aa r X i v : . [ phy s i c s . s o c - ph ] J un Two novel immunization strategies for epidemic control in directedscale-free networks with nonlinear infectivity
Wei Shi, Junbo Jia, Pan Yang, Xinchu Fu ∗ Department of Mathematics, Shanghai University, Shanghai 200444, China (Dated: June 5, 2018)In this paper, we propose two novel immunization strategies, i.e., combined immunization and dupleximmunization, for SIS model in directed scale-free networks, and obtain the epidemic thresholds forthem with linear and nonlinear infectivities. With the suggested two new strategies, the epidemicthresholds after immunization are greatly increased. For duplex immunization, we demonstrate thatits performance is the best among all usual immunization schemes with respect to degree distribution.And for combined immunization scheme, we show that it is more effective than active immunization.Besides, we give a comprehensive theoretical analysis on applying targeted immunization to directednetworks. For targeted immunization strategy, we prove that immunizing nodes with large out-degrees are more effective than immunizing nodes with large in-degrees, and nodes with both largeout-degrees and large in-degrees are more worthy to be immunized than nodes with only large out-degrees or large in-degrees. Finally, some numerical analysis are performed to verify and complementour theoretical results. This work is the first to divide the whole population into different types andembed appropriate immunization scheme according to the characteristics of the population, and itwill benefit the study of immunization and control of infectious diseases on complex networks.
Keywords : SIS model; Complex network; Combined immunization; Duplex immunization.
PACS numbers: 05.45.Ra, 05.10.-a
I. INTRODUCTION
Devising effective immunization schemes is very important for the prevention and control of infec-tious diseases and computer viruses. For implement of an immunization scheme, when a portion ofnodes (individuals) are immunized, those nodes can be thought as being removed from the networkand cannot infect or be infected by others, then the tolerance of the network will be strengthened,so it is very important to choose key nodes to be immunized. In the previous works, studies aremainly focused on immunization schemes for susceptible-infected-susceptible (SIS) models [1–4, 7, 9].The epidemic threshold characterizes the robustness of networks, and it signifies the critical valuefor a disease outbreak, when infection rate of an epidemic is beyond it, the epidemic will spread onthe network, and an epidemic vanishes while the infection rate is below it. To increase the thresh-old, designing effective immunization strategies is very important [20]. Up to now, many effectiveimmunization schemes have been proposed and studied, such as acquaintance immunization [17], ran-dom immunization [3], targeted immunization [3], active immunization [4], greedy immunization [18],dynamic immunization [19], and so on.Although many immunization strategies for epidemic models on complex networks have been studiedextensively [9, 13, 17, 20, 22, 23], most of them are based on undirected networks [3, 4]. Real-worldnetworks are closely related to directed networks, such as social networks, food webs, phone-callnetworks, the WWW [21], etc. The direction of nodes’ edges plays an important role in the study ofepidemic spread on networks. Diseases, viruses or information spread out via their out-going edgesand connect to others, while a susceptible node may be infected by its in-coming edges. Therefore,studying immunization strategies in directed networks is more practical and meaningful.Many directed networks, such as the WWW [21] and social networks, have power-law degree dis-tributions of the form: P ( k ) = C k − − γ and Q ( l ) = C l − − γ ′ , ∗ Corresponding author. Tel: +86-21-66132664; Fax: +86-21-66133292; E-mail: [email protected] where C and C are normalization constants to guarantee P Mk = m P ( k ) = 1 and P Nl = n Q ( l ) = 1. Herewe have 0 < γ, γ ′ ≤
1. Networks with power-law degree distributions are called scale-free. Here wesuppose that m is the minimal out-degree, M the maximal out-degree, n the minimal in-degree, and N the maximal in-degree in the network.In this paper, based on the heterogeneous mean-field theory [9] and degree distribution, we mainlystudy four immunization strategies for the SIS model on a directed network, including active im-munization, targeted immunization, combined immunization and duplex immunization. We obtainthe epidemic thresholds for these four immunization schemes, and compare effectiveness among themunder the same immunization rate. Our results show that the proposed duplex immunization strategyis the most effective scheme among all usual immunization schemes, including proportional immu-nization, acquaintance immunization, targeted immunization, active immunization, and the proposedcombined immunization. Besides, we divide the targeted immunization strategy into three cases indirected