aa r X i v : . [ m a t h . P R ] J a n Epidemics and vaccination on weighted graphs
Maria Deijfen ∗ November 2010
Abstract
A Reed-Frost epidemic with inhomogeneous infection probabilities on agraph with prescribed degree distribution is studied. Each edge ( u, v ) inthe graph is equipped with two weights W ( u,v ) and W ( v,u ) that representthe (subjective) strength of the connection and determine the probabilitythat u infects v in case u is infected and vice versa. Expressions for theepidemic threshold are derived for i.i.d. weights and for weights that arefunctions of the degrees. For i.i.d. weights, a variation of the so calledacquaintance vaccination strategy is analyzed where vertices are chosenrandomly and neighbors of these vertices with large edge weights arevaccinated. This strategy is shown to outperform the strategy where theneighbors are chosen randomly in the sense that the basic reproductionnumber is smaller for a given vaccination coverage. Keywords:
Reed-Frost epidemic, weighted graph, degree distribution,epidemic threshold, vaccination.AMS 2000 Subject Classification: 92D30, 05C80.
The Reed-Frost model is one of the simplest stochastic epidemic models. Itwas formulated by Lowell Reed and Wade Frost in 1928 (in unpublished work)and describes the evolution of an infection in generations. Each infected in-dividual in generation t ( t = 1 , , . . . ) independently infects each susceptibleindividual in the population with some probability p . The individuals thatbecome infected by the individuals in generation t then constitute generation t + 1 and the individuals in generation t are removed from the epidemic pro-cess. See [3] for a description of the asymptotic (as the population size growsto infinity) behavior of the process. ∗ Department of Mathematics, Stockholm University, 106 91 Stockholm.Email: [email protected]. n the original version, an infective individual infects each susceptible in-dividual in the population with the same probability. Realistically howeveran infective individual has the possibility to infect only those individuals withwhom she actually has some kind of social contact. The Reed-Frost model iseasily modified to capture this by introducing a graph to represent the socialstructure in the population and then let the infection spread on this graph.More precisely, an infective individual infects each neighbor in the graph in-dependently with some probability p .When analyzing epidemics on graphs, the graph is usually taken to beunweighted with respect to the infection, that is, transmission takes place alongall edges with the same probability. In this paper however, inhomogeneity willbe incorporated in the transmission probability by aid of weights on the edges.More precisely, each edge ( u, v ) in the graph is assigned two weights W ( u,v ) and W ( v,u ) that are assumed to take values in [0,1]. The probability that u infects v if u gets infected is then given by W ( u,v ) and vice versa. Note thatit may well be that W ( u,v ) = W ( v,u ) . We shall mainly consider i.i.d. weights,although we briefly treat weights that are determined by the degrees of thevertices in Section 3.To describe the underlying network, we shall use the so called configura-tion model [12, 13]. Once the graph has been generated, each edge is equippedwith two weights as described above. Basically, the configuration model takesa probability distribution with support on positive integers as input and gener-ates a graph with this particular degree distribution; see Section 2 for furtherdetails. The degree distribution is indeed an important characteristic of anetwork with a large impact on the properties of the network and it is there-fore desirable to be able to control this in a graph model. Furthermore, theconfiguration model exhibits short distances between the vertices, which is inagreement with empirical findings; see [10]. Epidemics on un-weighted graphsgenerated by the configuration model has previously been studied in [1, 2, 7].Related results have also appeared in the physics literature [14].An important quantity in epidemic modeling is the epidemic threshold,commonly denoted by R . It is defined as a function of the parameters ofthe model such that a large outbreak in the epidemic has positive probabilityif and only if R >
1. Expressions for R typically stems from branchingprocess approximations of the initial stages of the epidemic. It is well-knownfrom branching process theory that the process has a positive probability ofexploding if and only if the expected number of children of an individualexceeds 1. A natural candidate for R is hence the expected number of newcases caused by a typical infective in the beginning of the time course. For thisreason, the epidemic threshold is often referred to as the basic reproductionnumber.The main goal of the paper is to study how the epidemic threshold isaffected by vaccination strategies based on the edges weights. To this end,we assume that a perfect vaccine is available that completely removes vacci-2ated individuals from the epidemic process. The simplest possible vaccina-tion strategy, usually referred to as random vaccination, is to draw a randomsample from the population and then vaccinate the corresponding individuals.An alternative, known as acquaintance vaccination, is to choose individualsrandomly and then, for each chosen individual, vaccinate a random neigh-bor rather than the individual itself [8, 7]. The idea is that, by doing this,individuals with larger