SIS Epidemic Spreading with Correlated Heterogeneous Infection Rates
aa r X i v : . [ phy s i c s . s o c - ph ] A ug SIS Epidemic Spreading with Correlated Heterogeneous InfectionRates
Bo Qu a,1, , Huiijuan Wang a a Delft University of Technology
Abstract
The epidemic spreading has been widely studied in homogeneous cases, where each node mayget infected by an infected neighbor with the same rate. However, the infection rate between apair of nodes, which may depend on e.g. their interaction frequency, is usually heterogeneousand even correlated with their nodal degrees in the contact network. In this paper, we aim tounderstand how such correlated heterogeneous infection rates influence the epidemic spreadingon different network topologies. Motivated by real-world datasets, we propose a correlatedheterogeneous Susceptible-Infected-Susceptible (CSIS) model which assumes that the infectionrate β ij (= β ji ) between node i and j is correlated with the degree of the two end nodes: β ij = c ( d i d j ) α , where α indicates the strength of the correlation between the infection rates andnodal degrees, and c is selected so that the average infection rate is 1 in this work. In orderto understand the effect of such correlation on epidemic spreading, we consider as well thecorresponding uncorrected but still heterogeneous infection rate scenario as a reference, wherethe original correlated infection rates in our CSIS model are shuffled and reallocated to thelinks of the same network topology. We compare these two scenarios in the average fraction ofinfected nodes in the metastable state on Erd¨os-R´enyi (ER) and scale-free (SF) networks witha similar average degree. Through the continuous-time simulations, we find that, when therecovery rate is small, the negative correlation is more likely to help the epidemic spread andthe positive correlation prohibit the spreading; as the recovery rate increases to be larger than acritical value, the positive but not negative correlation tends to help the spreading. Our findingsare further analytically proved in a wheel network (one central node connects with each of thenodes in a ring) and validated on real-world networks with correlated heterogeneous interactionfrequencies. Keywords:
SIS model; Epidemic spreading; Complex networks
1. Introduction
Biological, social and communication systems can be represented as networks by consideringthe system components or individuals as nodes and the interactions or relations in between
Email addresses: [email protected] (Bo Qu), [email protected] (Huiijuan Wang)
Preprint submitted to Physica A September 29, 2018 odes as links. Viral spreading models have been used to model processes e.g. the propagationof information and epidemics on such networks or complex systems [1, 2, 3, 4]. The Susceptible-Infected-Susceptible (SIS) model is one of the most studied models. In the SIS model, at anytime t , the state of a node is a Bernoulli random variable, where X i ( t ) = 0 represents that node i is susceptible and X i ( t ) = 1 if it is infected. Each infected node infects each of its susceptibleneighbors with an infection rate β . The infected node can be recovered to be susceptible againwith a recovery rate δ . Both infection and recovery processes are independent Poisson processes.The ratio τ , β/δ is called effective infection rate. When τ is larger than the epidemic threshold τ c , the epidemic spreads out with a nonzero fraction of infected nodes in the metastable state.The average fraction of infected nodes y ∞ in the metastable state, ranging in [0 , y ∞ is, the more severely thenetwork is infected.In the classic SIS model, both the infection and recovery rates are assumed homogeneous,i.e. the infection rates are the same for all infected-susceptible node pairs and the recoveryrates are the same for all infected nodes. However, the infection rates which can be reflectedfrom, for example, the interaction frequencies, between nodes in real-world networks are usuallyheterogeneous and even dependent on the properties of the nodes. For examples, the number offlights between different pairs of airports in a month are different in the airline transportationnetwork, and the number of collaborated papers between different pairs of authors in a yearvary in a co-author network [5, 6]. The interaction freqeuncy is found to be correlated with thenodal degrees in e.g. airline transportation network and metabolic network [5, 7, 6]. Hence, weaim to understand the effect of correlated infection rate on the viral spreading in this work.We propose a correlated heterogeneous SIS (CSIS) model, in which the recovery rates arehomogeneous but the infection rate β ij (= β ji ) between node i and j is correlated with theirdegrees d i and d j in the way: β ij = c ( d i d j ) α (1)where α indicates the strength of the correlation and c is a constant to control the averageinfection rate to 1. The correlation strength α ≈ . α ≈ .
2. Preliminary
In this section, we first introduce the construction of network models and the heterogeneousinfection rates which are considered in our CSIS model. We then introduce the continuous-time simulation which is the main approach in this work. Finally, we introduce the mean-fieldapproximation of the SIS and CSIS model which will be further used in our theoretical analysisof the CSIS model on the wheel network in Section 4.
