OPTIMAL CONTROL ON THE SPREAD OF SLBS COMPUTER VIRUS MODEL

In this paper, we discuss an optimal control on the spread of SLBS computer virus model. Control strategy by installing antivirus on each subpopulation aims to minimize the number of infective computers (latent and breakingout) and the cost associated with the control. Optimal system condition is determined by using Pontryagin principle and is solved numerically by using Forward-Backward Sweep method in combination with the fourth order Runge-Kutta method. Our numerical simulations show that the strategy by installing antivirus on susceptible computers as preventive give a great influence on suppressing the spread of computer viruses. AMS Subject Classification: 49J15


Introduction
A computer viruses are malicious program that can replicate themselves and spread among computers.When user on computer network or use removable storage device, the computers are at hight risk of being infective by a computer viruses.During the breakingout period, virus may cause damage on whole computer system and give huge financial lose.However, antivirus software as a tool to agains computer virus has limited ability to solve it [1].
Considering the urgency of controlling the spread of computer viruses, many research formulate models of the spread of computer viruses and concerns with qualitative analysis of the model.Various model was adopted form classic epidemic infection disease model, such as SIR or SIRS model [2][3], SIE model [4], SEIR model [5], etc.However, computer viruses have the main characteristics that differentiate from biological virus, i.e., there are no infective computers that do not have infectivity.Therefore, the spread of computer viruses model should not have exposed computers (E).Moreover, infective computers are categorized as latent or breakingout which depends on condition of viruses stay in it [6].The latent period that begins when the virus enters the computer until just before the virus damages the computer system, whereas the breakingout period is when the virus begins to destroy the computer system until just before cleaned.
Lu-Xing and Xiaofan [1] has proposed a new epidemic model of computer virus where connected computers in this case are categorized into three subpopulations: susceptible computers (S ≡ S(t)), latent computers (L ≡ L(t)), and breakout computers (B ≡ B(t)).By assuming that every new connected computer is in virus free condition, the compartment diagram for the model is shown in Figure 1 and the model is written below, where µ, β 1 , β 2 , θ, γ 1 , γ 2 , δ, α are positive constants.In this model, susceptible computers connect to the network at constant rate µ.Each subpopulation disconnects from network with coefficient δ.Due to the interaction, the coefficients of infection from latent computers and breakingout computers to susceptible computers are denoted by β 1 and β 2 .The susceptible computers can be infective due to the influence of removable storage device with coefficient θ.Latent computers may break out and enter the B compartment with coefficient α.
Finally it is assumed that latent and breakingout computers are cured with coefficient γ 1 and γ 2 , respectively.Lu-Xing and Xiaofan [1] have shown that system (1) has a unique equilibrium, i.e., the endemic equilibrium which is globally asymptotically stable.This showed that there is no virus-free condition and the virus will always exist in the system.To minimize the number of infective (latent and breakout) computers, other strategies than constant curing have to be implemented to control the spread of computer viruses.Such strategies are usually combined with efforts to minimize the cost related to these strategies.Hence, an important tool in this case is optimal control theory.Lijuan et al. [6] has proposed a control strategy by installing antivirus on breakingout computers to minimize the number of breakingout computers and the costs associated with the control.They found that optimal control solution exists and effective for reducing the number of the breakout computers.However, their numerical simulations showed that the breakout computers still exist and the number of latent computers are increased.In this paper we introduce a different strategy to minimize the number of both latent and breakout computer including the costs associated with the control.Our strategy, the related theoretical optimal control problem and the method to solve this problem will be given in Section 2. In Section 4. numerical simulations are performed to show the influence of control variables.All results are summarized in Section 5.

