This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

Optimal Control of Epidemic Information Dissemination in Mobile Ad Hoc Networks Pin-Yu Chen and Kwang-Cheng Chen, Fellow, IEEE Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan Email : [email protected] and [email protected] Abstract—To facilitate reliable communications, efficacious signaling is of crucial importance for information dissemination control, especially in dynamic and infrastructureless networking paradigms such as mobile ad hoc networks (MANETs). Since data transportation much resembles the spread of epidemics, two signal distribution schemes, self healing and vaccine spreading, are proposed to analyze the information dynamics via epidemic modeling. We consider a more realistic scenario where the effect of signaling is proportional to the distribution time. Intuitively, early distribution leads to slight cure while late distribution fails to prevent the epidemic from outbreak. Optimal control theory and early-stage analysis are exploited to determine the optimal distribution time for timely control. Moreover, we demonstrate how signal distribution aids to minimize buffer occupancy in epidemic routing. This research therefore paves novel avenues to network defense and information dissemination.

I. I NTRODUCTION Due to the fact that information dissemination much resembles the spread of epidemics [1], [2], epidemic modeling [3] has been used in a profusion of areas to investigate the information dynamics in various networks of interest, including efficient routing design in communication networks [4]– [6], broadcasting protocol analysis in wireless sensor networks [7], Internet worm propagation [8]–[11], malware propagation in generalized social networks [12], and especially message passing in mobile ad hoc networks (MANETs) [13]–[15]. Traditionally, the susceptible-infected-recovered (SIR) model is utilized to analyze the state transitions of information dynamics. Analogously, a node is informed (receives the information) is as if it is compromised by an epidemic, and a node is not informed is as if it is still susceptible to the epidemic. Furthermore, a node enters the recovered state if the immunity mechanism (signaling) takes effect so that an infected node stops spreading the epidemic, or a susceptible node becomes a vaccinee against the epidemic. However, current approaches often make an implicit assumption that an infected node can potentially recover from the epidemic at the initial stage, which is in general inadequate to characterize the spread of epidemics without existing antidotes, such as a new type of Internet worm or malware. To the best of our knowledge, the tradeoffs between signal distribution and information dissemination control still remain open, and the task is further complicated in MANETs owing to the infrastructureless networking paradigm and the fact that mobility nurtures the spread of epidemics [2], [16]. In addition to devising efficacious strategy for epidemic control, the study of signal distribution also plays an essential

role in efficient routing design in MANETs. End-to-end communication is achieved by forwarding the replicated data from the source node to the encountered relay nodes to mitigate the delivery delay to the destination node. In order to save the buffer occupancy, Haas and Small [5] introduce the concept of disseminating “antipackets” to delete the replicated packets carried by the relay nodes once the destination node has received the packet. The system buffer occupancy has been shown to be proportional to the number of nodes carrying the replicated packets [4], which is exactly the number of the infected population in epidemic sense. We will demonstrate how signal distribution aids to minimize the buffer occupancy in epidemic routing as an innovative application. In this paper, by relating the malicious self-propagating codes or data packets in MANETs to an epidemic, we consider a more realistic scenario that the immunity mechanism is activated only if the signal (i.e., antidote) is distributed. It is natural to assume that the effect of signaling is proportional to the time instance the signal is distributed. In other words, since early distribution leads to slight cure while late distribution fails to suppress the epidemic, an efficacious and timely distribution strategy is a must to eradicate the epidemic. Incorporating the spread of epidemics and the effect of signal distribution, we formulate the state transitions among mobile nodes as state equations. Via optimal control theory [17], we aim to determine the optimal distribution time to minimize the potential cost. Two signal distribution schemes, self healing and vaccine spreading, are proposed to investigate the impacts on signal distribution time and epidemic control. Self healing refers to the reactive mechanism named self immunity where an infected node recovers when it receives the antidotes, whereas vaccine spreading refers to the proactive mechanism named cooperative immunity where an immune node participates in distributing vaccines to other susceptible nodes. Moreover, we also explore early-stage analysis for signal distribution to prevent the epidemic from outbreak, offering quick system monitoring and defense at the imminent phase. The rest of this paper is organized as follows. Sec. II introduces the system model of MANET and the state equations of self healing and vaccine spreading schemes. We leverage optimal control theory and early-stage analysis to determine the optimal signal distribution time in Sec. III and Sec. IV, respectively. The performance of signal distribution is evaluated in Sec. V. We demonstrate how signal distribution aids epidemic routing to minimize buffer occupancy in Sec.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

