ARTICLE IN RESS Jornal of Theoretical Biology 240 (2006) 400 418 www.elsevier.com/locate/yjtbi reicting epiemics on irecte contact networks Laren Ancel Meyers a,b,, M.E.J. Newman b,c, Babak orbohlol,e a Section of Integrative Biology an Institte for Celllar an Moleclar Biology, University of Texas at Astin, 1 University Station C0930, Astin, TX 78712, USA b External Faclty, Santa Fe Institte, 1399 Hye ark Roa, Santa Fe, NM 87501, USA c Center for the Sty of Complex Systems, University of Michigan, Ranall Laboratory, 500 E. University Ave., Ann Arbor, MI 48109-1120, USA Division of Mathematical Moeling, University of British Colmbia Centre for Disease Control, 655 West 12th Avene, Vancover, BC, Canaa V5Z 4R4 e Department of Health Care & Epiemiology, University of British Colmbia, Canaa Receive 11 May 2005; receive in revise form 16 Agst 2005; accepte 6 October 2005 Available online 21 November 2005 Abstract Contact network epiemiology is an approach to moeling the sprea of infectios iseases that explicitly consiers patterns of person-to-person contacts within a commnity. Contacts can be asymmetric, with a person more likely to infect one of their contacts than to become infecte by that contact. This is tre for some sexally transmitte iseases that are more easily caght by women than men ring heterosexal enconters; an for severe infectios iseases that case an average person to seek meical attention an thereby potentially infect health care workers (HCWs) who wol not, in trn, have an opportnity to infect that average person. Here we se methos from percolation theory to evelop a mathematical framework for preicting isease transmission throgh semi-irecte contact networks in which some contacts are nirecte the probability of transmission is symmetric between inivials an others are irecte transmission is possible only in one irection. We fin that the probability of an epiemic an the expecte fraction of a poplation infecte ring an epiemic can be ifferent in semi-irecte networks, in contrast to the rotine assmption that these two qantities are eqal. We frthermore emonstrate that these methos more accrately preict the vlnerability of HCWs an the efficacy of varios hospital-base containment strategies ring otbreaks of severe respiratory iseases. r 2005 Elsevier Lt. All rights reserve. Keywors: Epiemiology; Contact network; Directe graph; Infectios isease; Hospital transmission 1. Introction Many infectios iseases sprea throgh irect person-to-person contact. Respiratory-borne iseases like inflenza, tberclosis, meningococcal meningitis an SARS, sprea throgh the exchange of respiratory roplets between people in close physical proximity to each other. Sexally transmitte iseases like HIV, genital herpes, an syphilis sprea throgh intimate sexal contact. Explicit moels of the patterns of contact among inivials in a commnity, contact network moels, provie a powerfl approach for preicting an controlling the sprea of sch infectios iseases (Longini, 1988; Sattenspiel an Simon, 1988; Morris, 1995; Kretzschmar et al., 1996; Ball et al., 1997; Morris an Kretzschmar, 1997; Fergson an Garnett, 2000; Hethcote, 2000; Lloy an May, 2001; Newman, 2002; Saner et al., 2002; Keeling et al., 2003; Meyers et al., 2003; Meyers et al., 2005). This approach has provie insight into the impact of simltaneos sexal partners on HIV transmission (Morris an Kretzschmar, 1997) an effective pblic health strategies for controlling STDs (Kretzschmar et al., 1996) an mycoplasma pnemonia (Meyers et al., 2003), among others. Corresponing athor. Section of Integrative Biology an Institte for Celllar an Moleclar Biology, University of Texas at Astin, 1 University Station C0930, Astin, TX 78712, USA. Tel.: +1 512 471 4950; fax: +1 512 471 3878. E-mail aresses: larenmeyers@mail.texas.e (L.A. Meyers), mejn@mich.e (M.E.J. Newman), babak.porbohlol@bccc.ca (B. orbohlol). 0022-5193/$ - see front matter r 2005 Elsevier Lt. All rights reserve. oi:10.1016/j.jtbi.2005.10.004
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 401 (A) (B) (C) Fig. 1. Contact networks: (A) nirecte network; (B) bipartite network; an (C) semi-irecte network. The simplest form of contact network moel represents inivials as vertices an contacts as eges connecting appropriate vertices. The nirecte network epicte in Fig. 1A assmes that if vertices i an j share an ege, then the probability that i infects j given that i is infective an j is ssceptible is eqal to the probability that j infects i given that i is ssceptible an j is infective. There are many iseases for which this assmption oes not hol. For example, there may be as mch as a two-fol ifference between male-to-female an female-to-male HIV transmission efficiency with females mch more vlnerable than males (Nicolosi et al., 1994); health care workers (HCWs) an patients may have asymmetric transmission probabilities becase, perhaps, patients are more likely to have immne eficiencies or caregivers are more likely to be expose to boily flis ring meical proceres; mothers can transmit bloo-borne iseases to offspring in tero whereas there may be no opportnity for transmission in the reverse irection. We can moel sch asymmetries sing bipartite contact networks in which there are two classes of noes that transmit isease to each other at ifferent rates (Fig. 1B). Mathematical methos for preicting the sprea of isease on bipartite contact networks have been escribe in Ball et al. (1997) an Meyers et al. (2003). Asymmetry in isease transmission may also arise if the isease inflences inivial behavior. Dring an otbreak, infecte inivials may moify their typical patterns of interaction. In particlar, they may visit a hospital or clinic at which they come into contact with HCWs an other patients. Inivials that are not infecte, however, will likely have no contact with hospital personnel. Since we cannot know a priori which inivials will become infecte, we cannot easily captre sch conitional contacts in a simple network moel. Directe eges, in which transmission occrs only in one irection, provie a way aron this ifficlty (Fig. 1C). A irecte ege leaing from a member of the general poplation () to a HCW (H) reflects the following relationship: If is infecte, he or she will expose H with some probability; bt if H is infecte, he or she will have no contact with. Ths, contact network moels containing both irecte an nirecte eges (henceforth semi-irecte networks) can be se to moel commnity-base isease transmission in which there is a sbstantial one-way flow of isease from the general pblic into health care facilities. For respiratory iseases, preicting an controlling this flow is vital. Hospitals are
402 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 particlarly vlnerable becase of the freqent hospitalization of infecte inivials with serios illness, the high nmber of patients with pre-existing high acity co-morbiity incling those who are immnocompromise, an the close an mltiple physical contacts between infecte inivials, caregivers, visitors an other patients. For these reasons, a significant proportion of SARS transmission events occrre within hospitals (Avenano et al.; Varia et al., 2003). Unerstaning an containing hospital-base transmission is critical not only for the protection of sch inivials bt also for the prevention of commnity-wie sprea otsie the hospital. Here we evelop mathematical tools for preicting the sprea of isease an impact of intervention on semi-irecte networks an then apply these tools to assess the impact of hospital-base transmission an intervention on the fate of an otbreak. For part one, we se generating fnction methos to erive the probability an expecte emographic istribtion of otbreaks, with an withot pblic health intervention. This is an extension of both epiemiological theory previosly evelope for nirecte contact networks (Newman, 2002) an a general theory of ranom graphs containing only irecte eges (Newman et al., 2001). Many of the calclations are fnamentally eqivalent to branching process calclations, an it seems likely that some of the reslts presente here col be erive sing branching process methos as well (Jagers, 1975; Anersson, 1998). We show that in semi-irecte networks the probability of an epiemic an the expecte fraction of the poplation infecte ring sch an epiemic may be ifferent. In contrast, many conventional moels assme the eqality of these two epiemiological vales, an then se isease incience ata to inirectly estimate the probability of an epiemic (Anerson an May, 1991). Or analysis therefore sggests that this assmption may be invali for poplations with asymmetric contact patterns. For part two, we make epiemiological preictions sing a simple moel of rban contact patterns base on emographic ata from the city of Vancover, British Colmbia. By incorporating conitional contacts within health care settings, we more accrately assess the role of HCWs in isease transmission an containment. 