Physica A 388 (2009) 1228–1236

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Physica A

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Epidemic spreading in communities with mobile agentsJie Zhou ∗, Zonghua LiuInstitute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai, 200062, China

a r t i c l e i n f o

Article history:Received 22 June 2008Received in revised form 26 October 2008Available online 16 December 2008

Keywords:Social networksMobile agentsEpidemic spreadingRandom walk

a b s t r a c t

We propose a model of mobile agents to study the epidemic spreading in communitieswith different densities of agents, which aims to simulate the realistic situation of multiplecities. The model addresses the epidemic process from a community with threshold λc1less than the infection rate λ to a community with threshold λc2 larger than λ through bothdirect and indirect contacts. By both theoretic analysis and numerical simulationswe showthat it is possible to sustain the epidemic spreading in the community with λc2 throughcontact with another community, provided that the latter is connected with an infectedcommunity. This result suggests that for effectively controlling the epidemic spread, weshould also pay attention to the risk caused by the infection through indirect contact.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Epidemic spreading has been studied for a long time and recently attracted great interest because of the new discoveriesin complex networks [1–3]. One of the key problems in this field is to find effective approaches to protect humancommunities from infection. A number of methods have been proposed to solve this problem, especially the segregatemethod [4–8], which is to cut the connection between the infected people (communities) and non-infected ones. Thesegregate method is a good choice when we do not know the property of the diseases or we do not have an effectiveway to immunize ourselves. It is generally believed that a community is safe if its connections to the surrounding infectedcommunities are removed [4,6]. However, real situations are very complex. On one hand, there is a time delay betweenthe infection of an agent and the knowledge of its status by others, which causes some difficulties to timely figure outthe dangerous connections. On the other hand, the epidemic spread may go through a third one, i.e., a safe communitywith an epidemic threshold λc smaller than its current infection rate. As we know, a single individual, can temporarilyimmunize himself or cut all his connections with the outside to protect himself once he is in dangerous environment.However, for a community, it is hard to do that as it has a large amount of transportation/communication with the outsideto sustain its normal function. Considering the fact that connections with the outside will always bring some risk, in thispaper, we will build a dynamical community model and make a theoretical analysis to understand how large the risk is.This problem is significant as it will help the governor to make an optimal strategy to balance the risk and the necessarytransportation/communication with the outside.Firstly, we need to understand how epidemics spread between two connected communities. This problem has been

recently addressed in static community networks byNewman et al. [9,10] and Liu et al. [11]where the community structuresare fixed and the viruses/diseases can only spread from one community to another through the links between them.Considering the fact that human beings in a community are mobile agents [12–16] and each agent may travel to othercommunities by transport, the links in and between communities are dynamical but not static. Therefore, it is necessaryto investigate the epidemic spreading in dynamical communities. In this paper, a dynamical community model for the two

∗ Corresponding author. Tel.: +86 13671537224.E-mail addresses: zjie@ntu.edu.sg (J. Zhou), zhliu@phy.ecnu.edu.cn (Z. Liu).

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.12.014

J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236 1229

Fig. 1. Schematic illustration of the epidemic spreading in the dynamical community model where the small segments with arrow denote the movingindividuals and the bridge between the two communities represents the jumping of agents.

and three connected communities is presented. The model shows that: (1) There is a stabilized number of infected agentsafter a transient process in each community, which depends on its density of agents and the jumping probability betweentwo neighboring communities; (2) In the case of two connected communities, a parameter γ is introduced to represent theefficiency of the jumping agents to reproduce infectors. It is found that the ratio of the number of infected agents between thetwo communities is inversely proportional to the average degree of the safe community; (3) In the case of three communities,the ratio of the number of infected agents is inversely proportional to the product of degrees of the two safe communities.An epidemic process can be generally characterized by the SIS (susceptible-infected-susceptible) model [17–20] and the

SIR (susceptible-infected-refractory) model [21–26]. In the SIS model, an agent has two status: susceptible and infected. Asusceptible agent may become infected once it contacts an infected one. After a time step, the infected agent recovers andreturns to the susceptible state. In thismodel, an interesting result pointed out by Paster–Satorras and Vespignani is that thevirus could spread on the Internet andWWW even when the infection probabilities are vanishingly small [17]. While in theSIRmodel, an agent has three status: susceptible, infected, and refractory. The infected agent cannot go back to a susceptiblestatus but become refractory, which describes the phenomenon of long-term immunity. In this paper, we use the SIS model.The paper is organized as follows. In Section 2, we propose the dynamical community model. In Section 3, we discuss

the epidemic spreading between two connected networks. Both theoretic analysis and numerical simulations are given. InSection 4, we discuss the epidemic spreading through indirect contagion in three communities. We find that for properlychosen population densities, it is possible to sustain the epidemic in a third community through indirect contact. Finally,the conclusions are given in Section 5.