VI. Finally, Sec. VII concludes this paper. II. S YSTEM M ODEL Throughout this paper, we use the ordinary differential equations (ODEs) of SIR model as the state equations for control process. Zhang et al. [4] have proved that ODEs of SIR model can be derived as limits of Markovian models under a natural scaling as the number of nodes increases in MANETs. We denote I(t), S(t) and R(t) as the infected, susceptible and recovered population at time t, respectively, and N = I(t) + S(t) + R(t) is the total population. N nodes move around in a L×L square area with identical transmission range δ, and two nodes are able to communicate with each other if they are within the transmission range of each other. Only susceptible nodes are vulnerable to the epidemic and the recovered nodes are immune to the epidemic for good. In the subsequent paragraphs we formulate the state equations of self healing and vaccine spreading schemes. A. Self healing Upon the healing signal distribution (e.g., release of antivirus code or control packet), the infected nodes are able to recover from the epidemic with probability u(t), where  0, t < TD , (1) u(t) = t ≥ TD . f (TD ), TD is the time to launch the healing signal distribution, and f (TD ) ∈ [0, 1] is a positive and nondecreasing function of TD since late distribution contributes to strong cure. Before signal distribution, the variation of infection population is affected by the pairwise infection rate, number of encountered nodes and the fraction of susceptible nodes to be compromised, whereas the signal heals the epidemic with probability f (TD ) after time TD . The recovered population is the number of nodes which recover from infected state. The state equations of self healing scheme can be formulated as ⎧ ˙ = λη S(t)I(t) ˆ − u(t)I(t), ⎨ I(t) ˙ (2) R(t) = u(t)I(t), ⎩ I(t) + R(t) + S(t) = N,

equations of vaccine spreading can be formulated as ⎧ ˙ = λη S(t)I(t) ˆ − u(t)I(t), ⎨ I(t) ˙ ˆ R(t) = u(t)I(t) + φ(t)η S(t)R(t), ⎩ I(t) + R(t) + S(t) = N, where

 φ(t) =

0, t < TD , κ, t ≥ TD ,

(4)

and κ ∈ [0, 1] is the cooperative immunity coefficient which can be interpreted as the pairwise reliability or willingness to become a vaccinee. The nodes in recovered state distribute vaccine signals to the encountered susceptible nodes with successful delivery rate κ. A direct observation from (3) is that vaccine spreading proliferates the recovered population, and it therefore mitigates the growth of infected population. Moreover, comparing (2) with (3), self healing is a degenerate case of vaccine spreading when κ = 0, where cooperative immunity degenerates to self immunity. From branching process, a necessary condition to eradicate the epidemic is λη < f (TD ) + κ since R0 = f (Tλη D )+κ represents the basic reproductive number [3] which counts the number of secondary infected cases generated by one primary infected node after signal distribution. In other words, the epidemic can be controlled only for R0 < 1. III. O PTIMAL C ONTROL AND D ISTRIBUTION T IME Without loss of generality, we use the state equations of vaccine spreading in (3) to obtain the optimal control u∗ (t) since self healing is a special case of vaccine spreading when there is no cooperation. Leveraging optimal control theory [17], we aim to minimize the performance measure J, where  Tf 1 [I(t)]β + u2 (t) dt (5) J= 2 0 and u(t) ∈ [0, 1] takes its quadratic form. β > 0 represents the severeness of the epidemic and Tf is the completion time which is assumed to be free. Via optimal control theory, we write the Hamiltonian H as H (I(t), R(t), u(t), ΛI (t), ΛR (t))   1 ˆ − u(t)I(t) = [I(t)]β + u2 (t) + ΛI (t) λη S(t)I(t)  2  ˆ , (6) + ΛR (t) u(t)I(t) + φ(t)η S(t)R(t)

ˆ where S(t) = S(t)/N is the normalized susceptible population, λ is the pairwise infection rate which accounts for the successful delivery rate, and η = πδ 2 /L2 is the average number of encountered nodes per unit time.

from which the costate equations are

B. Vaccine Spreading

∂H Λ˙ ∗I (t) = − ∂I

Despite the self immunity scheme, a more aggressive approach to suppress the epidemic is cooperative signal distribution. The immune nodes consisting of nodes which recover from infected state and nodes inoculated with vaccines collaboratively participate in vaccinating susceptible nodes against the epidemic. A node thereby transitions from susceptible state to recovered state if it becomes a vaccinee, which can be implemented by forwarding vaccine signals. The state

(3)

N − 2I(t) − R(t) − u(t) − λη = −β[I(t)] N

R(t) − Λ∗R (t) u(t) − φ(t)η (7) N ∂H Λ˙ ∗R (t) = − ∂R

I(t) N − I(t) − 2R(t) ∗ − Λ∗R (t)φ(t)η (8) = ΛI (t)λη N N β−1

Λ∗I (t)



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with boundary conditions Λ∗I (Tf ) = Λ∗R (Tf ) = 0. Assuming that all of the state and costate variables are according to their values for the optimal control u∗ (t), we rewire the Hamiltonian in (6) with the switching function θ∗ (t)  Λ∗I (t)I(t) − Λ∗R (t)I(t) = [Λ∗I (t) − Λ∗R (t)]I(t) as