The scale-free (SF) model has been used to capture the scale-free nature of degree distribu-tion in real-world networks such as the Internet [17] and World Wide Web [18]: Pr[ D = d ] ∼ d − λ , d ∈ [ d min , d max ], where d min is the smallest degree, d max is the degree cutoff, and λ > λ is usually in the range [2 , λ = 2 . d min = 2, the natural degree cutoff d max = ⌊ N / ( λ − ⌋ [20] , and the size N = 10 . Hence, the average degree is approximately 4.The Erd¨os-R´enyi (ER) model [21] has also been taken into account. In an ER randomnetwork with N nodes, each pair of nodes is connected with a probability p independent of3 -6 -5 -4 -3 -2 -1 P r [ B = b ] b -3 P r [ B = b ] b SF, a =0.5SF, a =-0.5 (a) P r [ B = b ] b P r [ B = b ] b ER, a =0.5 ER, a =-0.5 (b) Figure 1: The distribution of the heterogeneous infection rates for (a) SF networks and (b) ERnetworks, where the parameter α = 0 . α = − . D = d ] = (cid:0) N − d (cid:1) p d (1 − p ) N − − d and the average degree is E [ D ] = ( N − p . For large N and constant E [ D ], the degree distribution is Poissonian: Pr[ D = d ] = e − E [ D ] E [ D ] d /d !. Weconsider the ER networks with the size N = 10 and the average degree E [ D ] = 4. Given the network topology, we build two heterogeneous infection-rate scenarios: 1) thecorrelated infection rates and 2) the uncorrelated or shuffled infection rates. In the scenarioof correlated infection rates, we assume that β ij = c ( d i d j ) α where α indicates the correlationstrength. We selected the constant c such that the average infection rate is 1, whereas weconsider different values of homogeneous recovery rates. In this case, the infection rate ofeach link is determined by the given network topology and α . In the scenario of uncorrelatedinfection rates, we shuffle the infection rates from all the links as generated in the first scenarioand redistribute them randomly to all the links. In this way, we keep the distribution of infectionrates but effectively remove the correlation between the infection rates and nodal degrees. Thehomogeneous infection rate is a special case of our heterogeneous infection rate constructionwhere α = 0. Clearly, in a homogeneous network where all the nodes have the same number ofneighbors, the infection rates are homogeneous in both scenarios for any α .As examples, we show the distribution of the heterogeneous infection rates when α = 0 . α = − . α > α < α could not be realistic.4or example, [5, 7, 6] suggest that α is around 0 . . α ≈ .
14 and − .
12 respectively in two real-world dataset as described in Section 5. Hence, wefocus mainly on the range of α ∈ [ − ,
1] in this paper, and discuss the extreme case when theabsolute value of α is large in Section 3.2. In this paper, we perform the continuous-time simulations of the CSIS model on both ERnetworks and SF networks (the heterogeneity of the network topology is thus taken into account)with N = 10000 nodes. Given a network topology, a recovery rate δ and a value of α , we carryout 100 iterations. In each iteration, we construct the network as described in Section 2.1. Wegenerate the heterogeneous infection rates as described in (1) for the scenario of the correlatedinfection rates and shuffle them for the scenario of uncorrelated infection rates. Initially, 10%of the nodes are randomly infected. Then the infection and recovery processes of SIS model aresimulated until the system reaches the metastable state where the fraction of infected nodesis unchanged for a long time. The average fraction y ∞ of infected nodes is obtained over 100iterations for both scenarios of the correlated and uncorrelated infection rates. Moreover, forsimplicity, we use y ∞ ,c and y ∞ ,u to denote the average fraction of infected nodes in the scenariosof correlated and uncorrelated infection rates respectively.A discrete-time simulation could well approximate a continuous process if a small time binto sample the continuous process is selected so that within each time bin, no multiple eventsoccur. However, a heterogeneous SIS model allows different as well large infection or recoveryrates, which requires even smaller time bin size and challenges the precision of a discrete-timesimulation. Hence, instead of performing discrete-time simulations, we further develop thecontinuous-time simulator for our CSIS model, based on the one firstly proposed by van deBovenkamp and described in detail in [22] for homogeneous infection rates. The N-Intertwined Mean-Field Approximation (NIMFA) is an advanced mean-field approx-imation of the SIS model that takes the network topology into account. The governing equationfor a node i in NIMFA for the classic SIS model with the homogeneous infection and recoveryrates is d v i ( t )d t = − δv i ( t ) + β (1 − v i ( t )) N X j =1 a ij v j ( t ) (2)where v i ( t ) is the infection probability of node i at time t , and a ij = 1 or 0 denotes if thereis a link or not between node i and node j . In the steady state, defined by d v i ( t )d t = 0 andlim t →∞ v i ( t ) = v i ∞ , we obtain the infection probability of node i : v i ∞ = 1 −