2. Derivations of epiemic qantities 2.1. Moeling the poplation an the isease In a semi-irecte network, each vertex (inivial) has an nirecte egree representing the nmber of nirecte eges joining the vertex to other vertices as well as both an in-egree an an ot-egree representing the nmber of irecte eges incoming from other inivials an otgoing to other inivials, respectively. The nirecte-egree an in-egree inicate how many contacts can sprea isease to the inivial, an ths is relate to the likelihoo that an inivial will become infecte ring an epiemic; an the nirecte-egree an ot-egree inicate how many contacts may be infecte by that inivial shol he or she become infecte, an ths is relate to the likelihoo that an inivial will contribte to an epiemic. The semi-irecte egree istribtion tells s the probability that a ranomly chosen inivial will have a particlar combination of an nirecte-egree, in-egree, an ot-egree. One can preict analytically the sprea of an infectios isease throgh a poplation given two basic inpts: the semiirecte egree istribtion an the probabilities of isease transmission along the eges of the network. Some pathogens, like smallpox, are highly contagios an will ths have a high probability of moving along an ege in the network (Bozzette et al., 2003). Other pathogens, like SARS, are less likely to be transmitte (X et al., 2004). For a given isease, the probability of transmission along a particlar ege will also epen on the health of the inivials lying at either en of the ege an the natre of their interaction with each other. In (Newman, 2002), Newman showe that, when the rate of transmission of a isease between pairs of inivials is assme to be an i.i.. ranom variable, the sprea of the isease epens only on the mean total probability of transmission between inivials, or transmissibility, an not on the inivial probabilities for specific pairs. We make se of this reslt here also, an henceforth consier only T an T, the average probability that an infectios inivial will transmit the isease to a ssceptible inivial with whom they have a irecte or nirecte contact, respectively. Note that average transmissibilities T an T vary from isease to isease bt are always in the range 0pT, T p1. We will also consier the simpler case where average transmissibility is the same for irecte an nirecte eges, that is, T ¼ T ¼ T. Sppose a isease begins to sprea throgh a poplation from a particlar vertex. In or moel, transmission will occr along each of the irecte an nirecte eges pointing ot of that vertex with probabilities T an T, respectively. If we keep track of every ege in the network along which isease is transmitte an call these occpie eges, then we can reconstrct the final size an istribtion of the otbreak. In particlar, the otbreak will incle exactly the set of all vertices that are connecte to the initial vertex along a continos path of occpie eges. Becase of its resemblance to bon percolation, this moel can be analyse sing mathematical methos from percolation theory (Newman, 2002; Saner et al., 2002; Meyers et al., 2003). In what follows, we erive exact soltions for the expecte size of an otbreak, the probability of a large-scale epiemic, the size of sch an epiemic, the risk to inivials as a fnction of their egree, an the impact of varios forms of intervention.
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 403 2.2. robability generating fnctions for semi-irecte networks In the theory of ranom irecte graphs evelope by Newman et al. (2001), one consiers the joint probability istribtion p jk that a ranomly chosen vertex has in-egree j an ot-egree k. Then one efines a generating fnction Fðx; yþ whose coefficients are the probabilities in this istribtion: Fðx; yþ ¼ X jk p jk x j y k (1) from which many properties of the network can then be erive. Aopting a similar approach for or semi-irecte networks, we consier the joint probability istribtion p jkm that a vertex has j incoming eges, k otgoing eges, an m nirecte eges. Then we efine a generating fnction G that generates this istribtion ths: Gðx; y; Þ ¼ X jkm p jkm x j y k m. (2) This fnction has the properties that Gð1; 1; 1Þ ¼ 1 (3) for any properly normalize p jkm, an z ¼ G ð1;0;0þ ð1; 1; 1Þ ¼ G ð0;1;0þ ð1; 1; 1Þ; z ¼ G ð0;0;1þ ð1; 1; 1Þ, (4) where z is the average in-egree an ot-egree of a vertex for irecte eges (the two mst necessarily be the same, since every otgoing irecte ege mst also be an incoming ege at some vertex) an z is the average egree of nirecte eges. The notation G ðr;s;vþ inicates ifferentiation of G with respect to its three argments r, s, an v times, respectively, so that, for example G ð0;0;1þ ¼ qg q ; Gð1;1;0Þ ¼ q2 G qxqy. (5) The excess egrees are the nmbers of each type of ege emerging from a vertex arrive at by following an ege, not incling the incoming ege. Henceforth, sbscript refers to following an nirecte ege in either irection an sbscripts an r refer to following a irecte ege in the esignate an reverse irection, respectively. The excess egrees are biase by the fact that eges are more likely to arrive at vertices with higher in- an nirecteegree, in irect proportion to that egree. Ths the istribtion of eges of the three types, incoming, otgoing, an nirecte, at a vertex reache by following a irecte ege in the esignate irection is jp jkm =Sjp jkm, an hence the excess egree istribtion is generate by H ðx; y; Þ ¼ jkm jp jkmx j 1 y k m jkm jp jkm ¼ 1 z G ð1;0;0þ ðx; y; Þ. (6) The generating fnction for the excess egree istribtion for a irecte ege in the reverse irection is H r ðx; y; Þ ¼ jkm kp jkmx j y k 1 m jkm kp jkm ¼ 1 z G ð0;1;0þ ðx; y; Þ. (7) Similarly, the istribtion at a vertex reache by following an nirecte ege is generate by H ðx; y; Þ ¼ jkm mp jkmx j y k m 1 jkm mp jkm ¼ 1 z G ð0;0;1þ ðx; y; Þ. (8) We next moify these generating fnctions to consier the istribtion of occpie eges, that is, eges along which isease has been transmitte. In Appenix A.1, we erive the following probability generating fnction for the nmber of occpie eges of a vertex: Gðx; y; ; T ; T Þ ¼ Gð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ. (9) Similarly, the probability generating fnctions for the excess nmber of occpie eges, that is, the nmber of eges (excling the arrival ege) emanating from a vertex arrive at by following a ranomly chosen ege, are given by H ðx; y; ; T ; T Þ ¼ H ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ, (10) H r ðx; y; ; T ; T Þ ¼ H r ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ, (11)
404 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 H ðx; y; ; T ; T Þ ¼ H ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ. (12) 2.3. reicting the fate of a small otbreak In general, percolation theory escribes the behavior of connecte grops of vertices in a ranom graph. We se the methos of percolation theory to preict the size of the infecte clster, that is, the nmber of vertices reache via isease transmission along the eges in the network. For a fixe network of contacts, there typically exists a threshol transmission rate below which only small, finite-size otbreaks occr an above which large-scale epiemics (comparable to the size of the entire network) are possible. We begin by eriving the vale of the epiemic threshol an the expecte size of small otbreaks below the threshol. These calclations assme that milly contagios iseases sprea in a tree-like fashion, casing only short transmission chains that o not loop back on themselves. We relax this assmption later, when we trn to iseases that lie above the epiemic threshol. Let s enote the nmber of vertices containe in a small otbreak that begins at a ranomly selecte vertex. We now introce symbols g an h which shol not be confse with G an H (above), respectively. Let g(w; T, T ) be the generating fnction for the istribtion of otbreak sizes: gðw; T ; T Þ ¼ X s s ðt ; T Þw s, (13) where s (T, T ) is probability that a single initial case sparks an otbreak of size s at the specifie average transmissibilities. To solve for the average vale of s, we first evalate the size of an otbreak t that begins with a transmission event along a ranomly chosen ege. If that ege is irecte, then the set of vertices reache by occpie eges can be represente in graphical form as in the top row of Fig. 2. There are many possible otcomes: the isease oes not sprea along the ege, it spreas along the ege bt no frther, it spreas along the ege an then sbseqently along another irecte ege, it spreas along the ege an then sbseqently along an nirecte ege, it spreas along the original ege an then sbseqently along two ifferent irecte eges emanating from the same vertex, etc. We will constrct recrsive eqations to consier all possibilities. We efine a new generating fnction h (w; T, T ), which generates the probability istribtion of t ths: h ðw; T ; T Þ ¼ X t Q t ðt ; T Þw t, (14) where Q t (T, T ) is the probability that an otbreak beginning from a ranomly chosen ege in the network will be size t. Fig. 2 illstrates that h (w;t, T ) satisfies a recrsive conition of the form h ðw; T ; T Þ ¼ wh ð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ, (15) where h (w) is the corresponing generating fnction for nirecte eges, which itself satisfies a conition of the form h ðw; T ; T Þ ¼ wh ð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ (16) as epicte in the bottom row of Fig. 2. The self-consistent soltions of Eqs. (15) an (16) give the istribtion of t, given efinitions (10) an (12) of H an H. It follows that s, the size of an otbreak starting from a ranomly chosen vertex, is istribte accoring to gðw; T ; T Þ ¼ wgð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ. (17) + + + + + + + +... + + + + + + + +... Fig. 2. Ftre transmission iagram. When isease is transmitte along a irecte (top) or nirecte (bottom) ege, we can consier all possible patterns of ftre transmission. Starting from a irecte ege, for example, the isease may not sprea along the ege, it may sprea along the ege bt no frther, it may sprea along the original ege an then sbseqently along another irecte ege, it may sprea along the original ege an then sbseqently along an nirecte ege, it may sprea along the original ege an then sbseqently along two ifferent irecte eges emanating from the same vertex, etc. We constrct recrsive eqations to consier all possible otcomes beginning from a single irecte or nirecte ege.
