Optimal and Nonlinear Dynamic Countermeasure under a Node-Level Model with Nonlinear Infection Rate

This paper mainly addresses the issue of how to effectively inhibit viral spread by means of dynamic countermeasure. To this end, a controlled node-level model with nonlinear infection and countermeasure rates is established. On this basis, an optimal control problem capturing the dynamic countermeasure is proposed and analyzed. Specifically, the existence of an optimal dynamic countermeasure scheme and the corresponding optimality system are shown theoretically. Finally, some numerical examples are given to illustrate the main results, from which it is found that (1) the proposed optimal strategy can achieve a low level of infections at a low cost and (2) adjusting nonlinear infection and countermeasure rates and tradeoff factor can be conductive to the containment of virus propagation with less cost.


Introduction
In order to study the long-term behavior of computer virus and suppress viral spread macroscopically, a large number of dynamical models have been proposed in the past few decades (for the related references, see, e.g., [1][2][3][4][5][6][7][8][9][10][11]).From the perspective of the division scale of computers on networks, these models can be roughly divided into two categories: compartment-level models and node-level models.
Compartment-level models are those models that regard computers having the same state as an object to study.This work can be traced back to the 1980s.The first compartment-level model is proposed by Kephart and White [1], who followed the suggestions recommended by Cohen [12] and Murray [13].Since then, multifarious propagation models have been developed (see, e.g., [14][15][16][17][18][19][20][21][22]).It is worth noticing that Zhu et al. [6] proposed the original compartment-level SICS (susceptible-infected-countermeasured-susceptible) model with linear static countermeasure based on the CMC (Countermeasure Competing) strategy presented by Chen and Carley [23].However, compartment-level models ignore the effect of network eigenvalue on viral spread.Consequently, node-level models are considered.
Inspired by the above-mentioned work and based on the compartment-level SICS model, this paper considers a controlled node-level SICS model.Different from the conventional node-level models, this paper mainly addresses the issue of how to effectively distribute dynamic countermeasure by optimal control strategy (for the related references of optimal models, see, e.g., [29][30][31][32][33]).An optimal control problem is proposed and the existence of an optimal control is proved.The corresponding optimality system is also derived.Finally, some numerical examples are made, from which it can be seen that the proposed optimal strategy can achieve a low level of infections at a low cost.
The subsequent materials of this paper are organized as follows.Sections 2 and 3 formulate the controlled node-level model and analyze the optimal control problem, respectively.Numerical examples are provided in Section 4. Finally, Section 5 closes this work.

The Controlled Node-Level Model
In this paper, the propagation network of computer virus and countermeasure is represented by a graph  = (, ) with  nodes labelled 1, 2, . . ., , where each node and edge stand for a computer and a network link, respectively.Thus, the graph  can be described by its adjacency matrix A = [  ] × , where   = 0.
For convenience, two important functions, which will be used in the sequel, are defined as follows: Clearly,   () ≤ ∑        ().

The Optimal Control Problem
As   () +   () +   () ≡ 1, 1 ≤  ≤ , system (5) can be reduced to the following system: with initial condition ( 1 (0) , . . .,   (0) ,  1 (0) , . . .,   (0)) where Then system (9) can be written in matrix notation as with initial condition x(0) ∈ Ω.Now, the objective is to find a control variable u(⋅) ∈  so as to minimize both the prevalence of infected computers and the total budget for dynamic countermeasure during the time period [0, ].That is, the following optimal control problem needs to be solved: subject to system (12), where is the Lagrangian and  = ( 1 , . . .,   )  is a tradeoff factor based on the control effect and control cost of dynamic countermeasure.

Existence of an Optimal
Control.First, a lemma, which plays a critical role afterwards, is introduced.
Lemma 1 (see [34,35]).We have an optimal control problem with x(0) ∈ Ω, where Ω is positively invariant for system (15).The problem has an optimal control if the following six conditions hold simultaneously.
In order to prove the existence of an optimal control, six lemmas, one for each condition in Lemma 1, should be proved.
Proof.Substituting u ≡ u fl (, . . .,  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  )  into system (12), one can get the uncontrolled system: with x(0) ∈ Ω.Then the function f(x, u) is continuously differentiable, and Ω is positively invariant for the system.Hence, the claimed result follows from the Continuation Theorem for differential equations [36].
Lemma 3. The admissible set  is convex.
Hence, the claimed result follows.
Lemma 4. The admissible set  is closed.
Proof.Let u = ( 1 , . . .,   )  be a limit point of  and be a sequence of points in  such that From the completeness of ( 2 [0, ])  , one can get Hence, the closeness of  follows from the observation that Lemma 5. f(x, u) is bounded by a linear function in x.
Proof.Note that the Hessian matrix of (x, u) with respect to u ∈  is as follows: For any  ∈ [0, ], H u () is real symmetric and its eigenvalues are all positive.Then, H u () is positive definite.Hence, the convexity of (x, u) follows by the result in [37].
Thus, the proof is complete.Now, it is time to examine the main result of this subsection.
Theorem 8.The optimal control problem (P) has a solution.
Proof.Lemmas 2-7 show that the six conditions in Lemma 1 are all met.Hence, the proof is complete.

The Optimality System.
In this subsection, a necessary condition for an optimal control of problem (P) is drawn.Theorem 9. Suppose u * (⋅) is an optimal control for problem (P) and ( * (⋅),  * (⋅))  is the solution to system (9) with u(⋅) = u * (⋅).Then, there exist functions  * 1 () and  * 2 (), 0 ≤  ≤ , 1 ≤  ≤ , such that with transversality conditions Furthermore, one can get Proof.The corresponding Hamiltonian is where Thus, system (26) follows by direct calculations.As the terminal cost is unspecified and the final state is free, the transversality conditions hold.By using the optimality condition one can obtain that, for 0 ≤  ≤  and for 1 ≤  ≤ , either or  *  () =  or  *  () = .Hence, the proof is complete.
By combining the above discussions, one can get the optimality system for problem (P) as follows: with (I(0), C(0))  ∈ Ω and () = 0.

