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We establish that the saddle-point defense strategy has a simple threshold-based structure that ought to facilitate its implementation in a localized manner in resource constrained wireless de- vices. Specifically, we prove that:

Theorem 7.2.2. For the saddle-point defense strategyuN_{(.) = (u}Nr_{(.), u}Ni_{(.)), there exists timest}

1, t2,

0≤t1< T,0≤t2< T such that:

• uNr_{(t) =u}Nr

minfor0< t < t1, anduNr(t) =uNnormr fort1< t < T.

• uNi_{(t) = 1for0< t < t2,andu}Ni_{(t) = 0fort2< t < T.}

The overall strategy therefore has the following three phases: In the initialaggressive defense phase, i.e., during(0,min(t1, t2)), the susceptibles select the minimum possible reception rate, and the dispatchers transmit the patches whenever they are in contact with any other node. Thus, the quarantining is the most stringent, and the recovery most rapid during this phase. Then, in the interimwatchfulphase, i.e., during(min(t1, t2),max(t1, t2)), one of the defense con- trols subside while the other continues as before. Finally, in the terminal relaxedphase, i.e., in

(max(t1, t2), T), both defense controls subside, that is, the susceptibles select their normal recep- tion rate and the dispatchers do not transmit the patches. Thus, the QoS in data traffic is back to its normal value and the resource consumption overhead due to patch transmission ends.

Theorem 7.2.2 states that the control functions should be applied with most intensity at the beginning of the epidemic period. In retrospect, this makes intuitive sense. Note that the costs associated with patching and rate reductions depend only on the area beneath the functions uNi_{(t) andu}Nr_{(t). Thus, by keeping the areas the same and shifting a candidate function to}

earlier times, this portion of the cost does not change. Choosing higher values of the patching rate (largeruNi_{) and lower values of reception rates (smalleru}Nr_{) later (as opposed to earlier) will}

only let the number of infective and dead nodes increase. This is because each susceptible that is infected can serve the propagation of the infection. This increases the integrations of theκII(t)

andκDD(t)as well asKDD(T). Moreover, this leads to a reduced efficacy of the recovery process

- the latter happens because healing (of an infective node) is not as fast as the immunization (of a susceptible node) sinceπ≤1(equivalentlyβ2≤β1). Note that the proof of the Theorem exploits the latter property. Thus, irrespective of the killing strategy of the malware, the containment of the epidemic ought start as soon as possible, if it is worth taking any action at all. The fact that the defense actions should come to a complete halt (rather than a gradual decline) is however less direct to predict.

Note that the defense strategy always chooses either the maximum or the minimum values of the parameters except possibly in a set of measure zero (i.e., except possibly att1, t2). Such strategies are referred to asbang-bangin the control literature. The durations of the phases (i.e., the values of the threshold timest1, t2) and which defense subsides in the interim watchful pe- riod, depend on the damage coefficientsκI, κD, KD, κr, κi. We will shed more light on the latter

in Theorem 7.2.4 later. But first, we conclude this sub-section by proving Theorem 7.2.2.

Proof. The continuity and piecewise differentiability ofψNr_{(.), ψ}Ni_{(.)follows from those of the}

co-state functions. From the final conditions on the co-state functions, i.e., (7.2.1), ψNr_{(T) =}

−κr<0,ψNi(T) =R0>0.We show thatψNr(.)(ψNi(.), resp.) is a strictly decreasing (increasing,

resp.) function of time. Thus, each has at most one zero-crossing point in(0, T); denote these as t1, t2.IfψNr _(ψNi_{, resp.) has no zero crossing point in(0, T),t1= 0(t2= 0, resp.). Thus, from the}

continuity of theψ(.)functions, and from their terminal values, (i)ψNr_{(.)is negative in(t}

1, T) and positive in (0, t1), and (ii)ψNi(.)is positive in(t2, T)and negative in(0, t2).The theorem follows from (7.2.4) and (7.2.5). We prove the strict monotonicity ofψNr_(.),ψNi_{(.), using:}

Lemma 7.2.3. λS>0andλI > λS, λD≥0∀t,0< t < T.

The lemma is intuitive since the shadow prices (i.e., co-state variables) associated with the susceptibles and dead nodes ought to be positive, and also the shadow price associated with the infectives ought to be at least as high as that associated with susceptibles. However, as we

will see next, the proof requires detailed analysis of the state and co-state differential equations (7.1.1), (7.2.1) respectively, and is less direct.

Proof. SinceλD(T) =KD ≥0(from (7.2.1)), and _dtdλD≤0,λD ≥0. Now, for the rest, we argue

in two steps.

Step 1: λS(T) = 0andλI(T) =KI = 0,also:λ˙I(T)−λ˙S(T) = ˙λI(T) =−κI −KDuM(T)<0.

Therefore,∃ >0s.t. on(T− . . . T)we haveλS >0and(λI−λS)>0.

Step 2: Proof by contradiction. Let τ be such that: λS > 0,(λI −λS) > 0 on(τ . . . T) &

λS(τ) = 0ORλI(τ) =λS(τ).From the continuity of the co-state functions,(λI(τ)−λS(τ))≥0,

andλS(τ)≥0.

