Two-pathogen model with competition on clustered networks
Peter Mann, V. Anne Smith, John B. O. Mitchell, Simon Dobson
TTwo-pathogen model with competition on clustered networks
Peter Mann, ∗ V. Anne Smith, John B.O. Mitchell, and Simon Dobson
School of Computer Science, University of St Andrews, St Andrews, Fife KY16 9SX, United KingdomSchool of Chemistry, University of St Andrews, St Andrews, Fife KY16 9ST, United Kingdom andSchool of Biology, University of St Andrews, St Andrews, Fife KY16 9TH, United Kingdom (Dated: July 8, 2020)Networks provide a mathematically rich framework to represent social contacts sufficient for thetransmission of disease. Social networks are often highly clustered and fail to be locally tree-like. Inthis paper, we study the effects of clustering on the spread of sequential strains of a pathogen usingthe generating function formulation under a complete cross immunity coupling. We derive conditionsfor the epidemic threshold of the first strain and the threshold of coexistence of the second strain.We find that clustering has a dual affect on the first strain, reducing the epidemic threshold butalso decreasing the final outbreak size at large transmissibilities. Clustering reduces the coexistencethreshold of the second strain and its outbreak size. We apply our model to the study of multilayerclustered networks and observe the fracturing of the residual graph experimentally.
I. INTRODUCTION
Complex networks can be found across many differentareas of biology, medicine, the physical and computer sci-ences. Each network, empirical or synthetic, has a richand characteristic structure that exhibits large-scale net-work properties from local interactions. Amongst theseapplications, complex networks have proven to be excel-lent models of social networks. The nodes of the graphrepresent individuals while the edges that connect themrepresent points of contact.Of primary importance is the study of diseases spread-ing among the nodes transmitted through their contacts[1–4]. The nature of the disease transmission mechanismwill determine the topology of the contact network. Forinstance, it is expected that sexually transmitted diseaseswould spread among a population in quite a distinct man-ner to an airborne pathogen. Understanding how thestructure of the contact topology impacts the dynamicsof the disease is of the utmost importance for controlling,containing and mitigating the spread of the pathogen.The threat of a novel mutant strain invading an equi-librated system that has already experienced a diseaseis observed both in viral and bacterial dynamics. Wheninfection by one strain imparts perfect cross immunity tothe other strain, complex competition dynamics can leadto non-trivial threshold behaviour for each pathogen. Inthis limit, those nodes infected by the first strain cannotbe infected by the second. This case was first studiedby Newman [5, 6] for tree-like edges. Of particular inter-est is the threshold for coexistence of both strains in thenetwork.Perhaps the most fundamental network model is theErd¨os-R´enyi random graph, a member of the exponentialrandom graph ensemble with a constraint on the num-ber of edges within a given realisation. Random graphsare well studied within the network science communityusing a variety of mathematical tools. One such theoreti-cal framework, the so-called generating function formula-tion [7], has excellent ability to extract the properties ofdiseases, such as the number of individuals who become infected, spreading over such networks. This is achievedby an isomorphism between the spreading pathogen andthe bond percolation process. The latter, a model thattraces its roots to statistical mechanics, examines theprobability that each edge in the network transmits thedisease between two neighbours with transmission prob-ability T ∈ [0 , − T . Wecall edges that transmit the disease occupied , while thosethat do not are said to be unoccupied . Once all edgeshave been considered, the network may no longer be wellconnected by the occupied edges. Within the context ofthe isomorphism, the size of the giant connected compo-nent (GCC) among occupied edges represents the frac-tion of the network that becomes infected by the disease.The