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

Authors: Xulong Zhang; Chenquan Gan
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## 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.


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## Body

## 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 andnode-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. Sections2 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 graphG=V,E with N nodes labelled 1,2,…,N, where each node and edge stand for a computer and a network link, respectively. Thus, the graph G can be described by its adjacency matrix A=aijN×N, where aii=0.As was treated in the traditional SICS model [6], at any time all nodes in the graph G are divided into three groups: S-nodes (susceptible nodes are uninfected but have no immunity), I-nodes (infected nodes), and C-nodes (countermeasured nodes are uninfected and have temporary immunity due to the presence of countermeasures). Let Sit, Iit, and Cit denote the probability of node i being susceptible, infected, and countermeasured at time t, respectively. Then the vector (1)S1t,…,SNt,I1t,…,INt,C1t,…,CNtTprobabilistically captures the state of the network at time t.For convenience, two important functions, which will be used in the sequel, are defined as follows:(2)fit=∑jaijβjIjt1+m1Ijt,m1≥0,βj>0.Clearly, fit≤∑jaijβjIjt.(3)git=∑jaijγjtCjt1+m2Cjt,m2≥0,where γjt∈L20,T is a controllable rate, γ_≤γjt≤γ¯, 0≤t≤T; γ_ and γ¯ are positive constants, 0<γ_<γ¯<1.Now, a set of probabilistic assumptions on the state transition of nodei are made (see also Figure 1).(A1)
AnS-node i becomes infected at rate fit.(A2)
AnS- or I-node i becomes countermeasured at rate git.(A3)
AnI-node i becomes susceptible at a constant rate αi>0.(A4)
AC-node i loses immunity at constant rate θi>0.Figure 1
The transfer diagram of the controlled node-level SICS model.LetΔt be a very small time interval and oΔt be a higher-order infinitesimal. Assumptions (A1)–(A4) imply that the probabilities of state transition of node i satisfy the following relations: (4)Pr⁡iisinfectedattimet+Δt∣iissusceptibleattimet=fitΔt+oΔt,Pr⁡iiscountermeasuredattimet+Δt∣iissusceptibleattimet=gitΔt+oΔt,Pr⁡iiscountermeasuredattimet+Δt∣iisinfectedattimet=gitΔt+oΔt,Pr⁡iissusceptibleattimet+Δt∣iisinfectedattimet=αiΔt+oΔt,Pr⁡iissusceptibleattimet+Δt∣iiscountermeasuredattimet=θiΔt+oΔt.Invoking the total probability formulas and lettingΔt→0, the controlled node-level model (i.e., controlled node-level SICS model) can be derived.(5)dSitdt=αiIit+θiCit-fitSit-gitSit,dIitdt=-αiIit+fitSit-gitIit,dCitdt=-θiCit+gitSit+Iit,0≤t≤T,i=1,2,…,N,with initial condition (6)S10,…,SN0,I10,…,IN0,C10,…,CN0T∈Ω~,where (7)Ω~=S1,…,SN,I1,…,IN,C1,…,CNT∈R+3N∣Si+Ii+Ci=1,i=1,2,…,N.The admissible control set is(8)U=u·∈L20,TN∣γ_≤γi·≤γ¯,1≤i≤N,where u·=γ1·,…,γN·T.