11 + τ P Nj =1 a ij v j ∞ (3)5he trivial solution of (3) is v i ∞ = 0 for all nodes and indicates the absorbing state which willbe reached after an unrealistically long time for a real network that is not small in size. Beforereaching the absorbing state, there could exist the metastable state, indicated by the nonzerosolution of (3).The Laurent series of the steady-state infection probability is v i ∞ = 1 + ∞ X m =1 η m ( i ) τ − m (4)possesses the coefficients η ( i ) = − d i and η ( i ) = 1 d i (1 − N X j =1 a ij d j )and for m ≥
2, the coefficients obey the recursion η m +1 ( i ) = − d i η m ( i ) − N X j =1 a ij d j ! − d i m X k =2 η m +1 − k ( i ) N X j =1 a ij η k ( j )When the infection rates are heterogeneous, the governing equation becomesd v i ( t )d t = − δv i ( t ) + β ij (1 − v i ( t )) N X j =1 a ij v j ( t ) (5)The infection probability of a node follows v i ∞ = 1 − δδ + P Nj =1 β ij a ij v j ∞ (6)
3. Effect on the average fraction y ∞ of infected nodes As mentioned, the average fraction y ∞ of infected nodes in the metastable state indicateshow severe the network is infected. In this section, we explore how the average fraction y ∞ ofthe infected nodes depends on the parameter α in both of the two scenarios: correlated anduncorrelated infection rates. We mainly consider the influence of the correlation between theinfection rates and nodal degrees on epidemic spreading when the recovery rate varies and theabsolute value of α is in the range [ − , α is much larger. In this case, the influence of thecorrelation is independent from the value of the recovery rate. α ∈ [ − , δ and the parameter α vary. To support our explanations,6e then define an intermediate quantity and illustrate the effect of the correlation when therecovery rate is small and large respectively.The average fraction y ∞ of infected nodes as a function of the scale parameter α in bothER and SF networks are shown in Fig. 2. We employ different values of the recovery rate δ to illustrate the influence of the correlation on the epidemic spreading over different range ofrecovery rates, i.e. different prevalence of the epidemic.Our previous work [8] explored the SIS model with i.i.d. heterogeneous infection rates andshowed that the average fraction y ∞ of infected nodes tends to decrease as the variance of thei.i.d. infection rates increases. In this paper, the uncorrelated infection rates can be consideredas being independent. In this case, our previous results can be applied to derive the relationshipbetween the average fraction y ∞ of infected nodes and the parameter α as shown in Fig. 2: as α increases when α > α decreases when α <
0, the variance of the infection ratesincreases, hence y ∞ decreases in the scenario of uncorrelated infection rates. In the scenario ofcorrelated infection rates, though the peaks of y ∞ are not at α = 0 for both types of networks, y ∞ roughly decreases as the absolute value of α increases. This is because, as the absolutevalue of α increases, large infection rates are assigned to a small number of links, limiting thespread of the epidemic.From Fig. 2, we find that 1) the negative correlation ( α <
0) between the infection rates andthe degrees tends to enhance the epidemic spreading compared to the uncorrelated infection-rate scenario ( y ∞ ,c > y ∞ ,u ) when the recovery is small, but prohibit the spreading ( y ∞ ,c < y ∞ ,u )when the recovery rate is large; 2) the positive correlation ( α >
0) tends to enhance the epidemicspreading ( y ∞ ,c > y ∞ ,u ) when the recovery is large, but prohibit the spreading ( y ∞ ,c < y ∞ ,u )when the recovery rate is small.In the scenario of uncorrelated infection rates, the parameter α determines only the dis-tribution of the i.i.d. infection rates in a network, and the infection rate between any pair ofnodes is independent from their degrees. Compared to the scenario of uncorrelated infectionrates, a positive correlation between the infection rates and nodal degrees ensures that theinfection rates between nodes with larger degrees are also larger, whereas a negative correlationsuggests the other way around. Intuitively, we may think that if the nodes with larger degreescan be infected by larger infection rates, the infection probability of those nodes are higher andthose nodes can more effectively infect their neighbors as well. Hence, the positive correlationbetween the infection rates and nodal degrees seems to contribute to the epidemic spreading.This is indeed the case when the recovery rate δ is large, i.e. the prevalence of the epidemic islow.In contrast to this intuition, we observe that the positive correlation actually tends toprohibit the epidemic spreading when the recovery rate δ is small. This can be explained asfollows: when the recovery rate δ is small, i.e. the prevalence is high, the infection probabilitiesof the large-degree nodes are already high, then the increment of the infection rates between thelarge-degree nodes may not significantly increase the infection probabilities of these nodes, and7 .80.70.60.50.4 y -1.0 -0.5 0.0 0.5 1.0 a SF, d =0.5 Correlated Uncorrelated 8 (a) y -1.0 -0.5 0.0 0.5 1.0 a ER, d =0.5 Correlated Uncorrelated 8 (b) y -1.0 -0.5 0.0 0.5 1.0 a SF, d =2 Correlated Uncorrelated 8 (c) y -2 -1 0 1 2 a
8 ER, d =1 Correlated Uncorrelated (d) y -1.0 -0.5 0.0 0.5 1.0 a SF, d =5 Correlated Uncorrelated 8 (e) y -1.0 -0.5 0.0 0.5 1.0 a ER, d =2 Correlated Uncorrelated 8 (f) Figure 2: The average fraction y ∞ as a function of α for (a) SF networks with the recovery rate δ = 0 .
