Optimal Internet Worm Treatment Strategy Based on the Two-Factor Model Xiefei Yan and Yun Zou The security threat posed by worms has steadily increased in recent years. This paper discusses the application of the optimal and sub-optimal Internet worm control via Pontryagin’s maximum principle. To this end, a control variable representing the optimal treatment strategy for infectious hosts is introduced into the twofactor worm model. The numerical optimal control laws are implemented by the multiple shooting method and the sub-optimal solution is computed using genetic algorithms. Simulation results demonstrate the effectiveness of the proposed optimal and sub-optimal strategies. It also provides a theoretical interpretation of the practical experience that the maximum implementation of treatment in the early stage is critically important in controlling outbreaks of Internet worms. Furthermore, our results show that the proposed sub-optimal control can lead to performance close to the optimal control, but with much simpler strategies for long periods of time in practical use. Keywords: Optimal control, worm control, epidemic model, two-factor model, Pontryagin’s maximum principle, genetic algorithm. Manuscript received Feb. 10, 2007; revised Nov. 16, 2007. This work was supported by the National Natural Science Foundation of China (Grant nos. 60474078, 60574015 and 60304001). Xiefei Yan (phone: + 86 25 84315463, email: yanxiefei@hotmail.com) is with the School of Automation, Southeast University, Nanjing, China; the Department of Automation, Nanjing University of Science and Technology, Nanjing, China; the Postdoctoral Research Working Station of Wuxi Seamless Oil Pipe Co., Ltd., Wuxi, China; and the Postdoctoral Research Working Station of Wuxi National High-Tech Industrial Development Zone, Wuxi, China. Yun Zou (email: zouyun@vip.163.com) is with the Department of Automation, Nanjing University of Science and Technology, Nanjing, China. ETRI Journal, Volume 30, Number 1, February 2008 I. Introduction With the explosive growth of Internet applications, Internet worms have become a major problem for the security of Internet networks [1]. Worms, defined as autonomous programs that spread through computer networks by searching, attacking, and infecting remote computers automatically, have been developed since the first Morris worm appeared in 1988 [2]. In 2001, the Code Red and Nimda worms infected hundreds of thousands computers [3] and cost millions of dollars in losses [4]. In order to understand the general characteristics of worm propagation, epidemiological models have been employed in many studies [1], [4]-[12]. From 1991 to 1993, Kephart, White, and Chess of IBM performed a series of studies on viral infection based on epidemiology susceptible-infectioussusceptible (SIS) models [6]-[8]. This model is based on biological epidemiology and provides a qualitative understanding of the spread of viruses under the assumption that classical epidemic models are all homogeneous, which means that an infected host is equally likely to infect any other susceptible hosts. Based on the classical epidemic model, Zou and others [5] considered the two-factor model to study the propagation of the Code Red worm. Chen and others presented a discrete-time worm model that considers the patching and cleaning effect during worm propagation [9]. Wang and others [10] introduced an analytic model to capture the impact of underlying topology in computer viral propagation. Moreover, Wang and Wang [11] proposed a model extending the classical epidemic model by including two specific parameters, infection delay and user vigilance time. Kim and others [12] introduced an extension of the susceptible-infectious-removed (SIR) model to simulate worm propagation in two different Xiefei Yan et al. 81 network topologies. Qing and Wen [1] investigated the wormanti-worm (WAW) model in their survey on Internet worms. Almost all related studies have been concerned with the simulative and predictive functions of models, thus, providing insight into the dynamics of worm propagation. However, there are few studies that mathematically discuss dynamic strategies against worm propagation. Zou and others [4] provided a dynamic quarantine method based on the principle “assume guilty before proven innocent.” A dynamic quarantine system with a constant