2. The dynamical community model

Considering the fact that an agent can freely move in or between communities, whose neighbors change with time [12–16], we may conceive of a dynamical community model that contains the above features. Suppose an agent interacts onlywith its neighbors within radius r , i.e., there are links between those agents whose distance is smaller than 2r . Thus,the links are time dependent and the degree k of an agent is determined by the number of agents within the circle ofradius r . Also noticing that agents may sometimes travel to other communities, we let each agent have a probability pto jump to a neighboring community. In static community networks [9–11], this jumping behavior is equivalent to the linksbetween communities. Obviously, the difference between the static and dynamicalmodels is that these links are only for thechosen specific agents in the static community network but be possible for every agent in the dynamical community model.Therefore, the dynamical model is closer to reality. Considering the fact that the jumping process is much shorter than theinfection period, we assume that the status of the jumping agents will be kept when it jumps to another community. Basedon this analysis, our model consists of multiple communities with jumping agents between any two of them. Fig. 1 showsthe schematic illustration of two communities where each one is a square-shaped area with a periodic boundary condition,i.e., in a two-dimensional space. In contrast to the real world, these areas could represent different cities that have a longgeographical distance between them. People can move freely in their own city and travel to other cities by transport, suchas planes etc.Suppose there areN agentswhomovewith velocity v(t) in each community. Themodulus of the velocity of all the agents

v are the same in the whole spreading process while their directions are randomly distributed and updated stochastically intime. That is, each agent is a random walker with a probability p to travel to another community and the probability 1− pto stay in the community. As both A1 and A2 have the same population and the same jumping probability, their populationwill be always kept as N . When an agent travels to another community, its new position will be randomly chosen. On thecontrary, its position will be updated with the following relation if it moves within the same community

xi(t +∆t) = xi(t)+ vi(t)∆t, (1)

where xi(t) is the position of the ith agent in the two-dimensional space at time t , vi(t) = (v cos θi(t), v sin θi(t)),θi(t) = ξi(t), ξi(t) are N independent random variables chosen at each time with uniform probability in the interval[−π, π], and the velocity amplitude v is chosen as 0.03 in this paper. In the limit v → 0 the agents do not move and

1230 J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236

Fig. 2. Evolution of the number of infected nodes with p = 0 for four typical λ. The parameters are L1 = 0.5, L2 = 1, N = 1000, and r = 0.05, hence theepidemic thresholds are λc1 = 0.034 and λc2 = 0.125. The panel (a) represents the case of λ = 0.025 < λc1 in A1 , (b) the case of λ = 0.035 > λc1 in A1 ,(c) the case of λ = 0.12 < λc2 in A2 , and (d) the case of λ = 0.13 > λc2 in A2 .

the model becomes a static one. For v →∞ the agents become completely mixed between two updates. According to oursimulations, in a wide range of the velocities (0.003 < v < 0.3), the actual value of v does not affect the results.In the SIS model, nodes can be in two distinct states: susceptible or infected. As each agent has an interaction radius r , a

susceptible agent may be infected if there are infected agents within r . Suppose the susceptible node has a probability λ ofcontagion with each infected neighbor and has k neighbors in its interaction region, of which kinf are infected, then it willbecome infected with probability [1− (1− λ)kinf ]. At the same time, each infected node will become susceptible at rate µat each time step. To be brief, let us setµ = 1 since it only affects the definition of the time scale of the epidemic spreading.The basic notion in epidemiology is the epidemic threshold, λc , abovewhich the epidemic spreads and becomes endemic.

