Sexual Bridging Socially and Over Time: A Simulation Model Exploring the Relative Effects of Mixing and Concurrency on Viral Sexually Transmitted Infection Transmission

Doherty, Irene A. PhD, MPH*†; Shiboski, Stephen PhD‡; Ellen, Jonathan M. MD, MPH§; Adimora, Adaora A. MD, MPH*; Padian, Nancy S. PhD, MPH, MS†

Sexually Transmitted Diseases:
doi: 10.1097/01.olq.0000194586.66409.7a

Background: Sexual partnerships between people at higher and lower risk for sexually transmitted infections (STIs) (i.e., bridging) occur through dissortative mixing and concurrent partnerships, yet the relative effects of these network patterns on population STI spread are poorly understood.

Goal: Using a stochastic model, the authors investigated the impact of mixing and concurrency on the spread of a persistent viral STI.

Study Design: A total of 1050 populations were simulated of 1000 subjects over 400 weeks with varied concurrency levels and mixing patterns. STI prevalence and the average number of secondary transmissions per subject were analyzed with regression.

Results: Mixing had a greater impact on prevalence for all groups, whereas concurrency was significant for only the lowest activity group. Mixing patterns moderated the magnitude of concurrency’s impact on secondary transmissions.

Conclusions: Through connecting subgroups of differential risk, sexual mixing facilitates dissemination of STIs throughout a population. Concurrency expedites transmission by shortening the time between sexual contacts among infected and susceptible persons, particularly during the highly infectious period.

In Brief

A mathematical simulation of sexual networks demonstrated the relative effects of sexual mixing patterns and concurrency levels on viral sexually transmitted infection (STI) transmission. Mixing facilitates and concurrency expedites STI spread.

Author Information

From the *School of Medicine, University of North Carolina, Chapel Hill, North Carolina; the Departments of †Obstetrics/Gynecology and Reproductive Sciences and ‡Epidemiology and Biostatistics, University of California, San Francisco, California; and the §Department of Pediatrics, Johns Hopkins University, Baltimore, Maryland

The development of this model and the drafting of the manuscript were funded through grants from the National Institute of Allergy and Infectious Diseases (R01-AI48749 and 5 T32 AI007151-27). The authors thank Richard Rothenberg, Steve Selvin, Sevgi Aral, and Jennifer Smith for their helpful suggestions during various phases of this work.

Correspondence: Irene A Doherty, PhD, Postdoctoral Fellow, Division of Infectious Diseases, BioInformatics Bldg., 130 Mason Farm Road, CB 7030, UNC School of Medicine, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7030. E-mail:

Received for publication June 30, 2005, and accepted October 21, 2005.

Article Outline

THE EPIDEMIOLOGY OF SEXUALLY transmitted infections (STIs) is determined by a constellation of factors, including partnership dynamics and sexual networks. Within sexual networks, bridging generally refers to the formation of sexual partnerships between people of high risk for STIs with lower-risk individuals; bridging occurs in concurrent partnerships and through dissortative mixing (e.g., 1–6). Concurrent partnerships are a structural feature of networks defined by the timing of sexual encounters,7 whereas sexual mixing pertains to the types of linkages distributed within a network or population.8 Concurrent partnerships are a risk factor for acquiring infection9,10 and transmitting disease to others11,12 because they increase the likelihood of exposure to infected persons and shorten the time between sexual contacts. Mixing indicates whether people have sex with others like themselves (assortative mixing) or unlike themselves (dissortative mixing) relative to their own STI risk profile.8 Dissortative mixing between older men and female adolescents, for example, elevates the females’ infection risk.13–16 Thus, mixing operates as a social bridge, whereas concurrency operates as a temporal bridge.17

A complete understanding of the joint effects of mixing and concurrency on STI spread would require detailed information documenting the occurrence and timing of all events related to partnership formation and sexual activity within partnerships. Empiric network studies, however, are burdened by expense and implementation obstacles,18 so data describing these phenomena is very limited and rarely available. Infectious disease models, however, can provide qualitative insights into the properties of STI transmission dynamics under an array of scenarios.19 Transmission dynamics are expressed in mathematical terms either as rates (i.e., deterministic models) or event-driven probabilities (i.e., stochastic/discrete models).19 Several models have examined either the role of mixing8,20–26 or concurrency27–29 in STI transmission; other models accommodated both processes.30–33 Limitations of these other models include: restricting mixing to assortative patterns only31; limiting the variations in concurrency levels30; and defining mixing in terms of concurrency, rendering it impossible to uncouple the relative effect of each.33 One microsimulation, however, demonstrated that dissortative mixing among highly sexually active persons exerted the strongest impact on STI prevalence,32 a finding that underscores the need for investigation into the dissemination of STIs among subpopulations.34 Although these simulations have made important contributions to our understanding of STI transmission dynamics, the relative effects of mixing and concurrency remain poorly understood.