## 3. The Optimal Control Problem

AsSit+Iit+Cit≡1, 1≤i≤N, system (5) can be reduced to the following system:(9)dIitdt=-αiIit+fit1-Iit-Cit-gitIit,dCitdt=-θiCit+git1-Cit,0≤t≤T,i=1,2,…,N,with initial condition (10)I10,…,IN0,C10,…,CN0T∈Ω,where (11)Ω=I1,…,IN,C1,…,CNT∈R+2N∣Ii+Ci≤1,i=1,2,…,N.Then system (9) can be written in matrix notation as(12)dxtdt=fxt,ut,0≤t≤T,with initial condition x0∈Ω.Now, the objective is to find a control variableu·∈U so as to minimize both the prevalence of infected computers and the total budget for dynamic countermeasure during the time period 0,T. That is, the following optimal control problem needs to be solved: PMinimizeu∈U⁡Ju=∫0TLxt,utdtsubject to system (12), where (13)Lx,u=∑iIit+12εiγi2t,εi>0,is the Lagrangian and ε=ε1,…,εNT 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(14)Minimizeu∈U⁡Ju=∫0TLxt,utdtsubject to(15)dxtdt=fxt,ut,0≤t≤T,with x0∈Ω, where Ω is positively invariant for system (15). The problem has an optimal control if the following six conditions hold simultaneously. (C1)
There isu∈U such that system (15) is solvable.(C2)
U is convex.(C3)
U is closed.(C4)
fx,u is bounded by a linear function in x.(C5)
Lx,u is convex on U.(C6)
Lx,u≥c1u2ρ+c2 for some ρ>1, c1>0, and c2.In order to prove the existence of an optimal control, six lemmas, one for each condition in Lemma1, should be proved.Lemma 2.
There isu∈U such that system (9) or (12) is solvable.Proof.
Substitutingu≡u¯≔γ¯,…,γ¯︸NT into system (12), one can get the uncontrolled system: (16)dxtdt=fxt,u¯with x0∈Ω. Then the function fx,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 setU is convex.Proof.
Let(17)u1=γ11,…,γN1∈U,u2=γ12,…,γN2∈U,0<ξ<1.As L20,TN is a real vector space, one can obtain (18)1-ξu1+ξu2∈L20,TN.Then, the convexity of U follows by the observation that (19)γ_≤1-ξγi1+ξγi2≤γ¯,1≤i≤N.Hence, the claimed result follows.Lemma 4.
The admissible setU is closed.Proof.
Letu=γ1,…,γNT be a limit point of U and (20)un=γ1n,…,γNnT,n=1,2,…,be a sequence of points in U such that (21)un-u2≔∫0Tunt-ut2dt1/2<1n.From the completeness of L20,TN, one can get (22)limn→∞⁡un=u∈L20,TN.Hence, the closeness of U follows from the observation that (23)γ_≤γi=limn→∞⁡γin≤γ¯,1≤i≤N.Lemma 5.
f
x
,
u is bounded by a linear function in x.Proof.
Note that, for system (9) and for i=1,2,…,N, (24)-γ¯N24-αiIi≤dIidt≤-αiIi+∑jaijβjIj,-θiCi≤dCidt≤-θiCi+γ¯∑jaijCj.Thus, the claimed result follows.Lemma 6.
L
x
,
u is convex on U.Proof.
Note that the Hessian matrix ofLx,u with respect to u∈U is as follows:(25)HuL=∂2L∂γ12∂2L∂γ1∂γ2⋯∂2L∂γ1∂γN-1∂2L∂γ1∂γN∂2L∂γ2∂γ1∂2L∂γ22⋯∂2L∂γ2∂γN-1∂2L∂γ2∂γN⋮⋮⋱⋮⋮∂2L∂γN-1∂γ1∂2L∂γN-1∂γ2⋯∂2L∂γN-12∂2L∂γN-1∂γN∂2L∂γN∂γ1∂2L∂γN∂γ2⋯∂2L∂γN∂γN-1∂2L∂γN2=ε10⋯000ε2⋯00⋮⋮⋱⋮⋮00⋯εN-1000⋯0εN.For any t∈0,T, HuL is real symmetric and its eigenvalues are all positive. Then, HuL is positive definite. Hence, the convexity of Lx,u follows by the result in [37].Lemma 7.
L
x
,
u
≥
c
1
u
2
ρ
+
c
2 for some ρ>1, c1>0, and c2.Proof.
Letρ=2, c1=mini⁡εi/2, and c2=0. Then, Lx,u≥mini⁡εi/2×u22. Thus, the proof is complete.Now, it is time to examine the main result of this subsection.Theorem 8.
The optimal control problemP has a solution.Proof.
Lemmas2–7 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 problemP is drawn.Theorem 9.