Suppose we have a few infected agents as seeds in the beginning, then they can infect other susceptible nodes. After a finitetime, the infectionwill reach a steady state whichmay be zero for λ < λc and nonzero for λ > λc . As each agent is a randomwalker and its degree changes with time, a meaningful parameter is the average degree 〈k〉. Using L to represent the lengthof the square in Fig. 1, we have 〈k〉 = πr2N/L2. For guaranteeing the epidemic spreading, a seed should induce at least onesusceptible to become infected, which gives

λc ∼ 1/〈k〉. (2)

The condition of µ = 1 is utilized here. When the size of A1 differs from the size of A2, the densities of agents in the twocommunities of Fig. 1 will be different, so are their epidemic thresholds λc . For example, if we let the length of A1 be L1 andthe length of A2 be L2, we have λc1 ∼ L21/(πr

2N) and λc2 ∼ L22/(πr2N). Without the jumping between A1 and A2 (p = 0),

our numerical simulations have confirmed the λc phenomenon that the infection will continue when λ > λc but die whenλ < λc . Using I(t) to denote the number of infected nodes at time t , I(t) will be stabilized after a finite time, which is zerofor λ < λc and nonzero for λ ≥ λc . In this paper, we fix N = 1000, r = 0.05, L1 = 0.5 and L2 = 1 if without specificillustration. Hence, λc1 = 0.034 < λc2 = 0.125. Fig. 2 shows four specific situations with p = 0 where (a) and (b) representthe epidemic spreading in A1, (c) and (d) the case in A2, (a) and (c) for λ < λc , and (b) and (d) for λ > λc . Obviously, theepidemic will die for the cases of λ < λc in (a) and (c) but survive for the cases of λ > λc in (b) and (d). For convenience, wecall the community with λ > λc as infected community and the community with λ < λc as safe community.

3. Epidemic spreading between two connected communities

When p > 0, the situationwill be totally changed. It is even possible for the case of λ < λc to sustain an epidemic becauseof the jump. Let us analyze it in detail. Suppose the contagion rate λ for both the community A1 and the community A2 arethe same. The epidemic will survive in both A1 and A2 when λ > λc2 and die when λ < λc1, which are the trivial cases. Here

J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236 1231

we focus on the nontrivial case, i.e., λc1 < λ < λc2. Hence, A1 is the infected community and A2 is the safe community. Wewonder if the jump can sustain a non-zero epidemic in the safe community A2.Considering that each agent has the probability p to jump to another community, we divide each time step into two

sub-steps: infection and jumping. The infection process occurs in the sub-step of infection and then both the susceptibleand infected agents jump to another community in the sub-step of jumping. Thus, the total number of infected agents justbefore the time step t + 1 is

I ′1(t) = I1(t)(1− p)+ I2(t)p,

I ′2(t) = I2(t)(1− p)+ I1(t)p, (3)

where I1 and I2 represent the numbers of infected agents in A1 and A2, respectively. Eq. (3) indicates that there are I ′1(t)infected agents in A1 and I ′2(t) infected agents in A2 at time step t . In order to learn how a susceptible agent is infected,we take the case of A1 as an example. For each susceptible agent in A1, the probability for an infected agent to contact thesusceptible one isπr2/L21, which is assigned as q1. It can be easily inferred that the probability form infected agents to contact

the susceptible agent at the same time follows a binomial distribution, i.e.(I ′1(t)m

)qm1 (1− q1)

I ′1(t)−m. Then, the probability for

a susceptive agent to be infected will be∑I ′1(t)m=0

(I ′1(t)m

)qm1 (1− q1)

I ′1(t)−m(1− (1− λ)m). We have the same result for the caseof A2. Thus, the number of infected agents at the beginning of time step t + 1 in A1 and A2 are

I1(t + 1) = (N − I ′1(t))I ′1(t)∑m=0

(I ′1(t)m

)qm1 (1− q1)

I ′1(t)−m(1− (1− λ)m),

I2(t + 1) = (N − I ′2(t))I ′2(t)∑m=0

(I ′2(t)m

)qm2 (1− q2)

I ′2(t)−m(1− (1− λ)m), (4)

where q2 = πr2/L22. Eqs. (3) and (4) are based on a mean-field approximation, their correctness can be checked on thestatistical meaning. For a given λc1 < λ < λc2 and initial nonzero I1(0) and I2(0), Eq. (4) will go to a fixed point solution I∗1and I∗2 after the transient process. Of course, this solution depends on the jumping probability p. By letting I