quarantine time and a worm detection threshold was mathematically analyzed. The results show that in the dynamic quarantine system, a worm still propagates according to traditional epidemic models, but with a slower propagation speed and a higher epidemic threshold. Kreidl and others [13] presented a feedback control host-based autonomic defense system to protect the information and functionality of a server. Ram and others [14] studied a state space feedback control model to detect and control the spread of these viruses by measuring the number of connections an infected host makes. The objective of the mechanism is to slow down the spreading velocity of a worm by controlling (delaying) the total number of connections made by an infected host. As expected, the model showed that the sooner the infection is detected, the faster the reduction of the spreading velocity. Kim and others [15], [16] investigated an optimization model that takes into account the infection and treatment costs. As in many studies on the optimal control of epidemics [17]-[23], a control variable representing filtering treatment on infectious hosts was introduced into the classical SIS model. However, the classical model is not suitable for modeling propagation states of Internet worms. In the Internet, cleaning, patching, and filtering countermeasures against worms will remove both susceptible hosts and infectious hosts from circulation, but the classical model only accounts for the removal of infectious hosts. Moreover, the classical model assumes that the infection rate is constant, which is not true for a rampantly spreading Internet worm, such as the Code Red worm [5]. For these reasons, Zou and others [5] introduced the two-factor propagation model, which considers two factors. One factor is the dynamic countermeasures taken by ISPs and users; the other is the slowed down worm infection rate because rampant propagation of a worm causes congestion and troubles to some routers. This two-factor model is an extension and supplement of the classical model and more accurately reflects the propagation states of Internet worms [1]. In this study, we follow the idea of Kim and others about the optimal control of treatment costs for Internet worms [15], [16] and the works in epidemics control [21]-[23]. We further discuss the optimal strategy associated with the treatment of 82 Xiefei Yan et al. infectious hosts to prevent worm propagation based on the two-factor model. A sub-optimal solution with a much simpler strategy is also proposed. We introduce into this model one control variable representing the rate of treatment to remove infectious hosts from circulation by filtering or disconnection from Internet. The remainder of this paper is organized as follows. Section II describes the two-factor worm model with control. The analysis of optimization problems and the sub-optimal solution are presented in sections III and IV, respectively. In section V, we carry out a numerical simulation and briefly discuss the influences of some model parameters. Finally, the conclusions are summarized in section VI. II. Optimization of Two-Factor Worm Model In Zou’s two-factor worm propagation model [5], the host population is divided into four epidemiological classes: S, I, R, and Q (see Table 1). The model which incorporates control of infectious hosts is given by the following nonlinear system of differential equations: S (t ) = − β (t ) S (t ) I (t ) − μS (t )( I (t ) + R (t )) , (1) I(t ) = β (t ) S (t ) I (t ) − u (t ) I (t ) , (2) R (t ) = u (t ) I (t ) , (3) Q (t ) = μS (t )( I (t ) + R (t )) , (4) with S(0), I(0), R(0), and Q(0) given. In this model, human countermeasures result in the removal of both susceptible and infectious computers from circulation. During the course of Code Red propagation, an increasing Table 1. Notation of model parameters. Notation Definition S Number of susceptible hosts at time t I Number of infectious hosts at time t R Q I+R Number of removed hosts from the infectious population at time t Number of removed hosts from the susceptible population at time t Number of infected hosts at time t N Total number of hosts, N=S+I+R+Q β Infection rate at time t µ Removal rate of susceptible hosts ETRI Journal, Volume 30, Number 1, February 2008 number of people became aware of the worm and implemented some countermeasures: cleaning