(9)

Distinct from (3), the assumption S(t) ≈ N at early stages suggests that vaccine spreading also takes place after t ≥ TD since the encountered nodes are most likely to be susceptible. Therefore by (1) and (4), we obtain I(t) = I0 exp {[λη(1 − φ(t)ηt ) − u(t)]t}  I0 exp {ληt} , = I0 exp {[λη(1 − κηt ) − f (TD )]t} ,

t < TD , (15) t ≥ TD ,

where I0 is the initially infected population. The analytical H(I ∗ (t), R∗ (t), u(t), Λ∗I (t), Λ∗R (t)) form of performance measure J in (5) becomes 1  Tf = [I ∗ (t)]β + u2 (t) + η Sˆ∗ (t) [Λ∗I (t)λI ∗ (t) + Λ∗R (t)φ(t)R∗ (t)] 1 2 [I(t)]β + u2 (t) dt (16) J= ∗ 2 0 (10) − θ (t)u(t).  TD  Tf 1 = [I(t)]β dt + [I(t)]β + f 2 (TD ) dt Since the term [I ∗ (t)]β + 12 u2 (t) in (5) is convex with 2 0 TD respect to u(t), by Pontryagin’s minimum principle [18], the β I0 β I0 unconstrained optimal control U ∗ (t) that minimizes J is the (exp{ληβT } − 1) + = D ληβ [λη(1 − φ(t)ηt ) − f (TD )]β solution of the equation ∂H ∂u = 0, and from (10) we have  × (exp{[λη(1 − κηt ) − f (TD )]βTf } U ∗ (t) = θ∗ (t), (11) 1 − exp{[λη(1 − κηt ) − f (TD )]βTD }) + f 2 (TD )(Tf − TD ). i.e, U ∗ (t) can be obtained by solving the state and costate 2 equations (3), (7) and (8). Moreover, with the admissible The optimal distribution time when adopting early-stage analcontrol u(t) ∈ [0, 1], we have the constrained optimal control ysis can thus be obtained by ⎧ θ∗ (t) ≤ 0, ⎨ 0, ∗ (17) T D = arg min J. ∗ ∗ TD θ (t), θ∗ (t) ∈ (0, 1), (12) u (t) = ⎩ 1, θ∗ (t) ≥ 1, The advantage of the early-stage analysis is that it relaxes ∗ ∗ the need for solving the simultaneous ODEs for optimal where u (t) is a saturation function of U (t). control as discussed in Sec. III, but it may possess potential Note that discrepancy exists between the recovery probabilrisk by overestimating the spread of epidemic and signal ∗ ∗ ity in (1) and the constrained optimal control u (t). u (t) is distribution with the assumption that S(t) ≈ N at early stages. obtained by assuming that immediate reaction is permissible from the initial time 0, whereas in (1) we consider a more V. P ERFORMANCE E VALUATION realistic scenario where the state transition from infected state To demonstrate the tradeoffs between signal distribution and to recovered state takes place only after the signal distribution time TD . Considering practical implementation concerns, the the resulting information dynamics, we set the function f (TD ) function f (TD ) in (1) is often predetermined and it therefore in (1) to be fails to coincide with u∗ (t) in (12). To bridge the gap, the α }, (18) f (TD ) = min {1, c · TD ∗ is the optimal signal distribution optimal distribution time TD time that minimizes the performance measure J, i.e, where α is a nonnegative value which accounts for the effectiveness of the signal and c is a positive constant. The ∗ = arg min J. (13) TD TD effect of signaling has an power-law growth with respect to We will show that the solution in (13) provides efficacious distribution time. Regarding the simulation setup, N nodes are traversing in the square area in wrap-around condition via strategy for epidemic eradication in Sec. V. the L`evy walk mobility model [19], where the step length and pause time follow a power-law distribution with negative IV. E ARLY- STAGE A NALYSIS exponent, respectively. We set the length exponent l = 1.5 and the pause time exponent ϕ = 1.38, which fit the traceIn addition to optimal control approach for determining the based data of human mobility pattern collected in UCSD optimal distribution time, we provide early-stage analysis as a and Dartmouth [20]. Moreover, our previous research [2] has quick reference to system monitoring and defense at the imshown that the information dynamics in such MANET can be minent phase. At early stages before the outbreak of epidemic captured via epidemic modeling. (e.g., malware proliferation or data deluge), the state equations The information dynamics under self healing scheme is of vaccine spreading in (3) at time t can be approximated as shown in Fig. 1. The differences between the information a coupling regulator problem when S(t ) ≈ N , i.e., dynamics via optimal control u∗ (t) and the simulations via  ∗ R(t ) = exp{φ(t)ηt } − 1 ≈ φ(t)ηt , reside in the fact that in optimal signal distribution time TD (14) ˙ = [λη(1 − φ(t)ηt ) − u(t)]I(t). the simulation we take the effect of the function f (TD ) in I(t)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

100

35

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90

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∗ analysis with TD

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20

40 15

30 20

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Fig. 1. Information dynamics under self healing scheme. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.25, α = 2, β = 2, t = 1, Tf = 80, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 over 500 simulations.