∗

2 (t + 1) = I∗

2 (t)we have

I∗2 = (N − I∗

1p− I∗

2 (1− p))I∗2 (1−p)+I

∗1 p∑

m=0

(I∗2 (1− p)+ I

∗

1pm

)qm2 (1− q2)

I∗2 (1−p)+I∗1 p−m(1− (1− λ)m). (5)

In numerical simulations, we determine I∗1 and I∗

2 by checking the evolution of I1(t) and I2(t) and taking their stabilizedvalues. Then we take average on a number of realizations. For example, Fig. 3 shows the results for λ = 0.1 wherethe symbols represent the results of numerical simulations and the lines the results from Eq. (4). The inset is the ratiobetween the infectors of the two communities where the circles are from the numerical simulation and the lines from thesolution of Eq. (4). Obviously, the numerical simulations are consistent with the theoretical prediction very well. Actually,the monotonous increase of I∗2 with p can be seen more clearly. For the situation of small p and λ, the relation between I

∗

2and p can be approximately obtained by only keeping the first order of (1− λ)k as follows

I∗2 =λ〈k〉2I∗1p

1− λ〈k〉2(1− p), (6)

where 〈k〉2 = πr2N/L22 = q2N denotes the average degree in A2. This is also the relationship between I∗

1 and I∗

2 .How can the jumping agents induce the safe community A2 to sustain the epidemic? For understanding its mechanism,

let us analyze the function of jumping agents in detail. Without jumping agents, the initial seeds in A2 will die after thetransient time. While due to jumping agents, after the transient time, the infected agents I2(t + 1) comes partly from thejumping part I1(t)p and partly from the remaining part I2(t)(1−p)which is also determined by the previous jumping parts.Therefore, I2(t + 1) is completely caused by the jumping agents which includes both the part at time t and the part beforet . Notice that the slope of I∗2 in Fig. 3 decreases with p, indicating that I

∗

2 does not increase linearly with p. For the sake ofsimplicity, this point can be also seen from Eq. (6) where the slope

dI∗2dp=

λ〈k〉2I∗11− λ〈k〉2(1− p)

(1−

I∗2I∗1

)(7)

decreases with p, indicating the influence of p in the denominator of Eq. (6) makes the slope decrease. Therefore, theefficiency of the jumping agents in A2 to re-produce infectors depends on p. For measuring this efficiency, we introducea parameter

γ = I∗2 − pI∗

1 , (8)

1232 J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236

Fig. 3. Stabilized solution I∗1 and I∗

2 versus the jumping probability p where the symbols represent the results of numerical simulations and the lines theresults from Eq. (4), and the ‘‘circles’’ represent I∗1 and the ‘‘squares’’ I

∗

2 . The inset is the plot of I∗

1 /I∗

2 . The results of numerical simulations are averaged on100 realizations.

Fig. 4. γ versus p for λ = 0.1 and p0 ≈ 0.1 where the symbols represent the results of numerical simulations and the line the result from Eqs. (4) and (8).

which reflects the reproducing ability of the jumping agents. We find that there is an optimal p0, γ increases with p whenp < p0 and decreases with p when p > p0. Fig. 4 shows the result of λ = 0.1 and p0 ≈ 0.1. The decrease of γ for p > p0comes from the fact that larger pmakesmore infected I2 and less susceptibleN− I2, which results in that the infected agentsdo not have sufficient neighbors to be infected and thus reduce the reproducing ability. This parameter describes the extrainfectors induced by the original ones, thus itmay be useful in some potential applications. Take a hostile attack for example,how to organize the input of infectors to make the most extra victims with lowest cost, this parameter can tell the answer.Another factor which influences the infected number I∗2 in the safe community A2 is the average degree 〈k〉2, which is

determined by both the density of agents and the interaction radius r . The density can be reflected by the size L2 for a fixedN . In order to observe clearly, a simplified form of the ratio of I∗1 to I

∗

2 is gotten from Eq. (6):

I∗1I∗2=

1λ〈k〉2p

−1− pp

=L22

λπr2Np−1− pp

. (9)

It shows that for a fixed p, the ratio I∗1/I∗

2 is inverse proportional to the average degree of the safe community 〈k〉2,i.e., proportional to L22 and inverse proportional to r

2. For keeping λ = 0.1 in between [λc1, λc2], we let L2 increase from 1to 2.4 for r = 0.05 and let r change from 0.03 to 0.05 for L2 = 1. Our numerical simulations have confirmed the predictionEq. (9), see Fig. 5 for three typical p = 0.025, 0.05 and 0.1 where (a) represents the relationship between I∗1/I

∗

2 and L22 and

(b) I∗1/I∗

2 versus 1/r2.