degrees are vaccinated. We shall study a version ofacquaintance vaccination where, instead of vaccinating a random neighbor,the neighbor with the largest weight on its edge from the sampled vertex isvaccinated. In a human population, this correspond to asking individuals toname their closest friend (in some respect) instead of just naming a randomfriend. It is demonstrated that this is more efficient than standard acquain-tance vaccination, in the sense that the basic reproduction number with theweight based strategy is smaller for a given vaccination coverage.Throughout this paper we shall use the term “infection” to refer to thephenomenon that is spreading on the network. We remark that this does notnecessarily consist of an infectious disease spreading in a human population,but may also refer to other infectious phenomena such as a computer virusspreading in a computer network, information routed in a communicationnet or a rumor growing in a social media. In many of these situations theconnections are indeed highly inhomogeneous. Furthermore, depending onwhat type of spreading phenomenon that is at hand, the term vaccination canrefer to different types of immunization.Epidemics on weighted graphs have been very little studied so far andthere are few theoretical results. See however [14] for an approach based ongenerating function and [9, 15] for simulation studies. We mention also therecent work on first passage percolation on random graphs by Bhamidi et al.[4, 5, 6]. There, each edge in a graph generated according to the configurationmodel is equipped with an exponential weight and the length and weight ofthe weight-minimizing path between two vertices are studied. Interpretingthe weights as the traversal times for an infection, this can be related to thetime-dynamics of an epidemic.The rest of the paper is organized so that the graph model and the epidemicmodel are described in more detail in Section 2. In Section 3, expressions forthe epidemic thresholds are given and calculated for some specific weight dis-tributions. Section 4 is devoted to vaccination: In Section 4.1, a weight basedacquaintance vaccination strategy for weights with a continuous distributionis described and an expression for the epidemic threshold is derived. Section4.2 treats a strategy for a two-point weight distribution. The findings are sum-marized in Section 5, where also some directions for further work are given.We shall throughout refrain from giving rigorous details for the underlyingbranching process approximations, but instead focus on heuristic derivationsof the epidemic quantities. Indeed, what needs to be proved is basically thatthe branching process approximations hold long enough so that conclusions3or the branching processes are valid also for the epidemic processes. Thishowever is not affected by weights on edges (as long as these are not functionsof the structure of the graph) and hence rigorous details can presumably befilled in by straightforward modifications of the arguments in [7] (the degreebased weights mentioned in Section 3 might however require some more work). We consider a population of size n represented by n vertices. The graph rep-resenting the connections in the population is generated by the configurationmodel. To produce the graph, a probability distribution with support on thenon-negative integers is fixed and each vertex u is independently equippedwith a random number of half-edges D u according to this distribution. Thesehalf-edges are then paired randomly to create the edges in the graph, that is,first two half-edges are picked at random and joined, then another two half-edges are picked at random from the set of remaining half-edges and joined,etc. If the total number of half-edges is odd, a half-edge is added at a randomlychosen vertex to pair with the last half-edge.This procedure gives a multi-graph, that is, a graph where self-loops andmultiple edges between vertices may occur. If D has finite second momenthowever, there will not be very many of these imperfections. In particular,the probability that the resulting graph is simple will be bounded away from0 as n → ∞ ; see [7, Lemma 5.5] or [11, Theorem 7.10]. If D has finite secondmoment we can hence condition on the event that the graph is simple, andwork under this assumption. Another option is to erase self-loops and mergemultiple edges, which asymptotically does not affect the degree distribution if D has finite second moment; see [11, Theorem 7.9]. Henceforth we shall henceassume that D has finite second moment and ignore self-loops and multipleedges.When the graph has been generated, each edge ( u, v ) is assigned twoweights W ( u,v ) and W ( v,u ) that are assumed to take values in [0,1]. This canbe thought of as if each one of the half-edges that is used to create the edgeindependently receives a weight. The epidemic spread is initiated in that onerandomly chosen vertex is infected. This vertex constitutes generation 1. Theepidemic then propagates in that each vertex u in generation t ( t = 1 , , . . . ) in-fects each susceptible neighbor v independently with probability W ( u,v ) . Gen-eration t + 1 then consists of the vertices that are infected by the vertices ingeneration t and the vertices in generation t are removed from the epidemicprocess.We shall mainly restrict to the case where the weights are taken to beindependent. However, we mention also the possibility to let them be functionsof the degrees of the vertices: Independent weights.