networks. We prove that nodes with large out-degrees are more important than nodes withlarge in-degrees when targeted immunization is implemented. On the other hand, we demonstratethat the nodes with both large in-degrees and large out-degrees are more worthy to be immunized thannodes with only large in-degrees or large out-degrees for targeted immunization scheme. To illustrateand test the performance of the proposed immunization schemes, we present numerical simulations indirected BA network in Figs 1-3, the numerical results are in accordance with our theoretical results.The rest of the paper is organized as follows. In Section II, we establish an SIS model in a directednetwork and discuss epidemic thresholds with different infectivities. In Section III, we first study indetail the targeted immunization scheme in a directed network, then analyze the active immunizationin a directed network. Besides, we propose two novel immunization strategies, and calculate theepidemic thresholds for them, and compare their effectiveness with targeted immunization and activeimmunization. In Section IV, we present numerical simulations. Finally, in Section V, we concludethe paper. II. THE SIS MODEL IN DIRECTED NETWORKS
In this section, we investigate the SIS model on a directed network. Nodes of the directed networkare divided into two groups: Susceptible and Infected. Hereafter, we will denote a susceptible nodeby an S-node etc., for short. An S-node becomes infected at rate ν if it contacts with an infectedindividual, and an I-node may recover and become an S-node with probability δ . Previous works havedefined an effective spreading rate λ = νδ , where we take a unit recovery rate δ = 1. Let us denote thedensities of S- and I-nodes with in-degrees k and out-degrees l at time t by s k,l ( t ), ρ k,l ( t ), respectively,so we have s k,l ( t ) + ρ k,l ( t ) = 1 , where follows the joint probability distribution p ( k, l ). Then, the respective marginal probabilitydistribution of the out-degrees and in-degrees reads as P ( k ) = P l p ( k, l ) , Q ( l ) = P k p ( k, l ) , and their average degrees are h k i = P k,l kp ( k, l ) = P k kP ( k ) , h l i = P k,l lp ( k, l ) = P l lQ ( l ) . Then the SIS model can be written as the following ordinary differential equations: dρ k,l dt = λk (1 − ρ k,l ( t ))Θ( t ) − ρ k,l ( t ) . (1)Here we suppose that the connectivity of nodes is uncorrelated, then the probability of a randomlyselected outgoing link emanating form I-nodes at time t is given byΘ( t ) = P k,l ϕ ( k, l ) p ( k, l ) ρ k,l ( t ) P k,l lp ( k, l ) = P k,l ϕ ( k, l ) p ( k, l ) ρ k,l ( t ) h l i , (2)where ϕ ( k, l ) denotes the infectivity of a node with degrees ( k, l ).Now, we calculate the epidemic threshold for model (1). At the steady state, we have dρ k,l dt = 0 forall k and l , from (1) we have ρ k,l = λk Θ1 + λk Θ , substituting the above equation into (2) we obtain a self-consistency equation for Θ as follows:Θ = 1 h l i X k,l λϕ ( k, l ) p ( k, l ) k Θ1 + λk Θ ≡ f (Θ) . If this equation has a solution Θ > f (1) = 1 h l i X k,l λϕ ( k, l ) p ( k, l ) k λk < h l i X k,l lp ( k, l ) = 1 ,f ′ (Θ) = 1 h l i X k,l ϕ ( k, l ) p ( k, l ) λk (1 + λk Θ) > ,f ′′ (Θ) = 1 h l i X k,l ϕ ( k, l ) p ( k, l ) − λk (1 − δ ( k, l ))) (1 + λk Θ) < , therefore, a nontrivial solution exists only if df (Θ) d Θ (cid:12)(cid:12)(cid:12)(cid:12) Θ=0 > , (3)so we obtain the value of λ yielding the inequality (3) which defines the critical epidemic threshold λ c : λ c = h l ih ϕ ( k, l ) k i , (4)where the λ c is a critical value for the infection rate λ : If λ > λ c , the disease will break out and persiston this network; Otherwise, when λ < λ c , the disease will gradually peter out. Hence, it is very crucialto increase λ c on the network to prevent epidemic outbreak. We will give detailed analysis on this inSection III.From the equality (4), we can see that the infectivity ϕ ( k, l ) also affects the value of the threshold λ c , so ϕ ( k, l ) is also important for controlling the disease. For this reason, we give further study ofthe infectivity in the following subsection. A. The epidemic threshold for the SIS model with nonlinear infectivity
For the SIS model on undirected scale-free networks [4], ϕ ( k ) indicates the infectivity of a node withdegree k . Previously, it was assumed that the larger the node degree, the larger the value of ϕ ( k ),and in [9–12], the ϕ ( k ) is just equal to the node degree, that is, ϕ ( k ) = k , in this case, the epidemicthreshold λ c = 0 when networks’ size is sufficiently large. However, in [13–15], the authors pointedout that large node with large ϕ ( k ) is not always appropriate, so they assumed that ϕ ( k ) = A , where A is a constant, and they obtained a different epidemic threshold λ c = A , which is always positive.On the basis of this, in [16] authors proposed a new nonlinear infectivity ϕ ( k ) = ak α bk α , and analysisits threshold on finite and infinite networks.Here in a directed scale-free network, we think both out-degrees and in-degrees play an importantrole in infectivity ϕ ( k, l ). At the early stage of a disease transmission, a susceptible individual may getinfected through out-going edges of infected individuals (in-coming edges of itself), then the diseasespreads out of its out-going edges and connects to other susceptible nodes. When a susceptibleindividual has no in-coming edges, it cannot be infected by infected individuals; similarly, it will notinfect other susceptible individuals without out-going edges even if it was infected.Base on the analysis above, we give a nonlinear infectivity ϕ ( k, l ) in a directed scale-free networkas follows: ϕ ( k, l ) = al α bl α · ck β dk β , (5)where 0 ≤ α, β ≤ , a > b, c, d ≥