Supposeu∗· is an optimal control for problem P and I∗·,C∗·T is the solution to system (9) with u·=u∗·. Then, there exist functions λ1i∗t and λ2i∗t, 0≤t≤T, 1≤i≤N, such that(26)dλ1i∗tdt=-1+λ1i∗tαi+fi∗t+gi∗t-βi1+m1Ii∗t2∑jaijλ1j∗t1-Ij∗t-Cj∗t,dλ2i∗tdt=λ1i∗tfi∗t+λ2i∗tθi+gi∗t+γi∗t1+m2Ci∗t2∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t,0≤t≤T,i=1,2,…,N,with transversality conditions (27)λ1i∗T=λ2i∗T=0,i=1,2,…,N.Furthermore, one can get (28)γi∗t=max⁡min⁡Ci∗tεi1+m2Ci∗t∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t,γ¯,γ_,0≤t≤T,i=1,2,…,N.Proof.
The corresponding Hamiltonian is(29)HI,C,λ,u=∑iIi+12εiγi2+∑iλ1idIidt+∑iλ2idCidt,where λ1i, λ2i are undetermined, λ=λ11,…,λ1N,λ21,…,λ2NT.
According to the Pontryagin Minimum Principle [35], there exist functions λ1i∗t and λ2i∗t, 0≤t≤T, 1≤i≤N, such that (30)dλ1i∗tdt=-∂HI∗t,C∗t,λ∗t,u∗t∂Ii,dλ2i∗tdt=-∂HI∗t,C∗t,λ∗t,u∗t∂Ci,0≤t≤T,i=1,2,…,N.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 (31)HI∗,C∗,λ∗,u∗=minu∈U⁡HI∗,C∗,λ∗,u,one can obtain that, for 0≤t≤T and for 1≤i≤N, either (32)∂HI∗t,C∗t,λ∗t,u∗t∂γi=εiγi∗t-Ci∗t1+m2Ci∗t∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t=0or γi∗t=γ_ or γi∗t=γ¯. Hence, the proof is complete.By combining the above discussions, one can get the optimality system for problemP as follows: (33)dIitdt=-αiIit+fit1-Iit-Cit-gitIit,dCitdt=-θiCit+git1-Cit,dλ1itdt=-1+λ1itαi+fit+git-βi1+m1Iit2∑jaijλ1jt1-Ijt-Cjt,dλ2itdt=λ1itfit+λ2itθi+git+γit1+m2Cit2∑jaijIjtλ1jt-1-Cjtλ2jt,γit=max⁡min⁡Citεi1+m2Cit∑jaijIjtλ1jt-1-Cjtλ2jt,γ¯,γ_,0≤t≤T,i=1,2,…,N,with I0,C0T∈Ω and λT=0.


## 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(14)Minimizeu∈U⁡Ju=∫0TLxt,utdtsubject to(15)dxtdt=fxt,ut,0≤t≤T,with x0∈Ω, where Ω is positively invariant for system (15). The problem has an optimal control if the following six conditions hold simultaneously. (C1)
There isu∈U such that system (15) is solvable.(C2)
U is convex.(C3)
U is closed.(C4)
fx,u is bounded by a linear function in x.(C5)
Lx,u is convex on U.(C6)
Lx,u≥c1u2ρ+c2 for some ρ>1, c1>0, and c2.In order to prove the existence of an optimal control, six lemmas, one for each condition in Lemma1, should be proved.Lemma 2.
There isu∈U such that system (9) or (12) is solvable.Proof.
Substitutingu≡u¯≔γ¯,…,γ¯︸NT into system (12), one can get the uncontrolled system: (16)dxtdt=fxt,u¯with x0∈Ω. Then the function fx,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 setU is convex.Proof.
Let(17)u1=γ11,…,γN1∈U,u2=γ12,…,γN2∈U,0<ξ<1.As L20,TN is a real vector space, one can obtain (18)1-ξu1+ξu2∈L20,TN.Then, the convexity of U follows by the observation that (19)γ_≤1-ξγi1+ξγi2≤γ¯,1≤i≤N.Hence, the claimed result follows.Lemma 4.
The admissible setU is closed.Proof.