The weights are taken to be i.i.d. copies of a random4ariable W that takes values in [0,1]. The distribution of W can be defined inmany different ways: • As an intrinsic distribution on [0,1], for instance a uniform distributionor, more generally, a Beta distribution. • By letting N be an integer valued random variable, indicating for in-stance how many times a given vertex contacts a given neighbor duringsome time period, and then setting W d = 1 − (1 − p ) N , with p ∈ [0 , • By, similarly, letting X be a positive random variable, interpreted as the(subjective) strength of a connection, and then for instance setting W d = { X ≥ θ } for some θ ≥ W d = 1 − α X for α ∈ [0 , X could be interpreted as the resistance involved in a connection and W modeled as a decreasing function of X . Degree dependent weights.
The weights of an edge ( u, v ) could alsobe modeled as functions of D u and D v . We shall consider the case when W ( u,v ) = g ( D u ) for some function g that takes values in [0,1]. All outgoingedges from u hence have the same weight, and independent trials with thissuccess probability determine whether the edges are used to transmit infec-tion. With g increasing, this setup means that vertices with large degree have alarger probability of infecting their neighbors, for instance in that they tend tobe more active. With g decreasing, high degree vertices are instead less likelyto infect their neighbors, which might be the case for instance in a situationwhere high degree vertices have weaker bonds to their acquaintances. As mentioned in the introduction, expressions for epidemic thresholds usu-ally come from branching process approximations of the initial stages of anepidemic. As for epidemics on graphs, branching process approximations aretypically in force as soon as the graph is tree-like, that is, if with high proba-bility the graph does not contain short cycles. This means that the neighborsof a given infective in the beginning of the time course are susceptible withhigh probability and hence the initial stages of the generation process of infec-tives is well approximated by a branching process. Under the assumption thatthe degree distribution has finite second moment, the configuration model isindeed tree-like, allowing for such an approximation; see e.g. [10, 7] for details.The epidemic threshold is then given by the reproduction mean in the approx-imating branching process, which in turn is given by the expected number ofnew cases generated by an infective vertex in the beginning of the epidemic.When calculating this, one should not consider the initial infective, since this5ertex might be atypical, but rather an infective vertex in, say, the secondgeneration.Let { p k } k ≥ be the probabilities defining the degree distribution in theconfiguration model. Then the initial infective has degree distribution { p k } ,while the neighbors of this vertex have the size biased degree distribution { ˜ p k } defined by ˜ p k = kp k µ , where µ = P kp k denotes the mean degree. The infective vertices in thesecond (and later) generations hence have degree distribution { ˜ p k } . Denoteby e D a random variable with this distribution. Independent weights.
Consider an infected vertex in the second generation.One neighbor of this vertex must have transmitted the infection and can hencenot get reinfected, while the other neighbors are with high probability suscep-tible. The number of new cases generated by the vertex is hence distributedas ˜ D − X i =1 { neighbor i infected } . (1)If the weights are i.i.d. copies of W , then the mean of the indicators is γ := E [ W ] and we get R = γ E [ e D −
1] = γ (cid:18) µ + Var( D ) − µµ (cid:19) . In this case the epidemic threshold is hence the same as in a model withconstant infection probability p = γ ; see [2, 7]. Note that, for degree distri-butions with large variance, R can be large even if µ is small. Also note thatthe above reasoning remains valid in a situation where the weights W ( u,v ) and W ( v,u ) on a given edge are correlated, as long as the weights are independentbetween edges. In fact, as long as the transmission between separate links arei.i.d., the whole epidemic process is equivalent to a Reed Frost model (on theconfiguration model) with p = E [ W ]. Degree dependent weights.
Assume that W ( u,v ) = g ( D u ). Since W ( u,v ) does not depend on D v , the degree distribution of an infective in the secondgeneration is { ˜ p k } . Conditionally on its degree e D = ˜ d , the number of newcases generated by an infective in the second generation is Bin( ˜ d − , g ( ˜ d ))-distributed. It follows that R deg = E [( e D − g ( e D )] . Let R h denote the basic reproduction number for an epidemic with a homo-geneous infection probability given by the transmission probability E [ g ( ˜ D )]for a randomly chosen half-edge, that is, R h = E [ g ( e D )] E [ e D − . g is an increasing function we have R deg ≥ R h , due to the positivecorrelation between g ( e D ) and e D , while, if g is decreasing, then R deg ≤ R h .Another comparison that might be relevant is to relate R deg to the basicreproduction number for an epidemic with a homogeneous infection probabilitygiven by E [ g ( D )], that is, an epidemic where the infection probability g ( D )for a vertex with degree D is averaged over all possible degrees. The basicreproduction number in such an epidemic is given by R h = E [ e D − E [ g ( D )] . Example 3.1.