0. In Eq. (5), when k (or l ) is very small, we can simply regard ck β dk β (or al α bl α ) as 0, which means this node cannot infect others (or cannot be infected by others);and during the epidemic spreading process, the disease spreads out of infected individuals’ out-goingedges, so the in-degrees of susceptible nodes is relatively more important than the out-degrees at theearly stage, based on this, we choose a > a = 0; and here we divide the ϕ ( k, l ) into fourmain cases:(1) ϕ ( k, l ) = ac (1+ b )(1+ d ) when α = 0 , β = 0, which means infectivity is a constant;(2) ϕ ( k, l ) = al when α = 1 , b, c = 0;(3) ϕ ( k, l ) = al α when 0 < α < , b, c = 0;(4)if b, c = 0, then ϕ ( k, l ) = al α bl α · ck β dk β , and it becomes gradually saturated with the increasingof in-degree k and out-degree l . Finally, it will converges to a constant ϕ ( k, l ) = acbd .Substituting case (1) and case (2) into Eq. (4), we obtain two different epidemic thresholds asfollows: λ c = ac h l i (1+ b )(1+ c ) h k i and λ c = a h k i , which were partially studied in [5]; and with sufficientlylarge k and l , λ c = h ac ih (1+ b )(1+ c ) i and λ c = 0.When ϕ ( k, l ) = al α in case (3), we have h ϕ ( k, l ) k i = a P k kP ( k ) P l l α Q ( l ). By using a continuousapproximation, we obtain h kϕ ( k, l ) i = a Z ∞ m Z ∞ m k · k − − γ · l α · l − − γ ′ dkdl = a Z ∞ m k γ dk · Z ∞ m l γ ′ − α dl. (6)From the above equation, we can conclude that the (6) is bounded when α < γ ′ . As a result, theepidemic threshold is λ c = m γ − α γ (1 − αγ ′ ) a , (7)which is always positive regardless of the size of out-degrees and in-degrees. We believe this is ainteresting result, which is different from the result of a vanished threshold λ c = 0 given in [7].When ϕ ( k, l ) = al α bl α · ck β dk β , where we have 0 < α, β ≤ b, d = 0 in case(4), then h kϕ ( k, l ) i = ac X k,l p ( k, l ) k al α bl α · ck β dk β = Z ∞ n Q ( l ) al α bl α · Z ∞ m P ( k ) ck β dk β , (8)similar to the above analysis in case (3), we can find that h kϕ ( k, l ) i is always bounded, then we obtainthat the threshold λ c is always a positive value.Through analysis and calculation in this subsection, we obtain the different epidemic thresholds λ ic ( i = 1 , , ,
4) for the four cases, and compare them with previous results. In Figs. 1-3, wepresent numerical analysis for different infectivities ϕ ( k, l ), it clearly shows that the different infectivity ϕ ( k, l )’s value result in different thresholds; and with nodes’ infectivity grows, λ c become smaller andsmaller, so the network robustness against epidemics become weaker and weaker. III. THE SIS MODEL WITH IMMUNIZATION
From the analysis in Section II, we know that higher threshold λ c indicates better robustness againstthe outbreak of an epidemic on a network. Hence, designing an appropriate immunization strategyis important for effectively controlling the spread of the epidemic. And the SIS model is known as amore appropriate model than SIR model to study immunization schemes at the early time of epidemictransmission because the effects and recovery and death can be ignored, and this is the optimal timeto apply immunization strategies in order to prevent and control epidemic outbreaks. In this section,we study the SIS model with various immunization strategies and compare their effectiveness amongthem for the same average immunization rate. A. Targeted immunization in directed networks
The targeted immunization [6] is known as the best strategy on heterogeneous networks, but we stilllack a comprehensive understanding when applying it to directed networks. The traditional targetedimmunization [3, 6] on undirected networks is to pick up the nodes with connectivity k > κ toimmunize, such as Eq.(14) in [4]. In [5], Wang first studied the SIS model with targeted immunizationin directed networks, but only immunize nodes with large out-degrees.We realize that in the real-life systems with targeted immunization scheme, only select the largeout-degree’s nodes to immunize [5] may not always be appropriate. Beyond that, the in-degrees mayalso play a significant role in the epidemic immunization process; as we discussed in Section II, eventhe out-degrees of some nodes is very high, but those nodes may not always be infective with theirin-degrees are too small. Otherwise, the nodes of high in-degrees with low out-degrees may not beinfective as well.So here we divide targeted immunization schemes into three cases to further compare their effec-tiveness: (1) Immunize the nodes with k > π ; (2) Immunize the nodes with l > π ; and (3) Immunizethe nodes with k > π and l > π . Next in this subsection, we give a deep theoretical analysis ontarget immunization in directed networks under these three conditions, and to find an optimal onebeyond them. Here we considere ϕ ( k, l ) = A as a positive constant, we define the immunization rate δ ik,l (0 < δ ik,l ≤