compromised computers, patching or upgrading susceptible computers, setting up filters to block the worm traffic on firewalls or edge routers, or even disconnecting their computers from Internet. Therefore, the process of removing susceptible hosts is directly related to the number of infected hosts at time t. β (t ) = β 0 [1 − I (t ) N ]η is the decreased infection rate, not a constant rate, where β 0 is the initial infection rate. The large-scale worm propagation caused congestion and troubles to some Internet routers [5], [24], which slowed down the Code Red scanning process. The exponent η is used to adjust the infection rate sensitivity to the number of infectious hosts I(t), and η = 0 means a constant infection rate. We take η = 3 here. The control variable u(t) is a bounded Lebesgue integrable function [20]. It represents the rate of treatment for removing infectious hosts from circulation by filtering or disconnection from Internet at time t . Furthermore, from an epidemiological modeling point of view, the transfer rate u(t)I(t) corresponds to an exponential waiting time e-ut as the fraction that is still in the infectious class t units after entering class R and to 1 u as the mean waiting time [25]. Remark 1. The two-factor model is suitable for worms that propagate without the topology constraint [5]. The topology constraint means that an infectious host may not be able to directly reach and infect an arbitrary susceptible host. It needs to infect several hosts on the route to the target before it can reach the target. Most worms, such as Code Red, have no topology constraint. III. Optimal Control Problems (6) Ω Here Ω = {u ∈ L1 (0, t f ) | 0 ≤ a ≤ u ≤ b ≤ 1} , and a and b are fixed positive constants. Pontryagin’s maximum principle [27], [28] provides the necessary conditions for an optimal control problem. This principle converts (1)-(3), (5), and (6) into a problem of minimizing pointwise a Hamiltonian, H, with respect to u: H = BI (t ) + 3 C 2 u (t ) + ∑ λi f i , 2 i =1 (7) where fi (i=1, 2, 3) is the right hand side of the differential equation of (1)-(3), respectively. By applying Pontryagin’s maximum principle, we have the following adjoint equations: dλ1 ∂H = ρλ1 − , λ1 (t f ) = 0, dt ∂S # dλ3 ∂H = ρλ3 − , λ3 (t f ) = 0. dt ∂Q That is, ⎡ ⎛ ⎣⎢ ⎝ λ1 = ρλ1 + λ1 ⎢ β 0 ⎜1 − η ⎤ I ⎞ ⎟ I + μ ( I + R)⎥ N⎠ ⎦⎥ η λ2 = − B + ρλ2 − λ3u tf (5) subject to (1)-(3), where tf is the final time. This performance specification takes into account the cost of infection, the numbers of infectious hosts, and the cost of control. The total cost includes not only the cost for every infected host but also the cost of reduced system performance, increased network delay [15], organization, management, cooperation, and so on; hence, the cost function should be nonlinear. In this paper, a quadratic function is implemented to measure the control cost by referenced to Kim and others [15], [16] and many other studies in epidemics control [18]-[23]. Furthermore, the cost function is written with a discount factor e-ρt with ρ>0 to obtain ETRI Journal, Volume 30, Number 1, February 2008 J (u * ) = min J (u ) . I ⎞ ⎛ − λ 2 β 0 ⎜1 − ⎟ I , N ⎝ ⎠ The problem is to minimize the cost function C J (u ) = ∫ [ BI (t ) + u 2 (t )]e − ρt dt 2 0 higher benefit than the present in intertemporal model (See references in Kim and others [15] and some economic applications in [26] and [27]). The coefficients B and C are weight factors related to the size and importance of the two parts of the objective function. We seek to find an optimal control u*, such that η η −1 ⎡ ⎛ ⎤ I ⎞ I ⎞ SI ⎛ + λ1 ⎢ β 0 ⎜1 − ⎟ S − β 0η ⎜1 − ⎟ + μS ⎥ ⎝ N⎠ N ⎢⎣ ⎝ N ⎠ ⎥⎦ η η −1 ⎡ ⎛ ⎤ I ⎞ I ⎞ SI ⎛ − λ2 ⎢ β 0 ⎜1 − ⎟ S − β 0η ⎜1 − ⎟ − u ⎥, ⎝ N⎠ N ⎢⎣ ⎝ N ⎠ ⎥⎦ λ3 = ρλ3 + λ1 μS , (8) with transversality conditions λi (t f ) = 0 , i = 1, 2 , 3 . (9) By the bound in Ω = {u ∈ L1 (0, t f ) | 0 ≤ a ≤ u ≤ b ≤ 1} , the optimal control is given by Xiefei Yan et al. 83 1 ⎧ ⎫ u (t ) = min ⎨max{a, (λ2 − λ3 ) I }, b ⎬ , C ⎩ ⎭ (10) π =R In this work, the search for a global minimum of J ′(π ) is implemented by an extended method based on genetic algorithms [29]. This extension admits the constraints ti + Δ i = t i +1 , t N + Δ N = t f , ti ≥ 0 , Δ i ≥ 0, and u i ∈ Ω . which is derived from the condition ∂H = 0. ∂u Remark 2. Due to the a priori boundedness of the state and adjoint functions