First take D ∼ Po( µ ). It is not hard to see that then ˜ p k = p k − .Take g ( x ) = { x ≥ θ } . If W ( u,v ) = g ( D u ), this means that only vertices withdegree at least θ transmit the infection. We have E [ { e D ≥ θ } ] = P ( D ≥ θ − E [( e D − { e D ≥ θ } ] = X k ≥ θ ( k − kp k µ = X k ≥ θ µ k − ( k − e − µ = µ P ( D ≥ θ − . Hence R h = µ P ( D ≥ θ ) R h = µ P ( D ≥ θ − R deg = µ P ( D ≥ θ − . With g ( x ) = α x for α ∈ (0 , E [ α D ] = e − µ (1 − α ) and E [ α e D ] = αe − µ (1 − α ) . Furthermore E [( e D − α e D ] = X k ≥ ( k − α k kp k µ = α X k ≥ α k p k = α e − µ (1 − α ) . Hence R h = µe − µ (1 − α ) R h = µαe − µ (1 − α ) R deg = µα e − µ (1 − α ) . ✷ Example 3.2.
Now take a distribution with p k ∼ ck − . . In this case exactcomputations are out of reach but numerical values of the thresholds are easilyobtained. We give an example with g ( x ) = x − τ for τ ∈ [0 , τ , showing that R h >R h > R deg . The homogeneous epidemics remain supercritical (that is, their7 Figure 1: The basic reproduction numbers R h (solid line), R h (dashed line)and R deg (dash-dotted line) with g ( x ) = x − τ plotted against τ . The degreedistribution is a power law with exponent 3.5 and mean 4.reproduction numbers exceed 1) at τ = 1, while the epidemic with degreedependent weights becomes subcritical for τ close to 1. Indeed, the degreedependent epidemic is subcritical at τ = 1 for any degree distribution. ✷ For the remainder of the paper we shall restrict to the case with indepen-dent weights.
We now proceed to analyze a version of the so called acquaintance vaccinationstrategy. To this end, suppose that a perfect vaccine is available that preventsvaccinated vertices from participating in the epidemic process and that thisvaccine is distributed prior to the start of the epidemic. More precisely, firstwe generate the underlying graph and assign the edge weights, then we choosewhich vertices that are to be vaccinated and finally, when the vaccine has beendistributed, we analyze the epidemic spread among the unvaccinated vertices.The simplest vaccination scheme is to vaccinate each vertex independentlywith some probability v . We shall refer to this as uniform vaccination andwrite R U v for the corresponding basic reproduction number. By reasoning asin the case without vaccination and keeping in mind that only unvaccinatedvertices can be infected, it is not hard to see that R U v is obtained by multiplying8he expression for the case without vaccination with (1 − v ), that is, R U v = (1 − v ) γ (cid:18) µ + Var( D ) − µµ (cid:19) . (2)An alternative strategy, referred to as acquaintance vaccination, is to vaccinateneighbors of the chosen vertices rather than the vertices themselves; see [8].More precisely, each vertex is sampled independently with probability s and,for each sampled vertex, a randomly chosen neighbor is vaccinated. A neighborthat is chosen via more than one vertex is (of course) vaccinated only once. Thefact that two vertices can both pick a common neighbor to receive vaccinationmeans that the asymptotic fraction of vaccinated vertices v ( s ), referred toas the vaccination coverage , is smaller than s . In fact, in many cases it isnot possible to push an epidemic below criticality (that is, to obtain a basicreproduction number smaller than 1) even when s = 1. This motivates astrategy where a vertex can be sampled more than once and thereby havemore than one neighbor vaccinated.In [7], a strategy is analyzed where each vertex is sampled independentlya Po( β ) number of times and each time a vertex is sampled, a randomlychosen neighbor is vaccinated. When there are weights on the edges however,more efficient strategies are possible. Here we shall consider strategies where,rather than choosing neighbors randomly for vaccination, neighbors with largeweights on their edges from the sampled vertices are chosen. We treat the casewith independent directed weights. In Section 4.1, the weights are assumed tocome from a continuous distribution and Section 4.2 is devoted to a strategyfor two-point distributions. Assume that the directed edge weights are i.i.d. realizations from a continuousprobability distribution on [0,1]. Each vertex u is sampled independently aPo( β ) number of times. Write N u for the set of neighbors of a vertex u and,for i = 1 , . . . , |N u | , let v i ∈ N u be the vertex corresponding to the i :th largestelement in { W ( u,v ) : v ∈ N u } (note that v i is almost surely