1) by δ k,l = , k > π a, k = π , , k < π δ k,l = , l > π b, l = π , , l < π δ k,l = , k > π and l > π c, k = π and l = π , , otherwise (9)where 0 < a, b, c ≤
1, and P k,l δ ik,l p ( k, l ) = h δ ik,l i ( i = 1 , , , and h δ ik,l i are the average immunizationrates. Then the epidemic dynamics model (1) becomes dρ k,l dt = λk (1 − δ k,l )(1 − ρ k,l ( t ))Θ( t ) − ρ k,l ( t ) . (10)At the steady state, we have the condition dρ k,l dt = 0 for all k and l . So we can get from (10) that ρ k,l = λk (1 − δ k,l )Θ1 + λk (1 − δ k,l )Θ . Substituting these into Eq. 2, we obtain a self-consistency equation for Θ as follows:Θ = 1 h l i X k,l λϕ ( k, l ) p ( k, l ) k (1 − δ k,l )Θ1 + λk (1 − δ k,l )Θ ≡ f (Θ) , therefore, we can obtain the threshold for model (10):ˆ λ c = h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i . (11)Substituting (9) into (11) we obtain three epidemic thresholds with targeted immuniza-tion (1) (TGA), targeted immunization (2) (TGB) and targeted immunization (3) (TGC), respec-tively: ˆ λ c = h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i = h l i A ( h k i − h kδ k,l i ) , (12)ˆ λ c = h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i = h l i A ( h k i − h kδ k,l i ) , (13)ˆ λ c = h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i = h l i A ( h k i − h kδ k,l i ) . (14)Higher epidemic thresholds indicates better performance of immunization schemes, through threeepidemic thresholds (12) − (14), now we discuss the effectiveness of different targeted immunizationschemes by comparing these three values: h kδ k,l i , h kδ k,l i and h kδ k,l i , the bigger one corresponds tobetter effectiveness.Intuitively, we think the TGA is more effective than TGB, and the TGC is the optimal targetedimmunization scheme in directed networks. During the immunization process, the in-coming links ofthe immunized S-nodes comes from the I-nodes and S-nodes itself, therefore, when we implement atarget immunization on S-nodes, this in-coming links with l > π comes from S-nodes itself which areharmless but are immunized at the same time, so the TGB may less effective than TGA.In addition, as we explained in previous sections, the in-degrees also play a significant role in theimmunization process, so it would be better to immunize the nodes with both large in-degrees andlarge out-degrees, which will be further discussed in this subsection.Note that h kδ k,l i = X k,l kδ k,l p ( k, l ) = M X k = π kP ( k ) , h kδ k,l i = X k,l kδ k,l p ( k, l ) = N X l = π Q ( l ) M X k = m kP ( k ) , h kδ k,l i = X k,l kδ k,l p ( k, l ) = N X l = π Q ( l ) M X k = π kP ( k ) . Under the same average immunization rate, which we take the average, then δ k,l = δ k,l = δ k,l = h δ k,l i = h δ k,l i = h δ k,l i (15)= M X k = π P ( k ) = N X l = π Q ( l ) = M X k = π N X l = π P ( k ) Q ( l ) , through the above equations, we have h kδ k,l i − h kδ k,l i = M X k = π kP ( k ) − N X l = π Q ( l ) M X k = m kP ( k )= M X k = π kP ( k ) − (cid:0) π − X k = m kP ( k ) + M X k = π kP ( k ) (cid:1) M X k = π P ( k )= M X k = π kP ( k ) π − X k = m P ( k ) − π − X k = m kP ( k ) M X k = π P ( k )= M X k = π π − X k = m kP ( k ) P ( k ) − M X k = π π − X k = m kP ( k ) P ( k )= M X k = π π − X k = m P ( k ) P ( k ) (cid:0) k − k (cid:1) > , so it proves that the effectiveness of TGA is better than TGB.Next, we discuss the effectiveness between TGC and TGA,TGB. Here, for better comparison, weset π = h k i , π = h l i , and we find when π > h k i and π > ξ , where ξ is a positive constant (canfetch its value as h l i ), the TGC is better than TGA and TGB.Note that h kδ k,l i − h kδ k,l i = N X l = π Q ( l ) M X k = π kP ( k ) − N X l = π Q ( l ) M X k = m kP ( k )= M X k = π N X l = π Q ( l ) kP ( k ) − M X k = π N X l = π Q ( l ) P ( k ) M X k = m kP ( k )= M X k = π N X l = π Q ( l ) P ( k ) (cid:0) k − M X k = m kP ( k ) (cid:1) = M X k = π > h k i N X l = π Q ( l ) P ( k ) (cid:0) k − h k i (cid:1) > h kδ k,l i − h kδ k,l i = N X l = π Q ( l ) M X k = π kP ( k ) − M X k = π kP ( k )= M X k = π kP ( k ) M P k = π P ( k ) M P k = π P ( k ) − M X k = π kP ( k )= M P k = π kP ( k ) M P b k = π P ( b k ) − M P k = π P ( k ) M P b k = π b kP ( b k ) M P k = π P ( k )= M P k = π > h k i M P b k = π = h k i P ( k ) P ( b k ) (cid:0) k − b k (cid:1) M P k = π P ( k ) > , therefore, the epidemic thresholds of TGC is greater than TGA’s and TGB’s, this means that theperformance of TGC is better than TGA and TGB. Either the infectivity is linear or nonlinear, thisconclusion is always valid. Figs. 2(c)-(d) and Figs. 3(c)-(d) below show this in details.In this section, we divide the classic targeted immunization scheme into three cases in directednetworks, the discussion and comparison of those three cases are carried out in depth. Now we havegiven the analytical comparison of the three different epidemic thresholds (see (12) − (14)). We provethat the nodes with both large in-degrees and large out-degrees are more worthy to be immunizedduring target immunization process in directed networks. Besides, immunizing nodes with large out-degrees are more efficient than immunizing nodes with large in-degrees for targeted immunizationscheme. B. Active immunization in directed networks