Numerical Examples
In this section, the effectiveness of the optimal dynamic countermeasure will be verified by some numerical examples.For our purpose, three networks are considered: a synthetic small-world network (WS network [38]), a synthetic scale-free network (BA network [39]), and a partial Facebook network [40], with  = 150 nodes, respectively.The parameters of system (33) are set as   = 0.01,   = 0.004887 (the value of   comes from a report on some real infection probabilities in [41]),   = 0.02,   = 1,  = 0.01,  = 0.1, and  = 50, 1 ≤  ≤ , and the initial conditions are set as   (0) = 0.03 and   (0) = 0.01, 1 ≤  ≤ .The optimality system (33) is solved by invoking the backward-forward Euler scheme with step size 0.01.Here we have to point out that some parameter values are chosen hypothetically due to the unavailability of real world data.
Suppose u * () is an optimal control for problem (P) and x * () is a solution to the corresponding controlled system.Let  * () and  * () denote the average control and the proportion of infected nodes under u * (), respectively, where Example 1.Take a WS network with 150 nodes and 150 links as the propagation network.
Figure 2 exhibits the average control  * () and  * () under different control strategies.Table 1 gives the final proportion of infected nodes and the value of objective function  under different control strategies, where the value of static control u = 0.08895 is an average of several real curing probabilities reported in [42].From Figure 2 and Table 1, one can conclude that u * is indeed the optimal control strategy to minimize the objective function  and reduce virus prevalence to a low level simultaneously.dominant role in the suppression of virus diffusion, and (c) linear infection rate overestimates virus prevalence, which is in accordance with the result in [7]. Figure 5 depicts the final proportion of infected nodes  * () and the objective function (u * ) for varied  1 and  2 .From this figure, it can be seen that  is decreasing and increasing with respect to  1 and  2 , respectively, which makes a suggestion that enhancing  1 and diminishing  2 are beneficial to the containment of viral spread and reduce  to a low level simultaneously.
Figure 6 shows  * (),  * (),  * (), and (u * ) for different .From this figure, it is found that decreasing  is effective on the suppression of virus propagation and attains a lower (u * ) simultaneously, although it creates more control cost.This is in good agreement with the fact that when the control effect (i.e., to obtain a low level of infections) is given priority (i.e., with lower ), often the decision is made to spend enough control cost.Hence, the tradeoff factor  plays a critical role in the balance between control effect and control cost.   2 shows the values of  * () and (u) under different control strategies.Figures 8 and 9 depict  * () and  * () for different  1 and  2 , respectively.Figure 10 demonstrates  * () and (u * ) for different  1 and  2 .and  * () for different  1 and  2 , respectively.Figure 15 demonstrates  * () and (u * ) for varied  1 and  2 .Figure 16 depicts  * (),  * (),  * (), and (u * ) for different .
Most of the results concluded from this example are the same as those in Example 1 except the two phenomena listed as follows: (a) higher  2 increases  * (), which is contrary to the results in Figures 3(b) and 8(b), and (b)  1 has a negligible impact on  * () and  * ().This indicates that the network structure, to some extent, determines the control cost and virus diffusion.
Combining the above numerical examples, the main results are listed below.
(a) u * is indeed the optimal control strategy to minimize the objective function  and reduce the infections to a low level simultaneously.(e) Decreasing the tradeoff factor  is beneficial to the suppression of virus spread and obtains a lower (u * ) simultaneously, although it brings more control cost.
Additionally, the structure of network, to some extent, determines the virus prevalence and the control cost.Thus, we shall investigate how the network topology affects virus spreading and control cost in the next work.

Concluding Remarks
This paper has studied the issue of how to work out an optimal dynamic countermeasure for achieving a low level of infections with a low cost.In this regard, a controlled node-level SICS model with nonlinear infection rate has been established.Furthermore, an optimal control problem has been proposed.The existence of an optimal control and the corresponding optimality system have also been derived.Additionally, some numerical examples have been given to illustrate the main results.Specifically, it has been found that the proposed optimal countermeasure scheme can achieve a low level of infections at a low cost.
In our opinions, the next work could be made as follows.First, the quadratic cost functions may be generalized to some generic functions.Second, delays [43][44][45], pulses [46,47], and random fluctuations [15] may be incorporated to controlled node-level models.Last, but not least, it is worthy to carry out research on the impact of the network topology [9,25,48,49] on the dynamic countermeasure strategy.

Figure 1 :
Figure 1: The transfer diagram of the controlled node-level SICS model.

Figure 3
Figure 3 demonstrates the average control  * () for different  1 and  2 .From this figure, one can see that (a) enhancing  1 and  2 roughly reduces  * (), (b) the smaller  2 is, the longer  * () stays at , and (c)  2 has a more significant impact on  * () than  1 does.

Figure 4 displays
Figure 4  displays  * () for different  1 and  2 .From this figure, it can be seen that (a) lower  1 favors virus spreading, whereas lower  2 is conducive to the containment of virus prevalence, (b)  2 affects  * () more significantly than  1 does, which implies that dynamic countermeasure plays a

Example 2 .
Take a BA network with 150 nodes and 150 links as the propagation network.

Figure 11 exhibits
* (),  * (),  * (), and (u * ) for different .From them, one can get the same results in Example 1.So they are omitted here for brevity.Example 3. Take a partial Facebook network with 150 nodes and 603 links as the propagation network.
Table 3 gives the values of  * () and (u) under different control strategies.Figures 13 and 14 display * ()