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1 Eames, KTD (2007) Contact tracing strategies in heterogeneous populations. Epiemiology an infection, 135 (3). pp ISSN DOI: Downloae from: DOI: /S Usage Guielines Please refer to usage guielines at or alternatively contact Available uner license: Copyright the publishers

2 University of Warwick institutional repository: This paper is mae available online in accorance with publisher policies. Please scroll own to view the ocument itself. Please refer to the repository recor for this item an our policy information available from the repository home page for further information. To see the final version of this paper please visit the publisher s website. Access to the publishe version may require a subscription. Author(s): K. T. D. EAMES Article Title: Contact tracing strategies in heterogeneous populations Year of publication: 2007 Link to publishe version: /S Publisher statement: None

3 Epiemiol. Infect. (2007), 135, f 2006 Cambrige University Press oi: /s Printe in the Unite Kingom Contact tracing strategies in heterogeneous populations K. T. D. EAMES* Department of Biological Sciences an Mathematics Institute, University of Warwick, Coventry, UK (Accepte 25 April 2006; first publishe online 20 July 2006) SUMMARY Contact tracing is a well-establishe isease control measure that seeks to uncover cases by following chains of infection. This paper examines mathematical moels of both single-step an iterative contact tracing schemes an analyses the ability of these proceures to trace core groups an the sensitivity of the intervention to the timescale of tracing. An iterative tracing process is shown to be particularly effective at uncovering high-risk iniviuals, an thus it provies a powerful public health tool. Further targeting of tracing effort is consiere. When the population exhibits like-with-like (assortative) mixing the require effort for eraication can be significantly reuce by preferentially tracing the contacts of high-risk iniviuals; in populations where iniviuals have reliable information about their contacts, further gains in efficiency can be realize. Contact tracing is, therefore, potentially an even more potent tool than its present usage suggests. INTRODUCTION Contact tracing is a sophisticate control measure, applicable to a wie range of infectious iseases. It has been use to combat infections such as smallpox, SARS, tuberculosis, an a large number of sexually transmitte iseases (STDs) [1 9]. Unlike common control tools such as vaccination or public health campaigns, contact tracing is a multi-stage process potentially requiring a large investment of health-care provier time. The first stage of contact tracing requires the iagnosis of an infecte inex case; then any likely contacts of this inex case must be etermine, notifie, an treate; any contacts subsequently foun to be infecte become new inex cases an the process is repeate [1, 4, 9]. Because of the time an resources that contact tracing requires, * Author for corresponence: Dr K. T. D. Eames, Department of Biological Sciences an Mathematics Institute, University of Warwick, Gibbet Hill Roa, Coventry CV4 7AL, UK. ( K.Eames@warwick.ac.uk) mathematical moels can be useful in attempting to unerstan an explain when an whether contact tracing is a successful strategy. Previous moelling work has investigate the use of contact tracing in the context of a number of infectious iseases, in particular smallpox [5, 10 13] an STDs [6, 14 17]. In the latter instance contact tracing has been applie for many years whereas in the former it is uner consieration as an emergency measure to eal with a new outbreak of infection. It has been suggeste that, when symptoms can be easily ientifie an when contact tracing can be carrie out iteratively an rapily, it can be a highly effective measure [15 17]. In the case of STDs, contact tracing is particularly useful for uncovering asymptomatic cases (which can, nevertheless, be ientifie in the laboratory), thus proviing a way of treating iniviuals who woul otherwise remain infectious in the population [18]. The extent to which the iealize contact-tracing proceure escribe above is possible varies with