5, (b) ER networks with the recovery rate δ = 0 .
5, (c) SF networks with the recoveryrate δ = 2, (d) ER networks with the recovery rate δ = 1, (e) SF networks with the recoveryrate δ = 5 and (f) ER networks with the recovery rate δ = 2 in both scenarios of correlated ( ◦ )and uncorrelated ( (cid:3) ) infection rates.thus the infection probabilities of their neighbors may not be significantly increased. However,the negative correlation between the infection rates and nodal degrees leads to the higher8nfection rates between the small-degree nodes and effectively enhances the probabilities of thesmall-degree nodes compared to the scenario of the uncorrelated infection rates. Though theinfection probabilities of large-degree nodes decrease in this case, the large amount of small-degree nodes ensures that the overall infection is on average enhanced.To support our explanations, we define y ( d ) ∞ as the average infection probability of the nodeswith degree d , and y ( d ) ∞ ,c and y ( d ) ∞ ,u as that for the scenario of correlated infection rates anduncorrelated infection rates respectively. We show y ( d ) ∞ as a function of the degree d whenthe recovery rate is small, δ = 0 .
5, and the correlation parameter α = − . α = − .
6) between the infection ratesand nodal degrees indeed decreases the infection probabilities of large-degree nodes, but theinfection probabilities of the small-degree nodes are also significantly lifted. Furthermore, thenumber of small-degree nodes is much larger than that of large-degree nodes in SF networks,and those are similar in ER networks. To illustrate the combined effect above of the twoaspects, we define η ( d ) as (7), the product of the probability that a node has the degree d andthe difference between the average infection probability of the nodes with the degree d in thescenarios of correlated and uncorrelated infection rates: η ( d ) = ( y ( d ) ∞ ,c − y ( d ) ∞ ,u )Pr[ D = d ] (7)and y ∞ ,c − y ∞ ,u = P d = N − d =1 η ( d ). Note that a positive η ( d ) always indicates that y ∞ ,c > y ∞ ,u ,i.e. the correlation lifts the infection probability of nodes with the degree d compared to thescenario of the uncorrelated infection rates. As in the insets of Fig. 3, we plot η ( d ) as a functionof the degree d for both ER and SF networks. More plots of η as a function of the degree d are shown in Fig. 4 where the cases with small degrees are shown in the main figures andthose with relatively large degrees are shown in the insets. We find that in both networks thevalue of η is significantly large for the small-degree nodes and contributes more to a higherprevalence of the epidemic when the recovery rate is small and the correlation is negative asshown in Fig. 3, Fig. 4(c) and Fig. 4(d). The observation is consistent with our explanationabout why the negative correlation tends to help the epidemic spreading when the recoveryrate is small. In contrast, the observation, shown in Fig. 4(a) and Fig. 4(b), that the positivecorrelation does increase the infection probabilities of large-degree nodes but decreases those ofsmall-degree nodes more when the recovery rate is small, also supports our explanation abouthow the positive correlation prohibits the spreading when recovery rate is small.As the recovery rate becomes large thus the prevalence is low, the positive but not negativecorrelation, between the infection rates and nodal degrees may effectively enhance the infectionprobabilities of the large degree nodes comparing to the uncorrelated infection rates. When We still select α = − . α = 0 . α = − . α = 0 . δ = 5 and α = − . .950.900.850.800.750.70 y d -3 h d ( d ) SF, d =0.5, a =-0.6 Correlated Uncorrelated (a) y d -3 h d ER, d =0.5, a =-0.6 Correlated Uncorrelated ( d ) (b) Figure 3: In the main figures: the average infection probability y ( d ) ∞ of the nodes with degree d as a function of the degree d for (a) SF networks and (b) ER networks with the recovery rate δ = 2 and α = − . η as a function of d with the