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Materials and Methods

We developed a stochastic microsimulation that simultaneously varied mixing patterns and the prevalence of concurrency to investigate their joint effects on the spread of a persistent viral STI. As summarized in Table 1 and described subsequently, parameters developed for this model address population attributes, disease transmission, and partnership formation over 1050 microsimulations comprised of 1000 subjects over 400 time steps. The simulations subsequently served as independent observations that were analyzed with generalized linear regression.

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Population Characteristics

The model randomly distributed a closed population of 1000 subjects into three groups that represented low, moderate, and high levels of sexual activity. Because a small percentage of people tend to engage in high-risk sex,35,36 10% of the population (i.e., 100 subjects) were randomly assigned into the high-level group, 30% to the moderate-level group, and the remaining 60% were assigned into the low group. Subjects remained in their assigned group throughout the simulation.

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Infection Prevalence and Transmission

Baseline disease prevalence was 2% in the population overall, but it differed across subpopulations. Because the model assumed high sexual activity increases STI risk, the baseline prevalence of infection was 10% among the high, 2% among the moderate, and 1% among the low sexual activity groups. Infected subjects could transmit the virus to their susceptible partners throughout the simulation, but the probability of transmission changed from 15% to 0.5% to mimic a highly infectious initial period, lasting five time steps. Each time step represented 1 week, because it is most relevant for STI infectivity. To allow for differential transmission probabilities at the start of the simulation, subjects who were infected at baseline acquired their infection anywhere between zero to 200 time steps before the start of the simulation.

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Partnership Formation
Sexual Contact.

Subjects were allowed a maximum of 50 sex partners. Partnerships were kept short –a maximum of 200 time steps to loosely represent a more sexually active population such as young adults who may change partners more often than the “general” population. Although subjects could have several partners, they were limited to sexual contact with one sex partner per time step (a convention applied in stochastic37 and deterministic27 models). During the span of the relationship, sexual contact with the same partner could be repeated over multiple time steps yet contact did not necessarily occur at every time step. Once two persons were paired (as described subsequently), the future time steps at which they had sex were randomly chosen from future time steps available to both subjects.

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At the beginning of the simulation, no one was partnered. At each subsequent time step, the model randomly paired individuals from the pool of available (unpartnered) subjects for potential sexual partnership. The probability of an actual sexual union depended on the sexual activity level of both partners and, for the given simulation, the value of the mixing parameter. To distinguish degrees of sexual activity, the probabilities that pairs of low, moderate, and high sexual activity subjects formed a sexual union were LL = 15%, MM = 30%, and HH = 60%, respectively.

The mixing parameter, m, controlled mixing across groups. For example, if subjects from the low and high groups were paired, the probability that a sexual partnership formed was LH = (LL*HH)/m. When m = 1, the probability of partnership formation was (0.15*0.60)/1 = 0.09, whereas when m = 3, the probability was smaller at (0.15*0.60)/3 = 0.03. Thus, large values of m lowered the probability of mixing across groups to create networks characterized by assortative mixing, whereas small values of m increased the probability of dissortative mixing. Parameter values for m were randomly selected from ranges that produced various mixing patterns.

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Subjects were randomly selected to have either serial monogamous or concurrent partnerships. For those restricted to serial monogamy, once the span of a partnership was determined, they could not form partnerships with others within that timeframe. Subjects who were allowed to have concurrent partnerships, however, could be partnered with others at available time steps that were embedded within another partnership. This design allowed concurrent subjects to alternate sexual contact among multiple partners, including partners who remained serially monogamous.

To control population levels of concurrency, a proportion of subjects within each sexual activity group were randomly assigned to engage in concurrent partnerships. Three percentages were randomly selected, and because the model assumed that the prevalence of concurrency was greatest in the high-activity group, the high-activity group was assigned to the highest percent of concurrency, whereas the moderate- and low-activity groups were assigned, respectively, to the intermediate and lowest values. The parameter selection process was divided into three sets of simulations to create populations with little concurrency across all three activity levels to extensive concurrency in all three groups. Within each of these sets, mixing varied from assortative to dissortative extremes.

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Analysis of Simulations
Outcome Variables: Acquisition and Transmission.

The outcomes addressed both acquisition and transmission of a hypothetical viral STI. Disease acquisition was represented by the point prevalence of infection at the end (i.e., time step 400) of each simulation. Transmission is often quantified by the reproductive number, R0, defined as the average number of secondary infections estimated from the duration of disease, transmission probability, and the rate of partner change.38 However, in our model, the average number of secondary transmissions, T0, was directly obtained by averaging the number of times that each subject transmitted their infection to a susceptible individual; uninfected persons were included in the calculation. Prevalence and T0 were calculated for the population in addition to each sexual activity group.