expectation value of the GCC experiences a second-order phase transition as a function of T at some criticalvalue, T , c , known as the epidemic threshold. Prior to thethreshold, there is no GCC and only small componentsare connected.Social networks tend to contain a high density of tri-angles; connections between the neighbours of a node,also known as transitivity or clustering. Many mathe-matical models fail to describe the impact of clustering,which is well known to alter the properties of both bondpercolation and the epidemic outbreaks of a single dis-ease. Specifically, it can be shown that clustering reducesthe epidemic threshold for the disease to infect a finitefraction of the network as well as reducing the overalloutbreak size [8] for fixed mean degree. Miller [9, 10]conversely showed that clustering can also increase thethreshold when non-assortative networks are studied, aresult supported by [11].Clustering has been well studied in the context of thegenerating function formulation for a single strain; itrequires a generalisation of the generating function for-mulation to partition edges into distinct topological sets[8, 9]. The random clustered graphs we consider here arebuilt using the generalised configuration model [12–14].In this model, a vector of edge-topologies, τ , is defined;the simplest model consists of tree-like edges, denoted by ⊥ and triangles, denoted by ∆, such that τ = {⊥ , ∆ } . a r X i v : . [ phy s i c s . s o c - ph ] J u l Each node is assigned a stub-degree, k τ , for each topol-ogy in the topology set, τ ∈ τ . For instance, a nodeinvolved in 3 tree-like edges and 1 triangle has k ⊥ = 3and k ∆ = 2 and it should be clear that { k ∆ = 0 mod 2 } .During the network construction, the stubs are connectedtogether to create a random graph whose edge topologiesare distributed according to the assigned stub-degree.It is not clear, however, precisely how clustering im-pacts the spread of two cross immune pathogens spread-ing sequentially over a network. The subject has beenstudied before using percolation in the context of cliquerandom networks whereby each strain spreads on a par-ticular edge topology [15]. In this paper, we study theinfluence of clustering on the outbreak size of two sequen-tial pathogens spreading with a perfect cross-immunecoupling on a random clustered network. We will theninvestigate a method to consider these as simultaneousstrains that compete for hosts contemporaneously. Inthis instance, each pathogen has a mutual mitigating ef-fect on the spreading of the other and hence we use Gille-spie stochastic simulation to experiment this and com-pare to our model. II. SEQUENTIAL STRAIN MODEL WITHCLUSTERING
In this section, we introduce a two-strain model onclustered networks containing triangles in addition to thetree-like degrees. The second strain is assumed to tempo-rally separated from the first such as seasonal influenzaoutbreaks or a rare mutation in an equilibrated bacterialpopulation.
A. Strain-1
The generating function formulation [1, 7] rests uponthe degree distribution, p ( k ), the probability of choosinga node at random from the network of degree k . Whenthe network contains triangles, we introduce the joint de-gree distribution, p ( k ⊥ , k ∆ ), the probability of choosing anode at random from the network with k ⊥ tree-like edgesand k ∆ / p ( k ) from the jointdegree sequence as p ( k ) = ∞ (cid:88) k ⊥ =0 ∞ (cid:88) k ∆ =0 p ( k ⊥ , k ∆ ) δ k,k ⊥ + k ∆ (1)The joint probability distribution is generated by G ( z ⊥ , z ∆ ) = ∞ (cid:88) k ⊥ =0 ∞ (cid:88) k ∆ =0 p ( k ⊥ , k ∆ ) z ⊥ k ⊥ z ∆ k ∆ / (2)The probability of reaching a node of joint degree( k ⊥ , k ∆ ) by following a random tree-like edge back toa node is generated by G , ⊥ ( z ⊥ , z ∆ ) = 1 (cid:104) k ⊥ (cid:105) ∂G ∂z ⊥ (3) FIG. 1. The three triangles that a focal node may be con-nected to. (A) The focal node has two uninfected neighbours(green), neither of which are capable of transmitting infec-tion. (B) Both nodes are infected (red), but each direct edgefails to infect the focal node. (C) Only one neighbour is in-fected; however, it can infect the focal node by first infectingthe susceptible neighbour and then a further transmission tothe focal node.
