Correlated Infections: Quantifying Individual Heterogeneity in the Spread of Infectious Diseases

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1 American Journal of Epidemiology The Author Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of Public Health. All rights reserved. For permissions, please Vol. 177, No. 5 DOI: /aje/kws260 Advance Access publication: February 12, 2013 Practice of Epidemiology Correlated Infections: Quantifying Individual Heterogeneity in the Spread of Infectious Diseases C. Paddy Farrington*, Heather J. Whitaker, Steffen Unkel, and Richard Pebody * Correspondence to Dr. C. P. Farrington, Department of Mathematics and Statistics, Faculty of Mathematics, Computing and Technology, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom ( c.p.farrington@open.ac.uk). Initially submitted January 11, 2012; accepted for publication May 11, In this paper, we propose new methods for investigating the extent of heterogeneity in effective contact rates relevant to the transmission of infections. These methods exploit the correlations between ages at infection for different infections within individuals. The methods are developed for serological surveys, which provide accessible individual data on several infections, and are applied to a wide range of infections. We find that childhood infections are often highly correlated within individuals in early childhood, with the correlations persisting into adulthood only for infections sharing a transmission route. We discuss 2 applications of the methods: 1) to making inferences about routes of transmission when these are unknown or uncertain and 2) to estimating epidemiologic parameters such as the basic reproduction number and the critical immunization threshold. Two examples of such applications are presented: elucidating the transmission route of polyomaviruses BK and JC and estimating the basic reproduction number and critical immunization coverage of varicella-zoster infection in Belgium, Italy, Poland, and England and Wales. We speculate that childhood correlations stem from confounding of different transmission routes and represent heterogeneity in childhood circumstances, notably nursery-school attendance. In contrast, it is suggested that correlations in adulthood are route-specific. basic reprodfuction number; communicable diseases; correlations; disease transmission, infectious; frailty; heterogeneity; mass vaccination; serological tests Abbreviations: BKV, polyomavirus BK; CMV, cytomegalovirus; ESEN, European Sero-Epidemiology Network; JCV, polyomavirus JC; VZV, varicella-zoster virus. It has long been understood that the heterogeneity of a population with respect to factors that may enhance or inhibit the transmission of infections may influence the effectiveness of strategies to control such infections (1). In particular, estimates of the basic reproduction number and the critical immunization threshold that are derived without accounting appropriately for heterogeneity are likely to be biased. Therefore, allowing for individual heterogeneity in statistical and mathematical models of infectious disease is important. Such models often involve specifying contact rates between individuals. However, it is often difficult to decide what constitutes a contact and hence to specify what the relevant heterogeneities are, let alone measure them. When what constitutes a contact is clear, relevant heterogeneities can in principle be measured directly. This is the case for sexually transmitted infections, where contact means sexual contact and relevant heterogeneities include frequency of sexual contacts and rate of partner change. Heterogeneity can then be quantified explicitly through surveys of sexual behavior (2). However, for indirectly transmitted infections for example, infections transmitted by aerosol, by the fecal-oral route via contaminated food or water, or by fomites there are no contacts in any but a metaphorical sense. Additionally, while it may be known in broad terms which routes of transmission are involved, there is much less clarity about their relative importance. While some detailed studies of the relative importance of different transmission routes exist (3), these are uncommon. Often, all that can be done is to seek information on a proxy variable which might be expected to be correlated with relevant behavior. This is the rationale behind contact surveys using proxy measures such as having a 2-way conversation (4 6). Inevitably, 474