4 444 K. T. D. Eames circumstances an with the characteristics of the infection against which it is use. For instance, when there is no curative treatment available ientifie cases may nee to be quarantine rather than treate [13]. If there is no quick an reliable iagnostic test it may be necessary to isolate all contacts, whether infecte or not, until the emergence, or otherwise, of symptoms emonstrates their infection status [5, 10]. These time elays an the large numbers of patients involve make it ifficult to maintain chains of contact tracing an to trace quickly in such circumstances. In this paper we consier the factors that make contact tracing successful an examine the robustness of the intervention when these factors are remove. We focus specifically on the iterative properties of contact tracing an the spee at which contacts can be notifie. We consier the possibility of further targeting tracing efforts. In certain circumstances, by preferentially tracing particular contacts, the intervention effort require to eraicate infection can be reuce. METHODS S Fig. 1. An example of a simple mixing network. Iniviuals are represente by circles, partnerships by lines. The infection status of three iniviuals is inclue for illustration. The two highlighte iniviuals contribute 1 towars [SI], the number of susceptible infecte partnerships; along with the iniviual top right they comprise a SII triple. Contact tracing is intrinsically linke to networks of interactions within populations, such as sexual partnership networks for STDs [7, 19 21]. Transmission of infection takes place through inter-person connections [6, 7, 19 22] an these establishe links also allow the progress of infection to be trace. We efine a mixing network to be the set of all iniviuals an all links between iniviuals that coul allow transmission of infection a sexual mixing network I I woul contain information about sexual partnerships (Fig. 1). Contact tracing cannot be attempte without knowlege of the contacts of an iniviual, whether social or sexual, an obtaining this information is a vital step in the tracing process; we therefore use network moelling methos to stuy this intervention. We use pair-wise moels [23] as a robust metho of representing epiemics within a mixing network. Pair-wise moels treat connecte pairs of iniviuals as their basic variable an therefore lie between the more usual ranom mixing approaches [24, 25] an full stochastic simulations of complete networks [12, 26]. By explicitly moelling the essential unit of isease transmission an interaction between an infecte an a susceptible iniviual pair-wise moels are able to inclue the mixing behaviour observe in networks an can capture, to a large extent, the sprea of infection on networks. Although they cannot inclue much of the large-scale structure of networks, pair-wise moels can capture the local network structure, which is of primary importance in isease transmission, an they have been emonstrate to be accurate an aaptable tools [14]. Pair-wise moels can reaily be parameterize once the istribution of partnerships (who mixes with whom) in the population is known, an therefore o not require complex an time-consuming evaluation of complete mixing networks. Pair-wise moels retain the flexibility of more conventional approaches, an have been aapte to stuy heterogeneous populations, monogamous interactions, an contact tracing [14, 15, 17, 27, 28]. In this paper we consier simple infections in which iniviuals can be in one of two states: susceptible or infectious [24, 25]. Iniviuals become infecte at a rate t per infecte partner an recover at rate g, following which they are once again susceptible. This form of highly simplifie susceptible infecte susceptible (SIS) moel is appropriate for many common STDs such as chlamyia an gonorrhoea [24, 25], an provies a framework within which further complexities can be introuce. Births an eaths are ignore, an the mixing network is assume to remain fixe. We efine the infection parameter, r, via r=t/g. The notation use in this paper is liste in the Table. We enote by [S] the number of susceptible iniviuals an by [I] the number of infecte iniviuals, an consier how these numbers will change over time. [I] will ecrease owing to recovery an will increase when infection is transmitte. Within a mixing network, infection can only sprea between network