Similarly, the degree of the node reached by following arandom triangle edge to a node is G , ∆ ( z ⊥ , z ∆ ) = 1 (cid:104) k ∆ (cid:105) ∂G ∂z ∆ (4)In each case, (cid:104) k τ (cid:105) is the average τ -degree of a node whichis given by ∂ z τ G (1 , C is a metric that indicatesthe level of clustering in the network [8, 16]. It is givenby the following quotient C = 3 N ∆ N (5)where N ∆ is the number of triangles and N is the num-ber of connected triples. In terms of the above generatingfunctions and network size N , we have3 N ∆ = N (cid:18) ∂G ∂z ∆ (cid:19) (6) N = 12 N ∞ (cid:88) k =0 (cid:18) k (cid:19) p k (7)The probability that a node does not become infectedthrough its involvement in a tree-like edge (triangle) is g ⊥ ( g ∆ ). Each g τ is a function of u τ , the probabilitythat a neighbour is uninfected in a τ -site. These expres-sions are well-known for both tree-like and triangle edgetopologies. We construct g ⊥ ( u ⊥ ; T ) by summing the in-dependent probabilities that a given tree-like edge fails toinfect the focal node; this is either because the neighbour-ing node was uninfected by the disease with probability u ⊥ , or that it was infected but failed to transmit the dis-ease to the focal node with probability (1 − u ⊥ )(1 − T ).Together we have g ⊥ ( u ⊥ ; T ) = u ⊥ + (1 − u ⊥ )(1 − T ) (8)The g ∆ ( u ∆ ; T ) expression is slightly more complex toconsider due to the inter-neighbour connecting edge. Fora node that has η ∆ triangles (and therefore has triangledegree k ∆ = 2 η ∆ ), there are three ways to consider thefailure to infect the focal node as in Fig 1.Firstly, both neighbours can themselves be uninfectedwith probability u . Similarly, both neighbours couldhave been infected but both failed to transmit theirinfection to the focal node directly with probability[(1 − u ∆ )(1 − T )] ; in this case the inter-neighbour edge has no consequence on the final state of the focal node.However, in the case that one neighbour is infected, failsto transmit directly to the focal node and the othernode is initially uninfected (the probability of which is u ∆ (1 − u ∆ )(1 − T )), then the inter-neighbour edge canbe an avenue of infection back to the focal node. Thisfails to occur is 1 − T . Allowing there to be η ∆ trianglesaround the focal node we have g ∆ ( u ∆ ; T ) = (cid:18) η ∆ l (cid:19) [ u ] l (cid:18) η ∆ − lm (cid:19) [((1 − u ∆ )(1 − T )) ] m [2 u ∆ (1 − u ∆ )(1 − T )(1 − T )] η ∆ − l − m (9)The multiplication by two in the final term due to thesymmetry of the triangle. Each square bracket containsthe probability that the focal node remains uninfected inthe particular triangle it is considered to be a part of.To solve for the expected fraction of the network thatcontracts strain-1, S , we use fixed-point iteration to findeach u τ value as the solution to a self-consistent func-tional equation in u τ u τ = G ,τ ( g ⊥ , g ∆ ) (10)each equation converging on a solution in the unit inter-val. With these values, S can be found by solving S [ u ⊥ , u ∆ ; T ] = 1 − G ( g ⊥ , g ∆ ) (11)where the square brackets indicate the functional depen-dency of the GCC on u τ and the disease transmissionparameter, T . B. Strain-2
Once the first strain has passed through the network, afraction, S , of the nodes will have contracted it and con-sequently a fraction, 1 − S , remained uninfected. In thecase that nodes infected by strain 1 have perfect cross im-munity against further strains, then only those nodes inthe fraction 1 − S , termed the residual graph , can becomeinfected by the second strain. The threshold criterion forthe emergence of the second strain on unclustered ran-dom graphs has been solved previously by Newman. Wenow proceed to understand the role of clustering on thesecond strain.Setting the transmissibility of the second strain to T ,the probability that the second strain fails to infect anode chosen at random is comprised of the probabilitiesthat both the tree-like edges and the triangle edges eachfail to transmit