2 Correlated Infections 475 Table 1. Details of Surveys Included in a Study of Correlations Between Infections a Survey and Infection Pair (Location, Date) Age Range, years No. of Paired Samples Survey 1 (England and Wales, 1986) Mumps and rubella b ,585 Survey 2 (England and Wales, 1991) B19 and CMV ,331 B19 and rubella b ,700 CMV and rubella b B19 and BKV ,340 B19 and JCV ,340 BKV and CMV CMV and JCV BKV and rubella b JCV and rubella b BKV and JCV ,435 Survey 3 (England and Wales, ) Epstein-Barr virus and HSV ,893 Survey 4 (England and Wales, 1996) B19 and VZV ,747 B19 and HAV ,128 B19 and toxoplasma ,516 B19 and HPY ,829 HAV and VZV Toxoplasma and VZV ,226 HPY and VZV ,262 HAV and toxoplasma ,162 HAV and HPY ,263 HPY and toxoplasma ,632 Survey 5 (Belgium, ) B19 and VZV ,380 Survey 6 (Italy, ) B19 and VZV ,434 Survey 7 (Poland, ) B19 and VZV ,150 Abbreviations: B19, parvovirus B19; BKV, polyomavirus BK; CMV, cytomegalovirus; HAV, hepatitis A virus; HPY, Helicobacter pylori; HSV1, herpes simplex virus type 1; JCV, polyomavirus JC; VZV, varicella-zoster virus. a Data were obtained from 7 European serological surveys (18 21). b Males only. such an approach is approximate and may require post hoc adjustments (7). However, it can also provide important insights into transmission routes (8). Here we use a different approach to quantifying relevant heterogeneities, using correlations between infections in individuals. The rationale for the approach is as follows. If 2 infections are transmitted by a similar route, one might expect that the extent of heterogeneity in behavior relevant to the transmission of infection will be reflected by the strength of the correlation between the two infections. Thus, a person with a high activity level relevant to transmission of 2 infections (for example, eating habits or personal hygiene, in the case of fecal-oral infections, or a high rate of social interaction for infections transmitted by droplets) will be more likely to acquire both infections by a given age than a person with low activity levels. One benefit of the approach is that it does not require an explicit definition of what exactly such activity levels represent. This idea was first explored by Farrington et al. (9), and it has been applied to study the transmission of hepatitis B and C viruses (10) and to Epstein-Barr and herpes simplex type 1 viruses (11). Typically, blood samples collected from a defined population are tested for antibodies to several antigens. This gives rise to multivariate currentstatus data on several infections in the same individuals, with additional information on age, gender, etc. In this paper, we develop these ideas further. We use the correlations between different infections to gain a better understanding of how infections are transmitted, notably for infections with several possible transmission routes. We suggest that such correlations can help to elucidate likely transmission routes when these are not known. Further, the degree of heterogeneity inducing the correlation can be modeled, and this information can then be used to improve the estimates of epidemiologic parameters such as reproduction numbers, estimation of which typically accommodates only the effect of directly measured heterogeneities. In brief, we argue that correlations between infections open a window on individual behaviors which are difficult to measure, and induce heterogeneities of contacts which are difficult to define. We used recently developed methodology to take into account the fact that activity levels and hence the correlations they induce vary with age. This enabled us to characterize and quantify heterogeneities and how they evolved over time. We applied the methods to a wide variety of different data sets obtained in different serological surveys. We also explored 2 contrasting applications of this methodology: 1) to identification of routes of transmission of polyomavirus BK (BKV) and polyomavirus JC (JCV) and 2) to estimation of the basic reproduction number and critical immunization level for varicella-zoster virus (VZV) infection. STATISTICAL METHODS The statistical framework is described below in 4 subsections. The details are kept to a minimum; further details are provided in Web Appendix 1, available at oxfordjournals.org/. Incorporating age-dependent heterogeneity via frailty models Throughout, let x, y denote age. To begin with, consider a single infection, and suppose that age is the only measured attribute of that individual (the methods can readily