Letu=γ1,…,γNT be a limit point of U and (20)un=γ1n,…,γNnT,n=1,2,…,be a sequence of points in U such that (21)un-u2≔∫0Tunt-ut2dt1/2<1n.From the completeness of L20,TN, one can get (22)limn→∞⁡un=u∈L20,TN.Hence, the closeness of U follows from the observation that (23)γ_≤γi=limn→∞⁡γin≤γ¯,1≤i≤N.Lemma 5.
f
x
,
u is bounded by a linear function in x.Proof.
Note that, for system (9) and for i=1,2,…,N, (24)-γ¯N24-αiIi≤dIidt≤-αiIi+∑jaijβjIj,-θiCi≤dCidt≤-θiCi+γ¯∑jaijCj.Thus, the claimed result follows.Lemma 6.
L
x
,
u is convex on U.Proof.
Note that the Hessian matrix ofLx,u with respect to u∈U is as follows:(25)HuL=∂2L∂γ12∂2L∂γ1∂γ2⋯∂2L∂γ1∂γN-1∂2L∂γ1∂γN∂2L∂γ2∂γ1∂2L∂γ22⋯∂2L∂γ2∂γN-1∂2L∂γ2∂γN⋮⋮⋱⋮⋮∂2L∂γN-1∂γ1∂2L∂γN-1∂γ2⋯∂2L∂γN-12∂2L∂γN-1∂γN∂2L∂γN∂γ1∂2L∂γN∂γ2⋯∂2L∂γN∂γN-1∂2L∂γN2=ε10⋯000ε2⋯00⋮⋮⋱⋮⋮00⋯εN-1000⋯0εN.For any t∈0,T, HuL is real symmetric and its eigenvalues are all positive. Then, HuL is positive definite. Hence, the convexity of Lx,u follows by the result in [37].Lemma 7.
L
x
,
u
≥
c
1
u
2
ρ
+
c
2 for some ρ>1, c1>0, and c2.Proof.
Letρ=2, c1=mini⁡εi/2, and c2=0. Then, Lx,u≥mini⁡εi/2×u22. Thus, the proof is complete.Now, it is time to examine the main result of this subsection.Theorem 8.
The optimal control problemP has a solution.Proof.
Lemmas2–7 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 problemP is drawn.Theorem 9.
Supposeu∗· is an optimal control for problem P and I∗·,C∗·T is the solution to system (9) with u·=u∗·. Then, there exist functions λ1i∗t and λ2i∗t, 0≤t≤T, 1≤i≤N, such that(26)dλ1i∗tdt=-1+λ1i∗tαi+fi∗t+gi∗t-βi1+m1Ii∗t2∑jaijλ1j∗t1-Ij∗t-Cj∗t,dλ2i∗tdt=λ1i∗tfi∗t+λ2i∗tθi+gi∗t+γi∗t1+m2Ci∗t2∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t,0≤t≤T,i=1,2,…,N,with transversality conditions (27)λ1i∗T=λ2i∗T=0,i=1,2,…,N.Furthermore, one can get (28)γi∗t=max⁡min⁡Ci∗tεi1+m2Ci∗t∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t,γ¯,γ_,0≤t≤T,i=1,2,…,N.Proof.
The corresponding Hamiltonian is(29)HI,C,λ,u=∑iIi+12εiγi2+∑iλ1idIidt+∑iλ2idCidt,where λ1i, λ2i are undetermined, λ=λ11,…,λ1N,λ21,…,λ2NT.