same setting as the main figure.the recovery rate δ is large, for example, δ = 5 for SF networks and δ = 2 for ER networks inthis paper, the positive correlation leads to the increment of the infection probabilities of large-degree nodes and could further lift the probabilities of their small-degree neighbors which arelarge in number. The infection probabilities of small-degree nodes are reduced, but the infectionprobabilities are already low and the small-degree nodes have few neighbor to infect. Hence,the overall infection increases on average. The explanations are supported by Fig. 4(a) and Fig.4(b). The infection probabilities of the large-degree nodes increases for different recovery rateswhen α is positive, and the increment of the infection probabilities of large-degree nodes is alsolarger as the recovery rate increases. Thought the infection probabilities of the small-degreenodes may decrease when the correlation is negative, the increment of the infection probabilitiesof the large-degree nodes also lifts the infection probabilities of the small-degree nodes. As aresult, the infection probabilities of majority nodes are on average lifted when the recovery rate δ = 5 and δ = 2 for SF and ER networks respectively.For both ER and SF networks, the positive correlation between the nodal degrees and theinfection rates tends to enhance the spreading when the recovery rate is small, whereas thenegative correlation tends to help when the recovery rate is large. As the recovery rate δ increases from 0 and if the absolute value of α is small, we expect that there is a critical value δ c : when δ < δ c the negative correlation tends to enhance the spreading, otherwise ( δ > δ c ) thepositive correlation is likely to help the spreading. By the comparing Fig. 2(c) and Fig. 2(f),we can observe that δ c is larger in SF networks than ER networks. This difference is mainlycaused by that the prevalence in SF networks tends to be higher than that in ER networkswhen the recovery rate and the parameter α are the same and the positive correlation tends toenhance the epidemic spreading when the prevalence is low as we discussed.10 -3 -25-20-15-10-50 h d -6 h d SF, a =0.2 d =0.5 d =2 d =5 (a) -15x10 -3 -10-505 h d -3 h d ER, a =0.6 d =0.5 d =1 d =2 (b) -3 h d -800x10 -6 -600-400-200 h d SF, a =-0.2 d =0.5 d =2 d =5 (c) -3 h d -4x10 -3 -3-2-10 h d ER, a =-0.6 d =0.5 d =1 d =2 (d) Figure 4: The plot of η as a function of α for (a) SF networks and (b) ER networks withdifferent recovery rates. 11 .2. Extreme cases We then discuss the influence of the correlation between the infection rates and the nodaldegrees when the correlation is strong, i.e. the absolute value of α is large.When the absolute value of α is large, the variance of the infection rates is large as well. Inthis case, most links possess a small infection rate and few have a large infection rate. A largeproportion of the links have such a small infection rate that the infection processes driven by thesmall infection rate will hardly happen. The networks are actually filtered by the small infectionrates. The other small proportion of the large infection rates then mainly determine the overallinfection. In the scenario of the uncorrelated infection rates, the few large infection rates arerandomly distributed, thus cannot form a connected cluster. Compared to the scenario of theuncorrelated infection rates, the positive correlation ensures that the large infection rates aredistributed between the large-degree nodes which are more likely to connect with each other,forming a subgraph. A connected subgraph tends to help the spread of an epidemic. In theother way around, the negative correlation will almost surely stop the epidemic spreading sincethe few large infection rates distributed between the small-degree nodes could hardly form sucha connected cluster. In summary, the positive correlation enhances the epidemic spreadingwhereas the negative one prohibits when the absolute value of α is large.