Optimal Control Problem
In this paper we introduce control strategy u 1 ≡ u 1 (t), u 2 ≡ u 2 (t), and u 3 ≡ u 3 (t) that denote control for installing effective antivirus programs on susceptible, latent and breakout computers, respectively.Control u 1 is intended to provide protection for susceptible when interacting with infective computers or removable media.Control u 2 and u 3 are intended to repair infective computers (latent and breakingout).Due to limitation of antivirus abillity, we assume that by applying control u 3 for breakout computers at time t then there will be ηu 3 B breakout computers which will become susceptibles and (1 − η)u 3 B breakout computers will become latent, with η ∈ [0, 1].By applying such control strategies, model ( 1) can be written as, Control variables will be implemented during a time interval t ∈ [0, T ] and they are normalized such that the admissible set is given by, where u = (u 1 , u 2 , u 3 ).Our objective will be contructed with the following goal: 1. Minimized the number of latent computers and breakingout computers.2. Minimized the total cost for installing the antivirus program in each subpopulation.
Mathematically, the objective function is to minimize the following functional where v, w, x, y, and z are the weight constants subject to system of equations (2).To solve this optimal control problem we apply the Pontryagin minimum principle.For that, we define adjoint variables σ 1 ≡ σ 1 (t), σ 2 ≡ σ 2 (t), and σ 3 ≡ sigma 3 (t) for state S, L, and B respectively and consider the following Hamiltonian where σ = (σ 1 , σ 2 , σ 3 ) and f i are the right hand side of system (2), with transversal conditions σ i (T ) = 0, i = 1, 2, 3.
Proof.The adjoint equations and the transversality conditions are determined from the Hamiltonian (5).Differentiating the Hamiltonian (5) with respect to each state variable, i.e. ∂σ 1 ∂t = − ∂H ∂S , ∂σ 2 ∂t = − ∂H ∂L and ∂σ 3 ∂t = − ∂H ∂B , we obtain the adjoint system as follows In this system, there is no terminal cost and the final state is free.Therefore, the transversality conditions (bounded conditions) for the adjoint variables are σ i (T ) = 0, i = 1, 2, 3. Furthermore, the optimal conditions is obtained from the stationary conditions, i.e., Considering the feature of the admissible set U ad , see equation ( 3), we obtained u * 2 = min(max(0, u * 3 = min(max(0, Thus the optimal control solution is obtained by solving the following equation system,

Numerical Simulation
The optimality system (10) in this section is solved numerically using forwardbackward Sweep method in combination with the fourth order Runge-Kutta method, see [7] and [8] for the detail.For simulations, we take parameter values from [6] as shown in Tabel 1.The initial population is S 0 = 15, L 0 = 10, and B 0 = 10.The control variables will be implemented during t [0,T] with T = 100.We first study the influence of each control variable on system (2).For that aim, we carry out simulations by applying one control variable while two other control variables are set to be zero.The results of these simulations are shown in Figure 2.For comparison, we also plot the numerical solution without any control.It is found that without any control, the numerical solution is convergent to endemic equilibrium ( Ŝ, L, B)=(11.3709,14.3145, 14.3145).When a control strategy by installing antivirus only on susceptible computers, i.e.only u 1 is implemented, it is optimal to start this strategy at maximum rate.This strategy significantly influences the system which in this case the infective computers (both latent and breakout) are eradicated.At about t = 90, u 1 starts to decline.As a result, the number of latent and breakout computers start to increase due to the infection from removable media.If antivirus is installed only on latent computers or only on breakout computers, i.e., applying only u 2 or only u 3 , then the optimal control strategies are not at maximum rate.Such optimal control can reduce the number of latent or breakout computers but they are not as effective as u 1 .This is because the control strategy by installing antivirus only on latent computers or only on breakout computers can not prevent virus infection from removable storage.
In the second case, we implement all three control variables (u 1 , u 2 , and u 3 ) simultaneously, i.e., antivirus is installed on all susceptible, latent and breakout computers.The optimal solution for this case is shown in Figure 3.At the beginning, control u 1 is not at maximum rate due to u 2 and u 3 are also applied.Then, control u 1 is optimal to be implemented at maximum rate, while u 2 and u 3 are set to be zero.This optimal strategy leads to increasing the number of susceptible computers significantly and the number of latent and breakout computers decrease to zero.Since the control variables are stop at t = 100, u 1 starts to decrease at about t = 90.With decreasing u 1 , latent computers start to appear again due to the infection from removable media so that it is optimal to control the network by targeting the latent and breakout computers.Since all control has to be stop at t = 100, while the infection from removable media cannot be stop, the number of latent and breakout computers is getting bigger at the end of control.Figure 2 and Figure 3 show that applying only u 1 and combining all three control variable (u 1 , u 2 , and u 3 ) give relatively the same result.This indicates that to suppress the spread of computer viruses is enough by installing antivirus only on susceptible computers as prevention.

Conclusions
In this paper we have discussed an optimal strategy to eradicate virus infection on SLBS computer virus model by installing antivirus program on all subpopulation (susceptible, latent and breakout computers).The optimal control strategy is meant to minimize the number of latent and breakout computers as well as the cost associated with installing the antivirus program.The optimal problem is solved using Pontryagin's principle and the obtained optimality system is solved numerically using forward-backward sweep method.Based on the numerical simulations, it is found that installing antivirus program only on sus-ceptible computers as a preventive is the most effective strategy in suppressing the spread of computer viruses.

Figure 1 :
Figure 1: Compartment of the spread of vomputer virus.

Table 1 :
Parameter values for the simulations