We first prove that(λI(τ)−λS(τ))> 0.Suppose not. Then, λI(τ) = λS(τ). Thus: λ˙I(τ)−

˙

λS(τ) =−κI+λIβ2uNi−(λD−λI)uM−λSβ1uNi=−κI−λSuNi(β1−β2)−(λD−λI)uM.Here, (i)

the first term is strictly negative12_{, (ii) the second term is negative becauseλ}

S(τ)≥0andβ2≤β1 and (iii) the third term is negative because of (7.2.6). Thus, λ˙I(τ)−λ˙S(τ) > 0. But, then both

λI(τ) =λS(τ), and(λI−λS)>0on(τ . . . T)can not happen. Thus,(λI(τ)−λS(τ))>0.

Now, suppose λS(τ) = 0. λ˙S(τ) = −(λI −λS)β0uNrI|τ < 0. The last inequality follows

since (λI(τ)−λS(τ)) > 0, β0 > 0, uNr ≥ uNminr > 0and I(τ) > 0(lemma 7.1.1). This again contradicts the assumptions thatλS(τ) = 0andλS >0on(τ . . . T). Thus,λS(τ)6= 0, and hence

λS(τ)>0.

Strict monotonicity ofψNr_(.)

We show thatψ˙Nr_{(t)is strictlynegativeat allt∈(0, T)}13

˙

ψNr ₌ ∂

∂tψ

Nr _{= ( ˙λ}

I−λ˙S)β0IS+ (λI−λS)β0IS˙ + (λI−λS)β0IS˙ 12_{Negative in this proof is distinguished from strictly negative.}

which after replacement and simplification yields ˙ ψN r β0IS =−κI−(λD−λS)u M _−β 1λIuNi+β2λSuNi =−κI−(λD−λI)uM −(λI −λS)uM −(β1−β2)λIuNi−(λI−λS)β2uNi

From (7.2.6), lemma 7.2.3 and sinceκI >0,β1≥β2,uM(t)≥0, uNi(t)≥0at allt, the right hand side is negative. The result follows sinceβ0>0andS(t)>0, I(t)>0at allt(lemma 7.1.1).

Strict monotonicity ofψNi_(.) ˙ ψNi ₌ ∂ ∂tψ Ni _{= ( ˙λ} I−λ˙S)β0IS+ (λI−λS)β0IS˙ + (λI−λS)β0IS˙ ⇒ ˙ ψNi β0I =κIβ2+β2u M_λ D+β0β1SuNr(λI−λS)

The R.H.S is positive from lemma 7.2.3 and sinceκI >0, β0>0. Thus,ψ˙Ni >0sinceβ0>0and I(t)>0at allt(lemma 7.1.1).

In the next theorem, we show that under a sufficient condition, the quarantining period (by reduction of communication rates) ends before the immunization/healing effort is stopped. This is in accordance with our intuition that the primary use of quarantining is buying timefor the recovery process in the network.

Theorem 7.2.4. Lett1andt2be as defined in Theorem 7.2.2. Ifκr≥

β0S0

β2 , then eithert2= 0ort1< t2. Note that the condition of the theorem is quite intuitive. For instance, ifβ0 = β2, this con- dition is satisfied whenκr ≥1. Recall that the coefficients of the costs were rescaled so that the

coefficient of the cost of patching is normalized. Thus, this means when the instantaneous cost of per unit reduction of communication rates (quarantining) is no less than per unit patching. The 13_{partial derivative w.r.t time, only because of the dependence also on the initial values for the states. Otherwise,tis} the only independent variable.

other parameters inκr ≥

β0S0

β2 refer to the cases of relatively small initial susceptible pool, and a relatively fast healing rate. Intuitively, when the healing rate is slow compared to the propaga- tion rate of the infection, it might not be prudent to relax the communication rates of the nodes to normal soon, which is compatible with the condition of the theorem.

Proof. Recall from the proof of Theorem 7.2.2 thatψN r_(t)andψN i_{(t)each have at most one zero-}

crossing point andψN r(t)terminates in a negative, andψN i(t)terminates in a positive value at T. We now show that the value ofψNi_{(t)at apotentialzero-crossing point ofψ}Nr_{(t), is strictly}

negative. ψN r= (λI−λS)β0IS−κr= 0⇒λI = κr β0IS+λS, then: ψN i=R0−λSβ1R0S−λIβ2R0I=R0−λSβ1R0S−β2R0I( κr β0IS +λS) =R0(1− β2 β0 κr S )−λSβ1R0S−λSβ2R0I

According to lemmas 7.1.1 and 7.2.3, the last two terms are strictly negative. The first term is negative following the condition of the theorem (i.e., κr ≥

β2S0

β0 ) and the fact thatS is a non-increasing function of time. Similarly, at a potential zero-crossing point ofψN i_{(t), we have}

ψN r_{(t) = −λ} S(− β0β1 β2 S2_{−β0IS) +} β0S β2

−κr, which is strictly negative again following lem-

mas 7.1.1 and 7.2.3 and the sufficient condition of the theorem. The theorem follows from the continuity ofψN i(t)andψN r(t)and by referring to (7.2.4) and (7.2.5).

Note that the case oft2= 0occurs whenψN idoes not have a zero-crossing point and is hence non-negative throughout[0, T]. In this case, the immunization/healing effort is never launched.