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 405 Consier now the average size of an otbreak starting from a ranom vertex /ss, which is given by hsi ¼ X s s s ðt ; T Þ ¼ g 0 ð1; T ; T Þ, (18) where the prime enotes ifferentiation with respect to the first variable. In Appenix A.2, we erive the following expression for /ss in terms of the isease transmissibility an the pgf s for the egree istribtion an excess egree istribtion: hsi ¼ 1 þ T G ð0;1;0þ ð1 T ðh ð0;0;1þ ð1 T H ð0;1;0þ H ð0;0;1þ ÞÞ þ T G ð0;0;1þ ð1 T ðh ð0;1;0þ Þð1 T H ð0;0;1þ Þ T T H ð0;0;1þ H ð0;1;0þ H ð0;1;0þ ÞÞ, (19) where the argments of all generating fnctions are set to (1,1;1). Ths, we can preict the expecte size of an otbreak given the semi-irecte egree istribtion an transmissibilities T an T of the isease. If average transmissibilities along irecte an nirecte eges are eqal (T ¼ T ¼ T), then the expecte size of the otbreak is given by hsi ¼ 1 þ TGð0;1;0Þ ð1 TðH ð0;0;1þ ð1 TH ð0;1;0þ H ð0;0;1þ ÞÞ þ TG ð0;0;1þ ð1 TðH ð0;1;0þ Þð1 TH ð0;0;1þ Þ T 2 H ð0;0;1þ H ð0;1;0þ H ð0;1;0þ ÞÞ. (20) The expression for /ss iverges when the enominator in Eq. (19) is zero, an only preicts the expecte size of the otbreak when the enominator is greater than zero. Ths the eqation ð1 T H ð0;1;0þ Þð1 T H ð0;0;1þ Þ T T H ð0;0;1þ H ð0;1;0þ ¼ 0 (21) marks the phase transition at which the size of an otbreak first becomes extensive. Solving Eq. (21) for a given 0pT p1, we erive the critical transmissibility T c at which a large-scale epiemic becomes possible: T c ¼ H ð0;0;1þ T ðh ð0;1;0þ 1 T H ð0;1;0þ H ð0;0;1þ Similarly, for some 0pT p1, the critical vale is efine by T c ¼ H ð0;1;0þ T ðh ð0;1;0þ 1 T H ð0;0;1þ H ð0;0;1þ H ð0;0;1þ H ð0;1;0þ Þ. (22) H ð0;0;1þ H ð0;1;0þ Þ. (23) Ths, there is a line efine by (22) an (23) of transmissibility vales, below which we expect only small otbreaks of expecte size /ss an above which an epiemic is possible. If average transmissibility is the same for irecte an nirecte eges, then there is a single critical transmissibility: T c ¼ ðhð0;1;0þ þ H ð0;0;1þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ ðh ð0;1;0þ þ H ð0;0;1þ Þ 2 4ðH ð0;1;0þ H ð0;0;1þ H ð0;0;1þ H ð0;1;0þ Þ 2ðH ð0;1;0þ H ð0;0;1þ H ð0;0;1þ, (24) H ð0;1;0þ Þ whichever vale is positive. We call {T c, T c } (or T c ) the epiemic threshol. By ifferentiating Eqs. (6) an (8) an sbstitting the reslts into Eqs. (22) or (24) one can express the epiemic threshol in terms of the nerlying strctre of the contact network. 2.4. A simple example We se these formlas to preict the sprea of isease on a simple network in which all three egree istribtions are oisson with mean in-egree an ot-egree of z an mean nirecte egree of z. The pgf for the egree istribtion is given by Gðx; y; Þ ¼ X jkm z j e z j!! z k e z k! z m e z m! x j y k m ¼ e z ðxþy 2Þþz ð 1Þ. (25) The excess egree pgf s for this network are ientical to the original egree istribtion, that is, H ðx; y; Þ ¼ H r ðx; y; Þ ¼ H ðx; y; Þ ¼ Gðx; y; Þ. Therefore the expecte size of an epiemic is efine by hsi ¼ 1 1 T z T z. (26)
406 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 200 150 T =0.001 T =0.01 T =0.1 T =0.2 <s> 100 50 (A) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 T T 1/z No Epiemic Epiemic possible Epiemic threshol (B) T 1/z Fig. 3. Simple semi-irecte network. (A) The expecte size of a small otbreak as a fnction of T an T for a oisson semi-irecte network with oisson parameters z ¼ 2 an z ¼ 3. (B) The epiemic threshol for a oisson semi-irecte network with oisson parameters z an z. We plot /ss for a oisson semi-irecte network in Fig. 3A. By setting the enominator eqal to zero, we fin an epiemic threshol line of T c z þ T c z ¼ 1 (27) as epicte in Fig. 3B. 2.5. robability an size of a large-scale epiemic When the transmissibility of a isease is larger than the epiemic threshol, then Eq. (19) no longer inicates the size of the infecte sbpoplation. This is becase transmission is so rampant that the chains of transmission are likely to loop back pon themselves, ths violating the assmption nerlying the calclations epicte in Fig. 2. When we are above the epiemic threshol, in the region in which epiemics can occr, we wol like to know two qantities: the probability that a large-scale epiemic occrs an the fraction of inivials that are infecte in that case. These qantities are eqivalent to S in an S ot the fraction of vertices from which an extensive nmbers of others can be reache by following occpie eges an the fraction of vertices containe in sch an extensive interconnecte grop, respectively. In the langage of percolation, these are the giant strongly connecte component (GSCC) pls the giant incomponent (GIN) an the GSCC pls the giant ot-component (GOUT) efine by occpie eges. Fig. 4 illstrates the component strctre of semi-irecte networks. The relative size of the region shae in vertical lines inicates the probability that any single infection will lea to a wie-sprea epiemic, an the relative size of the region shae in horizontal lines inicates the expecte fraction of the poplation that will become infecte ring sch an epiemic. To calclate the typical size of a large-scale epiemic, we make se of the following argment. All vertices in the GSCC an GOUT are reachable from an extensive nmber of others (those in the GSCC an GIN), an all vertices that are not in these components are not reachable from an extensive nmber of others. We can calclate from how many vertices a ranomly chosen vertex is reachable by following occpie eges backwars from that vertex an fining the reslting
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 407 Tenrils GIN GSCC GOUT Fig. 4. Strctre of a semi-irecte network. The largest set of vertices for which yo can move between any two by following eges in the correct irection is the giant strongly connecte component (GSCC). The set of vertices not containe in the GSCC that can be reache by following eges in the correct irection from the GSCC is calle the giant ot-component (GOUT). The set of vertices not containe in the GSCC from which the GSCC can be reache by following eges in the correct irection is calle the giant in-component (GIN). Vertices that are not in the GSCC, GIN, or GOUT bt can either be reache from the GIN or can reach the GOUT are in the tenrils of the network. component. This is precisely the reverse of the calclation we performe in the previos section, an allows s to erive the fraction of the graph containe in the largest occpie component, that is, the size of a large epiemic. In Appenix A.3, we erive the following expression for the fraction of the poplation infecte ring a large-scale epiemic: S size ¼ 1 X jkm p jkm ð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m, (28) where a an b are soltions of jkm a ¼ kp jkmð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m jkm kp, (29) jkm b ¼ jkm mp jkmð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m 1 jkm mp. (30) jkm In most cases (28) is not solvable in close form, bt once we have the egree istribtion (the p jkm s) an transmissibilities T an T it can be solve nmerically by simple iteration, starting from appropriate initial vales. As iscsse in Appenix A.3, one can similarly calclate the probability that an infection at a ranomly chosen vertex will lea to a large-scale epiemic (S prob ). This qantity is eqal to the size of the GSCC pls the GIN an is given by S prob ¼ 1 X jkm p jkm ð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m, (31) where a an b are soltions of a ¼ b ¼ jkm jp jkmð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m jkm jp, (32) jkm jkm mp jkmð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m 1 jkm mp. (33) jkm Note that a an b in Eqs. (31) (33) are the probabilities that infection at a vertex at the en of a ranomly selecte irecte an nirecte ege (respectively) will not spark a large-scale epiemic. If average transmissibility is the same for irecte an nirecte eges, then simply sbstitte the single transmissibility vale T for T an T in Eqs. (28), (29), (30), (31), (32), an (33). These basic epiemiological qantities the epiemic threshol an the fate of otbreaks on either sie of the threshol have been erive previosly for completely irecte (Newman, 2002) an completely nirecte networks (Schwartz et al., 2002). We provie these formlae in Appenix A.4. 