3 476 Farrington et al. Table 2. Main Routes of Transmission for Selected Infections Infectious Agent Main Route(s) of Transmission First Author, Year (Reference No.) Cytomegalovirus Intimate mucosal contact Heymann, 2008 (32) Epstein-Barr virus Oropharyngeal via saliva Heymann, 2008 (32) Helicobacter pylori Fecal-oral, oral-oral, foodborne Heymann, 2008 (32) Hepatitis A virus Fecal-oral, foodborne Heymann, 2008 (32) Herpes simplex virus type 1 Oropharyngeal via saliva Heymann, 2008 (32) Mumps virus Airborne, droplets or direct contact Heymann, 2008 (32) Parvovirus B19 Close contact with respiratory secretions Melegaro, 2011 (8); Heymann, 2008 (32) Toxoplasma gondii Foodborne, oral ingestion of feline feces Heymann, 2008 (32) Rubella virus Droplets or direct contact, aerosol Heymann, 2008 (32); Banatvala, 2004 (33) Varicella-zoster virus Close contact with respiratory secretions, airborne or droplets Melegaro, 2011 (8); Heymann, 2008 (32) be extended to include others). Suppose that all of that individual s unmeasured attributes or behaviors which are relevant to the transmission of this infection at age x may be Table 3. Associations (w) Between Paired Infections According to Likelihood of a Shared Transmission Route a Infection Pair w 95% CI Shared Main Route of Transmission Likely HPY and toxoplasma , HAV and HPY , HAV and toxoplasma , B19 and VZV (Poland) , Epstein-Barr virus and HSV , B19 and VZV (England and , Wales) Mumps and rubella b , B19 and VZV (Belgium) , B19 and VZV (Italy) , B19 and rubella b,c , Shared Main Route of Transmission Unlikely B19 and toxoplasma , HAV and VZV , B19 and HAV , B19 and HPY , B19 and CMV , Toxoplasma and VZV , CMV and rubella b,c , HPY and VZV , Abbreviations: B19, parvovirus B19; CI, confidence interval; CMV, cytomegalovirus; HAV, hepatitis A virus; HPY, Helicobacter pylori; HSV1, herpes simplex virus type 1; VZV, varicella-zoster virus. a Data were obtained from 7 European serological surveys (18 21). b Males only. c Ages 11 years. described by an activity level, which is a positive random variable u x with density f x (u x ) and mean 1; for simplicity, we shall assume that u x is a deterministic function of x and a finite set of age-independent random variables. The variance of u x, gðxþ ¼Varfu x g; thus represents the degree of unmeasured individual heterogeneity in the population at age x. Our aim is to estimate gðxþ for a range of infections and use these estimates to make inferences about the epidemiology of the infections. Let β 0 (x; y) denote the average effective contact rate between an individual of age x and an individual of age y in this population. This is the contribution of a typical infectious individual of age y to the instantaneous rate of infection of a typical susceptible individual of age x; here typical means average with respect to unmeasured heterogeneities. We extend this notion to encompass activity levels by denoting β(x, u x ; y, v y ) the contribution of an infectious individual of age y and activity level v y to the instantaneous rate of infection of a susceptible individual of age x with activity level u x. To make further progress, we assume that bðx; u x ; y; v y Þ¼u x b 0 ðx; yþv y : ð1þ This model is an elaboration of one first proposed by Coutinho et al. (12) and implemented by Farrington et al. (9). The assumption that individual activity levels combine multiplicatively as in equation 1 is a form of proportional mixing (13). Now let λ(x, u x ) denote the force of infection exerted on an individual of age x and activity level u x. It follows from equation 1 that lðx; u x Þ¼u x l 0 ðxþ; ð2þ where λ 0 (x) is the baseline force of infection. This defines a frailty model for the force of infection, with age-varying multiplicative frailty u x (9, 14).