the strain. In analogy to the first dis-ease, we define the probability h ⊥ to be the probabilitythat a tree-like edge remains unoccupied following bothstrains and introduce v ⊥ is the probability that a neigh-bouring node at the end of a tree-like contact does nothave disease 2. The probability that a node with k tree-like contacts has precisely l ≤ k susceptible neighboursfollowing disease 1 of which m ≤ l also failed to contractdisease 2 is given by h ⊥ ( u ⊥ , v ⊥ ; T , T ) = (cid:18) kl (cid:19)(cid:18) lm (cid:19) [ u ⊥ v ⊥ ] m [ u ⊥ (1 − v ⊥ )(1 − T )] l − m [(1 − u ⊥ )(1 − T )] k − l (12)Similarly, the probability, h ∆ , that a focal node involvedin a triangle fails to become infected is given by the prob-ability that each avenue of infection fails, as consideredfor the first disease in Eq 9. Defining v ∆ to be the proba-bility that a node involved in a triangle, that is also in theresidual graph of the first strain, remains uninfected dur-ing the second epidemic, we now examine each bracketin Eq 9. In the first case, both nodes are uninfected with strain-1 with probability u . To remain uninfected with strain-2, these nodes must fail to transmit to the focal node.This can occur in three distinct ways: either both neigh-bours fail to contract strain-2, v , or they both havedisease-2 but fail to transmit, ((1 − v ∆ )(1 − T )) , or fi-nally, one remains uninfected with strain-2 and the otherfails directly to infect with probability 2 v ∆ (1 − v ∆ )(1 − T ).Next, in the case when the residual structure containsboth an infected and an uninfected node, there are onlytwo ways that the focal node can remain uninfected by strain-2. These are the probability that the neighbour re-mains uninfected, v ∆ , or is infected but fails to transmit,(1 − v ∆ )(1 − T ). Together, these terms can be writtenas h ∆ ( u ∆ , v ∆ ; T , T ) = (cid:18) ηl (cid:19) [ u ] l (cid:18) lj (cid:19) [ v ] j (cid:18) l − ji (cid:19) [2 v ∆ (1 − v ∆ )(1 − T )(1 − T )] i [((1 − v ∆ )(1 − T )) ] l − j − i (cid:18) η − lm (cid:19) [2 u ∆ (1 − u ∆ )(1 − T )(1 − T )] m (cid:18) mf (cid:19) [ v ∆ ] f [(1 − v ∆ )(1 − T )] m − f [((1 − u )(1 − T )) ] η − l − m (13)Upon application of the binomial theorem this expression becomes h ∆ ( u ∆ , v ∆ ; T , T ) = [ u [ v + 2 v ∆ (1 − v ∆ )(1 − T )(1 − T ) + [(1 − v ∆ )(1 − T )] ]+ [2 u ∆ (1 − u ∆ )(1 − T )(1 − T )[ v ∆ + (1 − v ∆ )(1 − T )]] + [((1 − u ∆ )(1 − T )) ] (14)Despite the length of this equation, the interpretationis simple, we spread strain-2 according to the triangleformula of Eq 9 in the case that the residual motif is atriangle (motif (A) in Fig 1), we spread according to thetree-like expression when the residual triangle has onlyone neighbour in the residual graph (motif C) and finally,we do not spread strain-2 in the case that the motif iscompletely part of the GCC of strain-1 (motif B) .We can generate v τ by writing self-consistent expres-sions, this time however, dividing by the prior probabil-ity that the neighbour does indeed belong to the residualgraph, which is simply u τ . v τ = G ,τ ( h ⊥ , h ∆ ) /u τ (15)The expectation value for the probability that a ran-domly chosen node fails to be infected by either strainis A = G ( h ⊥ , h ∆ )1 − S (16)where we have divided by the prior probability of be-longing to the residual graph of disease 1. The fractionof the residual network that belongs to the outbreak ofthe second strain is then given by S [ u τ , v τ ; T , T ] = (1 − A )(1 − S ) (17)The complete prescription is as follows: we use Eq 10 tocompute u τ ∀ τ ∈ τ , we can then use Eq 11 to computethe epidemic outbreak size of the first strain. With theseingredients we calculate v τ ∀ τ ∈ τ using Eq 15 beforefinalising the calculation of the second outbreak fractionwith Eq 17.A numerical example of the both strains can be seenin plot (C) of Fig 2 for varying clustering coefficients. Asthe clustering coefficient increases the epidemic threshold of the first strain decreases from T ,c = 0 . T ,c ≈ .