2.6. A simple example Compartmental epiemiological moels assme that the probability an expecte size of an epiemic are always eqal (Anerson an May, 1991; Hethcote, 2000). While this is tre for nirecte networks, these two vales can be ifferent in irecte an semi-irecte networks. We emonstrate this sing three ifferent networks: (N1) a completely nirecte
408 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 oisson network with mean egree z (where z is an even integer) that has generating fnction G N1 ðxþ ¼ X z j e z x j ¼ e zðx 1Þ, (34) j! j (N2) a semi-irecte network with a oisson istribtion of nirecte eges of mean egree z=2, a oisson in-egree istribtion of mean z=2, an a reglar ot-egree istribtion in which every vertex has an ot-egree of exactly z=2 that has generating fnction G N2 ðx; y; Þ ¼ X ðz=2þ j e z=2 ðz=2þ m e z=2 x j y z=2 m j! m! jm ¼ e ðz=2þðxþ 2Þ y z=2 an (N3) a flly irecte network with a oisson in-egree istribtion of mean z, an a reglar ot-egree istribtion in which every vertex has an ot-egree of exactly z that has generating fnction G N3 ðx; yþ ¼ X z j e z x j y z ¼ e zðx 1Þ y z. (36) j! j These three networks have the same total nmber of incoming an otgoing contacts since every nirecte ege incles two incoming an two otgoing contacts while every irecte ege incles a single incoming contact an a single otgoing contact. We se ifferent in- an ot-egree istribtions to emonstrate the ineqality of S prob an S size, becase semi-irecte an irecte networks with ientical in- an ot-egree istribtions have an eqal-size GIN an GOUT an therefore eqal vales of S prob an S size. All three networks share the same epiemic threshol of T c ¼ 1=z. Above the threshol, the probabilities an expecte sizes of epiemics in these networks are preicte by the following eqations: S N1 ¼ S N2size ¼ S N3size ¼ 1 e zð 1ÞT where ¼ e zð 1ÞT, (37) S N2prob ¼ 1 ð1 þ ða 1ÞTÞ z=2 e z=2ða 1ÞT where a ¼ ð1 þ ða 1ÞTÞ z=2 e z=2ða 1ÞT, (38) S N3prob ¼ 1 ð1 þ ðw 1ÞTÞ z ; where w ¼ ð1 þ ðw 1ÞTÞ z. (39) Fig. 5 illstrates these preictions for two sets of networks (z ¼ 4 an 8). For each set of networks, all three share the same expecte size of an epiemic. The probability of an epiemic is ientical to the expecte size of an epiemic in the nirecte network, mch larger than the expecte size in the completely irecte network, an at an intermeiate vale in the semi-irecte network. Or particlar choice of in- an ot-egree istribtions yiels networks with GIN larger than GOUT. If we reverse these two istribtions, then GOUT wol be larger than GIN, an therefore, the expecte size of the epiemic wol be larger than the probability of an epiemic. 2.7. Initial conitions We can refine or preictions if we know something abot the behavior of patient zero the first case of isease in a poplation. Sppose, for instance, that we know patient zero has ot-egree k an nirecte-egree m. The probability that he or she will spark a large-scale epiemic is jst the probability that transmission of the isease along at least one of the eges emanating from patient zero will lea to an epiemic. For any one of its k ot eges an m nirecte eges, the probability that the isease is not transmitte along the ege is 1 T an 1 T, respectively. As efine in Eqs. (32) an (33), a an b are the probabilities that an otbreak traveling along a given irecte or nirecte ege will sprea to only a local component of the poplation. Ths the probability that isease is transmitte along one of the k þ m eges bt oes not procee from there into a fll-blown epiemic is T a for a irecte ege or T b for an nirecte ege, an the overall probability that patient zero will spark an epiemic is given by e km ¼ 1 ð1 T þ T aþ k ð1 T þ T bþ m. (40) The probability that an otbreak of size N will lea to a large-scale epiemic is 1 Q N i¼1 ð1 e k i m i Þ, where k i is the otegree an m i is the nirecte-egree of inivial i. This is jst one mins the probability that none of the N infecte inivials sparks an epiemic. If we know the nmber of crrent cases bt not their contact patterns, then or best estimate for the probability of an epiemic is calclate similarly, with each of the ð1 e ki m i Þ s replace with the probability that a typical infecte inivial oes not spark an epiemic. Sch an inivial was infecte either along a irecte ege with a priori probability z =ðz þ z ) or along an nirecte ege with a priori probability z =ðz þ z ). The nmber of ð35þ
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 409 1 0.9 S 0.8 0.7 0.6 0.5 0.4 z = 8 z = 4 0.3 0.2 0.1 S N1 = S N2size = S N3size S N3prob SN2prob 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Fig. 5. Epiemiological preictions for nirecte, irecte an semi-irecte networks. The probability of an epiemic an expecte fraction of the poplation infecte ring an epiemic for three classes networks: (N1) a completely nirecte oisson network with mean egree z; (N2) a semi-irecte network with oisson nirecte an in-egree istribtions of mean egree z=2, an with every vertex having an ot-egree of exactly z=2; an (N3) a completely irecte network with a oisson in-egree istribtion of mean egree z, an with every vertex having an ot-egree of exactly z. For each of these networks, we plot the preicte probability an size of an epiemic (S) as a fnction of the average transmissibility of the isease (T). S N1 is both the expecte magnite an probability of an epiemic in network N1 an S N2size an S N2prob (S N3size an S N3prob ) are the expecte magnite an probability of an epiemic in N2 (N3). The left an right three lines correspon to networks with z ¼ 8 an 4, respectively. T eges throgh which the inivial can start an epiemic is given by the excess egree generating fnctions H an H, an the probability that one of those eges will not give rise to an epiemic is 1 T þ T a for a irecte ege an 1 T þ T b for an nirecte ege. Ths the probability that a typical infecte inivial oes not start an epiemic is given by z H ð1; ð1 T þ T aþ; ð1 T þ T bþþ þ z H ð1; ð1 T þ T aþ; ð1 T þ T bþþ z þ z, (41) an the probability that an otbreak of size N sparks an epiemic is given by 1 z H ð1; ð1 T þ T aþ; ð1 T þ T bþþ þ z H ð1; ð1 T þ T aþ; ð1 T þ T bþþ N (42) z þ z where a an b are as escribe by Eqs. (32) an (33). Simplifying slightly, we can rewrite Eq. (42) as! N jkm 1 jp jkm ð1 T þt aþk ð1 T þt bþ m þ jkm mp jkm ð1 T þt aþk ð1 T þt bþ m 1. (43) jkm ðjþmþp jkm Appenix A.4 provies the analogos eqations for completely irecte an completely nirecte networks. 2.8. Inivial risk an intervention The likelihoo that an inivial of in-egree j an nirecte-egree m will be infecte ring an epiemic is eqal to one mins the probability that none of his or her j þ m contacts will transmit the isease to him or her. The probability that a contact oes not transmit the isease is eqal to the probability that the contact was infecte, bt i not transmit the isease, 1 T for a contact along a irecte ege an 1 T for a contact along an nirecte ege, pls the probability that the contact was not infecte in the first place, T a for a contact along a irecte ege or T b for a contact along an nirecte ege, where a an b are as efine by Eqs. (29) an (30). Ths, a ranomly chosen vertex of in-egree j an nirecte-egree m will become infecte with probability n jm ¼ 1 ð1 T þ T aþ j ð1 T þ T bþ m. (44)