According to the Pontryagin Minimum Principle [35], there exist functions λ1i∗t and λ2i∗t, 0≤t≤T, 1≤i≤N, such that (30)dλ1i∗tdt=-∂HI∗t,C∗t,λ∗t,u∗t∂Ii,dλ2i∗tdt=-∂HI∗t,C∗t,λ∗t,u∗t∂Ci,0≤t≤T,i=1,2,…,N.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 (31)HI∗,C∗,λ∗,u∗=minu∈U⁡HI∗,C∗,λ∗,u,one can obtain that, for 0≤t≤T and for 1≤i≤N, either (32)∂HI∗t,C∗t,λ∗t,u∗t∂γi=εiγi∗t-Ci∗t1+m2Ci∗t∑jaijIj∗tλ1j∗t-1-Cj∗tλ2j∗t=0or γi∗t=γ_ or γi∗t=γ¯. Hence, the proof is complete.By combining the above discussions, one can get the optimality system for problemP as follows: (33)dIitdt=-αiIit+fit1-Iit-Cit-gitIit,dCitdt=-θiCit+git1-Cit,dλ1itdt=-1+λ1itαi+fit+git-βi1+m1Iit2∑jaijλ1jt1-Ijt-Cjt,dλ2itdt=λ1itfit+λ2itθi+git+γit1+m2Cit2∑jaijIjtλ1jt-1-Cjtλ2jt,γit=max⁡min⁡Citεi1+m2Cit∑jaijIjtλ1jt-1-Cjtλ2jt,γ¯,γ_,0≤t≤T,i=1,2,…,N,with I0,C0T∈Ω and λT=0.


## 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 N=150 nodes, respectively. The parameters of system (33) are set as αi=0.01, βi=0.004887 (the value of βi comes from a report on some real infection probabilities in [41]), θi=0.02, εi=1, γ_=0.01, γ¯=0.1, and T=50, 1≤i≤N, and the initial conditions are set as Ii0=0.03 and Ci0=0.01, 1≤i≤N. 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.Supposeu∗t is an optimal control for problem P and x∗t is a solution to the corresponding controlled system. Let γ∗t and I∗t denote the average control and the proportion of infected nodes under u∗t, respectively, where (34)γ∗t=1N∑iγi∗t,I∗t=1N∑iIi∗t.Example 1.
Take a WS network with 150 nodes and 150 links as the propagation network.Figure2 exhibits the average control γ∗t and I∗t under different control strategies. Table 1 gives the final proportion of infected nodes and the value of objective function J 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 J and reduce virus prevalence to a low level simultaneously.Table 1
I
∗
(
T
) and J under different control strategies with m1=m2=2 for Example 1.
u
=
u
∗
u
=
0.01
u
=
0.04
u
=
0.08995
u
=
0.1
I
∗
(
T
)
0.0089
0.0283
0.0231
0.0072
0.0053
J
(
u
)
172.55
218.97
211.61
181.14
177.27Figure 2
γ
∗
t and I∗t under different control strategies with m1=m2=2 for Example 1.
(a)
γ
∗
t
(b)
I
∗
tFigure3 demonstrates the average control γ∗t for different m1 and m2. From this figure, one can see that (a) enhancing m1 and m2 roughly reduces γ∗t, (b) the smaller m2 is, the longer γ∗t stays at γ¯, and (c) m2 has a more significant impact on γ∗t than m1 does.Figure 3
γ
∗
t for different m1 and m2 for Example 1.
(a)
m
2
=
5
(b)
m
1
=
5Figure4 displays I∗t for different m1 and m2. From this figure, it can be seen that (a) lower m1 favors virus spreading, whereas lower m2 is conducive to the containment of virus prevalence, (b) m2 affects I∗t more significantly than m1 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].Figure 4
I
∗
t for different m1 and m2 for Example 1.
(a)
m
2
=
5
(b)
m
1
=
5Figure5 depicts the final proportion of infected nodes I∗T and the objective function Ju∗ for varied m1 and m2. From this figure, it can be seen that J is decreasing and increasing with respect to m1 and m2, respectively, which makes a suggestion that enhancing m1 and diminishing m2 are beneficial to the containment of viral spread and reduce J to a low level simultaneously.Figure 5
The final proportion of infected nodesI∗T and the objective function Ju∗ for different m1 and m2 for Example 1.
(a)
m
2
=
5
(b)
m
1
=
5Figure6 shows γ∗t, I∗t, I∗T, and Ju∗ for different ε. From this figure, it is found that decreasing ε is effective on the suppression of virus propagation and attains a lower Ju∗ 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 ε), often the decision is made to spend enough control cost. Hence, the tradeoff factor ε plays a critical role in the balance between control effect and control cost.Figure 6
γ
∗
t, I∗t, I∗T, and Ju∗ for different ε with m1=m2=5 for Example 1.