41. The overall epidemic size at T = 1 is reduced as afunction of increasing clustering coefficient. Therefore, inthis experiment, clustering is seen to have a dual effecton the outbreak of strain-1 depending on T ; clusterednetworks can expect an epidemic at lower T , but alsoexpect fewer people to become infected. Setting T = 1,the total outbreak size of the second strain decreases asa function of increased clustering. C. R The R value, also known as the reproductive ratio ofa disease, is a quantity used in epidemiology to representthe number of infections that the average node in thenetwork will cause. When the disease has a low trans-missibility T ≤ T ,c , we do not expect that an epidemicwill occur throughout the entire network, in other words,the infections fizzle out over time. In these cases the R value is less than unity. R = 1 marks the thresh-old for which the epidemic infects a macroscopic fractionof the population and at this value the transmissibilityexperiences a critical point, T = T ,c . Under the bondpercolation isomorphism, a GCC of occupied edges formsin the network at and after this bond occupancy proba-bility. The critical transmissibility of the first strain canbe found by applying the Molloy-Reed criterion to theconfiguration model [9] (cid:18) dg ⊥ du ⊥ (cid:104) k ⊥ − k ⊥ (cid:105)(cid:104) k ⊥ (cid:105) − R (cid:19) (cid:18) dg ∆ du ∆ (cid:104) k − k ∆ (cid:105)(cid:104) k ∆ (cid:105) − R (cid:19) = dg ⊥ du ⊥ dg ∆ du ∆ (cid:104) k ⊥ k ∆ (cid:105) (cid:104) k ⊥ (cid:105)(cid:104) k ∆ (cid:105) (18)where each derivative is evaluated at the point u τ = 1.Each bracket on the left hand side can be used to in- C T c (A) -only-only ( , ) C T * (B) -only-only ( , ) T T T O u t b r ea k f r a c t i on (C) Strain 1Strain 2 C = 0.0 C = 0.1 C = 0.2 C = 0.3 FIG. 2. The percolation properties of the 2-strain model overclustered Poisson networks with clustering coefficient, C , andfixed average degree µ + 2 ν = 2 of tree-like and triangles,respectively. (A) The epidemic threshold of strain-1 (solid)as a function of C . The critical thresholds for a GCC to ex-ist solely among tree-like edges (small dash) or triangle edges(long dash) from Eq 18 are plotted in (A). When C = 0 wehave ν = 0 indicating the threshold is T ,c = 1 /
2, while at C = 1 / µ = 0 and hence find the critical thresholdas the root of T + 2 T − T ,c ≈ .
41. Similaranalysis in plot (B) shows the coexistence threshold, T ∗ , as afunction of increasing clustering coefficient from Eq 19. Alsoplotted in (B) is the difference T δ = T ,c − T ∗ between theepidemic and coexistence thresholds. After a short increase, T δ sharply falls with increased C , reducing the permissibletransmissibilities of strain 1 that allow the coexistence withstrain-2. (C) The expected epidemic size of each strain. Scat-ter points indicate experimental results of bond percolationon a network of size N = 40000 with 70 repetitions. Solidlines represent the theoretical predictions of Eqs 11 and 17for each strain. vestigate if a GCC occurs among the edges of a giventopology; or, the entire expression can be used to deter-mine of the entire network is connected, irrespective ofthe edge-type, see plot (A) in Fig 2. It is clear from thisplot that clustering increases the interval T ∈ [ T , c , T . In thecase of clustered networks, we find the condition to begiven by (cid:18) ∂h ⊥ ∂v ⊥ (cid:104) k ⊥ − k ⊥ (cid:105)(cid:104) k ⊥ (cid:105) − R (cid:19) (cid:18) ∂h ∆ ∂v ∆ (cid:104) k − k ∆ (cid:105)(cid:104) k ∆ (cid:105) − R (cid:19) = ∂h ⊥ ∂v ⊥ ∂h ∆ ∂v ∆ (cid:104) k ⊥ k ∆ (cid:105) (cid:104) k ⊥ (cid:105)(cid:104) k ∆ (cid:105) (19)The derivatives are evaluated at the point v τ = 1; how-ever we must find the point ( T ∗ , u ∗ τ ) that satisfies this.As with the first strain, the outbreak size of the secondpathogen among only the tree-like or the triangle edgescan be found by examining each bracket on the left handside of Eq 