4. The wheel network
In this section, we further consider a special topology – the wheel network. We are going toprove that, compared to the uncorrelated heterogeneous infection rate, the negative correlationbetween the infection rates and the degrees tends to help the epidemic spreading in a large wheelnetwork when the recovery rate is small, whereas the positive correlation tends to contributeto the epidemic spreading when the recovery rate is large.In a wheel network, m side nodes compose a ring, i.e. node i connects with node i + 1( i = 1 , , , ..., m −
1) and node m connects with node 1, and all the m side nodes connectwith one central node – node 0. In this section, we consider a large enough wheel network, andwithout loss of the generality, we still set both the homogeneous infection rate and the averageof heterogeneous infection rates to be 1, and tune the recovery rate.In the scenario of the uncorrelated infection rates, the infection rates are actually i.i.d.In our previous work, we found that, compared to the homogeneous infection rate, the i.i.d.heterogeneous infection rates always reduce the overall infection if the epidemic can spread out.We further verify this conclusion for the wheel network by simulations as shown in Appendix A,where we plot the average fraction of infection nodes as a function of α for the scenario ofuncorrelated infection rates and different values of the recovery rates. That is to say, the averagefraction of infected nodes reaches the maximum when the infection rates are homogeneous, i.e. α = 0 in the scenario of the uncorrelated infection rates. If the average fraction of infectednodes in the scenario of the correlated infection rates is larger than that when the infection12ates are homogeneous, then the correlation enhances the epidemic spreading compared to theuncorrelated case.We first consider the homogeneous infection rate in a wheel network, where the infectionrates are homogeneous thus the same for all links. The infection probability v ∞ of the centralnode is v ∞ = 1 −
11 + mτ v i ∞ = 1 − δδ + mv i ∞ (8)where v i ∞ is the infection probability of a side node, which is the same for all the side nodes.The infection probability of the side node is v i ∞ = 1 −
11 + 2 τ v i ∞ + τ v ∞ = 1 − δδ + 2 v i ∞ + v ∞ (9)By solving (8) and (9), we obtain the positive solution v ∞ = m − δ + 2 δ + δ (cid:16) ( δ − m +2 δ + √ ( δ − δ +9) m − (4 δ +12 δ ) m +4 δ (cid:17) m δ + m (10)and v i ∞ = (1 − δ ) m − δ + p ( δ − δ + 9) m − (4 δ + 12 δ ) m + 4 δ m (11)We consider a large m and a constant δ . In this case, v ∞ ≈ v i ∞ ≈ − δ + √ δ − δ + 94 (12)The fraction of infected nodes is then y ∞ = mv i ∞ + v ∞ m + 1 ≈ v i ∞ Now we consider the wheel network where the infection rates are correlated as definedbefore in (1). There are two kinds of infection rates in a wheel network: 1) the infection rate β between the central node and a side node, i.e. β ∼ (3 m ) α ; 2) the infection rate β betweena pair of connected side nodes, i.e. β ∼ (3 ∗ α . Since we consider a large m , we find β ≫ β if α >
0, whereas β ≪ β if α <
0. When the correlation between the infection rates andthe degree is positive, the infection rates β ≈ β ≈ β ≈ β ≈ when α > y ∞ ≈ v i ∞ = 22 + δ (13)When α <
0, the average fraction of infected nodes y ∞ = v i ∞ = 1 − δ Equation (13) can be similarly derived by solving (3). Equation (14) can be derived by applying the Laurent series as in (4). α = 0 in the scenario of the correlated infection rates indicates the homogeneousinfection rate which is the same as α = 0 in the scenario of the uncorrelated infection rates.We have shown that the average fraction of infected nodes reaches the maximum when α = 0in the scenario of the uncorrelated infection rates, so we further compare the average fractionof infected nodes as shown in (12) when the infection rates are homogeneous i.e. α = 0 withthat as shown in (14) when α < α > δ <
2, the average fraction y ∞ of infected nodes is higher if α < α = 0. Hence, the negative correlation between the infection rates and nodal degrees helpsthe epidemic spreading if the recovery rate is small, i.e. δ <
2. Similarly, if we compare (12)and (13), we find that when the recovery rate δ > α > α = 0, and conclude that the positive correlation between the infectionrates and nodal degrees contributes to the epidemic spreading if the recovery rate is large.The theoretical results are consistent with our previous conclusions: when the recovery rate issmall, the negative correlation tends to help the epidemic spreading, and as the recovery rateincreases to be larger than a critical value, i.e. 2 in this case, the positive correlation enhancesthe spreading.