4 Correlated Infections 477 Figure 1. Association (^w) between ages at infection for selected pairs of infectious agents. The dots represent empirical values, and the lines show smoothed trends. Top left: Helicobacter pylori (HPY) and toxoplasma (TOX) in England and Wales (EW), 1996; top right: hepatitis A virus (HAV) and HPY in EW, 1996; center left: HAV and TOX in EW, 1996; center right: parvovirus B19 (B19) and varicella-zoster virus (VZV) in Poland, ; bottom left: Epstein-Barr virus (EBV) and herpes simplex virus type 1 (HSV1) in EW, ; bottom right: B19 and VZV in EW, Paired serological survey data Consider 2 infections, labeled 1 and 2, conferring lasting immunity and for which long-lived serological markers are known. Serological tests on a blood sample collected at age x will determine whether the individual from which the sample was collected is seropositive or seronegative for each infection. Suppose further that individual activity levels u x are relevant to transmission of both infection 1 and infection 2. This will occur, in particular, if the two infections are transmitted by the same route. The forces of infection in an individual of age x with shared activity level u x are then l 1 ðx; u x Þ¼u x l 01 ðxþ and l 2 ðx; u x Þ¼u x l 02 ðxþ; ð3þ the subscripts 1 and 2 referring to infections 1 and 2. Since the same frailty term u x is shared by the two infections, equation 3 defines a shared frailty model (14). The test results obtained at age x can be as follows: seronegative for both infections, which occurs with probability denoted S 00 (x); seronegative for infection 1 and seropositive for infection 2, which occurs with probability S 01 (x); seronegative for infection 2 and seropositive for infection 1, with probability S 10 (x); and seropositive for both infections, with probability S 11 (x). So far, we have ignored variation with calendar time. This may be important, in particular, for infections transmitted via the fecal-oral route, owing to improvements in hygiene and sanitation over time. However, valid inferences about the shape of gðxþ may be obtained from a single survey even when the baseline forces of infection decline with calendar time (see Web Appendix 2). Displaying correlations between infections using bivariate serological survey data The extent of heterogeneity in the population at age x of relevance to the transmission of both infections of interest can be estimated from the strength of association in the 2 2 tables of counts (n 00x, n 01x, n 10x, n 11x ), using the same

5 478 Farrington et al. Figure 2. Association (^w) between ages at infection for selected pairs of infectious agents. The dots represent empirical values, and the lines show smoothed trends. Top left: mumps (MUM) and rubella virus (RUB; males only) in England and Wales (EW), 1986; top right: parvovirus B19 (B19) and varicella-zoster virus (VZV) in Belgium, ; center left: B19 and VZV in Italy, ; center right: B19 and RUB (males aged 11 years) in EW, 1991; bottom left: B19 and toxoplasma (TOX) in EW, 1996; bottom right: hepatitis A virus (HAV) and VZV in EW, notations as those for the cell probabilities. The odds ratio varies with age even when there is no age-specific heterogeneity, and so can be misleading. We use another measure, denoted w(x), whose properties approximate those of the cross-ratio function (15, 16). The value w(x) = 0 corresponds to independence; w(x) > 0 corresponds to positive association, notably that resulting from heterogeneity, and w(x) < 0 corresponds to negative association, as may arise owing to cross-immunity. We also use the following summary measure of association across age groups x =1,2,, M: w ¼ P M x¼1 p x^wðxþ 1 P M x¼1 p ; VarðwÞ ¼P M x x¼1 p ; ð4þ x where the circumflex denotes the estimated value of w(x) and p x is its (estimated) precision, that is, the reciprocal of its variance. Zeroes in the 2 2 tables of counts at each age x were handled recursively as follows. When one of the 4 margins of the table was zero, we combined it with the data for age x 1 and allocated the average of the ages for the combined table. For tables with zero counts but 4 nonzero margins, we added 0.5 to all 4 cells. Models for age-dependent heterogeneity The baseline forces of infection λ 01 (x) and λ 02 (x) are estimated using piecewise constant functions. Our interest centers on the frailty term and its variance gðxþ. Our basic model for the frailty is of the form u x ¼f1 þðw 1 1ÞhðxÞgw 2 ; ð5þ where w 1 and w 2 are independent γ-distributed random variables with mean 1 and variances g 1 and g 2, respectively, and h(x) is a deterministic function, typically of the form hðxþ ¼expð ðrxþ 2 Þ: ð6þ Note that E(u x ) = 1. These models were introduced in the paper by Farrington et al. (17). The rationale for their use