α

2.5

3

3.5

Fig. 3. Optimal distribution time via optimal control under different (α,β) configurations. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.25, κ = 0.2, Tf = 200, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 .

70

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ˆ I(t)

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0

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Fig. 2. Information dynamics under vaccine spreading scheme. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.25, κ = 0.1, α = 2, β = 2, t = 1, Tf = 80, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 over 500 simulations.

Fig. 4. Optimal distribution time via early-stage analysis under different (α,β) configurations. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.25, κ = 0.2, t = 1, Tf = 200, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 .

∗ (18) into consideration. The solution of TD in (13) is shown to be efficacious for epidemic eradication since it minimizes the performance measure J which is proportional to the area of the ∗ infected population up to time Tf . Moreover, the solution T D from early-stage analysis has a similar impact on the epidemic eradication but with slower decaying infected population com∗ due to early-stage approximation. On the other pared with TD hand, the information dynamics under vaccine spreading are further suppressed with the aid of vaccine spreading as shown in Fig. 2, where the growth of vaccinees effectively decelerates the spread of epidemic, even for small κ (κ = 0.1). In Fig. 3, the optimal distribution time via optimal control decreases with the increase of α and β since intuitively early distribution is more efficacious if the effectiveness of the signal (α) is stronger, or if we are facing a severe epidemic (larger β). Moreover, we observe that vaccine spreading also contributes to early distribution by launching the cooperative immunity mechanism to create more vaccinees against the epidemic. Similar results can be found for early-stage analysis as shown

in Fig. 4. When α is small, the assumption that S(t) ≈ N at early stages may not hold due to slight cure, leading to the pessimistic outcome of late distribution. When α is large, the overestimation of infection speed leads to the optimistic outcome of early distribution. We plot the relative difference ∗ ∗ ∗ − T ξ = (TD D )/TD in Fig. 5. Compared with optimal control theory, ξ < 0 refers to late distribution while ξ > 0 refers to early distribution for early-stage analysis. VI. A PPLICATION : M INIMIZING B UFFER O CCUPANCY IN E PIDEMIC ROUTING In this section, we demonstrate how to utilize the signal distribution for efficient routing design in epidemic routing. The control signal serves as the role to eliminate the unnecessary packets from the buffer, i.e., a node no longer relays the packet to the encountered node if it receives the control signal. This mechanism plays an essential role in MANETs since the buffer size is often limited. From [4], the average

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

distribution time for timely control. Simulations and analytical results support that the epidemic can be eradicated with efficacious healing signal distribution under self healing scheme, while vaccine spreading scheme further suppresses the spread of epidemic by delivering vaccine signals to susceptible nodes. As one application, we demonstrate that self immunity and cooperative immunity have significant impacts on minimizing buffer occupancy for epidemic routing in MANETs. Consequently, this paper reveals novel insights for optimal control of information dissemunation toward efficient and robust networking deployments in MANETs.

0.8 0.6 0.4 0.2

β = 1, self heaing β = 2, self heaing

0

ξ

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−0.6 −0.8 −1

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Fig. 5. Relative difference of optimal distribution time under different (α,β) configurations. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.25, κ = 0.2, t = 1, Tf = 200, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 .

4000 self healing 3500

vaccine spreading

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1 ∗ 2 TD

E[Q]

2500 ∗ 2TD

2000 1500 1000 ∗ TD

500 0

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Fig. 6. Average buffer occupancy in epidemic routing. N = 1000, L = 1000, I0 = 1, δ = 2, λ = 0.5, κ = 0.8, α = 3, β = 1, Tf = 100, ΛI (0) = 15, ΛR (0) = 10 and c = 10−3 .

buffer occupancy can be evaluated as  Tf E[Q] = λ I(t)dt,

(19)

0

which is exactly a special case of the performance measure J when β = 1. In epidemic routing, the signal distribution time is associated with the time-to-live (TTL) setup, self healing is employed by receiving the ACK packet from the destination node, and vaccine spreading is employed by delivering the ACK packet to susceptible nodes. Fig. 6 demonstrates that the optimal distribution time greatly reduces the buffer occupancy, while improper distribution time contributes to tremendous buffer occupancy, leading to inept routing in MANETs. VII. CONCLUSION In this paper, we investigate the tradeoffs between signal distribution and information dissemination control in MANETs via epidemic modeling, where optimal control theory and early-stage analysis are exploited to determine the optimal

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