unique, since theweights are assumed to come from a continuous distribution). Then, if u issampled i ≤ D u times, the neighbors v , . . . , v i are vaccinated. If i ≥ D u , allneighbors of u are vaccinated and, if D u = 0, no action is taken. This will bereferred to as weight based acquaintance vaccination.To derive an expression for the vaccinated fraction v ( β ) of the population,let V ∼ Po( β ) represent the number of times that a given vertex is sampled.The probability that a randomly chosen vertex u is not chosen for vaccinationby a given neighbor with degree k is given by r k = k − X i =0 P ( V = i ) (cid:18) − ik (cid:19) . (3)9ince the neighbors of u have degree distribution { ˜ p k } , the probability that u avoids being chosen for vaccination by a given neighbor equals α = X k ≥ r k ˜ p k . If u has degree j , then the probability that u is not vaccinated is α j . Thedegree distribution of u is { p j } and, since the fraction of unvaccinated verticescoincides with the probability that a randomly chosen vertex is not vaccinated,we obtain v ( β ) from the equation1 − v ( β ) = X j ≥ α j p j . Next, to identify the epidemic threshold in a population vaccinated accordingto the weight based acquaintance strategy, we shall employ a branching processapproximation of the initial stages of the epidemic. The process however isslightly more complicated than in the case with uniform vaccination, andthis is because the knowledge that an edge has not been used for vaccinationcarries information of the degrees of the corresponding vertices. The processis analogous to the one used in [7]. To describe it, say that a directed edge( u, w ) is used for vaccination if u is sampled and chooses the neighbor w forvaccination. Furthermore, conditionally on the weight, a directed edge is saidto be open for transmission if a bernoulli trial with success probability givenby the weight of the edge results in a success. A directed edge that is not usedfor vaccination and that is open for transmission is called dangerous .An “individual” in the branching process now consists of an unvaccinatedvertex u along with a dangerous outgoing edge ( u, w ). The individual thengives birth to a new individual if the vertex w is unvaccinated and in turn has adangerous edge ( w, w ′ ) pointing out from it. Note that an unvaccinated vertexcan hence give rise to several individuals (if it has more than one outgoingdangerous edge) or no individuals at all (if it does not have any outgoingdangerous edges). Furthermore, the individuals reproduce independently.It is not hard to see that the epidemic has a positive probability of takingoff if and only if the above branching process has a positive probability ofexploding: With positive probability the initial infective in unvaccinated andwith positive probability it has at least one dangerous out-edge. The propaga-tion of the epidemic from the vertices that are hit by these dangerous edges isthen approximated by the above branching process. To find an expression forthe reproduction mean of the process, consider a given unvaccinated vertex u along with an outgoing dangerous edge ( u, w ). How many new individualsdoes this give rise to? First, the degree distribution of w , which is size biased,is now affected also by the information that the edge ( w, u ) has not been usedfor vaccination. Write A for the latter event and P A and E A for probability10nd expectation respectively conditionally on A . We get P A ( D w = k ) = r k ˜ p k P r k ˜ p k = r k ˜ p k α , where r k is defined in (3). If w has degree k , then the probability that w is notchosen for vaccination by any of its other k − u that,by assumption, does not have a dangerous edge to w ) is α k − . Write H for thenumber of dangerous edges ( w, w ′ ) with w ′ = u . Note that, conditionally onthe degree of w , the event that w is unvaccinated (which carries informationon the in-weights of w ) does not affect H (which is determined by the out-weights of w ). The reproduction mean of the branching process is hence givenby R W β = X k ≥ P A ( D w = k ) α k − E A,k [ H ] , (4)where E A,k [ H ] := E A [ H | D w = k ] (below we use P A,k to denote the corre-sponding probability and P k to denote probability conditional only on that D w = k ). It remains to quantify this expectation. Clearly H is affected by thenumber of times V w that w is sampled to name a neighbor in the vaccinationprocedure, which in turn is affected by the information