Optimal Control Strategy for SLBRS with Two Control Inputs

: Computer virus a�acks result in signiﬁcant losses each year, drawing considerable a�ention from enterprises, governments, academic institutions, and various other sectors. Researchers have proposed various approaches to ﬁght against computer viruses, including antivirus software and internet ﬁrewalls. In this paper, we focus on investigating computer virus transmission from the perspective of mathematical modeling. Our main contributions in this paper are threefold: (1) we improve the classical SLBRS model by incorporating cure rates, eﬀectively capturing the dynamics of computer network maintenance; (2) we introduce an optimal control system within the SLBRS framework, with the dual objectives of minimizing network detoxiﬁcation costs and reducing the proportion of broken-out nodes; and (3) by employing Pontryagin’s Maximum Principle, we establish the existence and uniqueness of an optimal control strategy for the proposed control system. Furthermore, we perform numerical simulations to demonstrate the eﬀectiveness of our theoretical analyses.


Introduction
In the era of big data, information exchanges have escalated in our daily lives, thereby amplifying the potential for computer virus propagation within computer networks.Computer virus threats result in substantial losses to enterprises, governments, academic fields, and various sectors worldwide every year (For instance, notable cases like the Stuxnet virus in 2010 and the Ransomware virus in 2017 inflicted tens of billions of dollars within the finance, education, energy, and other sectors).A diverse range of measures must be undertaken to combat computer viruses, including the implementation of firewalls and antivirus software.Nevertheless, delving into the propagation mechanisms of computer viruses is of heightened urgency compared to merely eradicating them from specific networks.Over the past few decades, various ordinary differential equations have been proposed to model the propagation mechanisms of computer viruses.Usually, computers within a network are categorized into distinct compartments: susceptible nodes-S(t), infected nodes-I(t), quarantine nodes-Q(t), recovery nodes-R(t), broken-out nodes-B(t), latent nodes-L(t), etc.In accordance with specific scenarios, certain interdependencies among these compartments are established, leading to the nomenclature of the model based on the compartments involved, i.e., SI, SEIR, SEIRQ, and so on.
In 1991,  These early computer virus models are inspired extensively by the results concerning epidemiological virus models, which share a common assumption that an infected computer in a latent state will not transmit a virus to other computers.However, this assumption does not always hold true for computer virus transmission.In fact, when it comes to computer viruses, the following phenomenon occur:  [5]).The detailed relationships among the involved compartments are illustrated in Figure 1.When faced with a computer virus a ack, computer users often adopt different protective measures to fight against it(see Reference [6]).The following are some examples: (c) Some users promptly execute antivirus software to eliminate the virus once they become aware of the threat.If the damage is extensive and recovery is not feasible, users may resort to reinstalling the operating system.This scenario often occurs in broken-out nodes.(d) Other users might a empt to clear the virus proactively, even if they are not certain whether the virus is present on their computers.This behavior is typically observed in latent nodes.
Take (c) and (d) into account, people usually make the assumption that breaking-out computers can be cured with specific cure rates 1  and 3  (either through executing antivirus software or by reinstalling the system), and similarly, latent computers can be cured with a certain cure rate 2  (by using antivirus softwares).The graded cure rates γ , γ , and γ are integrated into the SLBRS model, resulting in a SLBRS model with cure rates.Figure 2 can be represented by a system of differential equations, as follows: where γ , γ , and γ represent the cure rates (usually γ > γ > γ ); α, β, σ, p, and μ represent the conversion rates (non-negative constants); and S(t), L(t), B(t), and R(t) represent the compartments in the network: susceptible nodes, latent nodes, broken-out nodes, recovery nodes.The intensity of computer virus a acks fluctuates over time, and the countermeasures adopted by computer users also vary (with computer users bolstering protection during hacking threats and vice versa).Hence, a more realistic approach is to hypothesize that the included cure rates are time-varying.We replace the cure rate 1   by a Lebesgue measurable function 1 u (t) .
Figure 3 can be represented by the following system of differential equations: Naturally, the inclusion of more control inputs gains a heightened level of manipulation over the system.We replace the cure rates 1   and 2  with Lebesgue measurable functions 1 u (t) and 2 u (t) (the other cases are similar and, thus, we omit the details in this paper).Figure 4 can be represented by the following system of differential equations: Research on computer virus models has primarily focused on modeling, stability analysis, and related simulations.As research progresses, various mathematical tools have been applied to the analysis of computer virus models, particularly control theory (see [7][8][9][10][11]).In this paper, we treat the time-varying cure rates 1 u (t) and 2 u (t) as control input functions and investigate the optimal control problem for SLBRS.The rest of this paper is outlined as below: In Section 2, we establish the theoretical foundations encompassing the existence and uniqueness of the optimal control strategy, and the necessary conditions for optimal control.
In Section 3, we provide a numerical simulation to demonstrate the effectiveness of the theoretical analyses presented in Section 2.
In Section 4, we draw the conclusion according to the theoretical analyses and the numerical experiment.u (t) and 2 u (t) .