410 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 When a single inivial of egrees j, k, m lowers the likelihoo of transmission to or from himself or herself (by wearing a face mask in the case of an air-borne isease, for example) from T an T to f T an f T (0pf, f p1), then the expressions for the likelihoo of casing an epiemic an becoming infecte ring an epiemic become e f km ¼ 1 ð1 f T þ f T aþ k ð1 f T þ f T bþ m, (45) n f jm ¼ 1 ð1 f T þ f T aþ j ð1 f T þ f T bþ m, (46) where a, b, a, an b are as in Eqs. (29), (30), (32), an (33). Note that these two qantities are ifferent for some semi-irecte networks, whereas they are always ientical for nirecte networks (Appenix A.4). 3. A case sty in hospital-base transmission of respiratory isease 3.1. The contact networks We have previosly evelope a metho to simlate rban contact networks base on emographic ata for the city of Vancover, British Colmbia (Statistics Canaa, 2001; BC Stats, 2002; Centre for Health Sevices an olicy Research, 2002; Vancover School Boar, 2002; BC Stats, 2003; Meyers et al., 2005). Using the egree istribtion from a contact network moel containing 10,000 hosehols (25,000 inivials), we preict the fate of an otbreak for a spectrm of respiratory-borne iseases for which hospitalization is likely. As reporte in (Meyers et al., 2005), the nirecte-egree istribtion is roghly exponential. The in-egree an ot-egree istribtions are solely etermine by the flow of infecte people into health care facilities. In this moel, we make the simple assmption that each non-hcw member of the poplation has three irecte eges pointing to ranomly chosen HCWs in his or her local hospital. Ths a typical inivial has ot-egree of three an in-egree of zero; an a typical HCW has ot-egree of zero an in-egree ranging from 409 to 530. Becase the moe of transmission (respiratory-borne) is the same for irecte an nirecte eges in this network, we assme that there is a single average transmissibility across the entire network, that is, T ¼ T ¼ T. 120 1 0.9 Nmber infecte <s > 100 80 60 40 20 T c T c robability of expecte size of an an epiemic S 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 robability an Expecte Size (Unirecte) robability (Semi-irecte) Expecte Size (Semi-irecte) 0 0 0.02 0.04 T 0 0.1 0.2 0.3 0.4 0.5 T Fig. 6. Epiemiological preictions on nirecte an semi-irecte contact networks. This graph shows the expecte size of small otbreaks below the epiemic threshol (left), an the probability an expecte size of a large-scale epiemic above the epiemic threshol (rate) for iseases with varios transmission rates (T) spreaing throgh an rban contact network. The preictions for the semi-irecte an nirecte networks are shown in black an gray, respectively.
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 411 (This assmption wol not be appropriate for a isease in which irecte an nirecte eges represente ifferent moes of transmission.) Using the formlae erive above, we calclate the epiemic threshol for this particlar contact network, the expecte size of an otbreak for iseases below the epiemic threshol, an the probability an expecte size of an epiemic for iseases above the epiemic threshol (Fig. 6). The inclsion of one-irectional isease transmission from the general pblic into health care settings significantly increases the vlnerability of a poplation. The epiemic threshol is lowere from T c ¼ 0:0322 to 0.0278. The lower the transmissibility of the isease, the more prononce the impact of hospitalbase transmission. For iseases close to the epiemic threshol, the probability of an epiemic in the more realistic semiirecte network is more than oble that of the simpler nirecte network. Note that if an epiemic oes ense, the expecte size of an epiemic is almost ientical for the two contact networks. We can also preict the role of HCWs in the sprea of isease an the impact of intervention. There are two basic categories of intervention (Meyers et al., 2005; orbohlol et al., 2005). Contact recing interventions moify the basic patterns of interaction. Within hospitals, for example, sspecte cases are isolate in negative pressre rooms an the nmber of caregivers attening to sch patients is limite. For the poplation at large, pblic health officials may implement qarantines an travel restrictions. Sch interventions can be moele by removing appropriate eges from the contact network. Vaccination prior to an otbreak, which entails removing a vertex an all of its eges from the contact network, is the extreme form of sch interventions. Transmission recing interventions like the se of facemasks, srgical gowns, an han washing lower the probability of infecting existing contacts. Dring an otbreak of a new infectios isease, the patient bren to hospitals may be so severe that health care officials cannot reasonably lower the nmber of contacts between HCWs an patients. Instea, as with SARS, they often implement strict hygienic precations that lower transmissibility (Le et al., 2003; McDonal et al., 2004). Fig. 7 illstrates the impact of varios levels of transmission recing interventions within hospitals. Here we assme that the average transmission rate along irecte eges only (T ) is rece. This moels hygienic precations taken by HCWs while treating sspecte cases of the isease. If a HCW becomes infecte, the threat remains high becase of a large nmber of robability of an epiemic 1 0.8 0.6 0.4 0.2 0 0% 25% 50% 95% Transmission rection Fig. 7. Hospital-base intervention. The probability of a large-scale epiemic ecreases as the HCWs se increasingly strict hygienic precations for a isease originally above the epiemic threshol, T ¼ T ¼ 0:1. The x-axis gives the percent rection in transmissibility along irecte eges pointing from members of the general pblic to HCWs. 1 robability of becoming infecte 0.8 0.6 0.4 0.2 Commnity Non-HCW HCW 0 0% 25% 50% 95% Transmission rection Fig. 8. Inivial precations. An inivial can lower the probability that he or she will become infecte ring an epiemic by taking measres that limit transmission. The x-axis gives the percent rection in transmissibility between the inivial an all of his or her irecte an nirecte contacts for a isease originally above the epiemic threshol, T ¼ T ¼ 0:1. The average likelihoo of infection across the entire poplation is shown in black bars. The benefit of intervention is mch greater for members of the general pblic (gray bars) than for HCWs (white bars).