5 Contact tracing strategies 445 Table. List of notation Notation Definition g, t Recovery rate, transmission rate per infectious contact r Infection parameter,=t/g [S], [I] Number of susceptible, infecte iniviuals [SI] Number of susceptible infecte partnerships within the network (numbers of other partnership types represente similarly) k Neighbourhoo size, i.e. number of partners in the mixing network [SSI] Number of susceptible susceptible infecte triples (numbers of other triple types represente similarly) f, f c Fraction of contacts trace, critical fraction to eraicate infection [T ] Number of iniviuals from which tracing is taking place c, c m Rate of tracing, rate of tracing from iniviuals with m neighbours a x1 Duration of the contact tracing process R 0 Basic reprouctive ratio: number of seconary cases generate by one inex case in a wholly susceptible population M Total number of partnerships from each risk group E, h Available tracing effort, fraction of effort assigne to the high-risk group neighbours. We enote by [SI] the number of connections within the network between a susceptible an an infecte iniviual; it is only along these connections that infection can be transmitte. We can now form ifferential equations for the evolution of the numbers of susceptible an infecte iniviuals [23]: [S ]=xt[si ]+g[i ], (1) t [I ]=t[si ]xg[i ]: (2) t To be iterate, we must also calculate how [SI], the number of susceptible infecte pairs, changes over time; [SI] increases when infection is introuce into a susceptible susceptible pair (which can take place if one of the susceptibles has an infecte partner) or if recovery occurs within an infecte infecte pair; [SI] ecreases if the susceptible iniviual becomes infecte either from within the pair or by an external partner or if the infecte iniviual recovers. We can now write a ifferential equation escribing the ynamics of [SI]: [SI ]=t([ssi ]x[isi ]x[si ])+g([ii ]x[si ]): (3) t Here [SSI] is the number of triples within the network consisting of a central susceptible iniviual with a susceptible an an infecte partner, an [ISI] similarly. Within this equation they relate to infection entering the pair of interest from outsie. The three pair terms in this equation relate to within-pair processes (infection or recovery). See Figure 1 for an illustration of these efinitions. As with the equations for /t[s] an /t[i], new terms, the triples, have been introuce. We coul continue the process, moelling these in terms of triples an quaruples, but this woul rapily become unfeasible, requiring both a large number of equations an a great eal of ata to parameterize. Instea, in orer to close the system, triples are evaluate in terms of pairs an singles using the moment closure approximation [ABC ] (kx1)[ab][bc], (4) k[b] where A, B, an C can represent either S or I. k is the neighbourhoo size (i.e. the number of contacts) of the central type-b iniviual [23]. The approximation is erive as follows: an [ABC] triple requires an [AB] pair; the B iniviual in any such pair has (kx1) further contacts. Out of a total of k[b] contacts of type-b iniviuals within the population, [BC] are with type-c iniviuals, thus any one contact has a probability of [BC]/k[B] of being with a type-c iniviual. Combining these gives the approximation. The system can easily be extene to consier populations containing iniviuals with a range of neighbourhoo sizes [14]; the complete set of equations is given in the Appenix. In this paper we aapt this set of equations to consier two istinct contact tracing approaches: single-step an iterative tracing. In the former case a proportion of the partners of inex cases is treate

6 446 K. T. D. Eames Equilibrium prevalence Tracing=0% Tracing=10% 0 1 Tracing=25% Tracing=50% r Fig. 2. Equilibrium prevalence plotte against the infection parameter, r, for a range of tracing fractions, f, in a singlestep tracing moel, g=1 throughout, with r varie by altering t. A heterogeneous network was use to parameterize the moel, with iniviuals of neighbourhoo size ranging from 1 to 13. Neighbourhoo size, k r Fig. 3. The region in parameter space for which single-step tracing can eraicate infection in a homogeneous network with uniform neighbourhoo size k, shown in grey. Above the shae region infection persists whatever the tracing fraction; below the shae region persistence is impossible even when there is no tracing. concurrently with the inex patient, thus resulting in the recovery of pairs of iniviuals; this is an appropriate moel for certain STD situations when patients an their partners atten a clinic together, but oes not allow infecte contacts to become new inex cases. The secon moel inclues the iterative behaviour of tracing, with the infecte partners of any iagnose cases being sought for treatment an further tracing; in the iterative moel, contact tracing behaves as a hyperparasite of infection, spreaing through the infecte portion of the mixing network. CONTACT TRACING Single-step contact tracing In this moel, briefly introuce elsewhere [14], some fraction, f, of the contacts of an inex case receive treatment at the same time as the inex patient. Thus, a recovery event