and the resulting Lipschitz structure of the ordinary differential equations (ODEs), we obtain the uniqueness of the optimal control for small tf [20]. The uniqueness of the optimal control pair follows from the uniqueness of the optimality system, which consists of (1)-(3), (8), and (9) with characterization (10). There is a restriction on the length of the time interval in order to guarantee the uniqueness of the optimality system. This restriction on the length of the time interval is due to the opposite time orientations of (1)-(3), (8), and (9). The state problem has initial values, and the adjoint problem has final values. This restriction is very common in control problems [18], [20]-[23]. Remark 3. The problem previously described is a two-point boundary value problem (TPBVP), with specified initial condition for state equations (1)-(3) and terminal boundary conditions (9) for adjoint equation (8). It can be numerically solved by the multiple shooting method [19], [30], [31]. IV. A Sub-optimal Solution In this section, a sub-optimal solution is obtained by genetic algorithms following the ideas of [19] and [23]. Let the class of admissible controls be restricted to the collection of functions of type ⎧⎪u i u (t ) = ⎨ ⎪⎩ 0 if t ∈ I i , (11) otherwise, where u i ∈ Ω are constants, and Ii are closed intervals [t i , ti + Δ i ] , such that I i ∩ I j = ∅ if i ≠ j . Therefore, the restricted class admissible controls consist of pulses of height ui and width Δ i , starting at time ti. In order to simplify the actual policy implementation, we assume that the two controls are of pulsed form. Let u(t)=uN(t) be admissible controls with N pulses, characterized by (t1 , Δ1 , u 1 , t 2 , Δ 2 , u 2 , " , t N , Δ N , u N ) . Therefore, the suboptimal control problem is to minimize J ′(t1 , Δ1 , u , t 2 , Δ 2 , u , " , t N , Δ N , u ) := J [u (⋅)] , 1 2 N N where J [⋅,⋅] is the same as defined in (5). Let π = (t1 , Δ1 , u , t 2 , Δ 2 , u , " , t N , Δ N , u ) ∈ R 1 84 Xiefei Yan et al. 2 N for the case of N pulses. The sub-optimal control problem is simply min J ′(π ) . 3N V. Numerical Simulation In this section, we investigate the numerical solution of the optimal and sub-optimal control in the two-factor worm model. The optimal control is obtained by solving the optimality conditions, consisting of twelve ODEs from the state and adjoint equations. This TPBVP problem is solved using the improved multiple shooting method (see [19], [30], and [31]) as follows: Step 1. Select n nodes to divide the interval [0, tf], so that 0 = t1 ≤ t 2 ≤ " ≤ t n = t f . Step 2. Choose yi := [ x(t i ), λ (t i )] , i = 1, ", n . Step 3. Integrate (1) and (5) for each time interval [t i , ti +1 ) using yi as the initial conditions and obtaining y (ti −1 ) = [ x(ti −1 ), λ (ti −1 )] . Step 4. Compute h := [hi ] , where hi := yi − y (ti ) . Step 5. If h is sufficiently close to a minimum value, stop; otherwise, modify the initial conditions yi for the next iteration (using, for instance, the Newton-Raphson algorithm to make hi := yinew − y (t i ) = 0 ) and go to step 3. Table 2. Baseline values of parameters Parameters Values N 1 million I(0) 9989N/10,000 R(0) N/1,000 S(0) N/10,000 β0 0.35/N η 3 µ 0.06/N ρ 0.01 Final time, tf 140 hours Time step duration, dt 1 hour Upper bound for control 0.50 Lower bound for control 0.05 Weight associated with I:B 5 Weight associated with u:C 600 3N ETRI Journal, Volume 30, Number 1, February 2008 Figure 1 shows the time dependent optimal and sub-optimal control laws for the case in which C=600. The control u is plotted as a function of time. To minimize the number of infectious hosts (I), the optimal control stays at its upper bound for about 39 days and then steadily decreases to its lower bound over the remaining simulation time. The sub-optimal control law stays at the upper bound during the first time interval. In fact, at the beginning of the simulation time, both the optimal control and sub-optimal controls remain at their upper bounds in order to remove as many infectious hosts as possible. The steady decrease to the lower bound in the optimal control case and maintenance at a proper value in the sub-optimal case are determined by the balance between the cost of the worm and the control. Comparisons of the number of infectious hosts under the 0.50 Optimal control Sub-optimal control 0.45 Control