviour for various noise intensity values. e system is highly oscillatory at higher values of the noise intensity, which is captured by computer simulations, particularly from Figures 10-14.

Conclusions
We proposed a four compartmental model for understanding the complexity and exploitation created by virus and their impacts on networks. With this aim, we proposed an SLBS model consisting of S(t), L(t), B(t), and P(t). Under certain conditions, the local stability of all equilibrium points is investigated. e delay parameter was set, and we established the occurrence of a Hopf bifurcation as it crossed a crucial point by both analytical and numerical analysis. We also used the centre manifold theorem and normal form theory to investigate the properties of the Hopf bifurcation. We performed numerical simulation tests under various scenarios with appropriate sample values to support the theoretical findings.
Furthermore, this article investigates the prospects of infected node eradication and patched node persistence in a computer network. e proposed model exhibits rich dynamics for various studies like delay and Hopf bifurcation analysis. is model exhibits effective rich dynamics in the presence of delay particularly after reaching a certain value as τ � 30.50. It is very clearly shown in the numerical findings that the system bifurcates and exhibits its dynamics at τ � 30.650 � τ * . Phase portrait figures are drawn for various combinations under different delay parameter values, which are clearly presented the delay dynamics of the system. Delay dynamics exhibited by the system by both ways, analytically and numerically are captured and presented greatly.
We also focussed on the impact of additive white noise in the proposed system. We introduced noise intensities as stochastic parameters to the system and studied the stochastic model by linearizing the model using the perturbations technique and applying Fourier Transform. By finding out the noise intensities under certain constraints, the system should attain its steadiness as per the values of distinct noise parameters. Analytical results are checked with numerical simulations with appropriate example values. Notable numerical observations are discussed based on the various noise intensity values. e proposed model exhibits rich dynamics for various intensity values numerically and analytically, which is captured in stochastic analysis.
Future Scope: we can go for more deterministic graphs with parameter variations in terms of Sensitivity analysis is one of the approaches which allows to study and draw some interesting results. We can reconstruct this model in partial differential equations, including diffusive parameters, to catch the spatiotemporal dynamics more innovatively.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.