The classic active immunization [4] is to immunize its neighbors with degrees k ≥ κ on undirectedscale-free networks, here we will generalize it to directed networks and calculate its epidemic threshold.Then, the epidemic dynamics model becomes dρ k,l dt = λk (1 − ρ k,l ( t ))Θ( t ) − (1 + δ k,l ) ρ k,l ( t ) , (16)where δ k,l = P k kP ( k ) h k i δ k,l , and δ k,l is defined in (9).By adopting dρ k,l dt = 0 leads toΘ = λ Θ h l i X k,l kϕ ( k, l ) p ( k, l ) λk Θ + 1 + δ k,l ≡ f (Θ);therefore, we obtain the epidemic threshold˜ λ c = h l i P k,l (1 + δ k,l ) − kϕ ( k, l ) p ( k, l ) . Note that δ k,l = X k kP ( k ) h k i δ k,l = h kδ k,l ih k i , we have ˜ λ c = h kl i + h l ih kδ k,l ih k ϕ ( k, l ) i . (17)Compare (11) with (17), we haveˆ λ c − ˜ λ c = h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i − h kl i + h l ih kδ k,l ih k ϕ ( k, l ) i = h kl ih ϕ ( k, l ) kδ k,l i − h klϕ ( k, l ) ih kδ k,l i + h l ih kδ k,l ih ϕ ( k, l ) kδ k,l i (cid:0) h k ϕ ( k, l ) i − h ϕ ( k, l ) kδ k,l ih k ϕ ( k, l ) i (cid:1) . (18)To better compare the target immunization and active immunization, we set the same average immu-nization rate δ k,l = δ k,l , therefore, h kδ k,l ih k i = h δ k,l i . (19)From (18) and (19), we haveˆ λ c − ˜ λ c = h kl ih ϕ ( k, l ) kδ k,l i − h klϕ ( k, l ) ih k ih δ k,l i + BA , (20)where A = h k ϕ ( k, l ) i − h ϕ ( k, l ) kδ k,l ih k ϕ ( k, l ) i , and B = h l ih kδ ak,l ih ϕ ( k, l ) kδ k,l i .Note that h klϕ ( k, l ) ih k ih δ k,l i = h kl ih ϕ ( k, l ) kδ k,l i + σ ′ . There may be appropriate small π , π , suchthat σ ′ is relatively smaller than B . So we can obtain that ˆ λ c > ˜ λ c , which means that the targetimmunization scheme is more effective than active immunization under the same average immunizationrate, Fig. 1(a) below illustrates this conclusion. C. Combined immunization
In this section we propose a new immunization scheme, combined immunization: Choose a sus-ceptible node and immunize its neighbors whose in-degrees l > κ , and choose an infected node toimmunize its neighbors whose out-degrees k > κ at the same time. Then the epidemic dynamicsmodel becomes: dρ k,l dt = λk (1 − ρ k,l ( t ))(1 − δ l )Θ( t ) − (1 + δ k ) ρ k,l ( t ) , (21)where δ l = X l lQ ( l ) h l i δ l , δ k = X k kP ( k ) h k i δ k ,δ l = , l > κ d, l = κ , , l < κ δ k = , k > κ e, k = κ , , k < κ and 0 < d, e ≤ dρ k,l dt = 0, than substitute it into (2), model (21) leads toΘ = λ Θ h l i X k,l kϕ ( k, l ) p ( k, l )(1 − δ l ) λk (1 − δ l )Θ + 1 + δ k ≡ f (Θ) . So the epidemic threshold for model (21) is¯ λ c = h l i P k,l (1 + δ k ) − kϕ ( k, l ) p ( k, l )(1 − δ l ) . Due to δ l = P l lQ ( l ) h l i δ l = h lδ l ih l i , δ k = P k kP ( k ) h k i δ k = h kδ k ih k i .We obtain that ¯ λ c = h l i ( h k i + h kδ k i ) h k ϕ ( k, l ) i ( h l i − h lδ l i ) . (22)Compare (22) with (4), we have¯ λ c = λ c + h l ih kδ k i + h kl ih lδ l ih k ϕ ( k, l ) i ( h l i − h lδ l i ) > λ c . (23)This means that the combined immunization scheme we propose here is indeed effective. Next wewill compare the new immunization scheme with the active immunization scheme, through (22) and(17), we have ¯ λ c − ˜ λ c = h l i ( h k i + h kδ k i ) h k ϕ ( k, l ) i ( h l i − h lδ l i ) − h kl i + h l ih kδ k,l ih k ϕ ( k, l ) i = h l ih kδ k i + h kl ih lδ l i + h l ih lδ l ih kδ k,l i − h l ih kδ k,l ih k ϕ ( k, l ) i ( h l i − h lδ l i ) . (24)0Setting the immunization rate as the same, so we have δ l + δ k = δ k,l , where δ k,l is defined in Section III B and δ l , δ k are defined in Section III C. Therefore, h lδ l ih l i + h kδ k ih k i = h kδ k,l ih k i . (25)From (24) and (25) we obtain that¯ λ c − ˜ λ c = h lδ l i (cid:0) h l ih kδ k i + h k ih lδ l i (cid:1) h k ϕ ( k, l ) i ( h l i − h lδ l i ) > , (26)as it is obvious that h l i > h lδ l i .That is to say, under the same average immunization rate, the combined immunization is moreeffective than the active immunization discussed in Section III B. Fig. 1(b) below illustrates thisconclusion.Considering (23) and (14), under the same average immunization h lδ l ih l i + h kδ k ih k i = h δ k,l i , we have¯ λ c = h l i ( h k i + h kδ k i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) h k lϕ ( k, l ) i ( h l i − h lδ l i ) ˆ λ c . Note that h l i ( h k i + h kδ k i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) > , h k lϕ ( k, l ) i ( h l i − h lδ l i ) > . So ¯ λ c = Λˆ λ c , where Λ is a positive constant, which means that the combined immunization scheme iscomparable in effectiveness to the targeted immunization scheme discussed in Section III A. D. Duplex immunization