may involve more than one iniviual leaving the infecte class. Similar single-step approaches have been suggeste by other authors [5, 6, 10, 13], particularly in contexts where there is no simple an rapi iagnostic test available an where chains of tracing are, therefore, not seen. It is most applicable in circumstances where formal contact tracing proceures are not carrie out but when two iniviuals in a partnership seek treatment together. The governing equations for single-step tracing are: [S ]=xt[si ]+g[i ]+fg[ii ], (5) t [I ]=t[si ]xg[i ]xfg[ii ], (6) t the extra recovery term resulting from those occasions when two iniviuals are treate at once. As above, the system is close by moelling the ynamics of pairs within the network an applying the moment closure approximation (see Appenix for the complete set of equations). As shown in Figure 2, the aitional recovery through contact tracing results in a reuction in equilibrium prevalence. It might be hope that by sufficiently increasing the tracing fraction single-step tracing can eraicate infection altogether. However, this is not necessarily the case. For example, in a homogeneous population, in which each iniviual has k contacts, eraication is only possible rk(rk 2 xrkx rx2k)<1. As Figure 3 shows, there is only a relatively small region of parameter space in which single-step tracing can rive infection from the system. When contact tracing is restricte to a single step it can only have a limite effect on the impact of a pathogen. Iterative contact tracing The more usual form of contact tacing, as carrie out by GUM clinics, is an iterative process: any infecte

7 Contact tracing strategies 447 Equilibrium prevalence Tracing=0% Tracing=10% Tracing=25% Tracing=50% r Fig. 4. Equilibrium prevalence plotte against the infection parameter, r, for a range of tracing fractions in an iterative tracing moel, g=1 throughout, with r varie by altering t. The tracing fraction is given by c/(c+a). The same network was use as in Figure 2. contacts of an inex case are treate as new inex cases, an their contacts are also sought. In this way contact tracing spreas through the population, following chains of infection. To represent iterative contact tracing this moel, introuce in ref. [15], inclues an extra tracing class (enote T): following treatment iniviuals enter this tracing class an their contacts are trace. Any successfully trace contacts likewise enter the tracing class. We assume that tracing takes place at rate c an that the tracing process has a uration a x1, after which iniviuals return to the susceptible class. The governing equations for numbers in each class are: [S]=xt[SI ]+a[t ], (7) t [I ]=t[si ]xc[it ]xg[i ], (8) t [T ]=c[it ]+g[i ]xa[t ]: (9) t The c[it] terms represent contact tracing from iniviuals in the tracing class, T, leaing to the removal of infecte iniviuals into the tracing class. Once again, the ynamics of pairs are moelle explicitly, making use of the moment closure approximation (see Appenix for the complete set of equations). In this moel tracing occurs iteratively, with each inex case being-capable of generating further inex cases [16, 17, 28]; the tracing class behaves as a hyperparasite within the system, passing between infecte iniviuals. We see from Figure 4 that iterative tracing is an effective control measure leaing to a large reuction in infection prevalence. Comparing Figures 2 an 4 shows that iterative tracing has a much more ramatic impact than single-step tracing. As has been shown previously [15, 16], when the tracing process is rapi compare with the ynamics of infection (c an a are much larger than t an g) the critical tracing fraction the fraction of contacts, f c, that must be trace to eraicate infection is given by f c =1x1/R 0, where R 0 is the basic reprouctive ratio the average number of new cases generate by an infecte iniviual in an otherwise susceptible population. This relationship hols in a wie range of scenarios, incluing heterogeneous populations an asymptomatic infections [15], an has a relatively simple explanation: for eraication, each infection must generate no more than one untrace case, i.e. (1xf c )R 0 <1 Þ f c >(1x1/R 0 ). Immeiately a ifference between single-step an iterative tracing emerges: in the latter case it is always possible to eraicate infection. Part of the power of contact tracing comes from its ability to target interventions towars the most high-risk parts of the network; contact tracing involves surveying the partners of infecte cases an thus it tens to focus on high-risk iniviuals. Such iniviuals, especially