(a)
γ
∗
t
(b)
I
∗
t
(c)
I
∗
T and Ju∗Example 2.
Take a BA network with 150 nodes and 150 links as the propagation network.Figure7 displays γ∗t and I∗t under different control strategies. Table 2 shows the values of I∗T and Ju under different control strategies. Figures 8 and 9 depict γ∗t and I∗t for different m1 and m2, respectively. Figure 10 demonstrates I∗T and Ju∗ for different m1 and m2. Figure 11 exhibits γ∗t, I∗t, I∗T, and Ju∗ for different ε. From them, one can get the same results in Example 1. So they are omitted here for brevity.Table 2
I
∗
(
T
) and J under different control strategies with m1=m2=2 for Example 2.
u
=
u
∗
u
=
0.01
u
=
0.04
u
=
0.08895
u
=
0.1
I
∗
(
T
)
0.0132
0.0288
0.0208
0.0111
0.0097
J
(
u
)
169.84
219.58
202.91
185.27
185.65Figure 7
γ
∗
t and I∗t under different control strategies with m1=m2=2 for Example 2.
(a)
γ
∗
t
(b)
I
∗
tFigure 8
γ
∗
t for different m1 and m2 for Example 2.
(a)
m
2
=
5
(b)
m
1
=
5Figure 9
I
∗
t for different m1 and m2 for Example 2.
(a)
m
2
=
5
(b)
m
1
=
5Figure 10
I
∗
T and Ju∗ for different m1 and m2 for Example 2.
(a)
m
2
=
5
(b)
m
1
=
5Figure 11
γ
∗
t, I∗t, I∗T, and Ju∗ for different ε with m1=m2=5 for Example 2.
(a)
γ
∗
t
(b)
I
∗
t
(c)
I
∗
T and Ju∗Example 3.
Take a partial Facebook network with 150 nodes and 603 links as the propagation network.Figure12 shows γ∗t and I∗t under different control strategies. Table 3 gives the values of I∗T and Ju under different control strategies. Figures 13 and 14 display γ∗t and I∗t for different m1 and m2, respectively. Figure 15 demonstrates I∗T and Ju∗ for varied m1 and m2. Figure 16 depicts γ∗t, I∗t, I∗T, and Ju∗ for different ε.Table 3
I
∗
(
T
) and J under different control strategies with m1=m2=2 for Example 3.
u
=
u
∗
u
=
0.01
u
=
0.04
u
=
0.08895
u
=
0.1
I
∗
(
T
)
0.0013
0.0504
0.0035
0.0008
0.0006
J
(
u
)
61.91
373.77
112.31
83.53
86.17Figure 12
γ
∗
t and I∗t under different control strategies with m1=m2=2 for Example 3.
(a)
γ
∗
t
(b)
I
∗
tFigure 13
γ
∗
t for different m1 and m2 for Example 3.
(a)
m
2
=
5
(b)
m
1
=
5Figure 14
I
∗
t for different m1 and m2 for Example 3.
(a)
m
2
=
5
(b)
m
1
=
5Figure 15
I
∗
T and Ju∗ for different m1 and m2 for Example 3.
(a)
m
2
=
5
(b)
m
1
=
5Figure 16
γ
∗
t, I∗t, I∗T, and Ju∗ for different ε with m1=m2=5 for Example 3.
(a)
γ
∗
t
(b)
I
∗
t
(c)
I
∗
T and Ju∗Most of the results concluded from this example are the same as those in Example1 except the two phenomena listed as follows: (a) higher m2 increases γ∗t, which is contrary to the results in Figures 3(b) and 8(b), and (b) m1 has a negligible impact on γ∗t and I∗t. 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 J and reduce the infections to a low level simultaneously.(b)
Linear infection rate overestimates the prevalence of virus.(c)
Enhancingm1 and diminishing m2 are conductive to the containment of viral propagation and reduce J to a low level simultaneously.(d)
m2 has more significant influences on γ∗t, I∗t, and Ju∗ than m1 does.(e)
Decreasing the tradeoff factorε is beneficial to the suppression of virus spread and obtains a lower Ju∗ 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.



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*Source: 2836865-2017-06-12.xml*