that ( w, u ) was notused for vaccination. Specifically, when D w = k, we have, for i = 0 , . . . , k − P A,k ( V w = i ) = P k ( A | V w = i ) P k ( V w = i ) P k ( A )= (cid:0) − ik (cid:1) P ( V w = i ) r k . Let W ( k ) j denote a random variable distributed as the j :th smallest in a collec-tion of k independent weight variables. If V w = i ( i = 1 , . . . , k − i largest weights are used for vaccination. The remaining k − i out-edges are dangerous with a probability given by the expectation oftheir weights. Note however that we do not want to count the edge to u ,whose weight indeed belongs to the k − i smallest since, by assumption, it isnot used for vaccination. The ordering of the weight on the edge to u amongthe remaining k − i out-weights is uniform on { , . . . , k − i } . We obtain E A,k [ H | V w = i ] = (cid:18) − k − i (cid:19) k − i X j =1 E [ W ( k ) j ] . (5)Note that, if V w = 0, then each one of the k − w to N w \{ u } isdangerous independently with probability γ , and the above expression reducesto ( k − γ . If V w ≥ k −
1, then all out-edges (except ( w, u )) are used forvaccination meaning that there are no dangerous out-edges. Hence E A,k [ H ] = k − X i =0 P A,k ( V w = i ) E A,k [ H | V w = i ] . j :th smallest observationin a collection of k i.i.d. variables with density f and distribution function F is given by f k,j ( x ) = Γ( k + 1)Γ( j )Γ( k + 1 − j ) ( F ( x )) j − (1 − F ( x )) k − j f ( x ) , (6)where Γ( · ) denotes the gamma function. For a given weight distribution F , themean can hence be calculated by aid of numerical integration. A particularlyeasy case is when the weights are uniform on [0,1]. Then W ( k ) j ∼ Beta( j, k + 1 − j )so that E [ W ( k ) j ] = j/ ( k + 1), and hence E A,k [ H | V w = i ] = (cid:18) − k − i (cid:19) k − i X j =1 jk + 1 = ( k − i − k − i + 1)2( k + 1) . (7)We now want to compare the epidemic threshold for the weight basedstrategy to the threshold for the standard acquaintance vaccination strategy,where neighbors are chosen randomly. In [7], the vaccination coverage v ( β )for the latter strategy is shown to be given by1 − v ( β ) = X j ≥ α j p j with α = P e − β/k ˜ p k . Furthermore, for a homogeneous infection probability p , the basic reproduction number is shown to be R A β = p X k ≥ ( k − α k − e − β/k ˜ p k . (8)Straightforward modifications of the arguments leading up to these expressionsreveals that they apply also for the inhomogeneous case with independentweights, with p replaced by the mean weight γ . Example 4.1.1.
Let the edge weights be uniformly distributed on [0 , γ = 1 / R W β in (4) is easily calculatedfor a given degree distribution { p k } . Figure 2 shows the basic reproductionnumber R W β plotted against the vaccination coverage v ( β ) when the degree dis-tribution is Po(6). The plot also shows the reproduction number for standardacquaintance vaccination and for uniform vaccination. For a given vaccinationcoverage, we have R U > R A > R W , although the difference between standard12cquaintance vaccination and the weight based strategy is quite small. Notehowever that, in practical situations also a small gain could be valuable: Thevaccination coverage required to push the epidemic threshold below 1 – therebypreventing large outbreaks – is referred to as the critical vaccination coverage .Clearly, when fighting an infectious disease in a large human population forinstance, even a very small decrease in the critical vaccination coverage mightimply large savings in terms of vaccination costs. ✷ Example 4.1.2.
Let the weights have a Beta distribution with parameters0.5 and 2.5; see Figure 3. In this case it is not possible to write down analyticalexpressions in closed form for R W but it is easily computed numerically. Fig-ure 4 shows the basic reproduction numbers plotted against the vaccinationcoverage when the degree distribution is Po(14). In this case the weight basedstrategy performs clearly better than the standard acquaintance vaccination.In particular, the critical vaccination coverage for uniform vaccination andstandard acquaintance vaccination is 0.58 and 0.53 respectively, while for theweight based strategy it is decreased to 0.47. The reason is that the weightdistribution is right-skewed: Most weights are small but there is a thick right-tail with large weights, and by getting rid of these large weights the mean inthe weight distribution is decreased more than in the uniform case. ✷ Example 4.1.3.