19. The emergence of a GCC among the en-tire residual graph is found using the entire expression,according to (B) in Fig 2.From plot (B) in Fig 2, it is clear that the interval[0 , T ∗ ] which defines the region that strain-2 can existis reduced as T ∗ decreases as a function of increasing C . Comparison of plots (A) and (B) indicate that whileboth T , c and T ∗ fall with C , the interval [ T , c , T ∗ ], whichdefines the coexistence of each strain on the network, alsois reduced, since, T ∗ falls faster than T , c . This indicatesthat clustering has only positive effects on the outbreakof the second strain, pushing the total fraction of thepopulation affected down at any given T ; decreasing therange of values of T at which strain-2 can coexist withstrain-1 present; and finally, decreasing the largest valueof T at which strain-2 is found in the network, squeezingit to a smaller region of the phase space.We will now apply the 2-strain model to clustered mul-tilayer networks [13]. For simplicity, we consider a 2-layersystem comprised of tree-like edges in the first (orange)layer and triangle edges in the second (green) layer. Thetwo layers are connected via interlayer tree-like edges.The model is a tautological extension of the model pre-sented in section II; strain-2 spreading over the residualgraph created by the GCC of the bilayer networked sys-tem. Representing interlayer tree-like edges that an or-ange (green) node has as ⊥ og ( ⊥ go ), the vector of permis-sible topologies is given by τ o = {⊥ o , ⊥ og } for the orangelayer and τ g = { ∆ g , ⊥ go } for the green layer, respectively.Following [13, 17], each layer has its own G ,λ ( z ) equa-tion, and each element of the topology vectors has itsown G ,λ,τ ( z ) equation also, where λ ∈ { o , g } is a layerindex.As a numerical example consider the case where alledge topologies follow a Poisson distribution such thatthe number of τ edges is η τ then p or ( η ⊥ , η ⊥ , og ) = (cid:104) η ⊥ (cid:105) η ⊥ e −(cid:104) η ⊥ (cid:105) η ⊥ ! (cid:104) η ⊥ , og (cid:105) η ⊥ , og e −(cid:104) η ⊥ , og (cid:105) η ⊥ , og ! (20) FIG. 3. An example of the multilayer network used to in thenumerical example. The green layer consists solely of triangleswhile the orange layer is tree-like. Each layer is connected viaa few tree-like edges to allow the GCC to span the network. and p gr ( η ∆ , η ⊥ , og ) = (cid:104) η ∆ (cid:105) η ∆ e −(cid:104) η ∆ (cid:105) η ∆ ! (cid:104) η ⊥ , go (cid:105) η ⊥ , go e −(cid:104) η ⊥ , go (cid:105) η ⊥ , go ! (21)The expected outbreak size of the first epidemic on theorange layer is then S o = 1 − e g ⊥ ( (cid:104) η ⊥ (cid:105)− e g ⊥ , og ( (cid:104) η ⊥ , og (cid:105)− (22)while the green layer has S g = 1 − e g ∆ ( (cid:104) η ∆ (cid:105)− e g ⊥ , go ( (cid:104) η ⊥ , go (cid:105)− (23)The g τ equations for each are given by Eqs 8 and 9 for theintralayer tree-like and triangle edges, respectively. Theinterlayer tree-like connections have a subtle symmetrybreaking depending on which layer we consider the focalnode to belong. We define g ⊥ , og ( u ⊥ , go ; T ) = u ⊥ , go + (1 − u ⊥ , go )(1 − T ) (24)and g ⊥ , g0 ( u ⊥ , og ; T ) = u ⊥ , og + (1 − u ⊥ , og )(1 − T ) (25)since, each focal node depends on the other end being un-infected. Each u τ is then the solution to a self-consistentequation according to Eq 10. The outbreak of the second epidemic follows from sec-tion II B and in the Poisson case is S , o = 1 − e h ⊥ ( (cid:104) η ⊥ (cid:105)− e h ⊥ , og ( (cid:104) η ⊥ , og (cid:105)− (26)while the green layer has S , g = 1 − e h ∆ ( (cid:104) η ∆ (cid:105)− e h ⊥ , go ( (cid:104) η ⊥ , go (cid:105)− (27)We examine this system in Fig 4. The network is con-structed such that the clustering coefficient of the greenlayer is C = 1 / (cid:104) k ∆ (cid:105) = 6 while the or-ange layer is C = 0 with mean tree-like degree (cid:104) k ⊥ (cid:105) = 3 . T than the or-ange layer due to its