5. Real-world networks
As mentioned, the interaction frequency β ij between node i and node j in a real-worldnetwork can be considered as the infection rate between them and it has been found that β ij ∼ ( d i d j ) α in many networks. In this section, we choose two real-world networks as examplesto illustrate how their heterogeneous infection rates affect the spread of SIS epidemics on thesenetworks. We compare the average fraction y ∞ of infected nodes in the metastable state ofthe two networks in the two scenario: 1) the scenario of correlated infection rates, where eachnetwork is equipped with its original heterogeneous infection rates (but normalized so that theaverage infection rate is 1) as given in the dataset; 2) the scenario of uncorrelated infectionrates, where each network is equipped with the infection rates (normalized as well) in theoriginal dataset but the infection rates are shuffled and reassigned to each link. Our objectiveis to explore the relation between the infection rates and average fraction of infected nodes inthese 2 scenarios for both networks to verify our previous findings.The first network is the airline network (with 3071 nodes and 15358 links) where the nodesare the airports and the infection rate along a link is the number of flights between the twoairports. We construct this network and its infection rates from the dataset of openFlights .The other one is the co-author network (with 39577 nodes and 175692 links) where the nodesare the authors of papers, and the infection rate is the collaboration frequency depending onthe number of collaborated papers and the number of authors in those papers[23]. http://openflights.org/data.html -4 -3 -2 -1 P r [ D = d ] d Co-author Network Airline Network l = 2.5 l = 1.5 (a) < b i j > d i d j Airline Network a =0.14 (b) < b i j > d i d j Co-author Network a =-0.12 (c) Figure 5: (a) The degree distributions of two real-world networks: the airline and co-authornetworks. (b) The interaction frequency β ij as a function of the nodal degrees d i d j for theairline network. (c) The interaction frequency β ij as a function of the nodal degrees d i d j forthe co-author network.As shown in Fig. 5(a), the degree distributions of the airline network and co-author networkapproximately follow a power law with the slope λ = 1 . . h β ij i as a function of the product of the two nodal degrees d i d j in Fig. 5(b) and 5(c) for the airline and co-author networks respectively. We find that, roughly β ij ∼ ( d i d j ) α with α = 0 .
14 and α = − .
12 for the airline and co-author networks respectively.We first discuss the case when the recovery rate is small, i.e. δ ∈ [0 . ,
8] (as shown inFig. 6), and then the case when δ is large, i.e. δ ≥
15 (as shown in Fig. 7), since for bothnetworks the recovery rate δ ∈ [0 . ,
8] enables the high prevalence of the epidemic and δ ≥ y ∞ of infected nodes as a function of therecovery rate δ for both infection-rate scenarios is shown in in Fig. 6 when the recovery rate δ is small. We find that the positive correlation ( α = 0 .
14) between the infection rates and nodaldegrees in the airline network retards the spread of epidemics, whereas the negative correlation( α = − .
12) in the co-author network contributes in the other way around. The observations15 .70.60.50.40.3 y d Airline Network Correlated Uncorrelated (a) y d Co-author Network Correlated Uncorrelated (b)
Figure 6: The average fraction y ∞ as a function of the recovery rate δ for (a) the airline networkand (b) the co-author network in both scenarios of correlated ( ◦ ) and uncorrelated ( (cid:3) ) infectionrates. The recovery rate is in the range [0 . , y ∞ of infected nodesas a function of the recovery rate δ when the recovery rate is large is shown in Fig. 7. Wefind that the positive correlation α = 0 .
14 in the airline network (Fig. 7(a)) helps the epidemicspreading when the recovery rate is larger than 15, and the negative correlation α = − . δ ≤
35, that the overall infection is close to 0. The observations also agree with ourconclusion that the positive correlation between the infection rates and nodal degrees tends tohelp the epidemic spreading but not the negative correlation when the recovery rate is large.Hence, the simulation results of the CSIS model on the real-world networks for both the smalland large recovery rates agree with our previous conclusions.
6. Conclusion