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Independent Variables.

Mixing: An aggregate summary of mixing, the Q statistic,39 can be computed from the mixing matrix, which is derived from the proportion of sexual links within and across the three sexual activity groups. Perfect assortative mixing is represented by Q = 1, and when three risk categories are used, Q = −0.50 indicates complete dissortative mixing. Random mixing is evident for Q values near zero to indicate that sexual formation occurs independently of group membership. These simulations produced Q values between −0.36 and 0.86.

Concurrency: Concurrency was quantified as the overall proportion of subjects (independent of sexual activity group) randomly selected to engage in concurrent relationships, which ranged from 4% to 93% (standard deviation, 28%).

Confounding—Network Density: Elevated concurrency increased the availability of subjects for sexual unions and dissortative mixing increased the probability of partnership formation, both of which contributed to denser networks. As networks become more dense, disease transmission opportunities inevitably increase. Because this analysis focused on the relative roles of mixing and concurrency in the context of both dense and sparse networks, density was included in the regression equations. The total number of sexual links present at the end of each simulation represented network density (rather than a proportion as is conventional in social network analysis40).

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Regression Analyses.

Because disease prevalence was based on a distribution of rates of infected subjects over 1050 populations, Poisson regression was used.41 The average number of secondary infections (T0) was treated as a continuous outcome and modeled using linear regression. Stratified analyses by levels of sexual activity were also performed.

A series of nested multivariate regression models were fitted (in which one variable is removed from a larger model and was thus “nested”41). Regression model comparison tests were used to determine if the differences in residual deviance (a measure of relative improvement in fit) between the nested models were statistically significant. For the prevalence analysis, the likelihood ratio test (LRT) was applied to the Poisson regressions, whereas F-tests were applied to the linear analysis for T0. Because regression models were fitted to the entire “population” of individuals represented in all simulations, the interpretation of significance tests differs somewhat from applications to independent, random samples. However, the P values produced still provide useful indices of the relative contribution of each variable (mixing and concurrency) to the changes in each outcome (prevalence and T0).

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The 1050 simulations produced sexual networks ranging from 1218 to nearly 10,500 sexual links. Across all simulations, the outcomes, disease prevalence, and the average number of new secondary transmissions per subject (T0) were highly correlated (R2 = 0.98). Correlation, however, between the independent variables, mixing (i.e., Q) and concurrency (i.e., overall percent concurrent), was negligible (R2 = −0.02) (in contrast to another model,33 in which the two variables were inherently correlated).

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Disease Prevalence

The relationships between disease prevalence and mixing (Fig. 1A) and concurrency (Fig. 1B) are best shown graphically; simple least-square lines were plotted for each activity group to demonstrate the different effects for each sexual activity group. As mixing became more assortative (Fig. 1A), the prevalence of infection fell among the low- and moderate-activity groups and rose within the high-activity group. Conversely, as mixing became more dissortative, disease prevalence diminished in the high-activity group and rose in the other groups. Increases in the occurrence of concurrency (Fig. 1B) increased prevalence among all three sexual activity groups.

For the multivariate analysis of prevalence, interaction between mixing and concurrency was not present. Table 2 presents the results of the LRT that compared changes in model performance between the full regression model, including all three terms (mixing, concurrency, and density) and three separate restricted models, with each independent variable removed. Mixing contributed significantly to the regression models’ prediction of disease prevalence in the overall population, the low group, and moderate group (i.e., LRT P values of <0.05). For the high sexual activity group, the removal of Q (i.e., mixing) from the equation yielded a borderline result (P = 0.07). Network density was a significant predictor of disease prevalence for only the high-activity group (P = 0.042). Concurrency exerted borderline statistically significant effects in only the low-activity group (P = 0.055).

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T0: Average Number of Secondary Transmissions

For bivariate analysis, except for the high-activity group, network density (D) was most strongly associated with T0, with R2 values ranging between 48% and 66% (not shown). The proportion of concurrent subjects also produced large R2 values (36–49%, not shown), and the association between mixing and T0 was weakest (R2 values between 2% and 18%, not shown).

For multivariate analysis of T0, we applied the same statistical approach used in the prevalence analysis. Unlike the prevalence analysis, removal of interaction terms for mixing and concurrency produced statistically significant losses to the residual sums of squares, thereby rendering it impossible to isolate the independent effect on transmission of either mixing or concurrency.