6 Correlated Infections 479 Figure 3. Association (^w) between ages at infection for selected pairs of infectious agents. The dots represent empirical values, and the lines show smoothed trends. Top left: parvovirus B19 (B19) and hepatitis A virus (HAV) in England and Wales (EW), 1996; top right: B19 and Helicobacter pylori (HPY) in EW, 1996; center left: B19 and cytomegalovirus (CMV) in EW, 1991; center right: toxoplasma (TOX) and varicellazoster virus (VZV) in EW, 1996; bottom left: CMV and rubella virus (RUB; males aged 11 years) in EW, 1991; bottom right: HPY and VZV in EW, will be explained further below; briefly, w 1 represents heterogeneity in childhood, which evolves according to h(x), the parameter ρ governing the rate of decline of such heterogeneity, and w 2 represents heterogeneity in adulthood. For this model, the age-specific heterogeneity has variance gðxþ ¼hðxÞ 2 g 1 ð1 þ g 2 Þþg 2 : ð7þ Suppose that paired serological data (n 00x, n 01x, n 10x, n 11x ) are available at ages x = 1, 2,, M. A Dirichlet-multinomial model is used, to allow for overdispersion due to assay variability (11). The model parameters and hence the function gðxþ and the baseline forces of infection may be estimated by maximizing the log likelihood. Impact on reproduction number and critical vaccination threshold The methods of Farrington et al. (9) may readily be extended to cover the present, more general setting. Suppose for simplicity that an infection confers long-lasting immunity, has a short infectious period D, and is in endemic equilibrium in a population of size N with rectangular age structure on [0, L]. If u(x) represents individual heterogeneity at age x, with variance gðxþ, then the basic reproduction number of the infection is the dominant eigenvalue of the operatorðnd=lþ½1 þ gðxþšb 0 ðx; yþ. Increasing heterogeneity has the effect of increasing R 0 and the critical immunization threshold for vaccination close to birth, π c =1 R 0 1. DATA SOURCES We used data from 7 serological surveys. The data were collected as part of a long-standing program of serological surveillance in England and Wales (18); 2 European networks, the European Sero-Epidemiology Network (ESEN) (19) and ESEN2 (20); and the Europe-wide project POLYMOD (21). These surveys used serum residues of samples taken for diagnostic testing, excluding those from persons tested for the infections of interest and from

7 480 Farrington et al. potentially immunocompromised persons. Details on the tests used may be found in the references noted below. Survey 1, undertaken in 1986 in England and Wales (22), provided paired seroprevalence data on mumps virus and rubella virus, the latter restricted to males owing to the selective rubella vaccination program conducted in adolescent girls. Survey 2, undertaken in England and Wales in 1991, provided information on seroprevalence of parvovirus B19 (23), cytomegalovirus (CMV) (24), and rubella (18). Because universal rubella vaccination at age 15 months was introduced in the United Kingdom in , we restricted the analysis of the rubella data to males aged 11 years or more. The samples were also tested for antibodies to BKV and JCV (25); these data will be discussed below. Survey 3, undertaken in England and Wales in 1994 and 1995, provided paired data on seroprevalence of Epstein- Barr virus and herpes simplex virus type 1 (26). Survey 4, undertaken in England and Wales in 1996, provided information on seroprevalence of VZV (27), parvovirus B19 (28), Helicobacter pylori (29), and hepatitis A virus (30). The samples were also tested for toxoplasma infection using Toxoplasma gondii-specific immunoglobulin G by enzyme-linked immunosorbent assay (Captia Select Toxo-G ELISA; Trinity Biotech, Bray, Ireland) (unpublished data). Surveys 5 7 were undertaken in Belgium ( ), Italy ( ), and Poland ( ), respectively, and provided paired seroprevalence data on parvovirus B19 and VZV as part of the POLYMOD project (8, 21, 31). Details on all 7 surveys are shown in Table 1. Excluding the polyomaviruses for the time being, these surveys provided 18 sets of paired data for analysis. Table 2 shows the major routes of transmission for the 10 infections considered, based primarily on the Control of Communicable Diseases Manual (32). Rubella is generally regarded as transmitted by droplets, though aerosol transmission is also mentioned in the literature (33). Close contact may be involved in transmission of parvovirus B19 and VZV (8). Table 4. Associations (w) Between Paired Infections at Age 21 Years or More According to Likelihood of a Shared