412 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 ncontrolle nirecte contacts with other HCWs an patients who are hospitalize for other conitions. These measres wol therefore be more effective if they were extene to all HCW patient interactions. In sm, the se of transmission recing interventions by HCWs treating sspecte cases will protect the poplation only when they block transmission to hospital personnel entirely. In the absence of organize intervention, inivials may choose to take precations. Before mch was known abot SARS, HCWs mae inivial choices abot prevention, an later in the otbreak, some members of the general pblic volntarily wore facemasks (Tang an Wong, 2004). In Fig. 8, we show the personal impact of sch precations. On average, taking rastic transmission recing measres can significantly lower the probability of becoming infecte ring an epiemic. Yet HCWs are not nearly as protecte by sch inivial measres as are members of the general pblic. This stems from the sheer nmbers of potential contacts between HCWs an infecte patients. 4. Discssion We have erive a nmber of important epiemiological qantities for semi-irecte contact networks in which the average transmissibility can be ifferent for irecte an nirecte contacts. When there are, in fact, two ifferent transmission rates, the epiemic threshol becomes a line iviing the space of transmission rates into a region in which there are only small otbreaks that ie ot before reaching a sizable fraction of the poplation an another region in which an epiemic is possible (Fig. 3). Above the epiemic threshol, semi-irecte networks are more complicate than nirecte networks. When the inegree an ot-egree istribtions iffer, then so o the probability of an epiemic an the expecte incience shol one occr. We have illstrate the ifferences between nirecte, semi-irecte, an irecte networks sing three simple networks that share the same total nmber of contacts. The gap between the probability an expecte size of an epiemic is non-existent for the nirecte networks, qite large for the irecte network, an somewhere in between for the semiirecte network (Fig. 5). In aition to these fnamental epiemiological qantities, we have also calclate the probability of an epiemic as a fnction of the egree of the first case an the impact of control measres on the complying inivial an the poplation as a whole. We have applie these methos to sty the pivotal role of hospitals in the sprea of air-borne iseases throgh commnities. Worlwie otbreaks of SARS between November 2002 an May 2003 increase pblic awareness abot the evastating hman, economic an psychological impact of emerging infectios iseases. SARS probably emerge in Sothern China from an animal reservoir an was transmitte primarily throgh respiratory roplets an seconarily throgh aerosolize gastrointestinal secretions (Donnelly et al., 2003). From the beginning, SARS exhibite istinctive epiemiological patterns. Dring its initial for months of sprea in China, 32% of confirme cases were HCWs an 39% were foo hanlers (hence the hypothesis that cooking wil animals was the primary rote of SARS transmission into hman poplations), yet there were no cases among schoolchilren or hosewives (X et al., 2004). As SARS sprea ot from China, the fate of otbreaks was tightly linke to containment efforts within hospitals (Le et al., 2003; McDonal et al., 2004). For example, the first cases of SARS in Vancover an Toronto were infecte almost simltaneosly while staying in Hotel M in Hong Kong. Whereas the Toronto case sparke a sizeable otbreak that involve extensive hospital-base transmission, no seconary cases occrre from the initial Vancover case. The sccessfl containment in Vancover may have stemme from rigoros hospital precations. In particlar, the Vancover emergency room at which the first case soght treatment ha recently participate in an infection control ait that emphasize the importance of barrier precations for all acte onset respiratory infections (Worl Health Organization, 2003, 2004; Skowronski et al., 2005). In contrast, patient zero in Toronto ie at home as an niagnose case of SARS after infecting several relatives. The first case to arrive in a Toronto hospital (on March 7, 2003) was a secon-generation, locally acqire case. He was treate with neblize salbtamol in the emergency room, where he remaine for 18 h withot special precations. After 21 h, he was place in air-borne isolation in the ICU for possible tberclosis, an roplet an contact precations were not applie ntil his forth ay in the hospital. By the time he ie on March 13, he ha infecte several HCWs an thereby exacerbate the Toronto otbreak (otanen et al., 2003; Varia et al., 2003). Given the importance of hospitals to the transmission an control of iseases like SARS, we have evelope a mathematical framework that explicitly moels the flow of patients into hospitals ring otbreaks. In particlar, we emonstrate that a semi-irecte contact network can captre a conitional contact between a layperson an a HCW that only occrs if the layperson becomes infecte an goes to the hospital. We can rapily preict the sprea of isease throgh sch a network sing an extension of the methos evelope in (Newman, 2002) an (Meyers et al., 2005). When we a interactions between HCWs an infecte patients to or moel, the preicte epiemic threshol the critical transmission rate above which otbreaks may evolve into fll-blown epiemics ecreases an the risk of infection for HCWs ramatically increases. Frthermore, interventions targete at recing the likelihoo of transmission from patients an HCWs may significantly lower the likelihoo of an epiemic. Ths, moels that ignore hospital-base