J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236 1233

Fig. 5. How the parameters L2 and r influence the ratio I∗1 /I∗

2 for L1 = 0.5 and λ = 0.1where the lines with ‘‘squares’’, ‘‘circles’’, and ‘‘triangles’’ denote thecases of p = 0.025, 0.05 and 0.1, respectively, and (a) represents the relationship between I∗1 /I

∗

2 and L22 for r = 0.05 and (b) I

∗

1 /I∗

2 versus 1/r2 for L2 = 1.

Fig. 6. Schematic illustration of the epidemic spreading in a model of three communities, see text for the detailed explanation.

4. Epidemic spreading through an indirect contagion

After learning the mechanism of epidemic spreading in a model of two communities, let us move to the model of threecommunities, i.e., epidemic spreading through an indirect contagion. This situation occurs very often in reality. For example,when an epidemic outbreaks in a community, its surrounding communitieswill usually remove the linkswith it to keep theirsafety. While for those communities who do not connect directly to the infected one, they will not remove the connectionsto their surroundings. Suppose one neighbor of the infected community does not remove its links between them timely,how is the neighbor’s neighbor influenced by the infected community through an indirect way? To solve this problem, letus construct amodel of three communities as shown in Fig. 6 where A1 denotes the infected community with higher densityof agents, and A2 and A3 denote the safe communities with lower density of agents in the beginning. The jump is allowedbetween A1 and A2 and also between A2 and A3 but forbidden between A1 and A3. For simplicity, all the three communitieshave the same populationN = 1000 and the jumping probabilities p between A1 and A2 and between A2 and A3 are the same.We assume L1 < L2 = L3. Hence the density of agents in A1 is higher than that in A2 and A3, and the epidemic threshold λc1of A1 is smaller than λc2 of A2 and A3. As the discussed model of two communities, here we let the three communities havethe same contagion rate λ and are also interested in the case of λc1 < λ ≤ λc2.Performing a similar analysis on Eqs. (3) and (4) we have

I ′1(t) = I1(t)(1− p)+ I2(t)p,I ′2(t) = I2(t)(1− 2p)+ I1(t)p+ I3(t)p,I ′3(t) = I3(t)(1− p)+ I2(t)p;

I1(t + 1) = (N − I ′1(t))I ′1(t)∑m=0

(I ′1(t)m

)qm1 (1− q1)

I ′1(t)−m(1− (1− λ)m),

I2(t + 1) = (N − I ′2(t))I ′2(t)∑m=0

(I ′2(t)m

)qm2 (1− q2)

I ′2(t)−m(1− (1− λ)m),

I3(t + 1) = (N − I ′1(t))I ′3(t)∑m=0

(I ′3(t)m

)qm3 (1− q3)

I ′3(t)−m(1− (1− λ)m), (10)

1234 J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236

Fig. 7. Stabilized solution I∗1 , I∗

2 and I∗

3 versus the jumping probability p where the symbols represent the results of numerical simulations and the linesthe results from Eq. (10), and the ‘‘circles’’, ‘‘triangles’’, and ‘‘stars’’ denote I∗1 , I

∗

2 , and I∗

3 , respectively. The results of numerical simulations are averaged on100 realizations.

where q3 = πr2/L23. Starting from an initial condition I1(0) > 0 and I2(0) = I3(0) = 0, we may get the stabilized solutionI∗1 , I

∗

2 , and I∗

3 of Eq. (10) after the transient process. For small p and λ, a simplified form of I∗

2 can be arrived as

I∗2 =λ〈k〉2p(I∗1 + I

∗

3 )

1− λ〈k〉2(1− 2p)

≈λ〈k〉2pI∗1

1− λ〈k〉2(1− 2p), (11)

where I∗3 is neglected because of I∗

3/I∗

1 � 1. Similarly, a simplified form of I∗

3 can be written as

I∗3 =λ〈k〉3pI∗2

1− λ〈k〉3(1− p)