Transmission Route a Infection Pair w 95% CI Shared Main Route of Transmission Likely HPY and toxoplasma , HAV and HPY , HAV and toxoplasma , B19 and VZV b (Poland) , Epstein-Barr virus and HSV , B19 and VZV b (England and , Wales) Mumps and rubella c , B19 and VZV (Belgium) , B19 and VZV (Italy) , B19 and rubella c , Shared Main Route of Transmission Unlikely B19 and toxoplasma , HAV and VZV b , B19 and HAV , B19 and HPY , B19 and CMV , Toxoplasma and VZV b , CMV and rubella c , HPY and VZV b , Abbreviations: B19, parvovirus B19; CI, confidence interval; CMV, cytomegalovirus; HAV, hepatitis A virus; HPY, Helicobacter pylori; HSV1, herpes simplex virus type 1; VZV, varicella-zoster virus. a Data were obtained from 7 European serological surveys (18 21). b Ages 11 years. c Males only. RESULTS Descriptive analysis Summary values for the association parameter shown in equation 4 are presented in Table 3, stratified according to whether the main route of transmission is likely to be shared or not. This categorization was decided a priori on the basis of Table 2. Three main features emerge. First, associations are generally higher and significantly positive for pairs of infections sharing a major transmission route than for infections not sharing such a route. Second, the associations between infections not sharing a major route of transmission are nevertheless often positive, though seldom significantly so. Third, the associations between infections transmitted by the respiratory route tend to be lower than between those transmitted by other routes. These patterns suggest that correlations between infections contain information on transmission routes, though the case is perhaps less compelling for respiratory infections. However, Figure 4. Association (^w) between ages at infection for polyomavirus BK and polyomavirus JC, England and Wales, The dots represent empirical values, and the lines show smoothed trends.

8 Correlated Infections 481 Figure 5. Association (^w) between ages at polyomavirus BK and JC infection and selected other infections, England and Wales, The dots represent empirical values, and the lines show smoothed trends. Top left: parvovirus B19 (B19) and polyomavirus BK (BKV); top right: B19 and polyomavirus JC (JCV); center left: cytomegalovirus (CMV) and BKV; center right: CMV and JCV; bottom left: rubella virus (RUB) and BKV in males aged 11 years or more; bottom right: RUB and JCV in males aged 11 years or more. overall measures are crude, so we plotted the values of the association parameter φ(x) at each age x in Figures 1 3, where the infection pairs appear in the same order as in Table 3. The areas of the points within each graph are proportional to the precisions p x (see equation 4); the smooth lines are nonparametric precision-weighted estimates of trend. These plots show that, in childhood, there is a strong correlation between infections irrespective of route of transmission, declining with age. For infections transmitted by different routes, the association declines to zero in adulthood, whereas for infections transmitted by the same route, the association generally declines to some positive, constant value with some exceptions, particularly among respiratory infection pairs. This is shown in Table 4, which presents the association measure for ages 21 years or more (when data beyond age 20 are available) or ages 11 years or more (when data beyond age 20 are not available). The patterns of association can be interpreted in terms of the changes in heterogeneity in the population that induce correlations via shared frailty terms: the stronger the association, the greater the heterogeneity. Note also that the strength of the associations should not be interpreted in terms of the magnitude of the forces of infection but in terms of heterogeneities. Interpretation of the observed patterns is deferred to the Discussion. Inference about routes of transmission Our methods can be used to generate hypotheses about routes of transmission, when these are uncertain or unknown. The idea is to test a panel of serum samples for antibodies to the infection of interest and to several infections with a known route of transmission. A shared route of transmission may be suggested by a positive correlation in adulthood. We illustrate this idea with data on BKV and JCV (26). We found a strong negative correlation between BKV and JCV at younger ages, suggestive of cross-protection; this is reflected in the plot of association between the two infections (Figure 4). The lack of association after age 50 years may reflect reduced sensitivity of the test, resulting in misclassification of sera and thus bias towards the null (26).