laws 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0 20 40 60 80 Time (days) 100 120 Fig. 1. Optimal and sub-optimal control laws. ETRI Journal, Volume 30, Number 1, February 2008 140 Number of infectious hosts Under optimal control Under sub-optimal control 800 600 400 200 0 0 20 40 60 80 Time (days) 100 120 140 (a) 1000 Number of infectious hosts Remark 4. Obtaining the ideal weights is very difficult in practice. It requires a lot of work on data mining, analysis, and fitting. The weights in the simulations here are only of theoretical interest, and are used to illustrate the control strategies proposed in this paper. We will briefly discuss cases with various values of C later in this section. 1000 Under optimal control Under constant control (u≡0.40) 800 600 400 200 0 0 20 40 60 80 Time (days) 100 120 140 (b) 5 3.0 Number of infectious hosts In simulations, we studied the impact of the optimal process of removal of infectious host during the outbreak of the Code Red worm. The optimal control was computed using the above iterative algorithm with 70 nodes up to tolerance 10-5 for h . We set the upper bound of u to 0.50 according to our reasonable assumption that it takes an average of at least two hours to cure an infectious host (cleaning, patching, filtering, or even disconnecting from the Internet). The sub-optimal control was computed for the case in which N=3 and for specified time intervals: Δ1 = 40 , Δ 2 = 60 , and Δ 3 = 40 . Parameters such as population size for the genetic algorithm, generation gap, and others were adjusted ad hoc [19], [23]. The model parameters are from published data [5] (see Table 2). In addition, considering the two weight factors associated with I and u, we chose B=5 and C=600 to illustrate the optimal strategies. x 10 Under lower bound control (u≡0.05) Under constant control (u≡0.20) 2.5 2.0 1.5 1.0 0.5 0.0 0 20 40 60 80 100 120 140 Time (days) (c) Fig. 2. Number of infectious hosts under optimal control, suboptimal control, constant control, and lower bound control. optimal control, sub-optimal control, and the constant controls (u ≡ 0.20,0.40) throughout the simulated time are shown in Fig. 2. As expected, in the early phase of the breakout, keeping the control at its upper bound directly slows down the increase in the number of the infectious hosts. The optimal control is much more effective for controlling the outbreak and decreasing the duration of the epidemic, which is the time interval from the beginning until there is no emergence of new infectious hosts. Furthermore, Fig. 2(c) demonstrates that if the lower bound control (u ≡ 0.05) and constant control with u ≡ 0.20 were implemented throughout the simulation time, the number of infectious hosts reached about 2.61×105 and Xiefei Yan et al. 85 0.50 x 104 Under optimal control Under sub-optimal control 3.0 Cost 2.5 2.0 1.5 1.0 10 0.40 0.35 0.30 0.25 0.20 0.15 0.5 0 µ=0.02/N µ=0.06/N µ=0.18/N 0.45 3.5 Optimal control law 4.0 0 20 40 x 104 60 80 Time (days) (a) 100 120 0.10 140 0 20 40 60 80 Time (days) 100 120 140 Fig. 5. Comparison of optimal control laws for different values of µ. Under constant control (u≡0.40) 0.50 6 4 2 0 C=300 C=600 C=900 0.45 0 20 40 60 80 Time (days) (b) 100 120 140 Fig. 3. Cost of infection (a) under optimal and sub-optimal control and (b) under constant control. Optimal control law Cost 8 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0 20 40 60 80 100 120 140 Time (days) Fig. 6. Comparison of optimal control laws for different values of C. 0.50 implement than the optimal control. Moreover, the results demonstrate that, in order to obtain a minimum cost, the 0.40 maximum implementation of control in the early stage is critically 0.35 important in controlling the outbreak of an Internet worm. 0.30 Figures 4 to 6 illustrate how the optimal control strategies 0.25 β0=0.30/N depend on the parameters β 0 , μ , and C, respectively. The 0.20 β0=0.35/N value of β 0 , which denotes the transmission rate of infections, 0.15 β0=0.40/N varies from network to network, depending on many factors 0.10 0 20 40 60 80 100 120 140 including the security conditions. In Fig. 4, the control is Time (days) plotted as a function of time for the three different values Fig. 4. Comparison of optimal control laws for different values of β0. of β 0 : 0.30/N, 0.35/N, and 0.40/N. Other parameters for these