when they mix with each other to form high-risk core groups, ten to ominate the ynamics of infection; isease is concentrate in core groups, particularly when population prevalence is low [25, 29 32]. Contact tracing alters the istribution of infection within the population because iniviuals at highest risk of infection are also the most likely to be trace. We use the mean number of contacts of infecte iniviuals in a heterogeneous network as a measure of the influence of core groups the higher this number, the more infection is restricte to high-risk iniviuals plotte in Figure 5 against equilibrium prevalence to examine how contact tracing affects the istribution of infection. In all cases, as anticipate, the greater the prevalence, the less it is concentrate in the core groups. When tracing is consiere, the results iffer greatly between the two forms of contact tracing. Single-step tracing has little effect on the istribution of infection through the population, whereas iterative tracing greatly reuces the ominance of core groups. By repeately following links from infecte cases, iterative tracing results in intervention efforts naturally focusing on areas of the network where prevalence

8 448 K. T. D. Eames (a) Mean neighbourhoo size of infecte Equilibrium prevalence Tracing=0% Tracing=10% Tracing=25% Tracing=50% (b) Mean neighbourhoo size of infecte Equilibrium prevalence Tracing=0% Tracing=10% Tracing=25% Tracing=50% Fig. 5. Mean neighbourhoo size of infecte iniviuals plotte against equilibrium prevalence for a range of tracing fractions, using the simulation results shown in Figures 2 an 4. (a) Single-step tracing; (b) iterative tracing. Also shown (ashe line) is the mean neighbourhoo size of the population. is highest, allowing the most ominant iniviuals to be contacte an treate. Because iterative tracing effectively targets high-risk parts of the mixing network, the ability of core groups to sustain infection is reuce, greatly aiing the eraication of infection. By contrast, in single-step tracing the core groups retain much of their ominance an eraication is consequently much more ifficult. Another issue affecting the success of contact tracing is the spee at which tracing is carrie out; for instance, it is of little benefit to trace all partners of an inex case if tracing oes not occur until infection has alreay sprea many further steps through the population [13]. The expression above for the critical tracing fraction epens on tracing being carrie out before seconary cases have ha a chance to transmit infection; if tracing is slower we woul expect to have to trace more iniviuals to eraicate infection. Figure 6 emonstrates the larger tracing fraction require to eraicate infection as the spees of transmission an tracing become similar. Although eraication remains possible, the critical tracing fraction becomes so large that it will selom be attainable in practice. Thus, contact tracing requires both rapi iagnostic tests an capable an well-resource health services. Targete contact tracing We have note that contact tracing irects intervention efforts towars at-risk parts of the mixing network, the neighbourhoo of infecte iniviuals. We now examine the possibility of further targeting Critical tracing fraction Λ= 0 2 Λ= Λ=1 Λ= R 0 Fig. 6. Critical tracing fraction necessary to eraicate infection for the iterative tracing moel plotte against R 0 for a range of timescale separations. The timescale separation, L, is efine by the ratio of the tracing an infection timescales: g=1 an a=lrg throughout; R 0 an tracing fraction are varie by changing t an c respectively. interventions, thus avoiing expening effort on subsections of the population where infection is rare. To investigate the value of targete contact tracing we consier a simplifie population structure, consisting of two groups with ifferent mixing properties, one high-risk (with 10 contacts) an one low-risk (with four contacts). The numbers of iniviuals in each group are selecte so that each group has the same total number of contacts, M. We inclue variation in population mixing patterns by ajusting the assortativity of the population [19, 33]. Assortativity is a measure of the extent to which