In this paper, we study how the correlation between the infection rates and nodal degreesinfluences the epidemic spreading, compared to the uncorrelated case. By continuous-timesimulations of our CSIS model in different infection-rate scenarios and networks with differenttopology heterogeneities, we find that, when the recovery rate is small, i.e. the prevalence of theepidemic is high, the negative correlation between the nodal degree and the infection rates tendsto help the epidemic spreading. However, when the prevalence is high, the positive correlationis more likely to enhance the spreading. The validation on two real-world networks and theproof in the large wheel network agree with our conclusions. Our results shed light on that howthe epidemic spreads in the real-world could be far away from the simple classic models. Notonly the heterogeneity of infection rates but also the correlation between the heterogeneity of16 .120.100.080.060.040.020.00 y d Airline Network Correlated Uncorrelated (a) -3 y d Co-author Network Correlated Uncorrelated (b) Figure 7: The average fraction y ∞ as a function of the recovery rate δ for (a) the airline networkand (b) the co-author network in both scenarios of correlated ( ◦ ) and uncorrelated ( (cid:3) ) infectionrates. The recovery rate is large.infection rates and network topologies could be various and complicated. Our work is the firststep to study the correlated heterogeneous SIS model in heterogeneous networks. ReferencesReferences [1] D. J. Daley, J. Gani, J. M. Gani, Epidemic modelling: an introduction, Vol. 15, CambridgeUniversity Press, 2001.[2] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes incomplex networks, arXiv preprint arXiv:1408.2701.[3] R. Pastor-Satorras, A. Vespignani, Epidemics and immunization in scale-free networks,Handbook of graphs and networks: from the genome to the internet (2005) 111–130.[4] R. Pastor-Satorras, A. Vespignani, Epidemic dynamics and endemic states in complexnetworks, Physical Review E 63 (6) (2001) 066117.[5] A. Barrat, M. Barthelemy, R. Pastor-Satorras, A. Vespignani, The architecture of complexweighted networks, Proceedings of the National Academy of Sciences of the United Statesof America 101 (11) (2004) 3747–3752.[6] W. Li, X. Cai, Statistical analysis of airport network of china, Physical Review E 69 (4)(2004) 046106.[7] P. Macdonald, E. Almaas, A.-L. Barab´asi, Minimum spanning trees of weighted scale-freenetworks, EPL (Europhysics Letters) 72 (2) (2005) 308.178] B. Qu, H. Wang, Sis epidemic spreading with heterogeneous infection rates, arXiv preprintarXiv:1506.07293.[9] C. Buono, F. Vazquez, P. Macri, L. Braunstein, Slow epidemic extinction in populationswith heterogeneous infection rates, Physical Review E 88 (2) (2013) 022813.[10] V. M. Preciado, M. Zargham, C. Enyioha, A. Jadbabaie, G. J. Pappas, Optimal resourceallocation for network protection against spreading processes, IEEE T. Contr Syst. T. 1 (1)(2014) 99–108.[11] Z. Yang, T. Zhou, Epidemic spreading in weighted networks: an edge-based mean-fieldsolution, Physical Review E 85 (5) (2012) 056106.[12] X. Fu, M. Small, D. M. Walker, H. Zhang, Epidemic dynamics on scale-free networks withpiecewise linear infectivity and immunization, Phys. Rev. E 77 (3) (2008) 036113.[13] P. Van Mieghem, J. Omic, In-homogeneous virus spread in networks, arXiv preprintarXiv:1306.2588.[14] P. Van Mieghem, The N-intertwined SIS epidemic network model, Computing 93 (2-4)(2011) 147–169.[15] H. Wang, Q. Li, G. DAgostino, S. Havlin, H. E. Stanley, P. Van Mieghem, Effect of theinterconnected network structure on the epidemic threshold, Physical Review E 88 (2)(2013) 022801.[16] F. D. Sahneh, C. Scoglio, P. Van Mieghem, Generalized epidemic mean-field model forspreading processes over multilayer complex networks, IEEE/ACM Transactions on Net-working (TON) 21 (5) (2013) 1609–1620.[17] G. Caldarelli, R. Marchetti, L. Pietronero, The fractal properties of internet, EPL (Euro-physics Letters) 52 (4) (2000) 386.[18] R. Albert, H. Jeong, A.-L. Barab´asi, Internet: Diameter of the world-wide web, Nature401 (6749) (1999) 130–131.[19] A.-L. Barab´asi, R. Albert, Emergence of scaling in random networks, Science 286 (5439)(1999) 509–512.[20] R. Cohen, K. Erez, D. ben Avraham, S. Havlin,Resilience of the internet to random breakdowns, Physical Review Letters 85 (2000)4626–4628. doi:10.1103/PhysRevLett.85.4626 .URL http://link.aps.org/doi/10.1103/PhysRevLett.85.4626 [21] P. Erd˝os, A. R´enyi, On random graphs i., Publ. Math. Debrecen 6 (1959) 290–297.1822] C. Li, R. van de Bovenkamp, P. Van Mieghem, Susceptible-infected-susceptible model: Acomparison of n-intertwined and heterogeneous mean-field approximations, Phys. Rev. E86 (2) (2012) 026116.[23] M. E. Newman, The structure of scientific collaboration networks, Proceedings of theNational Academy of Sciences 98 (2) (2001) 404–409.
Appendix A. The wheel network with i.i.d. infection rates
As shown in Fig. A.8, we plot the average fraction y ∞ of infected nodes as a function of thecorrelation strength parameter α for the wheel network with m = 1000 side nodes in the scenarioof the uncorrelated heterogeneous infection rates. We employ the recovery rates δ = 0 . δ = 2and δ = 10. We find that no matter what value the recovery rate is, the homogeneous infectionrate ( α = 0) always leads to the highest overall infection. y -1.0 -0.5 0.0 0.5 1.0 a x - d =10 (right axis) Wheel d =0.5 d =2 (left axis) Figure A.8: The average fraction y ∞ as a function of the correlation strength parameter α fora wheel network with m = 1000 side nodes in the scenario of the uncorrelated infection rates.The recovery rate is δ = 0 . (cid:13) ), δ = 2 ( (cid:3) ) and δ = 10 ( ▽▽