Infected individuals play a vital part in the early stages of disease transmission, diseases spreadthrough out-going links of infected individuals to the in-coming links of susceptible individuals. There-fore the out-degree l is a key character during the early stage of a disease transmission. And all immu-nization strategies [9–13] consider the all nodes as a whole to implement the immunization strategies,so in this section we proposed an immunization strategy based on a partition of the out-degrees l ;we divide the population of all nodes with out-degrees l and in-degrees k into two parts: Nodes without-degrees exceeding a positive constant number L is considered as the first part T = { ( k, l ) | l > L } ;and T , the complement of T , as the second part. In T , we use the targeted immunization in Sec-tion III A, and in T , we use the combined immunization proposed in Section III B. It turns out that theeffectiveness of these two immunization strategies’ combination is more effective than both of them.We introduce a constant α ∈ [0 , T , so we have s k,l ( t ) + ρ k,l ( t ) = ( α, if ( k, l ) ∈ T , − α, if ( k, l ) ∈ T. The condition α = 1 implies the classic targeted immunization (1) in Section III A , while α = 0means the combined immunization proposed in Section III C, implementing two kinds of immunizationstrategies together, the model (1) becomes dρ k,l dt = ( λk (1 − b δ k )( α − ρ k,l ( t )) θ ( t ) − ρ k,l ( t ) , if ( k, l ) ∈ T ,λk (1 − α − ρ k,l ( t ))(1 − δ l ′ ) θ ( t ) − (1 + δ k ′ ) ρ k,l ( t ) , if ( k, l ) ∈ T. (27)1where δ l ′ = X l lQ ( l ) h l i δ l ′ , δ k ′ = X k kP ( k ) h k i δ k ′ , and b δ k = , k > η a ′ , k = η , , k < η δ l ′ = , l > η b ′ , l = η , , l < η δ k ′ = , k > η c ′ , k = η , , k < η where 0 < a ′ , b ′ , c ′ ≤
1, and η ≥ L .By letting dρ k,l dt = 0, than substitute it into (2), model (27) leads to a self-consistency equation:Θ = λ Θ h l i X k,l αλϕ ( k, l ) p ( k, l ) k (1 − b δ k )1 + λk (1 − b δ k )Θ + X k,l (1 − α ) kϕ ( k, l ) p ( k, l )(1 − δ l ′ ) λk (1 − δ l ′ )Θ + 1 + δ k ′ ≡ f (Θ) , therefore, we can obtain the epidemic threshold for model (27):ˇ λ c = h l i α (cid:0) h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i (cid:1) + h l i ( h k i + h kδ k ′ i )(1 − α ) h k ϕ ( k, l ) i ( h l i − h lδ l ′ i ) , (28)which is clearly greater than the epidemic threshold λ c obtained in (4), that means the immunizationscheme we proposed is indeed effective.In Section III A, we know that the TGC is more effective than TGA and TGB, and through Sec-tion III B, we find the combined immunization is more effective than the active immunization.We now compare the new immunization scheme, the duplex immunization, with the TGC and thecombined immunization to find the optimal one.Through (28) and (14), we haveˇ λ c − ˆ λ c = h l i α (cid:0) h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i (cid:1) + h l i ( h k i + h kδ k ′ i )(1 − α ) h k ϕ ( k, l ) i ( h l i − h lδ l ′ i ) − h l ih ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i = 1 A ∗ (cid:0) B ∗ + C ∗ − D ∗ (cid:1) , where A ∗ = α (1 − α ) h k ϕ ( k, l ) i ( h l i − h lδ l ′ i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) ,B ∗ = = h l i (1 − α ) h k ϕ ( k, l ) i ( h l i − h lδ l ′ i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) ,C ∗ = α ( h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i ) h l i ( h k i + h kδ k ′ i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) ,D ∗ = h l i (1 − α ) α h k ϕ ( k, l ) i ( h l i − h lδ l ′ i )( h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i ) . It is obvious that these four polynomials: A ∗ , B ∗ , C ∗ and D ∗ are greater than zero. For T , theaverage immunization rate is α h b δ k i ; For T , the average immunization rate is (1 − α )( h lδ l ′ ih l i + h kδ k ′ ih k i );hence, under the same average immunization rate for duplex immunization and TGC, we have(1 − α )( h lδ l ′ ih l i + h kδ k ′ ih k i ) + α h b δ k i = h δ k,l i . (29)Note that B ∗ − D ∗ = h l i (1 − α ) h k ϕ ( k, l ) i ( h l i − h lδ l ′ i ) (cid:0) h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i − α ( h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i ) (cid:1) . (30)Combining (29) and (30), we obtain that when α satisfies the condition: α < E ∗ <
1, thenˇ λ c > ˆ λ c , (31)2where E ∗ = ( h ϕ ( k, l ) k i − h ϕ ( k, l ) kδ k,l i ) / ( h ϕ ( k, l ) k i − h ϕ ( k, l ) k b δ k i ), which means that the duplex im-munization scheme is more effective than the target immunization scheme discussed in Section III A.With the similar analysis method above, we can verify that the duplex immunization scheme ismore effective than the combined immunization scheme discussed in Section III C. Fig. 1(a) belowshows ˇ λ c > ¯ λ c very clearly with the same average immunization rate δ = 0 .