9 Contact tracing strategies 449 similar iniviuals interact; the more assortative the population, the more the mixing is like with like. In this two-group moel mixing can range from completely assortative, when all links are within group, to completely isassortative, when all links are between iniviuals in ifferent groups. We note that high assortativity leas to high R 0 whereas prevalence is maximize at some intermeiate assortativity [24, 30, 33]. The first result arises because of the ominance of the high-risk group uring the early stages of an epiemic: the more coherent this group, the higher R 0. In contrast, prevalence is maximize when there is both a sufficiently istinct core group to sustain high levels of infection within itself an enough contact between the core an non-core for isease to be wiesprea in the general population. The more assortative the population the greater the segregation of the groups an thus the greater the unevenness of the istribution of infection. Tracing effort can be apportione in various ways from entirely irecte towars one group to entirely irecte towars the other. Use of a simplifie population allows all possible choices of intervention targeting to be teste simply by allowing the fraction of effort applie to the high-risk group to vary between zero an one. We look here at two means of targeting tracing: targeting accoring to the characteristics of the inex case, an targeting accoring to the characteristics of the contact. First we consier interventions that alter the rate at which tracing is carrie out from inex cases. We write c m as the rate at which iniviuals are trace from an inex case with m partners, an allow this parameter to iffer between the two groups, epening on how tracing-effort is apportione. Specifically, if the available effort, E, has a fraction h assigne to the high-mixing group, c high is increase by he/m (the enominator is in place since the effort is split between all links that might be trace from this group). When h=1 all effort is spent tracing the contacts of the high-risk group. Figure 7 shows the effort require to eraicate infection when contact tracing is targete compare to the effort require when tracing is uniformly applie. We see that in some cases there is very little ifference between the two approaches. However, targete contact tracing is worthwhile when the population is assortative, an the lower the pre-intervention level of infection the greater the improvement. It is optimal to irect tracing efforts preferentially towars the contacts of core group iniviuals. This can be Relative targete effort r=1/ r=1/3 r=1/ Within-group mixing Fig. 7. The effect of mixing pattern on the require effort to eraicate infection, using the iterative tracing moel in a simplifie population consisting of two types of iniviual. Relative targete effort is efine to be the effort neee when tracing is optimally targete ivie by the effort require when tracing is applie uniformly. Within-group mixing is efine to be the proportion of contacts that are between iniviuals in the same group. In this case, effort is targete accoring to the properties of the inex case. explaine as follows: for tracing from one group to be more beneficial than tracing from another, the groups must have partners with ifferent levels of infection. The prevalence in the neighbourhoo of an inex case will epen partly on the inex case itself, but will also epen on the properties of the iniviuals in the neighbourhoo if the neighbourhoo consists of high-risk iniviuals, for instance, then they are likely to have been infecte via other sources. Thus, targeting is likely to be useful when the population is non-ranomly mixe, i.e. either assortative or isassortative. Moreover, for it to be worthwhile to contact particular groups the prevalence of infection must iffer between groups. Therefore, although isassortative mixing allows for particular types of iniviual to be contacte, the relatively even istribution of infection means that targeting is of little worth. Assortatively mixe populations, however, isplay unevenly istribute infection, so core group iniviuals shoul be sought, an this can be achieve through tracing from other core group members. The fact that targeting becomes more beneficial at lower levels of pre-intervention prevalence supports this argument when prevalence is low the istribution of infection is more uneven. We can conclue that there are occasions when contact tracing shoul be targete. We have also seen instances when targeting is not worthwhile an,