=λ2〈k〉2〈k〉3p2I∗1

(1− λ〈k〉2(1− 2p))(1− λ〈k〉3(1− p)), (12)

where, 〈k〉2 and 〈k〉3 denote the average degree in A2 and A3, respectively. Eq. (12) shows that I∗3 increases monotonouslywith p. By calculating the slope of I∗3 we find that for p � 1, dI∗3/dp increases first and then decreases with p, which isdifferent from the monotonous decrease in Eq. (7). Our numerical simulations have confirmed this prediction. For fixedparameters L1 = 0.5, L2 = L3 = 1, r = 0.05, and λ = 0.1, Fig. 7 shows how I∗1 , I

∗

2 , I∗

3 change with p where the symbolsrepresent the results of numerical simulations, the lines indicate the results from Eq. (10), and the ‘‘circles’’, ‘‘triangles’’, and‘‘stars’’ denote I∗1 , I

∗

2 , and I∗

3 , respectively.For understanding how the density of agents and the interaction radius influence the infectors I∗3 , from Eq. (12) we have

I∗1I∗3=

1λ2p2〈k〉23

−2− 3pλp2〈k〉3

+(1− p)(1− 2p)

p2

=L43

λ2p2π2r4N2−(2− 3p)L23λp2πr2N

+(1− p)(1− 2p)

p2, (13)

where 〈k〉2 = 〈k〉3 is applied. It shows that for a fixed p, the ratio I∗1/I∗

3 is inverse proportional to the product of degrees ofthe two communities 〈k〉2〈k〉3, i.e., approximately proportional to L43 and inverse proportional to r

4. For keeping λ = 0.1 inbetween [λc1, λc2], we let L2 increase from 1 to 1.3 for r = 0.05 and let r change from 0.03 to 0.05 for L2 = L3 = 1. Ournumerical simulations have confirmed the prediction Eq. (13), see Fig. 8 for three typical p = 0.05, 0.075 and 0.1 where (a)represents the relationship between I∗1/I

∗

3 and L42 and (b) I

∗

1/I∗

3 versus 1/r4.

5. Discussions and conclusions

The surviving of epidemic in an SISmodel comes from the fact that the contagion rate λ is larger than its critical thresholdλc . The epidemic will die when λ < λc . Therefore, the number of infected agents cannot grow in the system with λ < λc .This resultmay lead amisunderstanding that the system A2withλ < λc2will naturally recover to the susceptible statewhenit is contacted by an infected community A1. Furthermore, A3, i.e., the neighbor of A2, will be more safe if A3 also satisfies

J. Zhou, Z. Liu / Physica A 388 (2009) 1228–1236 1235

Fig. 8. How the parameters L3 and r influence the ratio I∗1 /I∗

3 for L1 = 0.5 and λ = 0.1 where the ‘‘squares’’, ‘‘circles’’, and ‘‘triangles’’ denote the cases ofp = 0.05, 0.075 and 0.1, respectively, andwhere (a) represents the relationship between I∗1 /I

∗

3 and L43 for r = 0.05 and (b) I

∗

1 /I∗

3 versus 1/r2 for L2 = L3 = 1.

λ < λc3. However, our results show that it is possible to make A2 and then A3 to sustain an epidemic provided that λ islarger than the critical threshold λc1 of A1. This result may be helpful to understand the deep insight of epidemic spreadingin social networks.In conclusion, we have discussed the epidemic spreading in a communitymodel withmobile agents.We present amodel

to study themechanismof epidemic spreading in both two and three dynamical communities and to showhowa communityis influenced by an infected one which does not contact it directly. A parameter γ is introduced to quantify the efficiencyof jumping agents to reproduce infectors. We find that for two connected communities with different densities of agents, itis possible to sustain an epidemic spreading even when the contagion rate λ is smaller than its critical threshold, which isimpossible for an isolated community. While for the case of three communities, the safety of the third community cannotbe guaranteed even it does not directly contact the infected community. Our theoretical analysis and numerical simulationsmay be helpful in evaluating the cost for a community to contact the outside once an epidemic outbreaks.

Acknowledgements

This work was supported by the NNSF of China under Grant No. 10475027 and No. 10635040, by SPS under GrantNo. 05SG27 and NCET-05-0424, by Program for Changjiang Scholars and Innovative Research Team and by Ph.D. ProgramScholarship Fund of ECNU 2007.

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