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 413 transmission may nerestimate both the threat of an epiemic an the impact of control measres targete at protecting HCWs from the onslaght of patients they may face ring an otbreak. The extension of contact network epiemiology to semi-irecte graphs will allow s to bil an rapily analyse more realistic moels of infectios isease transmission when there are asymmetries in isease casing contacts. As or hospital example emonstrates, sch moels may provie important new insights into epiemiological patterns an pblic health strategy. Acknowlegments The athors wol like to thank Robert Brnham an Danta Skowronski of the University of British Colmbia Centre for Disease Control for fritfl iscssions. This work was spporte in part by grants from the Canaian Instittes of Health Research (CIHR) to the Canaian SARS Investigators that incles L.A.M an B.. (FRN: 67803), from the National Science Fonation (DEB-0303636) to L.A.M., an from the James S. McDonnell Fonation an the National Science Fonation (DMS-0234188 an DMS-0405348) to M.E.J.N. The Santa Fe Institte an CIHR spporte the working visits of B.., L.A.M., an M.E.J.N. Appenix A A.1. robability generating fnctions for the nmber of occpie (infecte) eges Following Newman (2002), we write the generating fnction for the nmber of occpie eges of a vertex in the form "!! # Gðx; y; ; T ; T Þ ¼ X1 X 1 X 1 X 1 X 1 X 1 j p jkm T a a¼0 b¼0 c¼0 j¼a k¼b m¼c a ð1 T k Þ j a T b b ð1 T m Þ!T k b c c ð1 T Þ m c x a y b c ¼ X "!! # X j j X k k p jkm ðt xþ a ð1 T Þ j a ðt yþ b Xm m k b ð1 T Þ!ðT Þ c ð1 T Þ m c. jkm a¼0 a b c¼0 c Applying the binomial formla ðða þ bþ n ¼ n k¼0 an k b k Þ, this generating fnction simplifies to Gðx; y; ; T ; T Þ ¼ X jkm b¼0 p jkm ð1 T þ xt Þ j ð1 T þ yt Þ k ð1 T þ T Þ m ð47þ ¼ Gð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ. ð48þ We similarly erive the probability generating fnction for the nmber of occpie eges (excling the arrival ege) emanating from a vertex arrive at by following a ranomly chosen ege: H ðx; y; ; T ; T Þ ¼ H ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ, (49) H r ðx; y; ; T ; T Þ ¼ H r ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ, (50) H ðx; y; ; T ; T Þ ¼ H ð1 þ ðx 1ÞT ; 1 þ ðy 1ÞT ; 1 þ ð 1ÞT Þ. (51) Eq. (48) implies that Gðx; y; ; 1; 1Þ ¼ Gðx; y; Þ, (52) Gð1; 1; 1; T ; T Þ ¼ Gð1; 1; 1Þ, (53) G ð1;0;0þ ð1; 1; 1; T ; T Þ ¼ T G ð1;0;0þ ð1; 1; 1Þ, (54) G ð0;1;0þ ð1; 1; 1; T ; T Þ ¼ T G ð0;1;0þ ð1; 1; 1Þ, (55) G ð0;0;1þ ð1; 1; 1; T ; T Þ ¼ T G ð0;0;1þ ð1; 1; 1Þ an similarly for H, H r, an H. (56)
414 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 A.2. Derivation of the size of a small otbreak As explaine in the text, the generating fnctions for the size of an otbreak beginning with a ranomly chosen ege are given by h ðw; T ; T Þ ¼ wh ð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ, (57) h ðw; T ; T Þ ¼ wh ð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ (58) for irecte an nirecte starting eges, respectively. Frthermore, the generating fnction for the size of an otbreak starting from a ranomly chosen vertex is given by gðw; T ; T Þ ¼ wgð1; h ðw; T ; T Þ; h ðw; T ; T Þ; T ; T Þ. (59) Here we erive the expecte size of sch an otbreak hsi ¼ X s s s ðt ; T Þ ¼ g 0 ð1; T ; T Þ, (60) where the prime enotes ifferentiation with respect to the first variable. Differentiating eqations (15), (16), an (17), we fin g 0 ð1; T ; T Þ ¼ 1 þ G ð0;1;0;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ þ G ð0;0;1;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ, (61) h 0 ð1; T ; T Þ ¼ 1 þ H ð0;1;0;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ þ H ð0;0;1;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ, (62) h 0 ð1; T ; T Þ ¼ 1 þ H ð0;1;0;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ þ H ð0;0;1;0þ ð1; 1; 1; T ; T Þh 0 ð1; T ; T Þ, (63) where we have mae se of the fact that h ð1; T ; T Þ ¼ h ð1; T ; T Þ ¼ 1. Solving Eqs. (62) an (63) simltaneosly we fin h 0 ð1; T ; T Þ ¼ h 0 ð1; T ; T Þ ¼ ð1 H ð0;1;0;0þ ð1 H ð0;1;0;0þ 1 H ð0;0;1;0þ Þð1 H ð0;0;1;0þ 1 H ð0;1;0;0þ Þð1 H ð0;0;1;0þ þ H ð0;0;1;0þ Þ H ð0;0;1;0þ þ H ð0;1;0;0þ Þ H ð0;0;1;0þ, (64) H ð0;1;0;0þ H ð0;1;0;0þ, (65) where the argments of all generating fnctions are set to (1,1;1;T,T ). Sbstitting these expressions into Eq. (61), we calclate the expecte size of an otbreak beginning at a ranom vertex: hsi ¼ 1 þ Gð0;1;0;0Þ ð1 H ð0;0;1;0þ ð1 H ð0;1;0;0þ þ H ð0;0;1;0þ Þ þ G ð0;0;1;0þ ð1 H ð0;1;0;0þ Þð1 H ð0;0;1;0þ Þ H ð0;0;1;0þ H ð0;1;0;0þ þ H ð0;1;0;0þ Þ. (66) Eqs. (52) (56) allow s to separate transmissibilities T an T from the semi-irecte egree istribtions as follows: hsi ¼ 1 þ T G ð0;1;0þ ð1 T ðh ð0;0;1þ ð1 T H ð0;1;0þ H ð0;0;1þ ÞÞ þ T G ð0;0;1þ ð1 T ðh ð0;1;0þ Þð1 T H ð0;0;1þ Þ T T H ð0;0;1þ where the argments of all generating fnctions are now set to (1,1;1). A.3. Derivation of the size an probability of a large epiemic H ð0;1;0þ H ð0;1;0þ ÞÞ, (67) Sppose we start at a ranomly chosen ege an move backwars along irecte eges an along nirecte eges, traversing each irecte an nirecte ege with probability T or T, respectively. Then the istribtion of the sizes of the reslting components is generate by h r (w; T, T ) (if starting from a ranom irecte ege) an h r (w; T, T ) (if starting from a ranom nirecte ege) which satisfy an h r ðw; T ; T Þ ¼ wh r ðh r ðw; T ; T Þ; 1; h r ðw; T ; T Þ; T ; T Þ (68) h r ðw; T ; T Þ ¼ wh ðh r ðw; T ; T Þ; 1; h r ðw; T ; T Þ; T ; T Þ. (69)
ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 415 It follows that the istribtion of components from which a ranomly chosen vertex (rather than ege) can be reache is generate by g r ðw; T ; T Þ ¼ wgðh r ðw; T ; T Þ; 1; h r ðw; T ; T Þ; T ; T Þ. (70) The fraction of the graph fille by vertices for which the corresponing component is finite in size is then given by g r (1;T, T ) an hence S ot ¼ 1 g r ð1; T ; T Þ giving S ot ¼ 1 Gða; 1; b; T ; T Þ, (71) where a h r ð1; T ; T Þ an b h r ð1; T ; T Þ are soltions of a ¼ H r ða; 1; b; T ; T Þ; b ¼ H ða; 1; b; T ; T Þ. (72) Translating into epiemiological terms, we can preict the size of a large-scale epiemic from the egree istribtion an transmissibility with S size ¼ S ot ¼ 1 X jkm p jkm ð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m, (73) where a an b are soltions of a ¼ jkm kp jkmð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m jkm kp, (74) jkm b ¼ jkm mp jkmð1 þ ða 1ÞT Þ j ð1 þ ðb 1ÞT Þ m 1 jkm mp. (75) jkm Similarly, one can calclate S in, the size of the GSCC pls the GIN, which is the fraction of vertices from which an extensive nmber of others can be reache. In epiemiology, this is S prob, the probability that a single ranomly place infection will spark a large-scale epiemic. By analogy with Eqs. (71) an (72), S in is given by where S in ¼ 1 Gð1; a; b; T ; T Þ, (76) a ¼ H ð1; a; b; T ; T Þ; b ¼ H ð1; a; b; T ; T Þ. (77) In terms of the egree istribtion an transmissibility of isease, Eqs. (76) an (77) become S prob ¼ S in ¼ 1 X jkm p jkm ð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m, (78) where a an b are soltions of a ¼ jkm jp jkmð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m jkm jp, (79) jkm b ¼ jkm mp jkmð1 þ ða 1ÞT Þ k ð1 þ ðb 1ÞT Þ m 1 jkm mp. (80) jkm A.4. Epiemic qantities for completely irecte an completely nirecte graphs A.4.1. Basic qantities The epiemic threshol, expecte size of a small otbreak, probability of a large scale epiemic an the expecte size of sch an epiemic have been erive previosly for both nirecte networks (Newman, 2002) an completely irecte networks (Schwartz et al., 2002). For irecte networks, the expecte size of a small otbreak is given by hsi ¼ 1 þ TGð0;1Þ ð1; 1Þ 1 TH ð0;1þ ð1; 1Þ (81)