10 450 K. T. D. Eames Relative targete effort r=1/ r=1/3 r=1/ Within-group mixing Fig. 8. The effect of assortativity on the require effort to eraicate infection. As in Figure 7, but here effort is targete accoring to the properties of the contact. inee, the effort involve in attempting to target interventions may be such as to negate any positive effect. Targeting is only noticeably beneficial in assortative populations an it is unclear whether human mixing networks, whether social or sexual, are sufficiently assortative for targete tracing to be recommene [1, 29, 34, 35]. An aequate answer epens on a far more thorough unerstaning of the patterns of population mixing an the limitations an constraints inherent in human interactions. In principle, ajusting the tracing rate accoring to the characteristics of the inex case shoul be achievable; inex cases are seen by the health services an can be interviewe to etermine their relevant properties. Alternatively, we can consier targeting tracing epening on the characteristics not of the inex but of the contact: for example by attempting preferentially to trace core group iniviuals. This requires that inex cases can be questione not only about the ientities an numbers of their contacts, but also about their contacts attributes. To work effectively, this approach requires that iniviuals have goo knowlege about their partners an that they are prepare to ivulge this information. To examine this option requires only the smallest of ajustments to the moel: c m becomes the rate with which iniviuals with m partners are trace. We see from Figure 8 that, once again, there are situations when this form of targeting is highly beneficial. As before, the less even the sprea of infection the greater the benefit of targeting contact tracing, but here we o not require such highly assortative populations to make targeting worthwhile. Previously, only in such cases was it possible to be know which iniviuals were being trace, but if tracing is carrie out accoring to the properties of contacts, then it is always possible to specifically target core groups. If it is possible to trace base on the properties of contacts then the range of circumstances in which significant savings can be mae is noticeably extene. DISCUSSION Contact tracing is potentially a potent isease control measure. To operate at its fullest efficiency it requires sophisticate an highly traine operatives, capable of gathering the necessary information, coorinating efforts, an contacting iniviuals. Contact tracing relies on networks of interactions for its success an therefore requires network-base moelling techniques to capture the tracing process. Here we have use pair-wise moels of epiemics on networks to examine single-step an iterative contact-tracing schemes. Through the construction of moels of contact tracing we have seen that to be successful contact tracing must be carrie out rapily an must act iteratively. When these conitions are met the impact on infection prevalence is great; the natural targeting of tracing efforts towars the at-risk neighbourhoo of infecte inex cases allows core group iniviuals to be uncovere an treate. The automatic targeting of contact tracing is a property of the intervention that emerges over a series of tracing steps. A singlestep tracing scheme is therefore unable to access core groups as effectively: the failure to trace the partners of all uncovere cases means that infection can be rapily reintrouce. Inee, it is only for a very limite range of infection parameters that single-step tracing can eraicate infection, whereas iterative contact tracing is much more broaly effective. Despite the targeting inherent in contact tracing, there are circumstances in which further focusing of the intervention significantly reuces the tracing effort require to eraicate infection. In assortative networks, where mixing tens to be between iniviuals with similar characteristics, great savings are possible. In such networks the istribution of infection is highly skewe towars the core group a lack of interaction between the groups means that prevalence in the general population is lower an since the contacts of core iniviuals ten also to be in the core, preferential tracing from the core group allows tracing to target the areas of the network where