9 482 Farrington et al. Table 5. Associations (w) of Selected Infections With Polyomaviruses BK and JC in Persons Aged Years a Infection No. of Cases Polyomavirus BK Polyomavirus JC w 95% CI w 95% CI Cytomegalovirus , , Parvovirus B , , Rubella virus (males) , , Abbreviation: CI, confidence interval. a Data were obtained from 7 European serological surveys (18 21). The plots of association of BKV and JCV with CMV, parvovirus B19, and rubella (the latter studied in males aged years, to ensure that they were unvaccinated) are shown in Figure 5. The odd appearance of the plots at low ages may be attributable to the negative correlation between BKV and JCV. A further problem is the decline in test sensitivity at older ages. To mitigate these effects while retaining sufficient data, we calculated summary values for the association parameter in adulthood for the age range years, using equation 7. These are shown in Table 5. These estimates suggest a possible positive association in this age range between CMV and JCV, but not between CMV and BKV, though the association data are sparse. The data also suggest a possible positive association between parvovirus B19 and BKV in adulthood, but none with JCV (associations in childhood are clearly affected by the strong negative association between BKV and JCV). There is no compelling evidence of any associations with rubella in males. These considerations suggest that the route of transmission of BKV may be shared with parvovirus B19, while that of JCV might be shared with CMV. These hypotheses, though very tentative and requiring confirmation from other serological surveys, are in line with those of Morris et al. (26). For completeness, Figure 6 shows the seroprevalence of the 4 infections, plotted to age 44 years owing to sparsity of data at older ages. The force of infection of BKV is greater than that of parvovirus B19 and is more akin to that Figure 6. Prevalence of antibodies to polyomavirus BK (top left), parvovirus B19 (top right), polyomavirus JC (bottom left), and cytomegalovirus (bottom right) by age (dots) and trend (lines), England and Wales, 1991.

10 Correlated Infections 483 Figure 7. Observed (dots) and fitted (lines) associations (^w) between times to infection for parvovirus B19 and varicella-zoster virus in Belgium ( ), Italy ( ), Poland ( ), and England and Wales (1996). of VZV (transmitted by a similar route as parvovirus B19). The force of infection of JCV is similar to that of cytomegalovirus. Note also the decline in measured antibodies to BKV at older ages. Estimating the heterogeneity in VZV transmission and its impact on key epidemiologic parameters We illustrate the impact of individual age-dependent heterogeneity on estimates of the basic reproduction number R 0 and the critical immunization threshold π c for VZV infection at 4 European locations: Belgium, Italy, Poland, and England and Wales. For this purpose, we explicitly model the heterogeneity using the model shown in equations 5 and 6; Web Appendix 3, including Web Table 1 and Web Figure 1, provides further details on model choice and fit. This model presumes 2 distinct sources of heterogeneity, represented by the frailty terms w 1 and w 2. The frailty w 2, of variance g 2, represents heterogeneity of behavior and individual circumstances related to transmission by the route specific to VZV, namely exchange of respiratory secretions. This route-related heterogeneity is presumed to remain present throughout life, though it becomes apparent only in adulthood. The frailty w 1, on the other hand, represents heterogeneities in childhood behavior and circumstances (such as nursery-school attendance) which are related to transmission of virtually any childhood infection. This childhood-related heterogeneity is presumed to decline with increasing age, as childhood behavior and circumstances evolve for example, through the learning of personal hygiene. To estimate the heterogeneity, we used paired data on parvovirus B19 and VZV from the 4 locations. The empirical and fitted associations are shown in Figure 7. The association is particularly strong in Poland, less so in the 3 other locations. In Poland and in England and Wales, there is more evidence of association at older ages as represented by the positive asymptote than in Belgium or Italy. These associations reflect the changing heterogeneity with age, shown in Figure 8, obtained from the estimated values of the parameters g 1, g 2, and ρ using equation 7. To estimate reproduction numbers and critical immunization thresholds, we used the social contact matrices described by Farrington et al. (34). The estimated values of R 0 and π c of VZV are shown in Table 6. The R 0 estimates, without allowing for additional heterogeneity, lie within the range 3 7, similar to those obtained in other studies using a variety of different methods (8, 31, 35). Our interest is primarily in the impact of heterogeneity, shown in the final 2 columns of Table 6. The impact is small for Belgium and Italy but is appreciable for England and Wales and substantial for Poland.