Finally, let the weights have the same Beta distribution as inthe previous example, but take the degree distribution to be a power-law withexponent 3.5 and the same mean 14 as in the Poisson distribution. Figure 5displays the basic reproduction numbers in this case. Again the weight basedstrategy performs better than standard acquaintance vaccination. In this casehowever, the most striking feature is the difference between the uniform vac-cination and the acquaintance based strategies: when the degree distributionis a power law, the basic reproduction number is pushed down very effectivelyby targeting high degree vertices. ✷ The finding in the previous section that the weight based strategy performswell for right-skewed weight distributions in the continuous case might leadone to suspect that the strategy is particularly useful for a “polarized” discretedistribution. In this section we analyze the simple case when the weights havea two-point distribution. As a motivation we can think of a network havingtwo types of directed transmission links, one that spreads an infection withhigh probability and one that does so only with a very small probability. Itwould then be natural to design a strategy that targets vertices with highlyinfectious connections.Assume that W ∈ { a, b } where a < b and write p a = P ( W = a ) and p b = P ( W = b ). The strategy is defined so that each vertex is sampledindependently with probability s and, for each sampled vertex u , its neighbors13 Figure 2: Basic reproduction numbers with U (0 ,
1) weights for a Po(6) de-gree distribution plotted against the vaccination coverage: the weight basedacquaintance strategy (solid line), the standard acquaintance vaccination(dashed line) and uniform vaccination (dash-dotted line).
Figure 3: A Beta density with parameter 0.5 and 2.5.14
Figure 4: Basic reproduction numbers with Beta(0.5,2.5) weights for a Po(14)degree distribution plotted against the vaccination coverage: the weight basedacquaintance strategy (solid line), the standard acquaintance vaccination(dashed line) and uniform vaccination (dash-dotted line).
Figure 5: Basic reproduction numbers with Beta(0.5,2.5) weights for power-law degree distribution with exponent 3.5 and mean 14 plotted against thevaccination coverage. 15ith weight b on their edge from u are vaccinated: Recall that N u denotes theset of neighbors of a vertex u and let N ( b ) u = { v ∈ N u : W ( u,v ) = b } , that is, N ( b ) u is the set of neighbors of u for which the weight on the edge( u, v ) attains the larger value b . Then, if u is sampled, the vertices in N ( b ) u arevaccinated. No action is taken if N ( b ) u is empty.To derive the vaccination coverage, note that the probability that a ran-domly chosen vertex in the graph is not chosen for vaccination by a givenneighbor equals α = 1 − sp b . As in the previous section we obtain the vaccination coverage from the equation1 − v ( s ) = X j ≥ α j p j . The derivation of the epidemic threshold is based on the same branching pro-cess as in the previous section, that is, an individual in the branching processconsists of an unvaccinated vertex u along with an outgoing edge ( u, w ) that isnot used for vaccination and that is open for transmission. To find an expres-sion for the reproduction mean R D s , which serves as the epidemic threshold,first note that in this case the degree distribution of vertex w is not affected bythe information that w did not chose u for vaccination (recall that the latterevent is denoted A ). Indeed, whether u is vaccinated or not if w is sampled isdetermined only by W ( w,u ) . Hence P A ( D w = k ) = ˜ p k .Conditionally on D w = k , the probability that w is not vaccinated via anyof its other k − u ) is given by α k − . We also needto determine the expected number of dangerous edges from w to vertices in N w \ { u } conditionally on that D w = k and on A (note that, conditionallyon the degree, the distribution of the number of dangerous edges from w is not affected by the information that w is not vaccinated). For this weneed the corresponding probability that w is sampled to name a neighbor forvaccination. With V w ∈ { , } denoting the number of times that w is sampledto name a neighbor, we get P A,k ( V w = 1) = P k ( A | V w = 1) P k ( V w = 1) P k ( A ) = p a sα =: ν. Note that this probability does not depend on k . If V w = 0, then the expectednumber of dangerous edges from w (to other vertices than u ) is ( k − γ . If V w = 1 on the other hand, then the neighbors reached by edges with thelarge weight are vaccinated. The expected number of remaining out-edgesfrom w (to other vertices than u ) is ( k − p a and each one of these is openfor transmission with probability a . The expected number of dangerous edges16rom w is hence ( k − ap a . Write R W2 s for the basic reproduction number withthe current vaccination strategy. We get R W2 s = ( νap a + (1 − ν ) γ ) X k ≥ ˜ p k α k − ( k − . (9)We now compare this to the epidemic threshold (2) for uniform vaccinationand, in particular, to the threshold (8) for the standard acquaintance vaccina-tion strategy. Example 4.2.1.
Figure 6 shows the basic reproduction numbers when thedegree distribution is Po(14) and the weight distribution is specified by P ( W =0 .
1) = 1 − P ( W = 1) = 0 . ✷ Example 4.2.2.
Figure 7 shows the basic reproduction numbers for the sameweight distribution as in the previous example when the degree distributionis a power-law with exponent 3.5 and mean 14. Again we see that the weightbased strategy is the most efficient. ✷ Example 4.2.3.