three cases are presented in Table 2. It shows that the control 5.04×104, respectively. In addition, we noticed that plays an increasing role as β 0 increases. This is an expected implementation of the constant control with u ≡ 0.40 can result because as β 0 increases, the number of new infectious also effectively control outbreaks (see Fig. 2(b)). hosts increases, too. On the contrary, the level of control However, let us investigate the cost incurred by this constant increases as μ or C decreases, as shown in Figs. 5 and 6, control, as shown in Fig. 3(b). Comparing this cost and the costs incurred by the optimal and sub-optimal controls (Fig. 3(a)), respectively. As μ decreases, the number of susceptible hosts the costs incurred by the optimal control and sub-optimal removed by users decreases, too. So the level of control control are much lower than that of the constant control increases due to the increase in the number of newly infected (u ≡ 0.40) . Moreover, the cost incurred by the sub-optimal hosts. Moreover, as C decreases, the cost of control decreases, control is very close to that incurred by the optimal control, so the level of control increases to play a more significant role despite the fact that the sub-optimal control is much easier to to remove more infectious hosts. Optimal control law 0.45 86 Xiefei Yan et al. ETRI Journal, Volume 30, Number 1, February 2008 VI. Conclusion To better prepare us against future Internet worms, optimal control theory has been applied for worm control in this study. One control variable representing the rate of treatment to remove infectious hosts from circulation has been incorporated into the optimization two-factor model. Simulation results give a theoretical interpretation to the practical experience that maximum implementation of treatment in the early stage is critically important in controlling outbreaks of Internet worms with a minimum cost. As was mentioned in [17], the ideal time-varying optimal strategy might not be easy to apply in practice. Nevertheless, it provides a reference basis on which to design practical quasi-optimal control strategies or policies and assess their effectiveness. However, the proposed sub-optimal control in this paper can provide much simpler strategies, which, as verified by numerical simulations, can also lead to performance very similar to that achieved by the optimal control. In practical implementations, it may be considerably easier to adopt sub-optimal strategies. References [1] S. Qing and W. Wen, “A Survey and Trends on Internet Worms,” Computers & Security, vol. 24, 2005, pp. 334-346. [2] E.H. Spafford, “The Internet Worm Incident,” 2nd European Software Engineering Conf. (ESEC), Coventry, United Kingdom, 1989. [3] D. Moore and C. Shannon, “Code-Red: A Case Study on the Spread and Victims of an Internet Worm,” Proc. the ACM SICGOMM Internet Measurement Workshop, France, Nov. 2002, pp. 273-284. [4] C.C. Zou, W. Gong, and D. Towsley, “Code Red Worm Propagation Modeling and Analysis under Dynamic Quarantine Defense,” Proc. the ACM Conf. Computer and Comm. Security, 2003, pp. 51-60. [5] C.C. Zou, W. Gong, and D. Towsley, “Code Red Worm Propagation Modeling and Analysis,” Proc. the 9th ACM Conf. Computer and Comm. Security, Washington DC, Nov. 2002, pp. 138-147. [6] J.O. Kephart and S.R. White, “Directed-Graph Epidemiological Models of Computer Viruses,” Proc. the IEEE Symp. Security and Privacy, 1991. [7] J.O. Kephart, D.M. Chess, and S.R. White, “Computers and Epidemiology,” IEEE Spectrum, vol. 30, no. 5, May 1993, pp. 2026. [8] J.O. Kephart and S.R. White, “Measuring and Modeling Computer Virus Prevalence,” Proc. the IEEE Symp. Security and Privacy, 1993. [9] Z. Chen, L. Gao, and K. Kwiat, “Modeling the Spread of Active ETRI Journal, Volume 30, Number 1, February 2008 Worms,” IEEE INFOCOM, 2003. [10] Y. Wang, D. Chakrabati, C. Wang, and C. Faloutsos, “Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint,” Proc. 22nd Int’l Symp. Reliable Distributed Systems, Oct. 2003. [11] Y. Wang and C. Wang, “Modeling the Effects of Timing Parameters on Virus Propagation,” Proc. the ACM Workshop on Rapid Malcode (WORM), Oct. 2003. [12] J. Kim, S. Radhakrishnan, and S.K. Dhall, “Measurement and Analysis of Worm Propagation on Internet Network Topology,” Proc. 13th Int’l Conf. Computer Comm. and Networks, 2004. [13] O. Kreidl