11 Contact tracing strategies 451 infection is most likely to be foun. In ranomly mixe networks, where the neighbourhoos of all iniviuals have broaly similar properties, there is little to be gaine from tracing preferentially from particular iniviuals. In isassortative networks prevalence is reasonably constant across the population so there are no obvious esirable targets for tracing. When tracing can be ajuste accoring to the characteristics of the target iniviual rather than the inex case, there is a greater range of populations in which eraication effort is reuce. In this case, core iniviuals can be specifically targete an so long as there is an uneven istribution of infection this proves beneficial. In particular, this form of tracing confers an avantage in ranomly mixe populations, not seen when tracing rates epen on the properties of the inex case alone. Although tracing accoring to the properties of the target is an attractive proposition, it is unclear to what extent it is practical. Stuies have suggeste that, in sexual mixing networks, iniviuals are not well-informe about their partners behaviour [36, 37], but, epenent on the biases present in iniviuals perceptions, there may be sufficient information for targete tracing to be feasible. Furthermore, there are cases where risk is highly correlate with characteristics such as occupation, ethnicity, or place of resience [20, 30, 32, 38], in which case relevant information about contacts may be more reaily available. Further network stuies are neee to clarify these issues, but it certainly appears that with goo network knowlege an consiere use of available resources traitional methos of isease control can be improve. APPENDIX SIS governing equations For completeness, the full set of pair-wise equations for an SIS type infection in a heterogeneous network is given below. In such a population, where the neighbourhoo size varies, iniviuals are labelle acccoring to both their infection status an their neighbourhoo size. Thus, [S m ] is the number of susceptible iniviuals with m contacts an [S m I n ] is the number of pairs consisting of a susceptible with m contacts an an infecte with n contacts. Other terms are efine analogously. All triples in the system are evaluate in terms of pairs an singles using the moment closure approximation: [A m B n C p ] [Am B n ][B n C p ](nx1) n[b n, (10) ] where A, B, an C can be either S or I. This approximation is erive from the same reasoning as outline in the main text. t [Sm ]= g[i m ]xt P 9 [S m I n ], n t [I m ]=xg[i m ]xt P [S m I n ], n t [Sm S n ]=xt P >= ([S m S n I q ]+[I q S m S n ])+g([s m I n ]+[I m S n ]), (11) q t [Sm I n ]= t P q t [I m I n ]= t P q ([S m S n I q ]x[i q S m I n ])xt[s m I n ]xg[s m I n ]+g[i m I n ], ([I m S n I q ]+[I q S m I n ])+t([s m I n ]+[I m S n ])x2g[i m I n ]: >; In orer to parameterize the system it is necessary to know the number of partnerships within the network between iniviuals of all neighbourhoo sizes; that is to say, we nee to know [mn], the number of pairs within the network consisting of one iniviual with m contacts an one with n contacts (ignoring infection status), for all values of m an n. This information can be obtaine by counting pairs within a specific mixing network or can be etermine to represent a population of interest. For example, in the targete tracing section we consier a population consisting of two groups, one with 10 contacts an one with 4; by setting [4 10]

12 452 K. T. D. Eames (the number of between-group contacts) appropriately we can obtain a population with a require level of between-group mixing. Governing equations for single-step contact tracing As escribe in the main text, the equations governing the ynamics of the numbers of susceptible an infecte iniviuals in the presence of single step contact tracing are [S]=xt[SI ]+g[i ]+fg[ii ], (12) t [I ]=t[si ]xg[i ]xfg[ii ], (13) t where f is the fraction of contacts that are trace. As in the tracing-free SIS case, we must form similar equations for the behaviour of the various pair types; reasoning as before gives [SS]=x2t[SSI ]+2g[SI ]+2fg[II ]+2fg[SII ], (14) t [SI ]=t([ssi ]x[isi ])xt[si ]xg[si ]+g(1xf)[ii ]+fg([iii ]x[sii ]), (15) t [II ]=2t[SI ]+2t[ISI ]x2g[ii ]x2fg[iii ]: (16) t Exactly the same moment closure approximation as use previously closes the system an allow it to be iterate. Governing equations for iterative tracing As escribe in the main text, the equations governing the ynamics of the numbers of susceptible, infecte, an tracing iniviuals in the presence of iterative contact tracing are [S]=xt[SI ]+a[t ], (17) t [I ]=t[si ]xc[it ]xg[i ], t (18) [T ]=c[it ]+g[i ]xa[t ], t (19) where c is the tracing rate an a x1 the tracing uration. Again, we form equations for the behaviour of pairs. [SS]=x2t[SSI ]+2a[ST ], (20) t [SI ]=t([ssi ]x[isi ])xt[si ]xg[si ]+a[it ]xc[sit ], (21) t [ST ]=xt[ist ]+g[si ]+a([tt ]x[st ])+c[sit ], (22) t [II ]=2t[ISI ]+2t[SI ]x2g[ii ]x2c[iit ], (23) t [IT ]=t[ist ]+g([ii ]x[it ])xa[it ]+c([iit ]x[tit ])xc[it ], (24) t [TT ]=2g[IT ]x2a[tt ]+2c[IT ]+2c[TIT ]: (25) t Exactly the same moment closure approximation as use previously closes the system an allow it to be iterate.

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