Optimal Control Problem
Let T be a pre-assigned constant.We define the admissible control set as follows:   To effectively restore a contaminated network, any undertaken measure should aim to minimize the proportion of infected computers (L(t) and B(t)) or reduce the cost of system maintenance.With this objective, we present the optimal control problem as follows: which is subject to (3).
Corresponding to (5), we define the Lagrangian as follows: And we define the Hamiltonian as follows:

Main Results and Their Proofs
Firstly, we demonstrate the existence of an optimal solution for the control system (3-5) based on the following theorem: Theorem 1.There exists an optimal control input subject to the control system (3)-( 5) with initial conditions: Proof.To establish the existence of an optimal solution for the control system (3)-( 5), it suffices to validate the following general conditions (see [12]): , , ,   with the transversal conditions (T) 0, (T) 0, (T) 0, (T) 0.
Furthermore, the optimal control inputs were determined by Proof.By differentiating the Hamiltonian, we obtain the following: Consequently, we can reformulate (14) into the co-state Equations ( 8)- (11).By deducing from the optimal conditions, we derive the control equations as follows: It follows from (15) and the admissible condition   Hence, we have successfully derived the state equations: and the co-state equations: with the initial condition and the transversal condition □

Numerical Example
In this section, we conduct numerical simulations to illustrate the influence of optimal control on the SLBRS model.

Algorithm
We employ an iterative algorithm based on the fourth-order Runge-Ku a method, which involves four distinct steps: Step

Simulation
We choose the following initial values: To ensure the stability of the system, we use the parameters presented in Table 1.In the visual representation, the red curves denote the states of SLBRS with constant cure rates, while the blue curves denote the states of SLBRS with control inputs.The results of the numerical experiment reveal the following: (1) The asymptotic stability of SLBRS is evident as both the red and blue curves converge towards a specific equilibrium.

Conclusions
In this paper, we investigate the SLBRS model from the perspectives of mathematical We point out that the control strategies significantly contribute to the restoration of a contaminated network.Taking the same initial condition (22) and using the same parameters (Table 1), we compute the equilibrium points of the aforementioned control systems (1)(2)(3).The results are outlined below (Table 2).We can conclude, based on the results in Table 2, that the nontoxic compartments S* and R* increase while the toxic compartments L* and B* decrease.This implies that our proposed measures are indeed effective in counteracting computer virus a acks.
The more control variables there are, the easier it becomes to achieve the desired control effects.Therefore, it is more challenging to study the optimal control problem associated with SLBRS with fewer control inputs.From the technical perspective, the analysis methods for the scenario with three control inputs are nearly identical to those for the scenario with two control inputs.Therefore, our research methods in this paper can be directly applied to solve the optimal control problem for SLBRS with three control inputs.

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by the author.Licensee MDPI, Basel, Swi erland.This article is an open access article distributed under the terms and conditions of the Creative Commons A ribution (CC BY) license (h ps://creativecommons.org/license s/by/4.0/).
(a) Once a computer is infected, it immediately gains the ability to spread the infection; (b) Recovered computers can acquire temporary immunity.In 2012, Yang, Zhang, Li et al. modified the classical SLBS model by incorporating a recovery compartment (R(t)) and used the SLBRS to model computer virus transmission, taking into account the observations in (a) and (b) (see Reference

( a )Theorem 2 .
The set of control and state variables is nonempty.(b) The admissible control set U is both closed and convex.(c) The right-hand side of the state system can be bounded by a linear function of the state variables.(d) The Lagrangian is concave on the admissible control set U, and there exist constants , B, u) c c (| u | +| u | ) , u (u , u ),   □ Next, by using Pontryagin's Maximum Principle, we deduce a necessary condition for the optimal control strategy based on the following theorem: Given an optimal control input , B , R of (3), there exist adjoint variables1 2 3 4

( 2 )
When controls are applied, the restoration of the contaminated network becomes evident: susceptible and recovered computers show an increased trend (Figures 5 and 6), whereas latent and breaking-out computers exhibit a decreased trend (Figures 7 and 8).

Table 1 .
Values of parameters in the numerical simulation.

Table 2 .
Comparison of stable values of the state variables in each model.