and T. Frazier, “Feedback Control Applied to Survivability: A Host-Based Autonomic Defense System,” IEEE Trans. Reliability, vol. 53, no. 1, 2004, pp. 148-166. [14] R. Dantu, J. Cangussu, and A. Yelimeli, “Dynamic Control of Worm Propagation,” Proc. the Int’l Conf. Information Technology: Coding and Computing (ITCC), 2004. [15] J. Kim, S. Radhakrishnan, and S.K. Dhall, “Optimal Control of Treatment Costs for Internet Worm,” http://www.wormblog.com/ 2005/03/optimal_control.html, Mar. 20, 2005. [16] J. Kim, S. Radhakrishnan, and J. Jang, “Cost Optimization in SIS Model of Worm Infection,” ETRI Journal, vol. 28, no. 5, 2006, pp. 692-695. [17] N.K. Gupta and R.E. Rink, “Optimal Control of Epidemics,” Mathematical Biosciences, vol. 18, 1973, pp. 383-396. [18] D. Kirschner, S. Lenhart, and S. Serbin, “Optimal Control of the Chemotherapy of HIV,” Journal of Mathematical Biology, vol. 35, 1997, pp. 775-792. [19] M. Antonio, L. Caetano, and T. Yoneyama, “Optimal and Suboptimal Control in Dengue Epidemics,” Optimal Control Applications and Methods, vol. 22, 2001, pp. 63-73. [20] E. Jung, S. Lenhart, and Z. Feng, “Optimal Control of Treatments in A Two-Strain Tuberculosis Model,” Discrete and Continuous Dynamical Systems-Series B, vol. 2, no. 4, 2002, pp. 473-482. [21] X. Yan and Y. Zou, “Optimal Quarantine and Isolation Control in SEQIJR SARS Model,” Proc. the 9th Int’l Conf. Control, Automation, Robotics, and Vision, Singapore, Dec. 2006. [22] X. Yan, Y. Zou and J. Li, “Optimal Quarantine and Isolation Strategies in Epidemics Control,” World Journal of Modeling and Simulation, vol. 3, no. 3, 2007, pp. 202-211. [23] X. Yan and Y. Zou, “Optimal and Sub-optimal Quarantine and Isolation Control in SARS Epidemics,” Mathematical and Computer Modelling, vol. 47, 2008, pp. 235-245. [24] L. Wang et al, “Observation and Analysis of BGP Behavior under Stress,” Internet Measurement Workshop, France, Nov. 2002. [25] H.W. Hethcote, “The Mathematics of Infectious Diseases,” SIAM Review, vol. 42, 2000, pp. 599-653. [26] Michael D. Intriligator, Mathematical Optimization and Economic Theory, Englewood Cliffs, N.J., Prentice-Hall, 1971. [27] S.P. Sethi and G. L. Thompson, Optimal Control Theory Applications to Management Science, Martinus Nijhoff Xiefei Yan et al. 87 Publishing, Boston, 1981. [28] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, 1962. [29] L. Davis, Genetic Algorithms and Simulated Annealing, Pitman, London, 1987. [30] R. Bulirsch and J. Stoer, Introduction to Numerical Analysis, Springer, Berlin, 1980. [31] H.J. Pesch, “Real Time Computation of Feedback Controls for Constrained Optimal Control Problems. Part 2: A Correction Method Based on Multiple Shooting,” Optimal Control Applications and Methods, vol. 10, 1989, pp. 147-171. Xiefei Yan received the BS degree in information and computational mathematics from the Nanjing University of Science and Technology in 2003, and the PhD degrees in automatic control engineering from the Nanjing University of Science and Technology in 2007. He was a Research Assistant in the Department of Building Service Engineering at the Hong Kong Polytechnic University in 2005. Currently, he is a Postdoctoral Researcher in School of Automation, Southeast University and Postdoctoral Research Working Station of Wuxi Seamless Oil Pipe Co., Ltd. and Postdoctoral Research Working Station of Wuxi National High-tech Industrial Development Zone, China. His current research interests include industrial control system, industrial control network, optimization, emergency control, epidemic control, and isolation control. Yun Zou received the BS degree in mathematics from the Northwestern University, Xian, China, in 1983, and the M.S. and PhD degrees in automatic control engineering from Nanjing University of Science and Technology, Nanjing, China, 1987, and 1990, respectively. He is currently a Professor of Electrical Engineering at Nanjing University of Science and Technology. His current research interests include differential-algebraic equation systems, two-dimensional systems, singular perturbations, transient stability of power systems, and power market. Dr. Zou is a Mathematical Reviewer of the journal Mathematical Reviews and a Member of American Mathematical Society. 88 Xiefei Yan et al. ETRI Journal, Volume 30, Number 1, February 2008

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