11 484 Farrington et al. Figure 8. Standard deviation of the frailty associated with varicella-zoster and parvovirus B19 infection by age in 4 European locations: Belgium ( ), Italy ( ), Poland ( ), and England and Wales (1996). DISCUSSION We have developed new methods for exploring unmeasured individual factors relevant to the transmission of infectious diseases. Our methods are based on interpreting correlations between infections in terms of heterogeneities. They are readily applied to paired data on the presence of antibodies measured in the same serum samples, for different infections. We find support for such an approach from the fact that, as expected, infections sharing a common mode of transmission tend to display positive correlations between ages at infection. Our approach substantially extends previously published methodology (9) on analyzing bivariate serological survey data, through a more appropriate representation of the association (16), better understanding and explicit modeling of age effects (17), and novel applications. As illustrated with polyomaviruses, one potential application of these ideas is to the reverse inference, namely to shed light on routes of transmission of infections for which these routes are uncertain, by examining correlations with infections with known transmission routes. A second application is to improve estimation of epidemiologic parameters, such as R 0 and π c. This requires explicit modeling of the heterogeneity, a natural framework for which is provided by age-dependent frailty modeling. We find that ignoring such individual heterogeneity risks underestimating the overall immunization level required for effective control. Our statistical approach suffered from 3 key weaknesses. First, a shared frailty can only represent part of the overall heterogeneity, as it does not incorporate sources of heterogeneity that are not shared between infections. Second, frailty models cannot identify the source of heterogeneity. Finally, our data only provided information on past infections and were not randomly sampled, which may have introduced some bias. For most of the infection pairs we studied, we found a strong association in childhood, usually declining with age either to zero (typically for infections without a shared route of transmission) or to a positive constant (typically for infections with a shared route). One interpretation is as follows. The routes of transmission considered here are confounded in childhood, owing to the nature of contacts made at young ages, which include close mixing involving much direct contact. At young ages, the population is highly heterogeneous with respect to family circumstances and nursery-school attendance. We speculate that at older ages, behavior and circumstances change so that transmission routes gradually become differentiated and common social factors (such as school attendance) intervene, so that the heterogeneity drops (though the force of infection typically increases during the school years). For infections transmitted by the same route, the association eventually reflects the heterogeneity in behaviors associated with transmission solely via that specific route. For infections transmitted by different routes, there is no common factor, and the heterogeneity drops to zero. The lower associations between respiratory infections in adulthood could be due to the lesser relevance of variations in individual behaviors to transmission of such infections for example, if aerosol spread is a major factor in transmission. It is possible to conceive of alternative explanations for the observed associations. One is individual variation in the development of children s immune systems, resulting in individual variation in systemic susceptibility rather than in Table 6. Impact of Heterogeneity on the Basic Reproduction Number R 0 and Critical Immunization Coverage π c Location Ignoring Heterogeneity Allowing for Heterogeneity R 0 π c R 0 π c Ratio of R 0 Values Relative Increase Odds Ratio for π c Values Belgium Italy Poland England and Wales

12 Correlated Infections 485 effective contact rates. Another is a selection effect induced by certain types of frailty distributions, though exhaustive study has not yielded any support for this so far (17). Finally, cross-reactions in the antibody assays used to test the samples could generate spurious dependencies, though such a consistent pattern as that seen in these data would perhaps be difficult to explain. In this paper, we have suggested that there is merit in analyzing serological data on different infections together. Further investigations are needed to validate the approach. ACKNOWLEDGMENTS Author affiliations: Department of Mathematics and Statistics, The Open University, Milton Keynes, United Kingdom (C. Paddy Farrington, Heather J. Whitaker, Steffen Unkel); and Health Protection Agency, London, United Kingdom (Richard Pebody). This work was supported by the Medical Research Council (Project Grant G ) and by a Royal Society Wolfson Research Merit Award to C.P.F. 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