Finally, Figure 8 displays the basic reproduction numbersfor the same power-law degree distribution as in the previous example but fora weight distribution specified by P ( W = 0 .
1) = 1 − P ( W = 1) = 0 .
5. In thiscase almost nothing is gained by using the weight based strategy comparedto standard acquaintance vaccination (the lines are almost aligned). The ex-planation for this is that, although the weight based strategy targets highlyinfective links, it does so more “locally” in the graph: Recall that all neighborswith large weight on their edges from a sampled vertex are vaccinated. Thismeans that, to achieve a given vaccination coverage, a much smaller sampleof vertices is required compared to standard acquaintance vaccination if theprobability of the larger weight is reasonably large; Figure 9 shows a plot forthe current example. Thus the weight based strategy affects fewer parts ofthe graph and this cancels the positive effect that lies in securing high riskconnections. However, the strategy does not perform worse than the standardacquaintance strategy. Hence the strategy is still more effective in the sensethat it requires a smaller sample of vertices to name neighbors for vaccinationto obtain a given vaccination coverage. In situations when there are costsassociated with selecting and communication with the sampled vertices, thismight be important. ✷ We have formulated and analyzed a model for epidemic spread on weightedgraphs, where the weight of an edge indicates the probability that it is used17
Figure 6: Basic reproduction numbers plotted against the vaccination coveragewith P ( W = 0 .
1) = 1 − P ( W = 1) = 0 . Figure 7: Basic reproduction numbers plotted against the vaccination coveragewith P ( W = 0 .
1) = 1 − P ( W = 1) = 0 . Figure 8: Basic reproduction numbers plotted against the vaccination coveragewith P ( W = 0 .
1) = 1 − P ( W = 1) = 0 . Figure 9: The fraction of the population that has to be sampled to vaccinate(at least one) neighbor(s) plotted against the resulting vaccination coverage forthe weight based strategy (solid line) and standard acquaintance vaccination(dashed line). 19or transmission. Expressions have been derived for the epidemic threshold,specifying when there is a positive probability for an epidemic to take off.The case with independent weights is analogous to the case with a constantinfection probability given by the mean weight. For degree dependent out-weights – which for instance makes it possible to model a situation where highdegree vertices infect their neighbors with a smaller probability – however thebehavior is different from a homogeneous epidemic.Furthermore, we have analyzed a version of the acquaintance vaccinationstrategy where neighbors of the sampled vertices reached by edges with largeweights are vaccinated. The selected vertices hence impose vaccination on theneighbor(s) that they have the strongest connection(s) to instead of a randomneighbor. Two versions of this strategy have been treated: one for continuousweight distributions and one for two-point distributions. In the examples wehave looked at, these strategies have been seen to outperform standard ac-quaintance vaccination, the difference being largest in cases where the weightdistribution is highly right-skewed. The reason why the weight based ac-quaintance strategies perform better than standard acquaintance vaccinationis that, in addition to removing the vaccinated neighbors, the ability to spreadthe epidemic is decreased also for the sampled vertices in that their high-weightconnections are secured.As for further work, there are numerous possibilities. In many situationsit would be desirable to allow for (typically positive) correlations between theweights W ( u,v ) and W ( v,u ) on a given edge, for instance one might want toassign only one weight per edge, specifying the probability of transmission inany direction. This leads to complications in the current analysis, basicallybecause the information that a vertex is unvaccinated then gives informationon the weights on the edges of its neighbors. Furthermore, the basic idea inacquaintance vaccination is that, by vaccinating neighbors of the sampled ver-tices, one reaches vertices with higher degree. A natural further developmentof this idea would be to vaccinate neighbors with maximal degree, that is,selected vertices are asked to identify their neighbor(s) with the largest degreeamong the neighbors (assuming that they have this information) and theseneighbors are then vaccinated. Unfortunately this seems to lead to compli-cated dependencies in the resulting epidemic process.We also mention that it would be interesting to investigate the final size ofthe epidemic. This is usually related to the probability of a large outbreak andquantified via an equation involving the generating function of the reproduc-tion distribution. For the vaccination strategies that we have considered here,this equation would involve the distribution (6) of order statistics and is hencepresumably complicated. But it would be interesting to study the final size byaid of simulation. Other possible continuations include investigating how theresults are affected by introducing clustering (triangles and other short cycles)in the underlying graph, to involve time-dynamic in the vaccination procedureand to generalize the model for the epidemic spread.20 cknowledgement. The author gratefully acknowledges the support fromThe Bank of Sweden Tercentenary Foundation.
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