Thus, to further examine the nature of this interaction, we conducted stratified analyses by type of mixing. To create the strata, the simulations were grouped by Q statistic values to indicate dissortative (Q <−0.10), proportionate (−0.10 < Q < 0.10), and assortative (Q >0.10) mixing. Average T0 values for each stratum were significantly different (analysis of variance: F <0.0001) at 0.062, 0.052, and 0.043 for dissortative, random, and assortative mixing, respectively (not shown). For each mixing pattern, Figure 2 displays the plot of concurrency values against T0. Least-squares lines were also added to illustrate that different mixing patterns either exacerbate or attenuate the effects of concurrency on T0. Concurrency had the strongest effects in dissortatively mixed populations.

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Several mathematical models have been developed to evaluate the effect of sexual network properties on STI transmission dynamics, yet few have been used to conduct an in-depth evaluation of the joint effects of concurrency and mixing. We created a microsimulation that built on other models and addressed some of their limitations to examine the relative effects of these sexual partnership dynamics on disease prevalence and transmission for a persistent viral STI such as HIV. Furthermore, because the effects of mixing and concurrency differed within sexual activity groups, stratified analyses were critical.

Our findings suggest that mixing generally exerted a greater influence on disease prevalence than concurrency, but the results differed across sexual activity groups. Concurrency levels only affected the prevalence of infection among those in lowest level of sexual activity, which is consistent with empiric findings,3–6 signaling important public health implications. For example, monogamous/low-risk women in developing countries such as Thailand, Cambodia, and areas of Africa were at increased risk for HIV if they were in a long-term relationship or marriage with men who concurrently partnered with commercial sex workers.3–6 These results underscore the need to consider factors that extend beyond individuals in future research and the development of interventions.

Concurrency also increased the average number of secondary infections per subject, across all subpopulations, but the impacts were either tempered or intensified by the population’s underlying mixing patterns. Consistent with another model,33 compared with assortatively mixed populations, concurrency had a more pronounced effect on dissortatively mixed populations, because dissortative mixing increased the probability of partnership formation across groups, which then consequently increased the opportunities for concurrent partnerships to form.

The somewhat counterintuitive observation of a significant interaction in regression models for T0, yet not for prevalence, can be partially explained by the fact that the Poisson models represent interactions on a multiplicative rather than an additive scale. Fundamentally, disease acquisition (prevalence) and disease transmission (T0) are conceptually distinct components of an infectious disease transmission system that provide complementary insights into the population ecology of STIs.42,43 Prevalence reflects the extent to which infection infiltrates a population. Conversely, T0 quantifies the frequency with which disease is transmitted to susceptible persons.

Therefore, higher prevalence indicated that disease reached/infiltrated the lower-risk, peripheral outskirts of the network, because dissortative partnering enabled more sexual contact to occur across subpopulations of heterogeneous risk. For populations characterized by assortative mixing, prevalence was reduced overall and in the low-activity group because infection was largely contained in the high-risk group, which accounted for only 10% of the population. Hence, these results demonstrate how dissortative mixing serves as a socially derived form of bridging that affects disease acquisition.17

Although related to prevalence, the extent of transmission depends on: who is available for partnership; who is susceptible; and most importantly, the infectivity at the time of sexual contact.44,45 As the probability of transmission drops over time, the highly infectious period may elapse before one sequential partnership dissolves and the next one begins. Thus, concurrent partnerships form temporal bridges that increase the risk of transmission to one’s partners, especially during the highly infectious period.17,45,46

This microsimulation has several limitations. For simplicity, we ignored gender, which means that an individual could potentially have sex with the other 999 persons in the population. The number of partnerships per subject, however, was limited to 50, thereby avoiding excessively dense networks. Without such restrictions, approximately 499,500 links are theoretically possible; these simulations generated networks of at most 10,500 links.

Other than network density, computation of other microstructural indicators (e.g., component sizes, k-cores, or clustering coefficients) was not possible within our model. Nonetheless, other models have shown that elevated concurrency results in larger components within sexual networks to propagate transmission,32,47 which support our findings, because the correlation between concurrency levels and network density was high (R2 = 0.83, not shown).

We also focused on persistent viral infections such as HIV. Because bacterial STIs resolve with treatment, the mechanisms of this model do not necessarily apply to bacterial infections.

The baseline prevalence and transmission probabilities were set artificially higher than that typically estimated.44,45 We also did not limit the mixing and concurrency levels to empiric estimates.1,48–53 Instead, the full range of combinations of low or high concurrency within populations with assortative to dissortative mixing patterns was generated. Generally, the modeling assumptions and the parameter values were chosen intentionally to exaggerate network formation and disease transmission processes. By simplifying these components of the process, we could focus on partnership dynamics.

Despite the shortcomings of this microsimulation, these analyses revealed how mixing and concurrency interact and how they can be used to assess potential STI spread. Specifically, these findings demonstrated that mixing facilitates dissemination of STIs by connecting subgroups of differential risk while concurrency expedites transmission by shortening the time between sexual contacts among infected and susceptible persons, particularly during the highly infectious period.

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