12, and this result appliesto all same δ .Besides, through [4, 5], we know the targeted immunization scheme is more effective than propor-tional immunization scheme and acquaintance immunization scheme for the same average immuniza-tion rate in directed networks. Hence, we reach the following conclusion: The duplex immunization weproposed has the best effectiveness comparing to all other usual immunization schemes with respectto degree distribution in directed scale-free networks.
E. A brief summary
In the previous section, we have discussed targeted, active, combined and duplex immunizationschemes, and calculated the thresholds for these strategies. By comparing the thresholds for differ-ent immunization strategies, we have conclude that the epidemic threshold of TGB (see Eq. (13)) isgreater than that of TGA (see Eq. (12)); Then we proved that the targeted nodes with both large in-degrees and large out-degrees (see Eq. (14)) are more worthy to be immunized in directed networks;We extended the traditional active immunization [4] into directed networks, analyzed its epidemicthreshold and compared its effectiveness with targeted immunization scheme and combined immu-nization scheme under the same average immunization rate; The proposed combined immunizationscheme is more effective than active immunization scheme, and it is comparable to the targeted im-munization scheme; And the performance of the duplex immunization scheme is the best among allusual immunization schemes discussed in this section.
IV. NUMERICAL ANALYSIS
In this section, we present the results of numerical simulations to further illustrate the above the-oretical analysis and show the effectiveness of different immunization schemes. We use the algorithmof Barab´asi and Albert [8] to generate a directed scale-free network with γ = 1 and γ ′ = 1. Here thejoint degree distribution is independent, we consider a population of 1000 individuals and take a unitrecovery rate.3 Different immunization under same average immunization rate δ Threshold λ I n f ec t e d d e n s i t y Duplex−immunization ( δ =0.12)Target−immunization (C) ( δ =0.12)Target−immunization (A) ( δ =0.12)Target−immunization (B) ( δ =0.12)Combined−immunization ( δ =0.12)No−immunization (a) Threshold λ I n f ec t e d d e n s i t y Compare combined immunization with active immunization
No−immunizationCombined−immunization ( δ =0.2736)Active−immunization ( δ =0.2736) (b) FIG. 1: (Color online) Comparison of the effectiveness of different immunization schemes under the same averageimmunization rate δ . (a) shows the thresholds among those five immunization schemes under the average immunizationrate δ = 0 .
12, the threshold λ c = 0 .
25 for no immunization, the threshold ¯ λ c = 0 .
27 for combined immunization, thethreshold ˆ λ c = 0 .
29 for TGB, the threshold ˆ λ c = 0 . λ c = 0 . λ c = 0 . δ = 0 . λ c = 0 .
32 for active immunization and ¯ λ c = 0 .
35 forcombined immunization.
In Fig. 1(a), we repeated the simulation above when the immunization schemes-targeted (1, 2, 3),combined and duplex-are implemented. We set the same average immunization rate for these fiveimmunization schemes to better compare their effectiveness with each other, and we implement ano-immunization curve for all immunization schemes to illustrate that all immunization schemes areeffective comparing to the case without any immunization. The average out-degree ( h k i ) and averagein-degree ( h l i ) for the generated network is 3 and 5, respectively, and k max = 82 , l max = 100. Here weset the infectivity as a constant ϕ ( k, l ) = 2. For TGA we choose π = 7 and a = 1, for TGB we choose π = 4 and b = 1, and for TGC we choose π = 4, π = 2 and c = 1. We can see in Fig. 1(a), underthe same average immunization rate δ = 0 .
