Abstract
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.
1. 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., [1–11]). 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., [14–22]). 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.
Node-level models are those models that regard a single computer as an object to investigate. The first node-level model (i.e., SIS (susceptible-infected-susceptible) model) is proposed by Van Mieghem et al. [7]. Since then, Sahneh and Scoglio [8] presented the node-level SAIS (susceptible-alert-infected-susceptible) model, and Yang et al. [9, 10] considered the node-level SLBS (susceptible-latent-breaking-susceptible) and SIRS (susceptible-infected-recovered-susceptible) models, respectively. Very recently, Gan [11] established the node-level SIES (susceptible-infected-external-susceptible) model. Besides, for the other related work about this topic, one can refer to [24–28] and the references cited therein.
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., [29–33]). 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.
2. The Controlled Node-Level Model
In this paper, the propagation network of computer virus and countermeasure is represented by a graph with nodes labelled , where each node and edge stand for a computer and a network link, respectively. Thus, the graph can be described by its adjacency matrix , where .
As was treated in the traditional SICS model [6], at any time all nodes in the graph are divided into three groups: -nodes (susceptible nodes are uninfected but have no immunity), -nodes (infected nodes), and -nodes (countermeasured nodes are uninfected and have temporary immunity due to the presence of countermeasures). Let , , and denote the probability of node being susceptible, infected, and countermeasured at time , respectively. Then the vector probabilistically captures the state of the network at time .
For convenience, two important functions, which will be used in the sequel, are defined as follows:Clearly, .where is a controllable rate, , ; and are positive constants, .
Now, a set of probabilistic assumptions on the state transition of node are made (see also Figure 1).(A1)An -node becomes infected at rate .(A2)An - or -node becomes countermeasured at rate .(A3)An -node becomes susceptible at a constant rate .(A4)A -node loses immunity at constant rate .
Let be a very small time interval and be a higher-order infinitesimal. Assumptions (A1)–(A4) imply that the probabilities of state transition of node satisfy the following relations:
Invoking the total probability formulas and letting , the controlled node-level model (i.e., controlled node-level SICS model) can be derived.with initial condition where
The admissible control set iswhere .
3. The Optimal Control Problem
As , , system (5) can be reduced to the following system:with initial condition where
Then system (9) can be written in matrix notation aswith initial condition .
Now, the objective is to find a control variable so as to minimize both the prevalence of infected computers and the total budget for dynamic countermeasure during the time period . That is, the following optimal control problem needs to be solved: subject to system (12), where is the Lagrangian and is a tradeoff factor based on the control effect and control cost of dynamic countermeasure.
3.1. 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 subject towith , where is positively invariant for system (15). The problem has an optimal control if the following six conditions hold simultaneously. (C1)There is such that system (15) is solvable.(C2) is convex.(C3) is closed.(C4) is bounded by a linear function in .(C5) is convex on .(C6) for some , , and .
In order to prove the existence of an optimal control, six lemmas, one for each condition in Lemma 1, should be proved.
Lemma 2. There is such that system (9) or (12) is solvable.
Proof. Substituting into system (12), one can get the uncontrolled system: with . Then the function is continuously differentiable, and 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.
Proof. Let As is a real vector space, one can obtain Then, the convexity of follows by the observation that Hence, the claimed result follows.
Lemma 4. The admissible set is closed.
Proof. Let be a limit point of and be a sequence of points in such that From the completeness of , one can get Hence, the closeness of follows from the observation that
Lemma 5. is bounded by a linear function in .
Proof. Note that, for system (9) and for , Thus, the claimed result follows.
Lemma 6. is convex on .
Proof. Note that the Hessian matrix of with respect to is as follows:For any , is real symmetric and its eigenvalues are all positive. Then, is positive definite. Hence, the convexity of follows by the result in [37].
Lemma 7. for some , , and .
Proof. Let , , and . Then, . Thus, the proof is complete.
Now, it is time to examine the main result of this subsection.
Theorem 8. The optimal control problem has a solution.
Proof. Lemmas 2–7 show that the six conditions in Lemma 1 are all met. Hence, the proof is complete.
3.2. The Optimality System
In this subsection, a necessary condition for an optimal control of problem is drawn.
Theorem 9. Suppose is an optimal control for problem and is the solution to system (9) with . Then, there exist functions and , , , such thatwith transversality conditions Furthermore, one can get
Proof. The corresponding Hamiltonian is where , are undetermined, .
According to the Pontryagin Minimum Principle [35], there exist functions and , , , such that 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 and for , either or or . Hence, the proof is complete.
By combining the above discussions, one can get the optimality system for problem as follows: with and .
4. 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 nodes, respectively. The parameters of system (33) are set as , (the value of comes from a report on some real infection probabilities in [41]), , , , , and , , and the initial conditions are set as and , . 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 is an optimal control for problem and is a solution to the corresponding controlled system. Let and denote the average control and the proportion of infected nodes under , 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 is an average of several real curing probabilities reported in [42]. From Figure 2 and Table 1, one can conclude that is indeed the optimal control strategy to minimize the objective function and reduce virus prevalence to a low level simultaneously.
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Figure 3 demonstrates the average control for different and . From this figure, one can see that (a) enhancing and roughly reduces , (b) the smaller is, the longer stays at , and (c) has a more significant impact on than does.
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Figure 4 displays for different and . From this figure, it can be seen that (a) lower favors virus spreading, whereas lower is conducive to the containment of virus prevalence, (b) affects more significantly than does, which implies that dynamic countermeasure plays a 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].
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Figure 5 depicts the final proportion of infected nodes and the objective function for varied and . From this figure, it can be seen that is decreasing and increasing with respect to and , respectively, which makes a suggestion that enhancing and diminishing are beneficial to the containment of viral spread and reduce to a low level simultaneously.
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Figure 6 shows , , , and for different . From this figure, it is found that decreasing is effective on the suppression of virus propagation and attains a lower 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.
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Example 2. Take a BA network with 150 nodes and 150 links as the propagation network.
Figure 7 displays and under different control strategies. Table 2 shows the values of and under different control strategies. Figures 8 and 9 depict and for different and , respectively. Figure 10 demonstrates and for different and . Figure 11 exhibits , , , and for different . From them, one can get the same results in Example 1. So they are omitted here for brevity.
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Example 3. Take a partial Facebook network with 150 nodes and 603 links as the propagation network.
Figure 12 shows and under different control strategies. Table 3 gives the values of and under different control strategies. Figures 13 and 14 display and for different and , respectively. Figure 15 demonstrates and for varied and . Figure 16 depicts , , , and for different .
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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 increases , which is contrary to the results in Figures 3(b) and 8(b), and (b) 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) is indeed the optimal control strategy to minimize the objective function and reduce the infections to a low level simultaneously.(b)Linear infection rate overestimates the prevalence of virus.(c)Enhancing and diminishing are conductive to the containment of viral propagation and reduce to a low level simultaneously.(d) has more significant influences on , , and than does.(e)Decreasing the tradeoff factor is beneficial to the suppression of virus spread and obtains a lower 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.
5. 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 [43–45], 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by Natural Science Foundation of China (Grant nos. 61572006 and 61503307), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant nos. KJ1500415, KJ1704080, and KJ1704081), and Doctoral Scientific Research Foundation of Chongqing University of Posts and Telecommunications (Grant nos. A2015-02 and A2016-10).