clustering. This causes the outbreakfraction of the first strain to show a double 2nd-ordertransition [13, 18]. We confirm the presence of a phasetransition by plotting the experimental second largestconnected component (SLCC), peaks in which indicatea critical point.Due to the different connectivity of each layer, theresidual graph also experiences two critical points. Weconfirm this by plotting the second largest residual con-nected component (SLRCC), peaks in which indicate thepresence of a phase transition in the residual network.The difference between the first peak in the SLCC andthe last peak in the SLRCC defines the transmissibilityrange that allows coexistence of each strain in the net-work. III. CONCLUSION
The study of disease spreading among human contactnetworks is of fundamental importance to society. Inparticular, the study of multiple sequential strains withthe presence of clustering can provide realistic models ofsocial interactions capable of pathogen transmission. Inthis paper, we have studied the problem of bond per-colation on the residual graph of clustered configurationnetworks created by a prior bond percolation process.This represents two strains spreading sequentially amonga population.We investigated the expected outbreak sizes of eachstrain of an epidemic as a function of the clustering co-efficient of the substrate contact network whose nodeshave fixed average degrees. We found that clustering re-duces the epidemic threshold, T ,c , of the first strain butdecreases the overall outbreak size at larger T values;therefore, having a dual effect on S parameterised by T .Clustering was found to reduce the maximum outbreaksize of the second strain. The largest value of T thatpermits the spreading of the second strain, T ∗ , is reducedby clustering. This indicates that increased clusteringforces the second strain to occupy a smaller region of themodels phase space. The phase region that permits the T O u t b r ea k f r a c t i on S e c onda r y ou t b r ea k f r a c t i on GCC (Theoretical)GRCC (Theoretical)GCC (Experimental)GRCC (Experimental)SLCC (Experimental)SLRCC (Experimental)
FIG. 4. The expected epidemic size of each strain on a Pois-son distributed clustered multilayer network with 2-layers. Inthis experiment, the orange layer has a clustering coefficientof C = 0 while the green layer is set to C = 1 /
3. Interlayertree-like edges have been added to allow the GCC to span theentire network. Scatter points indicate experimental resultsof bond percolation on a network of size N = 20000 with25 repeats. Solid lines represent the theoretical predictionsof Eqs. Also plotted is the SLCC and the SLRCC, peaks inwhich indicate a phase transition. From this plot we can seethat peaks in the SLCC and the SLRCC do not align witheach other, their separation defines the region of coexistencefor each strain. coexistence of each strain, given by the difference between T ,c and T ∗ , is also reduced due to clustering. Initially,this region broadens with the introduction of triangles tothe contact network ( T δ in plot (B) of Fig 2). However,the loss of tree-like edges causes the residual graph tofracture more than the original network with clusteringas shown by plotting T δ .We applied this model to the study of multilayer net-works providing a numerical example of a 2-layer system.We found that the presence of a double 2nd-order phasetransition in the GCC also creates a double 2nd-orderphase transition in the GRCC. This was supported byexamining the structure of the SLRCC as a function oftranmissibility. References ∗ [email protected][1] M. E. J. Newman, “Spread of epidemic disease on net-works,” Phys. Rev. E , vol. 66, p. 016128, Jul 2002.[2] B. Kerr, L. Danon, A. P. Ford, T. House, C. P. Jewell,M. J. Keeling, G. O. Roberts, J. V. Ross, and M. C.Vernon, “Networks and the epidemiology of infectiousdisease,”
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