12, ˆ λ c > ˆ λ c > ˆ λ c , which means the performance of TGC isbetter than TGA and TGB. And for the targeted immunizations on a directed scale-free network, toimmunize nodes with large out-degrees is more efficient than to immunize nodes with large in-degrees.Besides, we can obtain the epidemic threshold for duplex immunization ˇ λ c = 0 . λ c = 0 . T = { ( k, l ) | l > } , η = 17, a ′ = 1, η = 18, b ′ = 1, η = 10, c ′ = 1,and α can be calculated as 0 . δ = 0 . π = 9 for active immunization scheme, and κ = 15 , κ = 13 for combined immunization scheme.So we can illustrate the conclusion in Section III C (see Eq. 26), which means that the combinedimmunization scheme proposed in Section III C is more effective than the active immunization schemediscussed in Section III B for the same average immunization rate.4 Active imunization with different linear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . l No-immunization (a)
Combined imunization with different linear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . l No-immunization (b)
Targeted imunization (C) with different linear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . l No-immunization (c)
Threshold λ I n f ec t e d d e n s i t y Duplex imunization with different linear infectivity ϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . lϕ ( k,l ) = 0 . l No-immunization (d)
FIG. 2: (Color online) Different linear infectivity ϕ ( k, l )’s effects on four immunization schemes. We take linearinfectivity ϕ ( k, l ) = al , in (a) and (b), a = 0 . , . , . , . , .
6; in (c) and (d), a = 0 . , . , . , . , . In Fig. 2, we choose a linear infectivity ϕ ( k, l ) = al and set the same δ = 0 .
12. For active andcombined immunization schemes, we set a = 0 . , . , . , . , .
6, and for targeted immunization(c)and duplex immunization, we set a = 0 . , . , . , . , .
9. Still, in Figs. 2(a)-(d), we take the sameaverage immunization rate δ = 0 .
12. The Figs. 2(a)-(d) clearly show that with an increasing a , thethresholds of those four immunization schemes are increasing at the same time; on the other hand, withdifferent linear infectivity, the duplex immunization is still more effective than targeted immunization,and the combined immunization is still more effective than active immunization.In Fig. 3, we choose a nonlinear infectivity ϕ ( k, l ) = al α . For active and combined immunization,we set a = 0 . α = 0 . , . , . , . ,
09 and for targeted immunization(c) and duplex immunization,we set a = 0 . α = 0 . , . , . , . ,
09 . We choose δ = 0 .
12 in Figs. 3(a)-(d), which shows thesimilar property when infectivity is linear: when α increases, threshold increases. And by differentnonlinear infectivity, the thresholds changes faster than linear infectivity. Besides, with differentnonlinear infectivity for same immunization rate, the duplex immunization is more effective thantargeted immunization and the combined immunization is more effective than active immunization.5 Active immunization with different nonlinear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . No-immunization (a)
Threshold λ I n f ec t e d d e n s i t y Combined immunization with different nonlinear infectivity ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . No-immunization (b)
Targeted imunization (C) with different nonlinear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . No-immunization (c)
Duplex imunization with different nonlinear infectivity
Threshold λ I n f ec t e d d e n s i t y ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . ϕ ( k,l ) = 0 . l . No-immunization (d)
FIG. 3: (Color online) Different nonlinear infectivity ϕ ( k, l )’s effects on four immunization schemes. We take non-linear infectivity ϕ ( k, l ) = al α , in (a) and (b), a = 0 . α = 0 . , . , . , . , .
9; in (c) and (d), a = 0 . α = 0 . , . , . , . , . In addition, in Fig. 1, we present the contrast among different immunization strategies for thesame immunization rate and validate that the duplex immunization is more effective than targetedimmunization; the performance of combined immunization is better than active immunization. InFigs. 2-3, we use different linear and nonlinear infectivities ϕ ( k, l ) on active, combined, targeted andduplex immunization schemes, it is shown that with higher infectivity, the epidemic threshold isdramatically reduced; besides, the results of comparison between those immunization schemes are stillvalid with different linear and nonlinear infectivities. V. CONCLUSIONS AND DISCUSSIONS
In this paper, different immunization strategies for SIS models in directed scale-free networks withdifferent infectivities are studied, and we calculate the epidemic thresholds for different immunizationschemes, and obtained the following results:Firstly, the epidemic threshold λ c takes a positive value if ϕ ( k, l ) = al α and α < γ ′ in a finitenetwork with sufficiently large size; besides, when ϕ ( k, l ) = al α bl α · ck β dk β , λ c is always positive.Secondly, for the targeted immunization in directed networks, we prove that immunizing nodeswith large out-degrees are more effective than immunizing nodes with large in-degrees when targetedimmunization is implemented; on the other hand, we demonstrate that the nodes with both large in-degrees and large out-degrees are more worthy to be immunized during target immunization processthan nodes with only large in-degrees or large out-degrees.Thirdly, the duplex immunization we proposed has the best effectiveness comparing to all otherusual immunization schemes (e.g., proportional immunization, acquaintance immunization, targetedimmunization, active immunization, and combined immunization) for the same average immunizationrate. Besides, the performance of the combined immunization we proposed on disease control is betterthan active immunization.Finally, from realistic viewpoints, weighted networks and degree-correlated networks are more rea-sonable for epidemic immunization, and we expect that our work may be extended into these and6even multiplex and interconnected networks in our future research. Acknowledgements
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