416 ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 when the enominator on the right-han sie is greater than zero. This yiels an epiemic threshol of T c ¼ 1 H ð0;1þ ð1; 1Þ. (82) For transmissibility above T c, the expecte size of an epiemic is given by S size ¼ 1 X jk where v is the soltion to the eqation jk v ¼ kp jkð1 þ ðv 1ÞTÞ j jk kp jk p jk ð1 þ ðv 1ÞTÞ j, (83) an the probability that sch an epiemic will arise in the first place is given by S prob ¼ 1 X jk p jk ð1 þ ðw 1ÞTÞ k, (85) where w is the soltion to the eqation jk w ¼ jp jkð1 þ ðw 1ÞTÞ k jk jp. (86) jk For nirecte networks, the expecte size of a small otbreak is given by hsi ¼ 1 þ TG0 ð1þ 1 TH 0 ð1þ with an epiemic threshol of T c ¼ 1 H 0 ð1þ. (88) For T4T c, the probability an expecte size of an epiemic in an nirecte network are ientical an given by S ¼ S size ¼ S prob ¼ 1 X m p m ð1 þ ð 1ÞTÞ m, (89) where is the soltion to the eqation m ¼ mp mð1 þ ð 1ÞTÞ m 1 m mp. (90) m (84) (87) A.4.2. Initial conitions an inivial risk First consier a completely irecte network. If patient zero has ot-egree k, then the probability that he or she will spark a large-scale epiemic is given by e k ¼ 1 ð1 T þ TwÞ k, (91) where w is the soltion to (86). If he or she complies with an intervention that lowers the probability of transmission to others by a factor fð0pfp1þ, then the probability of sparking an epiemic is reces to e f k ¼ 1 ð1 ft þ ftwþk. (92) If there is an initial otbreak of N cases (of nknown egree), then the probability of a large-scale epiemic is given by 1 jk jp jk ð1 TþTwÞk jk jp jk! N, (93) where w is the soltion to (86). Dring a large-scale epiemic, the probability that a ranomly chosen vertex of in-egree j will become infecte is given by n j ¼ 1 ð1 T þ TvÞ j, (94)
where v is the soltion to (84). If he or she complies with an intervention that lowers the likelihoo of transmission from infecte contacts by a factor fð0pfp1þ, then the probability of infection is rece to n j ¼ 1 ð1 ft þ ftvþ j. (95) We next give the analogos formlae for nirecte networks. These were originally erive in Meyers et al. (2005). If patient zero has egree k, then the probability that he or she will spark a large-scale epiemic is given by e k ¼ 1 ð1 T þ TÞ k, (96) where is the soltion to (90). If he or she complies with an intervention that lowers the probability of transmission to others by a factor fð0pfp1þ, then the probability of sparking an epiemic is reces to e f k ¼ 1 ð1 ft þ ftþk. (97) If there is an initial otbreak of N cases (of nknown egree), then the probability of a large-scale epiemic is given by! N 1, (98) k kp k ð1 TþTÞk 1 k kp k ARTICLE IN RESS L.A. Meyers et al. / Jornal of Theoretical Biology 240 (2006) 400 418 417 where is the soltion to (90). Dring a large-scale epiemic, the probability that a ranomly chosen vertex of egree k will become infecte is given by n k ¼ 1 ð1 T þ TÞ k, (99) where is the soltion to (90). If he or she complies with an intervention that lowers the likelihoo of transmission from infecte contacts by a factor fð0pfp1þ, then the probability of infection is rece to n k ¼ 1 ð1 ft þ ftþ k. (100) References Anerson, R.M., May, R.M., 1991. Infectios Diseases of Hmans, Dynamics an Control. Oxfor University ress, Oxfor. Anersson, H., 1998. Limit theorems for a ranom graph epiemic moel. Ann. Appl. rob. 8, 1331 1349. Avenano, M., Derkach,., Swan, S. Clinical corse an management of SARS in health care workers in Toronto: a case series. CMAJ 168, 1649 1660. Ball, F., Mollison, D., Scalia-Tomba, G., 1997. Epiemics with two levels of mixing. Ann. Appl. rob. 7, 46. BC Stats, 2002. Labor force employment, nemployment, relate rates. http://www.bcstats.gov.bc.ca/ata/lss/labor.htm BC Stats, 2003. 2001 censs profile of British Colmbia s regions: Greater Vancover regional istrict, www.bcstats.gov.bc.ca Bozzette, S., Boer, R., Bhatnagar, V., Brower, J., Keeler, E., Morton, S., Stoto, M., 2003. A moel for a smallpox-vaccination policy. N. Engl. J. Me. 348, 416 425. Centre for Health Services an olicy Research, 2002. The British Colmbia Health Atlas, secon e. http://www.chspr.bc.ca/research/ healthatlas.htm#2ne Donnelly, C.A., Ghani, A.C., Leng, G.M., Heley, A.J., Fraser, C., Riley, S., Ab-Raa, L.J., Ho, L.-M., Thach, T.-Q., Cha,., Chan, K.-., Lam, T.-H., Tse, L.-Y., Tsang, T., Li, S.-H., Kong, J.H.B., La, E.M.C., Fergson, N.M., Anerson, R.M., 2003. Epiemiological eterminants of sprea of casal agent of severe acte respiratory synrome in Hong Kong. Lancet 1. Fergson, N.M., Garnett, G.., 2000. More realistic moels of sexally transmitte isease transmission ynamics: sexal partnership networks, pair moels, an moment closre. Sex Transm. Dis. 27, 600. Hethcote, H., 2000. Mathematics of infectios iseases. SIAM Rev. 42, 599. Jagers,., 1975. Branching rocesses with Biological Applications. Wiley, Lonon. Keeling, M.J., Woolhose, M.E., May, R.M., Davies, G., Grenfell, B.T., 2003. Moelling vaccination strategies against foot-an-moth isease. Natre 421, 136 142. Kretzschmar, M., van Dynhoven, Y.T., Sverijnen, A.J., 1996. Moeling prevention strategies for gonorrhea an Chlamyia sing stochastic network simlations. Am. J. Epiemiol. 144, 306 317. Le, D., Bloom, S., Ngyen, Q., Maloney, S., Le, Q., Leitmeyer, K., Bach, H., Reynols, M., Montgomery, J., Comer, J., Horby,., lant, A., 2003. Lack of SARS transmission among pblic hospital workers, Vietnam. Emerg. Infect. Dis. 10, 265 268. Lloy, A.L., May, R.M., 2001. Epiemiology. How virses sprea among compters an people. Science 292, 1316. Longini, I.M., 1988. A mathematical moel for preicting the geographic sprea of new infectios agents. Math. Biosci. 90, 367. McDonal, L., Simor, A., S, I., Maloney, S., Ofner, M., Chen, K., Lano, J., McGeer, A., Lee, M., Jernigan, D., 2004. SARS in healthcare facilities, Toronto an Taiwan. Emerg. Infect. Dis. 10, 777 781. Meyers, L.A., Newman, M.E.J., Martin, M., Schrag, S., 2003. Applying network theory to epiemics: control measres for Mycoplasma pnemoniae otbreaks. Emerg. Infect. Dis. 9, 204. Meyers, L.A., orbohlol, B., Newman, M.E.J., Skowronski, D.M., Brnham, R., 2005. Network theory an SARS: preicting otbreak iversity. J. Theor. Biol. 232, 71 81. Morris, M., 1995. Data riven network moels for the sprea of isease. In: Mollison, D. (E.), Epiemic Moels: Their Strctre an Relation to Data. Cambrige University ress, Cambrige, pp. 302 322.
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